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[Texi2html-cvs] texi2html ChangeLog TODO texi2html.init texi2ht...
|
From: |
Patrice Dumas |
|
Subject: |
[Texi2html-cvs] texi2html ChangeLog TODO texi2html.init texi2ht... |
|
Date: |
Fri, 09 Jan 2009 21:21:03 +0000 |
CVSROOT: /cvsroot/texi2html
Module name: texi2html
Changes by: Patrice Dumas <pertusus> 09/01/09 21:21:00
Modified files:
. : ChangeLog TODO texi2html.init texi2html.pl
examples : docbook.init html32.init roff.init xml.init
test/coverage/res/formatting: formatting.html
test/coverage/res/texi_formatting: formatting.passfirst
formatting.passtexi
formatting.texi
test/coverage/res/texi_imbrications: imbrications.passfirst
imbrications.passtexi
imbrications.texi
test/encodings/res/formatting_converted_to_utf8: formatting.html
test/encodings/res/formatting_fr: formatting.html
test/encodings/res/formatting_fr_icons: formatting.html
test/formatting/res/formatting_weird_quotes: formatting.html
test/formatting/res/texi_block_EOL: block_EOL.passfirst
block_EOL.passtexi
block_EOL.texi
test/formatting/res/texi_formatting: formatting.passfirst
formatting.passtexi
formatting.texi
test/layout/res/formatting_chm: formatting.html
formatting_1.html
formatting_2.html
formatting_3.html
formatting_4.html
formatting_5.html
formatting_abt.html
formatting_ovr.html
formatting_toc.html
test/layout/res/formatting_exotic: formatting.html
formatting_1.html
formatting_2.html
formatting_3.html
formatting_4.html
formatting_abt.html
formatting_ovr.html
formatting_toc.html
test/layout/res/formatting_makeinfo: chapter.html chapter2.html
formatting_abt.html
index.html
s_002d_002dect_002cion.html
subsection.html
subsubsection-_0060_0060simple_002ddouble_002d_002dthree_002d_002d_002dfour_002d_002d_002d_002d_0027_0027.html
test/layout/res/formatting_regions: formatting_regions.html
test/layout/res/texi_formatting_regions:
formatting_regions.passfirst
formatting_regions.passtexi
formatting_regions.texi
test/macros : tests.txt
test/macros/res/texi_cond: cond.passfirst cond.passtexi
cond.texi
test/manuals/res/texi_mini_ker: mini_ker.passfirst
mini_ker.passtexi mini_ker.texi
test/manuals/res/texi_texinfo: texinfo.passfirst
texinfo.passtexi texinfo.texi
test/misc/res/formatting_html32: formatting.html
test/nested_formats/res/texi_nested_formats:
nested_formats.passfirst
nested_formats.passtexi
nested_formats.texi
test/singular_manual/res/texi_singular: singular.passfirst
singular.passtexi
singular.texi
Log message:
* texi2html.pl, texi2html.init, examples/*: Handle the raw
formats during output formatting, not in the first pass.
CVSWeb URLs:
http://cvs.savannah.gnu.org/viewcvs/texi2html/ChangeLog?cvsroot=texi2html&r1=1.361&r2=1.362
http://cvs.savannah.gnu.org/viewcvs/texi2html/TODO?cvsroot=texi2html&r1=1.79&r2=1.80
http://cvs.savannah.gnu.org/viewcvs/texi2html/texi2html.init?cvsroot=texi2html&r1=1.175&r2=1.176
http://cvs.savannah.gnu.org/viewcvs/texi2html/texi2html.pl?cvsroot=texi2html&r1=1.258&r2=1.259
http://cvs.savannah.gnu.org/viewcvs/texi2html/examples/docbook.init?cvsroot=texi2html&r1=1.16&r2=1.17
http://cvs.savannah.gnu.org/viewcvs/texi2html/examples/html32.init?cvsroot=texi2html&r1=1.22&r2=1.23
http://cvs.savannah.gnu.org/viewcvs/texi2html/examples/roff.init?cvsroot=texi2html&r1=1.26&r2=1.27
http://cvs.savannah.gnu.org/viewcvs/texi2html/examples/xml.init?cvsroot=texi2html&r1=1.12&r2=1.13
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/coverage/res/formatting/formatting.html?cvsroot=texi2html&r1=1.17&r2=1.18
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/coverage/res/texi_formatting/formatting.passfirst?cvsroot=texi2html&r1=1.7&r2=1.8
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/coverage/res/texi_formatting/formatting.passtexi?cvsroot=texi2html&r1=1.7&r2=1.8
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/coverage/res/texi_formatting/formatting.texi?cvsroot=texi2html&r1=1.7&r2=1.8
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/coverage/res/texi_imbrications/imbrications.passfirst?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/coverage/res/texi_imbrications/imbrications.passtexi?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/coverage/res/texi_imbrications/imbrications.texi?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/encodings/res/formatting_converted_to_utf8/formatting.html?cvsroot=texi2html&r1=1.17&r2=1.18
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/encodings/res/formatting_fr/formatting.html?cvsroot=texi2html&r1=1.17&r2=1.18
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/encodings/res/formatting_fr_icons/formatting.html?cvsroot=texi2html&r1=1.17&r2=1.18
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/formatting/res/formatting_weird_quotes/formatting.html?cvsroot=texi2html&r1=1.18&r2=1.19
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/formatting/res/texi_block_EOL/block_EOL.passfirst?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/formatting/res/texi_block_EOL/block_EOL.passtexi?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/formatting/res/texi_block_EOL/block_EOL.texi?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/formatting/res/texi_formatting/formatting.passfirst?cvsroot=texi2html&r1=1.4&r2=1.5
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/formatting/res/texi_formatting/formatting.passtexi?cvsroot=texi2html&r1=1.4&r2=1.5
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/formatting/res/texi_formatting/formatting.texi?cvsroot=texi2html&r1=1.4&r2=1.5
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_chm/formatting.html?cvsroot=texi2html&r1=1.10&r2=1.11
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_chm/formatting_1.html?cvsroot=texi2html&r1=1.5&r2=1.6
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_chm/formatting_2.html?cvsroot=texi2html&r1=1.5&r2=1.6
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_chm/formatting_3.html?cvsroot=texi2html&r1=1.5&r2=1.6
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_chm/formatting_4.html?cvsroot=texi2html&r1=1.5&r2=1.6
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_chm/formatting_5.html?cvsroot=texi2html&r1=1.7&r2=1.8
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_chm/formatting_abt.html?cvsroot=texi2html&r1=1.5&r2=1.6
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_chm/formatting_ovr.html?cvsroot=texi2html&r1=1.7&r2=1.8
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_chm/formatting_toc.html?cvsroot=texi2html&r1=1.7&r2=1.8
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_exotic/formatting.html?cvsroot=texi2html&r1=1.15&r2=1.16
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_exotic/formatting_1.html?cvsroot=texi2html&r1=1.5&r2=1.6
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_exotic/formatting_2.html?cvsroot=texi2html&r1=1.11&r2=1.12
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_exotic/formatting_3.html?cvsroot=texi2html&r1=1.11&r2=1.12
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_exotic/formatting_4.html?cvsroot=texi2html&r1=1.10&r2=1.11
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_exotic/formatting_abt.html?cvsroot=texi2html&r1=1.7&r2=1.8
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_exotic/formatting_ovr.html?cvsroot=texi2html&r1=1.9&r2=1.10
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_exotic/formatting_toc.html?cvsroot=texi2html&r1=1.9&r2=1.10
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_makeinfo/chapter.html?cvsroot=texi2html&r1=1.10&r2=1.11
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_makeinfo/chapter2.html?cvsroot=texi2html&r1=1.12&r2=1.13
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_makeinfo/formatting_abt.html?cvsroot=texi2html&r1=1.6&r2=1.7
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_makeinfo/index.html?cvsroot=texi2html&r1=1.13&r2=1.14
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_makeinfo/s_002d_002dect_002cion.html?cvsroot=texi2html&r1=1.9&r2=1.10
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_makeinfo/subsection.html?cvsroot=texi2html&r1=1.9&r2=1.10
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_makeinfo/subsubsection-_0060_0060simple_002ddouble_002d_002dthree_002d_002d_002dfour_002d_002d_002d_002d_0027_0027.html?cvsroot=texi2html&r1=1.9&r2=1.10
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/formatting_regions/formatting_regions.html?cvsroot=texi2html&r1=1.17&r2=1.18
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/texi_formatting_regions/formatting_regions.passfirst?cvsroot=texi2html&r1=1.6&r2=1.7
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/texi_formatting_regions/formatting_regions.passtexi?cvsroot=texi2html&r1=1.6&r2=1.7
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/layout/res/texi_formatting_regions/formatting_regions.texi?cvsroot=texi2html&r1=1.6&r2=1.7
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/macros/tests.txt?cvsroot=texi2html&r1=1.6&r2=1.7
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/macros/res/texi_cond/cond.passfirst?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/macros/res/texi_cond/cond.passtexi?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/macros/res/texi_cond/cond.texi?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/manuals/res/texi_mini_ker/mini_ker.passfirst?cvsroot=texi2html&r1=1.2&r2=1.3
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/manuals/res/texi_mini_ker/mini_ker.passtexi?cvsroot=texi2html&r1=1.3&r2=1.4
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/manuals/res/texi_mini_ker/mini_ker.texi?cvsroot=texi2html&r1=1.2&r2=1.3
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/manuals/res/texi_texinfo/texinfo.passfirst?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/manuals/res/texi_texinfo/texinfo.passtexi?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/manuals/res/texi_texinfo/texinfo.texi?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/misc/res/formatting_html32/formatting.html?cvsroot=texi2html&r1=1.17&r2=1.18
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/nested_formats/res/texi_nested_formats/nested_formats.passfirst?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/nested_formats/res/texi_nested_formats/nested_formats.passtexi?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/nested_formats/res/texi_nested_formats/nested_formats.texi?cvsroot=texi2html&r1=1.1&r2=1.2
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/singular_manual/res/texi_singular/singular.passfirst?cvsroot=texi2html&r1=1.2&r2=1.3
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/singular_manual/res/texi_singular/singular.passtexi?cvsroot=texi2html&r1=1.3&r2=1.4
http://cvs.savannah.gnu.org/viewcvs/texi2html/test/singular_manual/res/texi_singular/singular.texi?cvsroot=texi2html&r1=1.2&r2=1.3
Patches:
Index: ChangeLog
===================================================================
RCS file: /cvsroot/texi2html/texi2html/ChangeLog,v
retrieving revision 1.361
retrieving revision 1.362
diff -u -b -r1.361 -r1.362
--- ChangeLog 9 Jan 2009 16:55:19 -0000 1.361
+++ ChangeLog 9 Jan 2009 21:20:00 -0000 1.362
@@ -2,6 +2,8 @@
* texi2html.pl: (dir) is added as up node only if automatic
directions are used.
+ * texi2html.pl, texi2html.init, examples/*: Handle the raw
+ formats during output formatting, not in the first pass.
2009-01-08 Patrice Dumas <address@hidden>
Index: TODO
===================================================================
RCS file: /cvsroot/texi2html/texi2html/TODO,v
retrieving revision 1.79
retrieving revision 1.80
diff -u -b -r1.79 -r1.80
--- TODO 8 Jan 2009 00:21:33 -0000 1.79
+++ TODO 9 Jan 2009 21:20:01 -0000 1.80
@@ -404,9 +404,6 @@
* add as a filter when things come from STDIN, and there is no file in
argument. If there is no setfilename, file name is stdin.info.
-* @FORMAT like @tex, @html... should be kept with -E and certainly handled
- in formatting function. Cf test/macros/out/texi_cond for a wrong output.
-
* texi2html doesn't understand `-o /dev/null'. It aborts with
*** /dev/ not writable
report from Werner LEMBERG.
Index: texi2html.init
===================================================================
RCS file: /cvsroot/texi2html/texi2html/texi2html.init,v
retrieving revision 1.175
retrieving revision 1.176
diff -u -b -r1.175 -r1.176
--- texi2html.init 8 Jan 2009 00:21:33 -0000 1.175
+++ texi2html.init 9 Jan 2009 21:20:01 -0000 1.176
@@ -12,7 +12,7 @@
# Afterwards, load the file with command-line
# option -init-file <your_init_file>
#
-# $Id: texi2html.init,v 1.175 2009/01/08 00:21:33 pertusus Exp $
+# $Id: texi2html.init,v 1.176 2009/01/09 21:20:01 pertusus Exp $
######################################################################
# The following variables can also be set by command-line options
@@ -3975,10 +3975,10 @@
'ifxml' => 0,
'ifhtml' => 0,
'ifdocbook' => 0,
- 'html' => 0,
- 'tex' => 0,
- 'xml' => 0,
- 'docbook' => 0,
+# 'html' => 0,
+# 'tex' => 0,
+# 'xml' => 0,
+# 'docbook' => 0,
'titlepage' => 1,
'documentdescription' => 1,
'copying' => 1,
@@ -6015,20 +6015,25 @@
{
my $style = shift;
my $text = shift;
- if ($style eq 'verbatim' or $style eq 'tex')
+ my $expanded = 1 if (grep {$style eq $_} @EXPAND);
+ if ($style eq 'verbatim' or ($style eq 'tex' and $expanded))
{
return "<pre class=\"$style\">" . &$protect_text($text) . '</pre>';
}
- elsif ($style eq 'html')
+ elsif ($style eq 'html' and $expanded)
{
chomp ($text);
return $text;
}
- else
+ elsif ($expanded)
{
- main::echo_warn ("Raw style $style not handled");
+ main::echo_warn ("Raw $style not handled especially, but expanded");
return &$protect_text($text);
}
+ else
+ {
+ return '';
+ }
}
# raw environment when removing texi (in comments)
@@ -6036,7 +6041,11 @@
{
my $style = shift;
my $text = shift;
+ if ($style eq 'verbatim' or grep {$style eq $_} @EXPAND)
+ {
return $text;
+ }
+ return '';
}
# This function formats a footnote reference and the footnote text associated
Index: texi2html.pl
===================================================================
RCS file: /cvsroot/texi2html/texi2html/texi2html.pl,v
retrieving revision 1.258
retrieving revision 1.259
diff -u -b -r1.258 -r1.259
--- texi2html.pl 9 Jan 2009 16:55:19 -0000 1.258
+++ texi2html.pl 9 Jan 2009 21:20:04 -0000 1.259
@@ -74,7 +74,7 @@
}
# CVS version:
-# $Id: texi2html.pl,v 1.258 2009/01/09 16:55:19 pertusus Exp $
+# $Id: texi2html.pl,v 1.259 2009/01/09 21:20:04 pertusus Exp $
# Homepage:
my $T2H_HOMEPAGE = "http://www.nongnu.org/texi2html/";;
@@ -3120,6 +3120,7 @@
}
}
+#Â don't set set_no_line_macro for raw EXPAND formats
foreach my $key (keys(%Texi2HTML::Config::texi_formats_map))
{
unless ($Texi2HTML::Config::texi_formats_map{$key} eq 'raw')
@@ -3129,6 +3130,20 @@
}
}
+#Â the remaining (not EXPAND) raw formats are set as 'raw' such that
+# they are propagated to formatting functions, but
+#Â they don't start paragraphs or preformatted.
+foreach my $raw (@raw_regions)
+{
+ if (!defined($Texi2HTML::Config::texi_formats_map{$raw}))
+ {
+ $Texi2HTML::Config::texi_formats_map{$raw} = 'raw';
+ $Texi2HTML::Config::format_in_paragraph{$raw} = 1;
+ set_no_line_macro($raw, 1);
+ set_no_line_macro("end $raw", 1);
+ }
+}
+
# handle ifnot regions
foreach my $region (keys (%Texi2HTML::Config::texi_formats_map))
{
@@ -10500,20 +10515,15 @@
if ($cline =~ /^(.*?)address@hidden([a-zA-Z][\w-]*)/o and ($2 eq
$tag))
{
$cline =~ s/^(.*?)(address@hidden)//;
- # we add it even if 'ignored', it'll be discarded when there is
- # the @end
+ # we add it even if 'ignored', it'll be discarded just below
+ # with the @end
add_prev ($text, $stack, $1);
my $end = $2;
my $style = pop @$stack;
- if ($style->{'text'} !~ /^\s*$/ or $state->{'arg_expansion'})
- # FIXME if 'arg_expansion' and also 'ignored' is true,
- # theoretically we should keep
- # what is in the raw format however
- # it will be removed later anyway
- {# ARG_EXPANSION
+ #Â if 'arg_expansion' and 'ignored' are both true text
+ # is ignored.
add_prev ($text, $stack, $style->{'text'} . $end) unless
($state->{'ignored'});
delete $state->{'raw'};
- }
next;
}
else
@@ -10749,63 +10759,6 @@
next if $macro_kept;
return if ($cline =~ /^\s*$/);
}
-# elsif ($macro eq 'definfoenclose')
-# {
-# die "Not here definfoenclose expansion";
-# # FIXME if 'ignored' or 'arg_expansion' maybe we could parse
-# # the args anyway and don't take away the whole line?
-#
-# # as in the makeinfo doc 'definfoenclose' may override
-# # texinfo @-commands like @i. It is what we do here.
-# if ($state->{'arg_expansion'})
-# {
-# add_prev($text, $stack, "address@hidden" . $cline);
-# return;
-# }
-# return if ($state->{'ignored'});
-# if ($cline =~ s/^\s+([a-z]+)\s*,\s*([^\s]+)\s*,\s*([^\s]+)//)
-# {
-# $info_enclose{$1} = [ $2, $3 ];
-# }
-# else
-# {
-# echo_error("Bad address@hidden", $line_nr);
-# }
-# return if ($cline =~ /^\s*$/);
-# $cline =~ s/^\s*//;
-# }
-# elsif ($macro eq 'include')
-# {
-# die "Not here include expansion";
-# if ($state->{'arg_expansion'})
-# {
-# add_prev($text, $stack, "address@hidden" . $cline);
-# return;
-# }
-# return if ($state->{'ignored'});
-# #if (s/^\s+([\/\w.+-]+)//o)
-# if ($cline =~ s/^(\s+)(.*)//o)
-# {
-# my $file_name = $2;
-# $file_name =~ s/\s*$//;
-# my $file = locate_include_file($file_name);
-# if (defined($file))
-# {
-# open_file($file, $line_nr);
-# print STDERR "# including $file\n" if $T2H_VERBOSE;
-# }
-# else
-# {
-# echo_error ("Can't find $file_name, skipping",
$line_nr);
-# }
-# }
-# else
-# {
-# echo_error ("Bad include line: $cline", $line_nr);
-# return;
-# }
-# return;
-# }
elsif ($macro eq 'value')
{
if ($cline =~ s/^{($VARRE)}//)
@@ -11890,6 +11843,7 @@
return if ($cline =~ /^\s*$/);
}
delete $state->{'raw'};
+ return if (($cline =~ /^\s*$/) and $state->{'remove_texi'});
next;
}
else
Index: examples/docbook.init
===================================================================
RCS file: /cvsroot/texi2html/texi2html/examples/docbook.init,v
retrieving revision 1.16
retrieving revision 1.17
diff -u -b -r1.16 -r1.17
--- examples/docbook.init 8 Jan 2009 00:21:35 -0000 1.16
+++ examples/docbook.init 9 Jan 2009 21:20:05 -0000 1.17
@@ -1448,7 +1448,8 @@
{
return docbook_add_id('screen').'>' . &$protect_text($text) .
'</screen>';
}
- elsif ($style eq 'docbook')
+ return '' unless (grep {$style eq $_} @EXPAND);
+ if ($style eq 'docbook')
{
chomp ($text);
return $text;
Index: examples/html32.init
===================================================================
RCS file: /cvsroot/texi2html/texi2html/examples/html32.init,v
retrieving revision 1.22
retrieving revision 1.23
diff -u -b -r1.22 -r1.23
--- examples/html32.init 27 Dec 2008 20:53:25 -0000 1.22
+++ examples/html32.init 9 Jan 2009 21:20:07 -0000 1.23
@@ -218,19 +218,22 @@
{
my $style = shift;
my $text = shift;
- if ($style eq 'verbatim' or $style eq 'tex')
+ my $expanded = 1 if (grep {$style eq $_} @EXPAND);
+ if ($style eq 'verbatim' or ($style eq 'tex' and $expanded))
{
return "<pre>" . &$protect_text($text) . '</pre>';
}
- elsif ($style eq 'html')
+ elsif ($style eq 'html' and $expanded)
{
+ chomp ($text);
return $text;
}
- else
+ elsif ($expanded)
{
warn "$WARN (bug) unknown style $style\n";
return &$protect_text($text);
}
+ return '';
}
# a whole menu
Index: examples/roff.init
===================================================================
RCS file: /cvsroot/texi2html/texi2html/examples/roff.init,v
retrieving revision 1.26
retrieving revision 1.27
diff -u -b -r1.26 -r1.27
--- examples/roff.init 14 Nov 2008 22:34:35 -0000 1.26
+++ examples/roff.init 9 Jan 2009 21:20:08 -0000 1.27
@@ -997,16 +997,18 @@
{
my $style = shift;
my $text = shift;
- if ($style eq 'verbatim' or $style eq 'tex' or $style eq 'html')
+ my $expanded = 1 if (grep {$style eq $_} @EXPAND);
+ if ($style eq 'verbatim' or ($expanded and ($style eq 'tex' or $style eq
'html')))
{
chomp ($text);
return ".(l M\n\\fR\\&\\f(CW" . &$protect_text($text) . "\\fR\n.)l\n" ;
}
- else
+ elsif ($expanded)
{
warn "$WARN (bug) unknown style $style\n";
return &$protect_text($text);
}
+ return '';
}
# This function formats a footnote reference and the footnote text associated
Index: examples/xml.init
===================================================================
RCS file: /cvsroot/texi2html/texi2html/examples/xml.init,v
retrieving revision 1.12
retrieving revision 1.13
diff -u -b -r1.12 -r1.13
--- examples/xml.init 27 Dec 2008 20:53:25 -0000 1.12
+++ examples/xml.init 9 Jan 2009 21:20:09 -0000 1.13
@@ -1157,7 +1157,8 @@
{
return '<verbatim xml:space="preserve">' . &$protect_text($text) .
'</verbatim>';
}
- elsif ($style eq 'xml')
+ return '' unless (grep {$style eq $_} @EXPAND);
+ if ($style eq 'xml')
{
chomp ($text);
return $text;
Index: test/coverage/res/formatting/formatting.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/coverage/res/formatting/formatting.html,v
retrieving revision 1.17
retrieving revision 1.18
diff -u -b -r1.17 -r1.18
--- test/coverage/res/formatting/formatting.html 8 Jan 2009 00:21:35
-0000 1.17
+++ test/coverage/res/formatting/formatting.html 9 Jan 2009 21:20:10
-0000 1.18
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/coverage/res/texi_formatting/formatting.passfirst
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/coverage/res/texi_formatting/formatting.passfirst,v
retrieving revision 1.7
retrieving revision 1.8
diff -u -b -r1.7 -r1.8
--- test/coverage/res/texi_formatting/formatting.passfirst 11 Nov 2008
00:52:26 -0000 1.7
+++ test/coverage/res/texi_formatting/formatting.passfirst 9 Jan 2009
21:20:12 -0000 1.8
@@ -586,12 +586,21 @@
formatting.texi(mymacro,38) in verbatim ''
formatting.texi(mymacro,38) @end verbatim
formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @xml
+formatting.texi(mymacro,38) <para> xml para </para> ''
+formatting.texi(mymacro,38) @end xml
formatting.texi(mymacro,38)
formatting.texi(mymacro,38) @html
formatting.texi(mymacro,38) html ''
formatting.texi(mymacro,38) @end html
formatting.texi(mymacro,38)
-formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @tex
+formatting.texi(mymacro,38) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,38) @end tex
+formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @docbook
+formatting.texi(mymacro,38) docbook ''
+formatting.texi(mymacro,38) @end docbook
formatting.texi(mymacro,38)
formatting.texi(mymacro,38) @majorheading majorheading
formatting.texi(mymacro,38)
@@ -1286,12 +1295,21 @@
formatting.texi(mymacro,42) in verbatim ''
formatting.texi(mymacro,42) @end verbatim
formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @xml
+formatting.texi(mymacro,42) <para> xml para </para> ''
+formatting.texi(mymacro,42) @end xml
formatting.texi(mymacro,42)
formatting.texi(mymacro,42) @html
formatting.texi(mymacro,42) html ''
formatting.texi(mymacro,42) @end html
formatting.texi(mymacro,42)
-formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @tex
+formatting.texi(mymacro,42) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,42) @end tex
+formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @docbook
+formatting.texi(mymacro,42) docbook ''
+formatting.texi(mymacro,42) @end docbook
formatting.texi(mymacro,42)
formatting.texi(mymacro,42) @majorheading majorheading
formatting.texi(mymacro,42)
Index: test/coverage/res/texi_formatting/formatting.passtexi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/coverage/res/texi_formatting/formatting.passtexi,v
retrieving revision 1.7
retrieving revision 1.8
diff -u -b -r1.7 -r1.8
--- test/coverage/res/texi_formatting/formatting.passtexi 11 Nov 2008
00:52:26 -0000 1.7
+++ test/coverage/res/texi_formatting/formatting.passtexi 9 Jan 2009
21:20:14 -0000 1.8
@@ -583,12 +583,21 @@
formatting.texi(mymacro,18) in verbatim ''
formatting.texi(mymacro,18) @end verbatim
formatting.texi(mymacro,18)
+formatting.texi(mymacro,18) @xml
+formatting.texi(mymacro,18) <para> xml para </para> ''
+formatting.texi(mymacro,18) @end xml
formatting.texi(mymacro,18)
formatting.texi(mymacro,18) @html
formatting.texi(mymacro,18) html ''
formatting.texi(mymacro,18) @end html
formatting.texi(mymacro,18)
-formatting.texi(mymacro,18)
+formatting.texi(mymacro,18) @tex
+formatting.texi(mymacro,18) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,18) @end tex
+formatting.texi(mymacro,18)
+formatting.texi(mymacro,18) @docbook
+formatting.texi(mymacro,18) docbook ''
+formatting.texi(mymacro,18) @end docbook
formatting.texi(mymacro,18)
formatting.texi(mymacro,18) @majorheading majorheading
formatting.texi(mymacro,18)
@@ -1290,12 +1299,21 @@
formatting.texi(mymacro,28) in verbatim ''
formatting.texi(mymacro,28) @end verbatim
formatting.texi(mymacro,28)
+formatting.texi(mymacro,28) @xml
+formatting.texi(mymacro,28) <para> xml para </para> ''
+formatting.texi(mymacro,28) @end xml
formatting.texi(mymacro,28)
formatting.texi(mymacro,28) @html
formatting.texi(mymacro,28) html ''
formatting.texi(mymacro,28) @end html
formatting.texi(mymacro,28)
-formatting.texi(mymacro,28)
+formatting.texi(mymacro,28) @tex
+formatting.texi(mymacro,28) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,28) @end tex
+formatting.texi(mymacro,28)
+formatting.texi(mymacro,28) @docbook
+formatting.texi(mymacro,28) docbook ''
+formatting.texi(mymacro,28) @end docbook
formatting.texi(mymacro,28)
formatting.texi(mymacro,28) @majorheading majorheading
formatting.texi(mymacro,28)
@@ -1997,12 +2015,21 @@
formatting.texi(mymacro,38) in verbatim ''
formatting.texi(mymacro,38) @end verbatim
formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @xml
+formatting.texi(mymacro,38) <para> xml para </para> ''
+formatting.texi(mymacro,38) @end xml
formatting.texi(mymacro,38)
formatting.texi(mymacro,38) @html
formatting.texi(mymacro,38) html ''
formatting.texi(mymacro,38) @end html
formatting.texi(mymacro,38)
-formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @tex
+formatting.texi(mymacro,38) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,38) @end tex
+formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @docbook
+formatting.texi(mymacro,38) docbook ''
+formatting.texi(mymacro,38) @end docbook
formatting.texi(mymacro,38)
formatting.texi(mymacro,38) @majorheading majorheading
formatting.texi(mymacro,38)
@@ -2698,12 +2725,21 @@
formatting.texi(mymacro,42) in verbatim ''
formatting.texi(mymacro,42) @end verbatim
formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @xml
+formatting.texi(mymacro,42) <para> xml para </para> ''
+formatting.texi(mymacro,42) @end xml
formatting.texi(mymacro,42)
formatting.texi(mymacro,42) @html
formatting.texi(mymacro,42) html ''
formatting.texi(mymacro,42) @end html
formatting.texi(mymacro,42)
-formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @tex
+formatting.texi(mymacro,42) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,42) @end tex
+formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @docbook
+formatting.texi(mymacro,42) docbook ''
+formatting.texi(mymacro,42) @end docbook
formatting.texi(mymacro,42)
formatting.texi(mymacro,42) @majorheading majorheading
formatting.texi(mymacro,42)
Index: test/coverage/res/texi_formatting/formatting.texi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/coverage/res/texi_formatting/formatting.texi,v
retrieving revision 1.7
retrieving revision 1.8
diff -u -b -r1.7 -r1.8
--- test/coverage/res/texi_formatting/formatting.texi 11 Nov 2008 00:52:26
-0000 1.7
+++ test/coverage/res/texi_formatting/formatting.texi 9 Jan 2009 21:20:15
-0000 1.8
@@ -584,12 +584,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
@@ -1291,12 +1300,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
@@ -1998,12 +2016,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
@@ -2699,12 +2726,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
Index: test/coverage/res/texi_imbrications/imbrications.passfirst
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/coverage/res/texi_imbrications/imbrications.passfirst,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/coverage/res/texi_imbrications/imbrications.passfirst 18 Aug 2008
18:02:13 -0000 1.1
+++ test/coverage/res/texi_imbrications/imbrications.passfirst 9 Jan 2009
21:20:16 -0000 1.2
@@ -18,6 +18,15 @@
imbrications.texi(,19)
imbrications.texi(,20) @cindex elem
imbrications.texi(,21)
+imbrications.texi(,22) @tex
+imbrications.texi(,23)
+imbrications.texi(,24) $$
+imbrications.texi(,25) \chi^2 = \sum_{i=1}^N
+imbrications.texi(,26) \left(y_i - (a + b x_i)
+imbrications.texi(,27) \over \sigma_i\right)^2
+imbrications.texi(,28) $$
+imbrications.texi(,29)
+imbrications.texi(,30) @end tex
imbrications.texi(,31)
imbrications.texi(,32) @node Second node
imbrications.texi(,33) @chapter second
@@ -113,6 +122,15 @@
imbrications.texi(,123)
imbrications.texi(,124) two line breaks
imbrications.texi(,125) Tex doesn't like math in @@example
+imbrications.texi(,127) @tex
+imbrications.texi(,128)
+imbrications.texi(,129) $$
+imbrications.texi(,130) \chi^2 = \sum_{i=1}^N
+imbrications.texi(,131) \left(y_i - (a + b x_i)
+imbrications.texi(,132) \over \sigma_i\right)^2
+imbrications.texi(,133) $$
+imbrications.texi(,134)
+imbrications.texi(,135) @end tex
imbrications.texi(,137) @cindex index in example
imbrications.texi(,138)
imbrications.texi(,139) Now a content within example
Index: test/coverage/res/texi_imbrications/imbrications.passtexi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/coverage/res/texi_imbrications/imbrications.passtexi,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/coverage/res/texi_imbrications/imbrications.passtexi 18 Aug 2008
18:02:13 -0000 1.1
+++ test/coverage/res/texi_imbrications/imbrications.passtexi 9 Jan 2009
21:20:17 -0000 1.2
@@ -18,6 +18,15 @@
imbrications.texi(,19)
imbrications.texi(,20) @cindex elem
imbrications.texi(,21)
+imbrications.texi(,22) @tex
+imbrications.texi(,23)
+imbrications.texi(,24) $$
+imbrications.texi(,25) \chi^2 = \sum_{i=1}^N
+imbrications.texi(,26) \left(y_i - (a + b x_i)
+imbrications.texi(,27) \over \sigma_i\right)^2
+imbrications.texi(,28) $$
+imbrications.texi(,29)
+imbrications.texi(,30) @end tex
imbrications.texi(,31)
imbrications.texi(,32) @node Second node
imbrications.texi(,33) @chapter second
@@ -113,6 +122,15 @@
imbrications.texi(,123)
imbrications.texi(,124) two line breaks
imbrications.texi(,125) Tex doesn't like math in @@example
+imbrications.texi(,127) @tex
+imbrications.texi(,128)
+imbrications.texi(,129) $$
+imbrications.texi(,130) \chi^2 = \sum_{i=1}^N
+imbrications.texi(,131) \left(y_i - (a + b x_i)
+imbrications.texi(,132) \over \sigma_i\right)^2
+imbrications.texi(,133) $$
+imbrications.texi(,134)
+imbrications.texi(,135) @end tex
imbrications.texi(,137) @cindex index in example
imbrications.texi(,138)
imbrications.texi(,139) Now a content within example
Index: test/coverage/res/texi_imbrications/imbrications.texi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/coverage/res/texi_imbrications/imbrications.texi,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/coverage/res/texi_imbrications/imbrications.texi 18 Aug 2008
18:02:14 -0000 1.1
+++ test/coverage/res/texi_imbrications/imbrications.texi 9 Jan 2009
21:20:18 -0000 1.2
@@ -19,6 +19,15 @@
@cindex elem
address@hidden
+
+$$
+\chi^2 = \sum_{i=1}^N
+\left(y_i - (a + b x_i)
+\over \sigma_i\right)^2
+$$
+
address@hidden tex
@node Second node
@chapter second
@@ -114,6 +123,15 @@
two line breaks
Tex doesn't like math in @@example
address@hidden
+
+$$
+\chi^2 = \sum_{i=1}^N
+\left(y_i - (a + b x_i)
+\over \sigma_i\right)^2
+$$
+
address@hidden tex
@cindex index in example
Now a content within example
Index: test/encodings/res/formatting_converted_to_utf8/formatting.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/encodings/res/formatting_converted_to_utf8/formatting.html,v
retrieving revision 1.17
retrieving revision 1.18
diff -u -b -r1.17 -r1.18
--- test/encodings/res/formatting_converted_to_utf8/formatting.html 8 Jan
2009 00:21:36 -0000 1.17
+++ test/encodings/res/formatting_converted_to_utf8/formatting.html 9 Jan
2009 21:20:19 -0000 1.18
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/encodings/res/formatting_fr/formatting.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/encodings/res/formatting_fr/formatting.html,v
retrieving revision 1.17
retrieving revision 1.18
diff -u -b -r1.17 -r1.18
--- test/encodings/res/formatting_fr/formatting.html 8 Jan 2009 00:21:36
-0000 1.17
+++ test/encodings/res/formatting_fr/formatting.html 9 Jan 2009 21:20:21
-0000 1.18
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/encodings/res/formatting_fr_icons/formatting.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/encodings/res/formatting_fr_icons/formatting.html,v
retrieving revision 1.17
retrieving revision 1.18
diff -u -b -r1.17 -r1.18
--- test/encodings/res/formatting_fr_icons/formatting.html 8 Jan 2009
00:21:36 -0000 1.17
+++ test/encodings/res/formatting_fr_icons/formatting.html 9 Jan 2009
21:20:22 -0000 1.18
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/formatting/res/formatting_weird_quotes/formatting.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/formatting/res/formatting_weird_quotes/formatting.html,v
retrieving revision 1.18
retrieving revision 1.19
diff -u -b -r1.18 -r1.19
--- test/formatting/res/formatting_weird_quotes/formatting.html 8 Jan 2009
00:21:37 -0000 1.18
+++ test/formatting/res/formatting_weird_quotes/formatting.html 9 Jan 2009
21:20:22 -0000 1.19
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/formatting/res/texi_block_EOL/block_EOL.passfirst
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/formatting/res/texi_block_EOL/block_EOL.passfirst,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/formatting/res/texi_block_EOL/block_EOL.passfirst 18 Aug 2008
18:03:13 -0000 1.1
+++ test/formatting/res/texi_block_EOL/block_EOL.passfirst 9 Jan 2009
21:20:23 -0000 1.2
@@ -41,15 +41,28 @@
block_EOL.texi(,42) @heading tex
block_EOL.texi(,43)
block_EOL.texi(,44) Block commands on a line
+block_EOL.texi(,45) @tex
+block_EOL.texi(,46) in block
+block_EOL.texi(,47) @end tex
block_EOL.texi(,48) end commands on a line.
block_EOL.texi(,49)
-block_EOL.texi(,53) Before the opening command end commands on a line.
+block_EOL.texi(,50) Before the opening command @tex
+block_EOL.texi(,51) in block
+block_EOL.texi(,52) @end tex
+block_EOL.texi(,53) end commands on a line.
block_EOL.texi(,54)
-block_EOL.texi(,57) Before the opening command after the closing command.
+block_EOL.texi(,55) Before the opening command @tex
+block_EOL.texi(,56) in block
+block_EOL.texi(,57) @end tex after the closing command.
block_EOL.texi(,58)
-block_EOL.texi(,62) Before the opening command . A symbol on a line.
+block_EOL.texi(,59) Before the opening command @tex
+block_EOL.texi(,60) in block
+block_EOL.texi(,61) @end tex
+block_EOL.texi(,62) . A symbol on a line.
block_EOL.texi(,63)
-block_EOL.texi(,66) Before the opening command . A symbol after the closing
command.
+block_EOL.texi(,64) Before the opening command @tex
+block_EOL.texi(,65) in block
+block_EOL.texi(,66) @end tex. A symbol after the closing command.
block_EOL.texi(,67)
block_EOL.texi(,68)
block_EOL.texi(,69) @heading verbatim
Index: test/formatting/res/texi_block_EOL/block_EOL.passtexi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/formatting/res/texi_block_EOL/block_EOL.passtexi,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/formatting/res/texi_block_EOL/block_EOL.passtexi 18 Aug 2008
18:03:13 -0000 1.1
+++ test/formatting/res/texi_block_EOL/block_EOL.passtexi 9 Jan 2009
21:20:25 -0000 1.2
@@ -41,15 +41,28 @@
block_EOL.texi(,42) @heading tex
block_EOL.texi(,43)
block_EOL.texi(,44) Block commands on a line
+block_EOL.texi(,45) @tex
+block_EOL.texi(,46) in block
+block_EOL.texi(,47) @end tex
block_EOL.texi(,48) end commands on a line.
block_EOL.texi(,49)
-block_EOL.texi(,53) Before the opening command block_EOL.texi(,53) end
commands on a line.
+block_EOL.texi(,50) Before the opening command @tex
+block_EOL.texi(,51) in block
+block_EOL.texi(,52) @end tex
+block_EOL.texi(,53) end commands on a line.
block_EOL.texi(,54)
-block_EOL.texi(,57) Before the opening command block_EOL.texi(,57) after the
closing command.
+block_EOL.texi(,55) Before the opening command @tex
+block_EOL.texi(,56) in block
+block_EOL.texi(,57) @end tex after the closing command.
block_EOL.texi(,58)
-block_EOL.texi(,62) Before the opening command block_EOL.texi(,62) . A symbol
on a line.
+block_EOL.texi(,59) Before the opening command @tex
+block_EOL.texi(,60) in block
+block_EOL.texi(,61) @end tex
+block_EOL.texi(,62) . A symbol on a line.
block_EOL.texi(,63)
-block_EOL.texi(,66) Before the opening command block_EOL.texi(,66) . A symbol
after the closing command.
+block_EOL.texi(,64) Before the opening command @tex
+block_EOL.texi(,65) in block
+block_EOL.texi(,66) @end tex. A symbol after the closing command.
block_EOL.texi(,67)
block_EOL.texi(,68)
block_EOL.texi(,69) @heading verbatim
Index: test/formatting/res/texi_block_EOL/block_EOL.texi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/formatting/res/texi_block_EOL/block_EOL.texi,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/formatting/res/texi_block_EOL/block_EOL.texi 18 Aug 2008 18:03:13
-0000 1.1
+++ test/formatting/res/texi_block_EOL/block_EOL.texi 9 Jan 2009 21:20:26
-0000 1.2
@@ -42,15 +42,28 @@
@heading tex
Block commands on a line
address@hidden
+in block
address@hidden tex
end commands on a line.
-Before the opening command end commands on a line.
+Before the opening command @tex
+in block
address@hidden tex
+end commands on a line.
-Before the opening command after the closing command.
+Before the opening command @tex
+in block
address@hidden tex after the closing command.
-Before the opening command . A symbol on a line.
+Before the opening command @tex
+in block
address@hidden tex
+. A symbol on a line.
-Before the opening command . A symbol after the closing command.
+Before the opening command @tex
+in block
address@hidden tex. A symbol after the closing command.
@heading verbatim
Index: test/formatting/res/texi_formatting/formatting.passfirst
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/formatting/res/texi_formatting/formatting.passfirst,v
retrieving revision 1.4
retrieving revision 1.5
diff -u -b -r1.4 -r1.5
--- test/formatting/res/texi_formatting/formatting.passfirst 1 Jan 2009
22:35:21 -0000 1.4
+++ test/formatting/res/texi_formatting/formatting.passfirst 9 Jan 2009
21:20:27 -0000 1.5
@@ -573,12 +573,21 @@
formatting.texi(mymacro,38) in verbatim ''
formatting.texi(mymacro,38) @end verbatim
formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @xml
+formatting.texi(mymacro,38) <para> xml para </para> ''
+formatting.texi(mymacro,38) @end xml
formatting.texi(mymacro,38)
formatting.texi(mymacro,38) @html
formatting.texi(mymacro,38) html ''
formatting.texi(mymacro,38) @end html
formatting.texi(mymacro,38)
-formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @tex
+formatting.texi(mymacro,38) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,38) @end tex
+formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @docbook
+formatting.texi(mymacro,38) docbook ''
+formatting.texi(mymacro,38) @end docbook
formatting.texi(mymacro,38)
formatting.texi(mymacro,38) @majorheading majorheading
formatting.texi(mymacro,38)
@@ -1154,12 +1163,21 @@
formatting.texi(mymacro,42) in verbatim ''
formatting.texi(mymacro,42) @end verbatim
formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @xml
+formatting.texi(mymacro,42) <para> xml para </para> ''
+formatting.texi(mymacro,42) @end xml
formatting.texi(mymacro,42)
formatting.texi(mymacro,42) @html
formatting.texi(mymacro,42) html ''
formatting.texi(mymacro,42) @end html
formatting.texi(mymacro,42)
-formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @tex
+formatting.texi(mymacro,42) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,42) @end tex
+formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @docbook
+formatting.texi(mymacro,42) docbook ''
+formatting.texi(mymacro,42) @end docbook
formatting.texi(mymacro,42)
formatting.texi(mymacro,42) @majorheading majorheading
formatting.texi(mymacro,42)
Index: test/formatting/res/texi_formatting/formatting.passtexi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/formatting/res/texi_formatting/formatting.passtexi,v
retrieving revision 1.4
retrieving revision 1.5
diff -u -b -r1.4 -r1.5
--- test/formatting/res/texi_formatting/formatting.passtexi 1 Jan 2009
22:35:22 -0000 1.4
+++ test/formatting/res/texi_formatting/formatting.passtexi 9 Jan 2009
21:20:28 -0000 1.5
@@ -570,12 +570,21 @@
formatting.texi(mymacro,18) in verbatim ''
formatting.texi(mymacro,18) @end verbatim
formatting.texi(mymacro,18)
+formatting.texi(mymacro,18) @xml
+formatting.texi(mymacro,18) <para> xml para </para> ''
+formatting.texi(mymacro,18) @end xml
formatting.texi(mymacro,18)
formatting.texi(mymacro,18) @html
formatting.texi(mymacro,18) html ''
formatting.texi(mymacro,18) @end html
formatting.texi(mymacro,18)
+formatting.texi(mymacro,18) @tex
+formatting.texi(mymacro,18) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,18) @end tex
formatting.texi(mymacro,18)
+formatting.texi(mymacro,18) @docbook
+formatting.texi(mymacro,18) docbook ''
+formatting.texi(mymacro,18) @end docbook
formatting.texi(mymacro,18)
formatting.texi(mymacro,18) @majorheading majorheading
formatting.texi(mymacro,18)
@@ -1158,12 +1167,21 @@
formatting.texi(mymacro,28) in verbatim ''
formatting.texi(mymacro,28) @end verbatim
formatting.texi(mymacro,28)
+formatting.texi(mymacro,28) @xml
+formatting.texi(mymacro,28) <para> xml para </para> ''
+formatting.texi(mymacro,28) @end xml
formatting.texi(mymacro,28)
formatting.texi(mymacro,28) @html
formatting.texi(mymacro,28) html ''
formatting.texi(mymacro,28) @end html
formatting.texi(mymacro,28)
-formatting.texi(mymacro,28)
+formatting.texi(mymacro,28) @tex
+formatting.texi(mymacro,28) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,28) @end tex
+formatting.texi(mymacro,28)
+formatting.texi(mymacro,28) @docbook
+formatting.texi(mymacro,28) docbook ''
+formatting.texi(mymacro,28) @end docbook
formatting.texi(mymacro,28)
formatting.texi(mymacro,28) @majorheading majorheading
formatting.texi(mymacro,28)
@@ -1746,12 +1764,21 @@
formatting.texi(mymacro,38) in verbatim ''
formatting.texi(mymacro,38) @end verbatim
formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @xml
+formatting.texi(mymacro,38) <para> xml para </para> ''
+formatting.texi(mymacro,38) @end xml
formatting.texi(mymacro,38)
formatting.texi(mymacro,38) @html
formatting.texi(mymacro,38) html ''
formatting.texi(mymacro,38) @end html
formatting.texi(mymacro,38)
-formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @tex
+formatting.texi(mymacro,38) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,38) @end tex
+formatting.texi(mymacro,38)
+formatting.texi(mymacro,38) @docbook
+formatting.texi(mymacro,38) docbook ''
+formatting.texi(mymacro,38) @end docbook
formatting.texi(mymacro,38)
formatting.texi(mymacro,38) @majorheading majorheading
formatting.texi(mymacro,38)
@@ -2328,12 +2355,21 @@
formatting.texi(mymacro,42) in verbatim ''
formatting.texi(mymacro,42) @end verbatim
formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @xml
+formatting.texi(mymacro,42) <para> xml para </para> ''
+formatting.texi(mymacro,42) @end xml
formatting.texi(mymacro,42)
formatting.texi(mymacro,42) @html
formatting.texi(mymacro,42) html ''
formatting.texi(mymacro,42) @end html
formatting.texi(mymacro,42)
-formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @tex
+formatting.texi(mymacro,42) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
+formatting.texi(mymacro,42) @end tex
+formatting.texi(mymacro,42)
+formatting.texi(mymacro,42) @docbook
+formatting.texi(mymacro,42) docbook ''
+formatting.texi(mymacro,42) @end docbook
formatting.texi(mymacro,42)
formatting.texi(mymacro,42) @majorheading majorheading
formatting.texi(mymacro,42)
Index: test/formatting/res/texi_formatting/formatting.texi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/formatting/res/texi_formatting/formatting.texi,v
retrieving revision 1.4
retrieving revision 1.5
diff -u -b -r1.4 -r1.5
--- test/formatting/res/texi_formatting/formatting.texi 1 Jan 2009 22:35:22
-0000 1.4
+++ test/formatting/res/texi_formatting/formatting.texi 9 Jan 2009 21:20:28
-0000 1.5
@@ -571,12 +571,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
@@ -1159,12 +1168,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
@@ -1747,12 +1765,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
@@ -2329,12 +2356,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
Index: test/layout/res/formatting_chm/formatting.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_chm/formatting.html,v
retrieving revision 1.10
retrieving revision 1.11
diff -u -b -r1.10 -r1.11
--- test/layout/res/formatting_chm/formatting.html 1 Jan 2009 22:35:23
-0000 1.10
+++ test/layout/res/formatting_chm/formatting.html 9 Jan 2009 21:20:29
-0000 1.11
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_chm/formatting_1.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_chm/formatting_1.html,v
retrieving revision 1.5
retrieving revision 1.6
diff -u -b -r1.5 -r1.6
--- test/layout/res/formatting_chm/formatting_1.html 11 Nov 2008 13:29:14
-0000 1.5
+++ test/layout/res/formatting_chm/formatting_1.html 9 Jan 2009 21:20:30
-0000 1.6
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_chm/formatting_2.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_chm/formatting_2.html,v
retrieving revision 1.5
retrieving revision 1.6
diff -u -b -r1.5 -r1.6
--- test/layout/res/formatting_chm/formatting_2.html 11 Nov 2008 13:29:15
-0000 1.5
+++ test/layout/res/formatting_chm/formatting_2.html 9 Jan 2009 21:20:30
-0000 1.6
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_chm/formatting_3.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_chm/formatting_3.html,v
retrieving revision 1.5
retrieving revision 1.6
diff -u -b -r1.5 -r1.6
--- test/layout/res/formatting_chm/formatting_3.html 11 Nov 2008 13:29:15
-0000 1.5
+++ test/layout/res/formatting_chm/formatting_3.html 9 Jan 2009 21:20:31
-0000 1.6
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_chm/formatting_4.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_chm/formatting_4.html,v
retrieving revision 1.5
retrieving revision 1.6
diff -u -b -r1.5 -r1.6
--- test/layout/res/formatting_chm/formatting_4.html 11 Nov 2008 13:29:15
-0000 1.5
+++ test/layout/res/formatting_chm/formatting_4.html 9 Jan 2009 21:20:31
-0000 1.6
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_chm/formatting_5.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_chm/formatting_5.html,v
retrieving revision 1.7
retrieving revision 1.8
diff -u -b -r1.7 -r1.8
--- test/layout/res/formatting_chm/formatting_5.html 8 Jan 2009 00:21:38
-0000 1.7
+++ test/layout/res/formatting_chm/formatting_5.html 9 Jan 2009 21:20:32
-0000 1.8
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_chm/formatting_abt.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_chm/formatting_abt.html,v
retrieving revision 1.5
retrieving revision 1.6
diff -u -b -r1.5 -r1.6
--- test/layout/res/formatting_chm/formatting_abt.html 11 Nov 2008 13:29:15
-0000 1.5
+++ test/layout/res/formatting_chm/formatting_abt.html 9 Jan 2009 21:20:32
-0000 1.6
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_chm/formatting_ovr.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_chm/formatting_ovr.html,v
retrieving revision 1.7
retrieving revision 1.8
diff -u -b -r1.7 -r1.8
--- test/layout/res/formatting_chm/formatting_ovr.html 8 Jan 2009 00:21:38
-0000 1.7
+++ test/layout/res/formatting_chm/formatting_ovr.html 9 Jan 2009 21:20:33
-0000 1.8
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_chm/formatting_toc.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_chm/formatting_toc.html,v
retrieving revision 1.7
retrieving revision 1.8
diff -u -b -r1.7 -r1.8
--- test/layout/res/formatting_chm/formatting_toc.html 8 Jan 2009 00:21:38
-0000 1.7
+++ test/layout/res/formatting_chm/formatting_toc.html 9 Jan 2009 21:20:34
-0000 1.8
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_exotic/formatting.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_exotic/formatting.html,v
retrieving revision 1.15
retrieving revision 1.16
diff -u -b -r1.15 -r1.16
--- test/layout/res/formatting_exotic/formatting.html 1 Jan 2009 22:35:23
-0000 1.15
+++ test/layout/res/formatting_exotic/formatting.html 9 Jan 2009 21:20:35
-0000 1.16
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_exotic/formatting_1.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_exotic/formatting_1.html,v
retrieving revision 1.5
retrieving revision 1.6
diff -u -b -r1.5 -r1.6
--- test/layout/res/formatting_exotic/formatting_1.html 11 Nov 2008 13:29:16
-0000 1.5
+++ test/layout/res/formatting_exotic/formatting_1.html 9 Jan 2009 21:20:36
-0000 1.6
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_exotic/formatting_2.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_exotic/formatting_2.html,v
retrieving revision 1.11
retrieving revision 1.12
diff -u -b -r1.11 -r1.12
--- test/layout/res/formatting_exotic/formatting_2.html 11 Nov 2008 13:29:16
-0000 1.11
+++ test/layout/res/formatting_exotic/formatting_2.html 9 Jan 2009 21:20:36
-0000 1.12
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_exotic/formatting_3.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_exotic/formatting_3.html,v
retrieving revision 1.11
retrieving revision 1.12
diff -u -b -r1.11 -r1.12
--- test/layout/res/formatting_exotic/formatting_3.html 8 Jan 2009 00:21:39
-0000 1.11
+++ test/layout/res/formatting_exotic/formatting_3.html 9 Jan 2009 21:20:37
-0000 1.12
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_exotic/formatting_4.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_exotic/formatting_4.html,v
retrieving revision 1.10
retrieving revision 1.11
diff -u -b -r1.10 -r1.11
--- test/layout/res/formatting_exotic/formatting_4.html 8 Jan 2009 00:21:39
-0000 1.10
+++ test/layout/res/formatting_exotic/formatting_4.html 9 Jan 2009 21:20:37
-0000 1.11
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_exotic/formatting_abt.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_exotic/formatting_abt.html,v
retrieving revision 1.7
retrieving revision 1.8
diff -u -b -r1.7 -r1.8
--- test/layout/res/formatting_exotic/formatting_abt.html 11 Nov 2008
13:29:16 -0000 1.7
+++ test/layout/res/formatting_exotic/formatting_abt.html 9 Jan 2009
21:20:37 -0000 1.8
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_exotic/formatting_ovr.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_exotic/formatting_ovr.html,v
retrieving revision 1.9
retrieving revision 1.10
diff -u -b -r1.9 -r1.10
--- test/layout/res/formatting_exotic/formatting_ovr.html 11 Nov 2008
13:29:16 -0000 1.9
+++ test/layout/res/formatting_exotic/formatting_ovr.html 9 Jan 2009
21:20:38 -0000 1.10
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_exotic/formatting_toc.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_exotic/formatting_toc.html,v
retrieving revision 1.9
retrieving revision 1.10
diff -u -b -r1.9 -r1.10
--- test/layout/res/formatting_exotic/formatting_toc.html 11 Nov 2008
13:29:16 -0000 1.9
+++ test/layout/res/formatting_exotic/formatting_toc.html 9 Jan 2009
21:20:39 -0000 1.10
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_makeinfo/chapter.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_makeinfo/chapter.html,v
retrieving revision 1.10
retrieving revision 1.11
diff -u -b -r1.10 -r1.11
--- test/layout/res/formatting_makeinfo/chapter.html 9 Jan 2009 00:14:49
-0000 1.10
+++ test/layout/res/formatting_makeinfo/chapter.html 9 Jan 2009 21:20:39
-0000 1.11
@@ -525,12 +525,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_makeinfo/chapter2.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_makeinfo/chapter2.html,v
retrieving revision 1.12
retrieving revision 1.13
diff -u -b -r1.12 -r1.13
--- test/layout/res/formatting_makeinfo/chapter2.html 8 Jan 2009 00:21:39
-0000 1.12
+++ test/layout/res/formatting_makeinfo/chapter2.html 9 Jan 2009 21:20:40
-0000 1.13
@@ -525,12 +525,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_makeinfo/formatting_abt.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_makeinfo/formatting_abt.html,v
retrieving revision 1.6
retrieving revision 1.7
diff -u -b -r1.6 -r1.7
--- test/layout/res/formatting_makeinfo/formatting_abt.html 11 Nov 2008
13:29:17 -0000 1.6
+++ test/layout/res/formatting_makeinfo/formatting_abt.html 9 Jan 2009
21:20:41 -0000 1.7
@@ -525,12 +525,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_makeinfo/index.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_makeinfo/index.html,v
retrieving revision 1.13
retrieving revision 1.14
diff -u -b -r1.13 -r1.14
--- test/layout/res/formatting_makeinfo/index.html 9 Jan 2009 00:14:49
-0000 1.13
+++ test/layout/res/formatting_makeinfo/index.html 9 Jan 2009 21:20:41
-0000 1.14
@@ -525,12 +525,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_makeinfo/s_002d_002dect_002cion.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_makeinfo/s_002d_002dect_002cion.html,v
retrieving revision 1.9
retrieving revision 1.10
diff -u -b -r1.9 -r1.10
--- test/layout/res/formatting_makeinfo/s_002d_002dect_002cion.html 8 Jan
2009 00:21:40 -0000 1.9
+++ test/layout/res/formatting_makeinfo/s_002d_002dect_002cion.html 9 Jan
2009 21:20:42 -0000 1.10
@@ -525,12 +525,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_makeinfo/subsection.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_makeinfo/subsection.html,v
retrieving revision 1.9
retrieving revision 1.10
diff -u -b -r1.9 -r1.10
--- test/layout/res/formatting_makeinfo/subsection.html 8 Jan 2009 00:21:40
-0000 1.9
+++ test/layout/res/formatting_makeinfo/subsection.html 9 Jan 2009 21:20:42
-0000 1.10
@@ -525,12 +525,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index:
test/layout/res/formatting_makeinfo/subsubsection-_0060_0060simple_002ddouble_002d_002dthree_002d_002d_002dfour_002d_002d_002d_002d_0027_0027.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_makeinfo/subsubsection-_0060_0060simple_002ddouble_002d_002dthree_002d_002d_002dfour_002d_002d_002d_002d_0027_0027.html,v
retrieving revision 1.9
retrieving revision 1.10
diff -u -b -r1.9 -r1.10
---
test/layout/res/formatting_makeinfo/subsubsection-_0060_0060simple_002ddouble_002d_002dthree_002d_002d_002dfour_002d_002d_002d_002d_0027_0027.html
8 Jan 2009 00:21:40 -0000 1.9
+++
test/layout/res/formatting_makeinfo/subsubsection-_0060_0060simple_002ddouble_002d_002dthree_002d_002d_002dfour_002d_002d_002d_002d_0027_0027.html
9 Jan 2009 21:20:42 -0000 1.10
@@ -525,12 +525,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/formatting_regions/formatting_regions.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/formatting_regions/formatting_regions.html,v
retrieving revision 1.17
retrieving revision 1.18
diff -u -b -r1.17 -r1.18
--- test/layout/res/formatting_regions/formatting_regions.html 8 Jan 2009
00:21:40 -0000 1.17
+++ test/layout/res/formatting_regions/formatting_regions.html 9 Jan 2009
21:20:43 -0000 1.18
@@ -524,12 +524,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
Index: test/layout/res/texi_formatting_regions/formatting_regions.passfirst
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/texi_formatting_regions/formatting_regions.passfirst,v
retrieving revision 1.6
retrieving revision 1.7
diff -u -b -r1.6 -r1.7
--- test/layout/res/texi_formatting_regions/formatting_regions.passfirst
11 Nov 2008 00:52:31 -0000 1.6
+++ test/layout/res/texi_formatting_regions/formatting_regions.passfirst
9 Jan 2009 21:20:44 -0000 1.7
@@ -578,12 +578,21 @@
formatting_regions.texi(mymacro,46) in verbatim ''
formatting_regions.texi(mymacro,46) @end verbatim
formatting_regions.texi(mymacro,46)
+formatting_regions.texi(mymacro,46) @xml
+formatting_regions.texi(mymacro,46) <para> xml para </para> ''
+formatting_regions.texi(mymacro,46) @end xml
formatting_regions.texi(mymacro,46)
formatting_regions.texi(mymacro,46) @html
formatting_regions.texi(mymacro,46) html ''
formatting_regions.texi(mymacro,46) @end html
formatting_regions.texi(mymacro,46)
-formatting_regions.texi(mymacro,46)
+formatting_regions.texi(mymacro,46) @tex
+formatting_regions.texi(mymacro,46) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$ ''
+formatting_regions.texi(mymacro,46) @end tex
+formatting_regions.texi(mymacro,46)
+formatting_regions.texi(mymacro,46) @docbook
+formatting_regions.texi(mymacro,46) docbook ''
+formatting_regions.texi(mymacro,46) @end docbook
formatting_regions.texi(mymacro,46)
formatting_regions.texi(mymacro,46) @majorheading majorheading
formatting_regions.texi(mymacro,46)
@@ -1277,12 +1286,21 @@
formatting_regions.texi(mymacro,49) in verbatim ''
formatting_regions.texi(mymacro,49) @end verbatim
formatting_regions.texi(mymacro,49)
+formatting_regions.texi(mymacro,49) @xml
+formatting_regions.texi(mymacro,49) <para> xml para </para> ''
+formatting_regions.texi(mymacro,49) @end xml
formatting_regions.texi(mymacro,49)
formatting_regions.texi(mymacro,49) @html
formatting_regions.texi(mymacro,49) html ''
formatting_regions.texi(mymacro,49) @end html
formatting_regions.texi(mymacro,49)
-formatting_regions.texi(mymacro,49)
+formatting_regions.texi(mymacro,49) @tex
+formatting_regions.texi(mymacro,49) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$ ''
+formatting_regions.texi(mymacro,49) @end tex
+formatting_regions.texi(mymacro,49)
+formatting_regions.texi(mymacro,49) @docbook
+formatting_regions.texi(mymacro,49) docbook ''
+formatting_regions.texi(mymacro,49) @end docbook
formatting_regions.texi(mymacro,49)
formatting_regions.texi(mymacro,49) @majorheading majorheading
formatting_regions.texi(mymacro,49)
Index: test/layout/res/texi_formatting_regions/formatting_regions.passtexi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/texi_formatting_regions/formatting_regions.passtexi,v
retrieving revision 1.6
retrieving revision 1.7
diff -u -b -r1.6 -r1.7
--- test/layout/res/texi_formatting_regions/formatting_regions.passtexi 11 Nov
2008 00:52:31 -0000 1.6
+++ test/layout/res/texi_formatting_regions/formatting_regions.passtexi 9 Jan
2009 21:20:44 -0000 1.7
@@ -581,12 +581,21 @@
formatting_regions.texi(mymacro,28) in verbatim ''
formatting_regions.texi(mymacro,28) @end verbatim
formatting_regions.texi(mymacro,28)
+formatting_regions.texi(mymacro,28) @xml
+formatting_regions.texi(mymacro,28) <para> xml para </para> ''
+formatting_regions.texi(mymacro,28) @end xml
formatting_regions.texi(mymacro,28)
formatting_regions.texi(mymacro,28) @html
formatting_regions.texi(mymacro,28) html ''
formatting_regions.texi(mymacro,28) @end html
formatting_regions.texi(mymacro,28)
-formatting_regions.texi(mymacro,28)
+formatting_regions.texi(mymacro,28) @tex
+formatting_regions.texi(mymacro,28) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$ ''
+formatting_regions.texi(mymacro,28) @end tex
+formatting_regions.texi(mymacro,28)
+formatting_regions.texi(mymacro,28) @docbook
+formatting_regions.texi(mymacro,28) docbook ''
+formatting_regions.texi(mymacro,28) @end docbook
formatting_regions.texi(mymacro,28)
formatting_regions.texi(mymacro,28) @majorheading majorheading
formatting_regions.texi(mymacro,28)
@@ -1289,12 +1298,21 @@
formatting_regions.texi(mymacro,39) in verbatim ''
formatting_regions.texi(mymacro,39) @end verbatim
formatting_regions.texi(mymacro,39)
+formatting_regions.texi(mymacro,39) @xml
+formatting_regions.texi(mymacro,39) <para> xml para </para> ''
+formatting_regions.texi(mymacro,39) @end xml
formatting_regions.texi(mymacro,39)
formatting_regions.texi(mymacro,39) @html
formatting_regions.texi(mymacro,39) html ''
formatting_regions.texi(mymacro,39) @end html
formatting_regions.texi(mymacro,39)
-formatting_regions.texi(mymacro,39)
+formatting_regions.texi(mymacro,39) @tex
+formatting_regions.texi(mymacro,39) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$ ''
+formatting_regions.texi(mymacro,39) @end tex
+formatting_regions.texi(mymacro,39)
+formatting_regions.texi(mymacro,39) @docbook
+formatting_regions.texi(mymacro,39) docbook ''
+formatting_regions.texi(mymacro,39) @end docbook
formatting_regions.texi(mymacro,39)
formatting_regions.texi(mymacro,39) @majorheading majorheading
formatting_regions.texi(mymacro,39)
@@ -1993,12 +2011,21 @@
formatting_regions.texi(mymacro,46) in verbatim ''
formatting_regions.texi(mymacro,46) @end verbatim
formatting_regions.texi(mymacro,46)
+formatting_regions.texi(mymacro,46) @xml
+formatting_regions.texi(mymacro,46) <para> xml para </para> ''
+formatting_regions.texi(mymacro,46) @end xml
formatting_regions.texi(mymacro,46)
formatting_regions.texi(mymacro,46) @html
formatting_regions.texi(mymacro,46) html ''
formatting_regions.texi(mymacro,46) @end html
formatting_regions.texi(mymacro,46)
-formatting_regions.texi(mymacro,46)
+formatting_regions.texi(mymacro,46) @tex
+formatting_regions.texi(mymacro,46) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$ ''
+formatting_regions.texi(mymacro,46) @end tex
+formatting_regions.texi(mymacro,46)
+formatting_regions.texi(mymacro,46) @docbook
+formatting_regions.texi(mymacro,46) docbook ''
+formatting_regions.texi(mymacro,46) @end docbook
formatting_regions.texi(mymacro,46)
formatting_regions.texi(mymacro,46) @majorheading majorheading
formatting_regions.texi(mymacro,46)
@@ -2693,12 +2720,21 @@
formatting_regions.texi(mymacro,49) in verbatim ''
formatting_regions.texi(mymacro,49) @end verbatim
formatting_regions.texi(mymacro,49)
+formatting_regions.texi(mymacro,49) @xml
+formatting_regions.texi(mymacro,49) <para> xml para </para> ''
+formatting_regions.texi(mymacro,49) @end xml
formatting_regions.texi(mymacro,49)
formatting_regions.texi(mymacro,49) @html
formatting_regions.texi(mymacro,49) html ''
formatting_regions.texi(mymacro,49) @end html
formatting_regions.texi(mymacro,49)
-formatting_regions.texi(mymacro,49)
+formatting_regions.texi(mymacro,49) @tex
+formatting_regions.texi(mymacro,49) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$ ''
+formatting_regions.texi(mymacro,49) @end tex
+formatting_regions.texi(mymacro,49)
+formatting_regions.texi(mymacro,49) @docbook
+formatting_regions.texi(mymacro,49) docbook ''
+formatting_regions.texi(mymacro,49) @end docbook
formatting_regions.texi(mymacro,49)
formatting_regions.texi(mymacro,49) @majorheading majorheading
formatting_regions.texi(mymacro,49)
Index: test/layout/res/texi_formatting_regions/formatting_regions.texi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/layout/res/texi_formatting_regions/formatting_regions.texi,v
retrieving revision 1.6
retrieving revision 1.7
diff -u -b -r1.6 -r1.7
--- test/layout/res/texi_formatting_regions/formatting_regions.texi 11 Nov
2008 00:52:31 -0000 1.6
+++ test/layout/res/texi_formatting_regions/formatting_regions.texi 9 Jan
2009 21:20:44 -0000 1.7
@@ -582,12 +582,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
@@ -1290,12 +1299,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
@@ -1994,12 +2012,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
@@ -2694,12 +2721,21 @@
in verbatim ''
@end verbatim
address@hidden
+<para> xml para </para> ''
address@hidden xml
@html
html ''
@end html
-
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$ ''
address@hidden tex
+
address@hidden
+docbook ''
address@hidden docbook
@majorheading majorheading
Index: test/macros/tests.txt
===================================================================
RCS file: /cvsroot/texi2html/texi2html/test/macros/tests.txt,v
retrieving revision 1.6
retrieving revision 1.7
diff -u -b -r1.6 -r1.7
--- test/macros/tests.txt 26 Nov 2008 19:09:04 -0000 1.6
+++ test/macros/tests.txt 9 Jan 2009 21:20:45 -0000 1.7
@@ -38,9 +38,11 @@
cond_xml cond.texi --init xml.init
cond_no-ifhtml_no-ifinfo_no-iftex cond.texi --no-ifhtml --no-ifinfo --no-iftex
cond_ifhtml_ifinfo_iftex cond.texi --ifhtml --ifinfo --iftex
+cond_info cond.texi --init info.init
defcondx_Dbar defxcond.texi -D bar
defcondx_Ubar defxcond.texi -U bar
macro-at macro-at.texi
+macro-at_info macro-at.texi --init info.init
value_in_pass0_macros value_in_pass0_macros.texi -init makeinfo.init
macros_in_pass0_macros macros_in_pass0_macros.texi -init makeinfo.init
node-expand node-expand.texi -init makeinfo.init
Index: test/macros/res/texi_cond/cond.passfirst
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/macros/res/texi_cond/cond.passfirst,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/macros/res/texi_cond/cond.passfirst 16 Dec 2008 11:18:41 -0000
1.1
+++ test/macros/res/texi_cond/cond.passfirst 9 Jan 2009 21:20:45 -0000
1.2
@@ -14,6 +14,9 @@
cond.texi(,22)
cond.texi(,26)
cond.texi(,27)
+cond.texi(,28) @tex
+cond.texi(,29) This is tex text.
+cond.texi(,30) @end tex
cond.texi(,31)
cond.texi(,35)
cond.texi(,37) This is ifnottex text.
Index: test/macros/res/texi_cond/cond.passtexi
===================================================================
RCS file: /cvsroot/texi2html/texi2html/test/macros/res/texi_cond/cond.passtexi,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/macros/res/texi_cond/cond.passtexi 16 Dec 2008 11:18:41 -0000
1.1
+++ test/macros/res/texi_cond/cond.passtexi 9 Jan 2009 21:20:46 -0000
1.2
@@ -14,6 +14,9 @@
cond.texi(,22)
cond.texi(,26)
cond.texi(,27)
+cond.texi(,28) @tex
+cond.texi(,29) This is tex text.
+cond.texi(,30) @end tex
cond.texi(,31)
cond.texi(,35)
cond.texi(,37) This is ifnottex text.
Index: test/macros/res/texi_cond/cond.texi
===================================================================
RCS file: /cvsroot/texi2html/texi2html/test/macros/res/texi_cond/cond.texi,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/macros/res/texi_cond/cond.texi 16 Dec 2008 11:18:42 -0000 1.1
+++ test/macros/res/texi_cond/cond.texi 9 Jan 2009 21:20:46 -0000 1.2
@@ -15,6 +15,9 @@
address@hidden
+This is tex text.
address@hidden tex
This is ifnottex text.
Index: test/manuals/res/texi_mini_ker/mini_ker.passfirst
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/manuals/res/texi_mini_ker/mini_ker.passfirst,v
retrieving revision 1.2
retrieving revision 1.3
diff -u -b -r1.2 -r1.3
--- test/manuals/res/texi_mini_ker/mini_ker.passfirst 2 Nov 2008 00:49:19
-0000 1.2
+++ test/manuals/res/texi_mini_ker/mini_ker.passfirst 9 Jan 2009 21:20:47
-0000 1.3
@@ -179,6 +179,9 @@
mini_ker.texi(,229) @enumerate
mini_ker.texi(,230) @item Cells which are elementary models and correspond to
evolution equations
mini_ker.texi(,231) such as:
+mini_ker.texi(,241) @tex
+mini_ker.texi(,242) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
+mini_ker.texi(,243) @end tex
mini_ker.texi(,245)
mini_ker.texi(,246) @noindent @math{d eta(t)/d t = g(eta(t),phi(t))}
mini_ker.texi(,247)
@@ -195,6 +198,11 @@
mini_ker.texi(,260)
mini_ker.texi(,261)
mini_ker.texi(,262) @item Transfers which are determined by constraint
equations such as:
+mini_ker.texi(,272) @tex
+mini_ker.texi(,273) $$
+mini_ker.texi(,274) \varphi(t) = f(\eta(t),\varphi(t))
+mini_ker.texi(,275) $$
+mini_ker.texi(,276) @end tex
mini_ker.texi(,278)
mini_ker.texi(,279) @noindent @math{phi(t) = f(eta(t),phi(t))}
mini_ker.texi(,280)
@@ -222,6 +230,12 @@
mini_ker.texi(,304)
mini_ker.texi(,305)
mini_ker.texi(,319)
+mini_ker.texi(,321) @tex
+mini_ker.texi(,322) $$\pmatrix{A & B\cr
+mini_ker.texi(,323) -C^+ & I-D\cr} \pmatrix{\delta \eta\cr
+mini_ker.texi(,324) \delta \varphi\cr} = \pmatrix{\Gamma\cr
+mini_ker.texi(,325) \Omega\cr}$$
+mini_ker.texi(,326) @end tex
mini_ker.texi(,328)
mini_ker.texi(,329) The blocks appearing in the Jacobian matrix are
constructed with partial derivative
mini_ker.texi(,330) of @math{f} and @math{g}, and with @math{\delta t}. From
this system the
@@ -286,6 +300,11 @@
mini_ker.texi(,389) @cindex TEF
mini_ker.texi(,390)
mini_ker.texi(,391) The general @acronym{TEF} system writes:
+mini_ker.texi(,402) @tex
+mini_ker.texi(,403) $$\eqalign{\partial_t \eta (t) &= g(\eta(t),\varphi(t))\cr
+mini_ker.texi(,404) \varphi(t) &= f(\eta(t),\varphi(t))\cr
+mini_ker.texi(,405) }$$
+mini_ker.texi(,406) @end tex
mini_ker.texi(,408)
mini_ker.texi(,409) @noindent @math{d eta(t)/d t = g(eta(t),phi(t))@*
mini_ker.texi(,410) phi(t) = f(eta(t),phi(t))}
@@ -295,6 +314,13 @@
mini_ker.texi(,416) model of Lotka-Volterra is used.
mini_ker.texi(,417) This model can be written in the following @acronym{TEF}
form:
mini_ker.texi(,418)
+mini_ker.texi(,436) @tex
+mini_ker.texi(,437) $$\left\{\eqalign{\partial_t \eta _{prey} &= a \eta
_{prey} - a \varphi _{meet} \cr
+mini_ker.texi(,438) \partial_t \eta _{pred} &= -c \eta _{pred} + c \varphi
_{meet}\cr}\right.$$
+mini_ker.texi(,439) @end tex
+mini_ker.texi(,440) @tex
+mini_ker.texi(,441) $$\varphi _{meet} = \eta _{prey}\eta _{pred}$$
+mini_ker.texi(,442) @end tex
mini_ker.texi(,444) @noindent @math{d eta_prey(t)/d t = a * eta_prey - a *
address@hidden
mini_ker.texi(,445) d eta_pred(t)/d t = -c * eta_pred +c * phi_meet}
mini_ker.texi(,446)
@@ -1007,15 +1033,31 @@
mini_ker.texi(,1183) The Jacobian matrix corresponding with:
mini_ker.texi(,1184) @c \varphi(t) &= f(\eta(t),\varphi(t))\cr
mini_ker.texi(,1185) @c \frac{\partial g(\eta(t),\varphi(t))}{\partial \eta(t)}
+mini_ker.texi(,1186) @tex
+mini_ker.texi(,1187) $$\partial_{\eta} g(\eta(t),\varphi(t));
+mini_ker.texi(,1188) $$
+mini_ker.texi(,1189) @end tex
mini_ker.texi(,1191) g_1(eta,phi);
mini_ker.texi(,1193) @item Bb
mini_ker.texi(,1194) The Jacobian matrix corresponding with:
+mini_ker.texi(,1195) @tex
+mini_ker.texi(,1196) $$\partial_{\varphi} g(\eta(t),\varphi(t));
+mini_ker.texi(,1197) $$
+mini_ker.texi(,1198) @end tex
mini_ker.texi(,1200) g_2(eta,phi);
mini_ker.texi(,1202) @item Bt
mini_ker.texi(,1203) The Jacobian matrix corresponding with:
+mini_ker.texi(,1204) @tex
+mini_ker.texi(,1205) $$\partial_{\eta} f(\eta(t),\varphi(t));
+mini_ker.texi(,1206) $$
+mini_ker.texi(,1207) @end tex
mini_ker.texi(,1209) f_1(eta,phi);
mini_ker.texi(,1211) @item D
mini_ker.texi(,1212) The Jacobian matrix corresponding with:
+mini_ker.texi(,1213) @tex
+mini_ker.texi(,1214) $$\partial_{\varphi} f(\eta(t),\varphi(t));
+mini_ker.texi(,1215) $$
+mini_ker.texi(,1216) @end tex
mini_ker.texi(,1218) f_2(eta,phi);
mini_ker.texi(,1220)
mini_ker.texi(,1221) @item aspha
@@ -1222,6 +1264,18 @@
mini_ker.texi(,1422) a chain of masselottes linked by springs and dumps is
bounded to a wall on the left,
mini_ker.texi(,1423) and open at right. The @acronym{TEF} formulation of the
problem is written in the phase space (position-shift, velocity)
mini_ker.texi(,1424) for node @math{k}, with bounding conditions:
+mini_ker.texi(,1457) @tex
+mini_ker.texi(,1458) $$\left\{\eqalign{\partial_t \eta _{k} ^{pos} &= \eta
_{k} ^{vel} \cr
+mini_ker.texi(,1459) \partial_t \eta _{k} ^{vel} &= ( \varphi_k ^{spr}
-\varphi _{k+1} ^{spr} + \varphi _{k} ^{dmp}-\varphi _{k+1} ^{dmp})\,/m_k
\cr}\right.$$
+mini_ker.texi(,1460) $$\left\{\eqalign{
+mini_ker.texi(,1461) \varphi_k ^{spr} &= -k_k (\eta _{k} ^{pos}- \eta _{k-1}
^{pos})\cr
+mini_ker.texi(,1462) \varphi_k ^{spr} &= -d_k (\eta _{k} ^{vel}- \eta _{k-1}
^{vel})
+mini_ker.texi(,1463) \cr}\right.$$
+mini_ker.texi(,1464) $$\left\{\eqalign{\eta ^{pos}_{0} &= 0\cr
+mini_ker.texi(,1465) \eta ^{vel}_{0} &= 0\cr
+mini_ker.texi(,1466) \varphi ^{spr}_{N+1} &= 0\cr
+mini_ker.texi(,1467) \varphi ^{dmp}_{N+1} &= 0\cr}\right.$$
+mini_ker.texi(,1468) @end tex
mini_ker.texi(,1470)
mini_ker.texi(,1471) States:@*
mini_ker.texi(,1472) @noindent @math{d position(t,k)/d t = velocity(t,k)@*
@@ -1595,6 +1649,10 @@
mini_ker.texi(,1852) Its derivative will have the following form:
mini_ker.texi(,1853)
mini_ker.texi(,1854)
+mini_ker.texi(,1863) @tex
+mini_ker.texi(,1864) $$\eqalign{\partial_x f^g &= g f^{g-1}\partial_x f + f^g
\log f\partial_x g\cr
+mini_ker.texi(,1865) &= f^{g-1}(g\partial_x f + f\partial_x g)\cr}$$
+mini_ker.texi(,1866) @end tex
mini_ker.texi(,1868)
mini_ker.texi(,1869) and is in the macros list already defined in:
@file{DERIVE_MAC}.
mini_ker.texi(,1870)
@@ -1712,6 +1770,11 @@
mini_ker.texi(,1992) @c in this section ($\mu$ elsewhere,
mini_ker.texi(,1993) and the observation function is noted @math{h}:
mini_ker.texi(,1994)
+mini_ker.texi(,1995) @tex
+mini_ker.texi(,1996) $$
+mini_ker.texi(,1997) \omega = h ( \eta , \varphi)
+mini_ker.texi(,1998) $$
+mini_ker.texi(,1999) @end tex
mini_ker.texi(,2001)
mini_ker.texi(,2002) @noindent @math{omega(t) = h(eta(t), phi(t))}
mini_ker.texi(,2003)
@@ -2284,6 +2347,11 @@
mini_ker.texi(,2571)
mini_ker.texi(,2572) The functionnal @math{J} to be optimised is defined as
mini_ker.texi(,2573)
+mini_ker.texi(,2582) @tex
+mini_ker.texi(,2583) $$
+mini_ker.texi(,2584) J = \psi[\eta(T),\varphi(T) ,h(T)] + \int_0 ^T
{l[\eta(\tau),\varphi(\tau),h(\tau)]}\, d\tau
+mini_ker.texi(,2585) $$
+mini_ker.texi(,2586) @end tex
mini_ker.texi(,2588) @noindent @math{J = psi(eta(T),phi(T),h(T)) + int_0^T
l(eta(tau),phi(tau),h(tau)) d tau}
mini_ker.texi(,2591)
mini_ker.texi(,2592) @cindex final cost
@@ -2465,6 +2533,13 @@
mini_ker.texi(,2768) stochastic perturbation on the state, and discrete noisy
observations.
mini_ker.texi(,2769) In the @acronym{TEF} this leads to:
mini_ker.texi(,2770)
+mini_ker.texi(,2781) @tex
+mini_ker.texi(,2782) $$\eqalign{
+mini_ker.texi(,2783) \partial_t \eta (t) &= g(\eta(t),\varphi(t)) + W(t)
\mu\cr
+mini_ker.texi(,2784) \varphi(t) &= f(\eta(t),\varphi(t))\cr
+mini_ker.texi(,2785) \omega(t) &= h ( \eta(t) , \varphi(t)) + \nu\cr
+mini_ker.texi(,2786) }$$
+mini_ker.texi(,2787) @end tex
mini_ker.texi(,2789)
mini_ker.texi(,2790) @noindent @math{d eta(t)/d t = g(eta(t),phi(t)) + W(t)
address@hidden
mini_ker.texi(,2791) phi(i) = f(eta(t),phi(t))@*
@@ -2487,6 +2562,23 @@
mini_ker.texi(,2810) transfers (equal to the states), but the error on the
state is of dimension
mini_ker.texi(,2811) 2. The 3 states are observed. The corresponding equations
read:
mini_ker.texi(,2812)
+mini_ker.texi(,2845) @tex
+mini_ker.texi(,2846) $$\left\{\eqalign{
+mini_ker.texi(,2847) \partial_t \eta_1 &= a_{11} \eta_1 + a_{12} \varphi_2 +
a_{13} \varphi_3 + W_{11} \mu_1 + W_{12} \mu_2\cr
+mini_ker.texi(,2848) \partial_t \eta_2 &= a_{21} \varphi_1 + a_{22} \eta_2 +
a_{23} \varphi_3 + W_{21} \mu_1 + W_{22} \mu_2\cr
+mini_ker.texi(,2849) \partial_t \eta_3 &= a_{31} \varphi_1 + a_{32} \varphi_2
+ a_{33} \eta_3 + W_{31} \mu_1 + W_{32} \mu_2
+mini_ker.texi(,2850) }\right.$$
+mini_ker.texi(,2851) $$\left\{\eqalign{
+mini_ker.texi(,2852) \varphi _1 &= \eta _1\cr
+mini_ker.texi(,2853) \varphi _2 &= \eta _2\cr
+mini_ker.texi(,2854) \varphi _3 &= \eta _3
+mini_ker.texi(,2855) }\right.$$
+mini_ker.texi(,2856) $$\left\{\eqalign{
+mini_ker.texi(,2857) \omega _1 &= \varphi _1 + \nu_1\cr
+mini_ker.texi(,2858) \omega _2 &= \eta _2 + \nu_2 \cr
+mini_ker.texi(,2859) \omega _3 &= \eta _3 + \nu_3
+mini_ker.texi(,2860) }\right.$$
+mini_ker.texi(,2861) @end tex
mini_ker.texi(,2862)
mini_ker.texi(,2864)
mini_ker.texi(,2865) Cells:@*
@@ -2944,6 +3036,11 @@
mini_ker.texi(,3326)
mini_ker.texi(,3327) The Singular value decomposition of a matrix is noted
mini_ker.texi(,3328)
+mini_ker.texi(,3329) @tex
+mini_ker.texi(,3330) $$
+mini_ker.texi(,3331) U w V^\dagger
+mini_ker.texi(,3332) $$
+mini_ker.texi(,3333) @end tex
mini_ker.texi(,3335)
mini_ker.texi(,3336) @noindent @math{U w V^t}
mini_ker.texi(,3337)
Index: test/manuals/res/texi_mini_ker/mini_ker.passtexi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/manuals/res/texi_mini_ker/mini_ker.passtexi,v
retrieving revision 1.3
retrieving revision 1.4
diff -u -b -r1.3 -r1.4
--- test/manuals/res/texi_mini_ker/mini_ker.passtexi 2 Nov 2008 00:49:19
-0000 1.3
+++ test/manuals/res/texi_mini_ker/mini_ker.passtexi 9 Jan 2009 21:20:47
-0000 1.4
@@ -207,6 +207,9 @@
mini_ker.texi(,229) @enumerate
mini_ker.texi(,230) @item Cells which are elementary models and correspond to
evolution equations
mini_ker.texi(,231) such as:
+mini_ker.texi(,241) @tex
+mini_ker.texi(,242) $$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
+mini_ker.texi(,243) @end tex
mini_ker.texi(,245)
mini_ker.texi(,246) @noindent @math{d eta(t)/d t = g(eta(t),phi(t))}
mini_ker.texi(,247)
@@ -223,6 +226,11 @@
mini_ker.texi(,260)
mini_ker.texi(,261)
mini_ker.texi(,262) @item Transfers which are determined by constraint
equations such as:
+mini_ker.texi(,272) @tex
+mini_ker.texi(,273) $$
+mini_ker.texi(,274) \varphi(t) = f(\eta(t),\varphi(t))
+mini_ker.texi(,275) $$
+mini_ker.texi(,276) @end tex
mini_ker.texi(,278)
mini_ker.texi(,279) @noindent @math{phi(t) = f(eta(t),phi(t))}
mini_ker.texi(,280)
@@ -250,6 +258,12 @@
mini_ker.texi(,304)
mini_ker.texi(,305)
mini_ker.texi(,319)
+mini_ker.texi(,321) @tex
+mini_ker.texi(,322) $$\pmatrix{A & B\cr
+mini_ker.texi(,323) -C^+ & I-D\cr} \pmatrix{\delta \eta\cr
+mini_ker.texi(,324) \delta \varphi\cr} = \pmatrix{\Gamma\cr
+mini_ker.texi(,325) \Omega\cr}$$
+mini_ker.texi(,326) @end tex
mini_ker.texi(,328)
mini_ker.texi(,329) The blocks appearing in the Jacobian matrix are
constructed with partial derivative
mini_ker.texi(,330) of @math{f} and @math{g}, and with @math{\delta t}. From
this system the
@@ -314,6 +328,11 @@
mini_ker.texi(,389) @cindex TEF
mini_ker.texi(,390)
mini_ker.texi(,391) The general @acronym{TEF} system writes:
+mini_ker.texi(,402) @tex
+mini_ker.texi(,403) $$\eqalign{\partial_t \eta (t) &= g(\eta(t),\varphi(t))\cr
+mini_ker.texi(,404) \varphi(t) &= f(\eta(t),\varphi(t))\cr
+mini_ker.texi(,405) }$$
+mini_ker.texi(,406) @end tex
mini_ker.texi(,408)
mini_ker.texi(,409) @noindent @math{d eta(t)/d t = g(eta(t),phi(t))@*
mini_ker.texi(,410) phi(t) = f(eta(t),phi(t))}
@@ -323,6 +342,13 @@
mini_ker.texi(,416) model of Lotka-Volterra is used.
mini_ker.texi(,417) This model can be written in the following @acronym{TEF}
form:
mini_ker.texi(,418)
+mini_ker.texi(,436) @tex
+mini_ker.texi(,437) $$\left\{\eqalign{\partial_t \eta _{prey} &= a \eta
_{prey} - a \varphi _{meet} \cr
+mini_ker.texi(,438) \partial_t \eta _{pred} &= -c \eta _{pred} + c \varphi
_{meet}\cr}\right.$$
+mini_ker.texi(,439) @end tex
+mini_ker.texi(,440) @tex
+mini_ker.texi(,441) $$\varphi _{meet} = \eta _{prey}\eta _{pred}$$
+mini_ker.texi(,442) @end tex
mini_ker.texi(,444) @noindent @math{d eta_prey(t)/d t = a * eta_prey - a *
address@hidden
mini_ker.texi(,445) d eta_pred(t)/d t = -c * eta_pred +c * phi_meet}
mini_ker.texi(,446)
@@ -1035,15 +1061,31 @@
mini_ker.texi(,1183) The Jacobian matrix corresponding with:
mini_ker.texi(,1184) @c \varphi(t) &= f(\eta(t),\varphi(t))\cr
mini_ker.texi(,1185) @c \frac{\partial g(\eta(t),\varphi(t))}{\partial \eta(t)}
+mini_ker.texi(,1186) @tex
+mini_ker.texi(,1187) $$\partial_{\eta} g(\eta(t),\varphi(t));
+mini_ker.texi(,1188) $$
+mini_ker.texi(,1189) @end tex
mini_ker.texi(,1191) g_1(eta,phi);
mini_ker.texi(,1193) @item Bb
mini_ker.texi(,1194) The Jacobian matrix corresponding with:
+mini_ker.texi(,1195) @tex
+mini_ker.texi(,1196) $$\partial_{\varphi} g(\eta(t),\varphi(t));
+mini_ker.texi(,1197) $$
+mini_ker.texi(,1198) @end tex
mini_ker.texi(,1200) g_2(eta,phi);
mini_ker.texi(,1202) @item Bt
mini_ker.texi(,1203) The Jacobian matrix corresponding with:
+mini_ker.texi(,1204) @tex
+mini_ker.texi(,1205) $$\partial_{\eta} f(\eta(t),\varphi(t));
+mini_ker.texi(,1206) $$
+mini_ker.texi(,1207) @end tex
mini_ker.texi(,1209) f_1(eta,phi);
mini_ker.texi(,1211) @item D
mini_ker.texi(,1212) The Jacobian matrix corresponding with:
+mini_ker.texi(,1213) @tex
+mini_ker.texi(,1214) $$\partial_{\varphi} f(\eta(t),\varphi(t));
+mini_ker.texi(,1215) $$
+mini_ker.texi(,1216) @end tex
mini_ker.texi(,1218) f_2(eta,phi);
mini_ker.texi(,1220)
mini_ker.texi(,1221) @item aspha
@@ -1250,6 +1292,18 @@
mini_ker.texi(,1422) a chain of masselottes linked by springs and dumps is
bounded to a wall on the left,
mini_ker.texi(,1423) and open at right. The @acronym{TEF} formulation of the
problem is written in the phase space (position-shift, velocity)
mini_ker.texi(,1424) for node @math{k}, with bounding conditions:
+mini_ker.texi(,1457) @tex
+mini_ker.texi(,1458) $$\left\{\eqalign{\partial_t \eta _{k} ^{pos} &= \eta
_{k} ^{vel} \cr
+mini_ker.texi(,1459) \partial_t \eta _{k} ^{vel} &= ( \varphi_k ^{spr}
-\varphi _{k+1} ^{spr} + \varphi _{k} ^{dmp}-\varphi _{k+1} ^{dmp})\,/m_k
\cr}\right.$$
+mini_ker.texi(,1460) $$\left\{\eqalign{
+mini_ker.texi(,1461) \varphi_k ^{spr} &= -k_k (\eta _{k} ^{pos}- \eta _{k-1}
^{pos})\cr
+mini_ker.texi(,1462) \varphi_k ^{spr} &= -d_k (\eta _{k} ^{vel}- \eta _{k-1}
^{vel})
+mini_ker.texi(,1463) \cr}\right.$$
+mini_ker.texi(,1464) $$\left\{\eqalign{\eta ^{pos}_{0} &= 0\cr
+mini_ker.texi(,1465) \eta ^{vel}_{0} &= 0\cr
+mini_ker.texi(,1466) \varphi ^{spr}_{N+1} &= 0\cr
+mini_ker.texi(,1467) \varphi ^{dmp}_{N+1} &= 0\cr}\right.$$
+mini_ker.texi(,1468) @end tex
mini_ker.texi(,1470)
mini_ker.texi(,1471) States:@*
mini_ker.texi(,1472) @noindent @math{d position(t,k)/d t = velocity(t,k)@*
@@ -1623,6 +1677,10 @@
mini_ker.texi(,1852) Its derivative will have the following form:
mini_ker.texi(,1853)
mini_ker.texi(,1854)
+mini_ker.texi(,1863) @tex
+mini_ker.texi(,1864) $$\eqalign{\partial_x f^g &= g f^{g-1}\partial_x f + f^g
\log f\partial_x g\cr
+mini_ker.texi(,1865) &= f^{g-1}(g\partial_x f + f\partial_x g)\cr}$$
+mini_ker.texi(,1866) @end tex
mini_ker.texi(,1868)
mini_ker.texi(,1869) and is in the macros list already defined in:
@file{DERIVE_MAC}.
mini_ker.texi(,1870)
@@ -1740,6 +1798,11 @@
mini_ker.texi(,1992) @c in this section ($\mu$ elsewhere,
mini_ker.texi(,1993) and the observation function is noted @math{h}:
mini_ker.texi(,1994)
+mini_ker.texi(,1995) @tex
+mini_ker.texi(,1996) $$
+mini_ker.texi(,1997) \omega = h ( \eta , \varphi)
+mini_ker.texi(,1998) $$
+mini_ker.texi(,1999) @end tex
mini_ker.texi(,2001)
mini_ker.texi(,2002) @noindent @math{omega(t) = h(eta(t), phi(t))}
mini_ker.texi(,2003)
@@ -2312,6 +2375,11 @@
mini_ker.texi(,2571)
mini_ker.texi(,2572) The functionnal @math{J} to be optimised is defined as
mini_ker.texi(,2573)
+mini_ker.texi(,2582) @tex
+mini_ker.texi(,2583) $$
+mini_ker.texi(,2584) J = \psi[\eta(T),\varphi(T) ,h(T)] + \int_0 ^T
{l[\eta(\tau),\varphi(\tau),h(\tau)]}\, d\tau
+mini_ker.texi(,2585) $$
+mini_ker.texi(,2586) @end tex
mini_ker.texi(,2588) @noindent @math{J = psi(eta(T),phi(T),h(T)) + int_0^T
l(eta(tau),phi(tau),h(tau)) d tau}
mini_ker.texi(,2591)
mini_ker.texi(,2592) @cindex final cost
@@ -2493,6 +2561,13 @@
mini_ker.texi(,2768) stochastic perturbation on the state, and discrete noisy
observations.
mini_ker.texi(,2769) In the @acronym{TEF} this leads to:
mini_ker.texi(,2770)
+mini_ker.texi(,2781) @tex
+mini_ker.texi(,2782) $$\eqalign{
+mini_ker.texi(,2783) \partial_t \eta (t) &= g(\eta(t),\varphi(t)) + W(t)
\mu\cr
+mini_ker.texi(,2784) \varphi(t) &= f(\eta(t),\varphi(t))\cr
+mini_ker.texi(,2785) \omega(t) &= h ( \eta(t) , \varphi(t)) + \nu\cr
+mini_ker.texi(,2786) }$$
+mini_ker.texi(,2787) @end tex
mini_ker.texi(,2789)
mini_ker.texi(,2790) @noindent @math{d eta(t)/d t = g(eta(t),phi(t)) + W(t)
address@hidden
mini_ker.texi(,2791) phi(i) = f(eta(t),phi(t))@*
@@ -2515,6 +2590,23 @@
mini_ker.texi(,2810) transfers (equal to the states), but the error on the
state is of dimension
mini_ker.texi(,2811) 2. The 3 states are observed. The corresponding equations
read:
mini_ker.texi(,2812)
+mini_ker.texi(,2845) @tex
+mini_ker.texi(,2846) $$\left\{\eqalign{
+mini_ker.texi(,2847) \partial_t \eta_1 &= a_{11} \eta_1 + a_{12} \varphi_2 +
a_{13} \varphi_3 + W_{11} \mu_1 + W_{12} \mu_2\cr
+mini_ker.texi(,2848) \partial_t \eta_2 &= a_{21} \varphi_1 + a_{22} \eta_2 +
a_{23} \varphi_3 + W_{21} \mu_1 + W_{22} \mu_2\cr
+mini_ker.texi(,2849) \partial_t \eta_3 &= a_{31} \varphi_1 + a_{32} \varphi_2
+ a_{33} \eta_3 + W_{31} \mu_1 + W_{32} \mu_2
+mini_ker.texi(,2850) }\right.$$
+mini_ker.texi(,2851) $$\left\{\eqalign{
+mini_ker.texi(,2852) \varphi _1 &= \eta _1\cr
+mini_ker.texi(,2853) \varphi _2 &= \eta _2\cr
+mini_ker.texi(,2854) \varphi _3 &= \eta _3
+mini_ker.texi(,2855) }\right.$$
+mini_ker.texi(,2856) $$\left\{\eqalign{
+mini_ker.texi(,2857) \omega _1 &= \varphi _1 + \nu_1\cr
+mini_ker.texi(,2858) \omega _2 &= \eta _2 + \nu_2 \cr
+mini_ker.texi(,2859) \omega _3 &= \eta _3 + \nu_3
+mini_ker.texi(,2860) }\right.$$
+mini_ker.texi(,2861) @end tex
mini_ker.texi(,2862)
mini_ker.texi(,2864)
mini_ker.texi(,2865) Cells:@*
@@ -2972,6 +3064,11 @@
mini_ker.texi(,3326)
mini_ker.texi(,3327) The Singular value decomposition of a matrix is noted
mini_ker.texi(,3328)
+mini_ker.texi(,3329) @tex
+mini_ker.texi(,3330) $$
+mini_ker.texi(,3331) U w V^\dagger
+mini_ker.texi(,3332) $$
+mini_ker.texi(,3333) @end tex
mini_ker.texi(,3335)
mini_ker.texi(,3336) @noindent @math{U w V^t}
mini_ker.texi(,3337)
Index: test/manuals/res/texi_mini_ker/mini_ker.texi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/manuals/res/texi_mini_ker/mini_ker.texi,v
retrieving revision 1.2
retrieving revision 1.3
diff -u -b -r1.2 -r1.3
--- test/manuals/res/texi_mini_ker/mini_ker.texi 2 Nov 2008 00:49:20
-0000 1.2
+++ test/manuals/res/texi_mini_ker/mini_ker.texi 9 Jan 2009 21:20:48
-0000 1.3
@@ -208,6 +208,9 @@
@enumerate
@item Cells which are elementary models and correspond to evolution equations
such as:
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@noindent @math{d eta(t)/d t = g(eta(t),phi(t))}
@@ -224,6 +227,11 @@
@item Transfers which are determined by constraint equations such as:
address@hidden
+$$
+\varphi(t) = f(\eta(t),\varphi(t))
+$$
address@hidden tex
@noindent @math{phi(t) = f(eta(t),phi(t))}
@@ -251,6 +259,12 @@
address@hidden
+$$\pmatrix{A & B\cr
+-C^+ & I-D\cr} \pmatrix{\delta \eta\cr
+\delta \varphi\cr} = \pmatrix{\Gamma\cr
+\Omega\cr}$$
address@hidden tex
The blocks appearing in the Jacobian matrix are constructed with partial
derivative
of @math{f} and @math{g}, and with @math{\delta t}. From this system the
@@ -315,6 +329,11 @@
@cindex TEF
The general @acronym{TEF} system writes:
address@hidden
+$$\eqalign{\partial_t \eta (t) &= g(\eta(t),\varphi(t))\cr
+\varphi(t) &= f(\eta(t),\varphi(t))\cr
+}$$
address@hidden tex
@noindent @math{d eta(t)/d t = g(eta(t),phi(t))@*
phi(t) = f(eta(t),phi(t))}
@@ -324,6 +343,13 @@
model of Lotka-Volterra is used.
This model can be written in the following @acronym{TEF} form:
address@hidden
+$$\left\{\eqalign{\partial_t \eta _{prey} &= a \eta _{prey} - a \varphi
_{meet} \cr
+\partial_t \eta _{pred} &= -c \eta _{pred} + c \varphi _{meet}\cr}\right.$$
address@hidden tex
address@hidden
+$$\varphi _{meet} = \eta _{prey}\eta _{pred}$$
address@hidden tex
@noindent @math{d eta_prey(t)/d t = a * eta_prey - a * address@hidden
d eta_pred(t)/d t = -c * eta_pred +c * phi_meet}
@@ -1036,15 +1062,31 @@
The Jacobian matrix corresponding with:
@c \varphi(t) &= f(\eta(t),\varphi(t))\cr
@c \frac{\partial g(\eta(t),\varphi(t))}{\partial \eta(t)}
address@hidden
+$$\partial_{\eta} g(\eta(t),\varphi(t));
+$$
address@hidden tex
g_1(eta,phi);
@item Bb
The Jacobian matrix corresponding with:
address@hidden
+$$\partial_{\varphi} g(\eta(t),\varphi(t));
+$$
address@hidden tex
g_2(eta,phi);
@item Bt
The Jacobian matrix corresponding with:
address@hidden
+$$\partial_{\eta} f(\eta(t),\varphi(t));
+$$
address@hidden tex
f_1(eta,phi);
@item D
The Jacobian matrix corresponding with:
address@hidden
+$$\partial_{\varphi} f(\eta(t),\varphi(t));
+$$
address@hidden tex
f_2(eta,phi);
@item aspha
@@ -1251,6 +1293,18 @@
a chain of masselottes linked by springs and dumps is bounded to a wall on the
left,
and open at right. The @acronym{TEF} formulation of the problem is written in
the phase space (position-shift, velocity)
for node @math{k}, with bounding conditions:
address@hidden
+$$\left\{\eqalign{\partial_t \eta _{k} ^{pos} &= \eta _{k} ^{vel} \cr
+\partial_t \eta _{k} ^{vel} &= ( \varphi_k ^{spr} -\varphi _{k+1} ^{spr} +
\varphi _{k} ^{dmp}-\varphi _{k+1} ^{dmp})\,/m_k \cr}\right.$$
+$$\left\{\eqalign{
+\varphi_k ^{spr} &= -k_k (\eta _{k} ^{pos}- \eta _{k-1} ^{pos})\cr
+\varphi_k ^{spr} &= -d_k (\eta _{k} ^{vel}- \eta _{k-1} ^{vel})
+\cr}\right.$$
+$$\left\{\eqalign{\eta ^{pos}_{0} &= 0\cr
+\eta ^{vel}_{0} &= 0\cr
+\varphi ^{spr}_{N+1} &= 0\cr
+\varphi ^{dmp}_{N+1} &= 0\cr}\right.$$
address@hidden tex
States:@*
@noindent @math{d position(t,k)/d t = velocity(t,k)@*
@@ -1624,6 +1678,10 @@
Its derivative will have the following form:
address@hidden
+$$\eqalign{\partial_x f^g &= g f^{g-1}\partial_x f + f^g \log f\partial_x g\cr
+ &= f^{g-1}(g\partial_x f + f\partial_x g)\cr}$$
address@hidden tex
and is in the macros list already defined in: @file{DERIVE_MAC}.
@@ -1741,6 +1799,11 @@
@c in this section ($\mu$ elsewhere,
and the observation function is noted @math{h}:
address@hidden
+$$
+\omega = h ( \eta , \varphi)
+$$
address@hidden tex
@noindent @math{omega(t) = h(eta(t), phi(t))}
@@ -2313,6 +2376,11 @@
The functionnal @math{J} to be optimised is defined as
address@hidden
+$$
+J = \psi[\eta(T),\varphi(T) ,h(T)] + \int_0 ^T
{l[\eta(\tau),\varphi(\tau),h(\tau)]}\, d\tau
+$$
address@hidden tex
@noindent @math{J = psi(eta(T),phi(T),h(T)) + int_0^T
l(eta(tau),phi(tau),h(tau)) d tau}
@cindex final cost
@@ -2494,6 +2562,13 @@
stochastic perturbation on the state, and discrete noisy observations.
In the @acronym{TEF} this leads to:
address@hidden
+$$\eqalign{
+\partial_t \eta (t) &= g(\eta(t),\varphi(t)) + W(t) \mu\cr
+\varphi(t) &= f(\eta(t),\varphi(t))\cr
+\omega(t) &= h ( \eta(t) , \varphi(t)) + \nu\cr
+}$$
address@hidden tex
@noindent @math{d eta(t)/d t = g(eta(t),phi(t)) + W(t) address@hidden
phi(i) = f(eta(t),phi(t))@*
@@ -2516,6 +2591,23 @@
transfers (equal to the states), but the error on the state is of dimension
2. The 3 states are observed. The corresponding equations read:
address@hidden
+$$\left\{\eqalign{
+\partial_t \eta_1 &= a_{11} \eta_1 + a_{12} \varphi_2 + a_{13} \varphi_3 +
W_{11} \mu_1 + W_{12} \mu_2\cr
+\partial_t \eta_2 &= a_{21} \varphi_1 + a_{22} \eta_2 + a_{23} \varphi_3 +
W_{21} \mu_1 + W_{22} \mu_2\cr
+\partial_t \eta_3 &= a_{31} \varphi_1 + a_{32} \varphi_2 + a_{33} \eta_3 +
W_{31} \mu_1 + W_{32} \mu_2
+}\right.$$
+$$\left\{\eqalign{
+\varphi _1 &= \eta _1\cr
+\varphi _2 &= \eta _2\cr
+\varphi _3 &= \eta _3
+}\right.$$
+$$\left\{\eqalign{
+\omega _1 &= \varphi _1 + \nu_1\cr
+\omega _2 &= \eta _2 + \nu_2 \cr
+\omega _3 &= \eta _3 + \nu_3
+}\right.$$
address@hidden tex
Cells:@*
@@ -2973,6 +3065,11 @@
The Singular value decomposition of a matrix is noted
address@hidden
+$$
+ U w V^\dagger
+$$
address@hidden tex
@noindent @math{U w V^t}
Index: test/manuals/res/texi_texinfo/texinfo.passfirst
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/manuals/res/texi_texinfo/texinfo.passfirst,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/manuals/res/texi_texinfo/texinfo.passfirst 18 Aug 2008 18:05:17
-0000 1.1
+++ test/manuals/res/texi_texinfo/texinfo.passfirst 9 Jan 2009 21:20:48
-0000 1.2
@@ -4058,6 +4058,9 @@
texinfo.texi(,4127)
texinfo.texi(,4128) Here are the four groups of chapter structuring commands:
texinfo.texi(,4129)
+texinfo.texi(,4130) @tex
+texinfo.texi(,4131) {\globaldefs = 1 \smallfonts}
+texinfo.texi(,4132) @end tex
texinfo.texi(,4133)
texinfo.texi(,4134) @multitable @columnfractions .19 .30 .29 .22
texinfo.texi(,4135) @item @tab
@tab @tab No new page
@@ -4069,6 +4072,9 @@
texinfo.texi(,4141) @item @code{@@subsection} @tab
@code{@@unnumberedsubsec} @tab @code{@@appendixsubsec} @tab
@code{@@subheading}
texinfo.texi(,4142) @item @code{@@subsubsection} @tab
@code{@@unnumberedsubsubsec} @tab @code{@@appendixsubsubsec} @tab
@code{@@subsubheading}
texinfo.texi(,4143) @end multitable
+texinfo.texi(,4144) @tex
+texinfo.texi(,4145) {\globaldefs = 1 \textfonts}
+texinfo.texi(,4146) @end tex
texinfo.texi(,4147)
texinfo.texi(,4148)
texinfo.texi(,4149) @node makeinfo top
@@ -7788,6 +7794,10 @@
texinfo.texi(,7977) @example
texinfo.texi(,7978) This is an example
texinfo.texi(,7979) @end example
+texinfo.texi(,7980) @tex
+texinfo.texi(,7981) % Remove extra vskip; this is a kludge to counter the
effect of display
+texinfo.texi(,7982) \vskip-3.5\baselineskip
+texinfo.texi(,7983) @end tex
texinfo.texi(,7984)
texinfo.texi(,7985) @noindent
texinfo.texi(,7986) This line is not indented. As you can see, the
@@ -10477,6 +10487,9 @@
texinfo.texi(,10873) @quotation
texinfo.texi(,10874) @defspec foobar (@var{var} address@hidden @var{to}
address@hidden) @address@hidden
texinfo.texi(,10875) @end defspec
+texinfo.texi(,10876) @tex
+texinfo.texi(,10877) \vskip \parskip
+texinfo.texi(,10878) @end tex
texinfo.texi(,10879) @end quotation
texinfo.texi(,10880)
texinfo.texi(,10881) @noindent
@@ -11506,6 +11519,11 @@
texinfo.texi(,11970) you are reading this in Info, you will not see the
equation that appears
texinfo.texi(,11971) in the printed manual.
texinfo.texi(,11976)
+texinfo.texi(,11977) @tex
+texinfo.texi(,11978) $$ \chi^2 = \sum_{i=1}^N
+texinfo.texi(,11979) \left(y_i - (a + b x_i)
+texinfo.texi(,11980) \over \sigma_i\right)^2 $$
+texinfo.texi(,11981) @end tex
texinfo.texi(,11982)
texinfo.texi(,11983) @findex ifhtml
texinfo.texi(,11984) @findex html
Index: test/manuals/res/texi_texinfo/texinfo.passtexi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/manuals/res/texi_texinfo/texinfo.passtexi,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/manuals/res/texi_texinfo/texinfo.passtexi 18 Aug 2008 18:05:17
-0000 1.1
+++ test/manuals/res/texi_texinfo/texinfo.passtexi 9 Jan 2009 21:20:49
-0000 1.2
@@ -4105,6 +4105,9 @@
texinfo.texi(,4127)
texinfo.texi(,4128) Here are the four groups of chapter structuring commands:
texinfo.texi(,4129)
+texinfo.texi(,4130) @tex
+texinfo.texi(,4131) {\globaldefs = 1 \smallfonts}
+texinfo.texi(,4132) @end tex
texinfo.texi(,4133)
texinfo.texi(,4134) @multitable @columnfractions .19 .30 .29 .22
texinfo.texi(,4135) @item @tab
@tab @tab No new page
@@ -4116,6 +4119,9 @@
texinfo.texi(,4141) @item @code{@@subsection} @tab
@code{@@unnumberedsubsec} @tab @code{@@appendixsubsec} @tab
@code{@@subheading}
texinfo.texi(,4142) @item @code{@@subsubsection} @tab
@code{@@unnumberedsubsubsec} @tab @code{@@appendixsubsubsec} @tab
@code{@@subsubheading}
texinfo.texi(,4143) @end multitable
+texinfo.texi(,4144) @tex
+texinfo.texi(,4145) {\globaldefs = 1 \textfonts}
+texinfo.texi(,4146) @end tex
texinfo.texi(,4147)
texinfo.texi(,4148)
texinfo.texi(,4149) @node makeinfo top
@@ -7835,6 +7841,10 @@
texinfo.texi(,7977) @example
texinfo.texi(,7978) This is an example
texinfo.texi(,7979) @end example
+texinfo.texi(,7980) @tex
+texinfo.texi(,7981) % Remove extra vskip; this is a kludge to counter the
effect of display
+texinfo.texi(,7982) \vskip-3.5\baselineskip
+texinfo.texi(,7983) @end tex
texinfo.texi(,7984)
texinfo.texi(,7985) @noindent
texinfo.texi(,7986) This line is not indented. As you can see, the
@@ -10524,6 +10534,9 @@
texinfo.texi(,10873) @quotation
texinfo.texi(,10874) @defspec foobar (@var{var} address@hidden @var{to}
address@hidden) @address@hidden
texinfo.texi(,10875) @end defspec
+texinfo.texi(,10876) @tex
+texinfo.texi(,10877) \vskip \parskip
+texinfo.texi(,10878) @end tex
texinfo.texi(,10879) @end quotation
texinfo.texi(,10880)
texinfo.texi(,10881) @noindent
@@ -11553,6 +11566,11 @@
texinfo.texi(,11970) you are reading this in Info, you will not see the
equation that appears
texinfo.texi(,11971) in the printed manual.
texinfo.texi(,11976)
+texinfo.texi(,11977) @tex
+texinfo.texi(,11978) $$ \chi^2 = \sum_{i=1}^N
+texinfo.texi(,11979) \left(y_i - (a + b x_i)
+texinfo.texi(,11980) \over \sigma_i\right)^2 $$
+texinfo.texi(,11981) @end tex
texinfo.texi(,11982)
texinfo.texi(,11983) @findex ifhtml
texinfo.texi(,11984) @findex html
Index: test/manuals/res/texi_texinfo/texinfo.texi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/manuals/res/texi_texinfo/texinfo.texi,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/manuals/res/texi_texinfo/texinfo.texi 18 Aug 2008 18:05:17 -0000
1.1
+++ test/manuals/res/texi_texinfo/texinfo.texi 9 Jan 2009 21:20:50 -0000
1.2
@@ -4106,6 +4106,9 @@
Here are the four groups of chapter structuring commands:
address@hidden
+{\globaldefs = 1 \smallfonts}
address@hidden tex
@multitable @columnfractions .19 .30 .29 .22
@item @tab @tab
@tab No new page
@@ -4117,6 +4120,9 @@
@item @code{@@subsection} @tab @code{@@unnumberedsubsec} @tab
@code{@@appendixsubsec} @tab @code{@@subheading}
@item @code{@@subsubsection} @tab @code{@@unnumberedsubsubsec} @tab
@code{@@appendixsubsubsec} @tab @code{@@subsubheading}
@end multitable
address@hidden
+{\globaldefs = 1 \textfonts}
address@hidden tex
@node makeinfo top
@@ -7836,6 +7842,10 @@
@example
This is an example
@end example
address@hidden
+% Remove extra vskip; this is a kludge to counter the effect of display
+\vskip-3.5\baselineskip
address@hidden tex
@noindent
This line is not indented. As you can see, the
@@ -10525,6 +10535,9 @@
@quotation
@defspec foobar (@var{var} address@hidden @var{to} address@hidden)
@address@hidden
@end defspec
address@hidden
+\vskip \parskip
address@hidden tex
@end quotation
@noindent
@@ -11554,6 +11567,11 @@
you are reading this in Info, you will not see the equation that appears
in the printed manual.
address@hidden
+$$ \chi^2 = \sum_{i=1}^N
+ \left(y_i - (a + b x_i)
+ \over \sigma_i\right)^2 $$
address@hidden tex
@findex ifhtml
@findex html
Index: test/misc/res/formatting_html32/formatting.html
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/misc/res/formatting_html32/formatting.html,v
retrieving revision 1.17
retrieving revision 1.18
diff -u -b -r1.17 -r1.18
--- test/misc/res/formatting_html32/formatting.html 8 Jan 2009 00:21:41
-0000 1.17
+++ test/misc/res/formatting_html32/formatting.html 9 Jan 2009 21:20:51
-0000 1.18
@@ -534,12 +534,10 @@
in verbatim ''
-
html ''
-
majorheading
chapheading
@@ -1259,7 +1257,6 @@
html ''
-
<a name="t_h-copying_majorheading"></a>
<h1> majorheading </h1>
@@ -1950,7 +1947,6 @@
html ''
-
<a name="majorheading"></a>
<h1> majorheading </h1>
@@ -2644,7 +2640,6 @@
html ''
-
<a name="majorheading-1"></a>
<h1> majorheading </h1>
Index: test/nested_formats/res/texi_nested_formats/nested_formats.passfirst
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/nested_formats/res/texi_nested_formats/nested_formats.passfirst,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/nested_formats/res/texi_nested_formats/nested_formats.passfirst
18 Aug 2008 18:06:05 -0000 1.1
+++ test/nested_formats/res/texi_nested_formats/nested_formats.passfirst
9 Jan 2009 21:20:52 -0000 1.2
@@ -70,11 +70,17 @@
nested_formats.texi(mymacro,233) in verbatim
nested_formats.texi(mymacro,233) @end verbatim
nested_formats.texi(mymacro,233)
+nested_formats.texi(mymacro,233) @xml
+nested_formats.texi(mymacro,233) <para> xml para </xml>
+nested_formats.texi(mymacro,233) @end xml
nested_formats.texi(mymacro,233)
nested_formats.texi(mymacro,233) @html
nested_formats.texi(mymacro,233) html
nested_formats.texi(mymacro,233) @end html
nested_formats.texi(mymacro,233)
+nested_formats.texi(mymacro,233) @tex
+nested_formats.texi(mymacro,233) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,233) @end tex
nested_formats.texi(mymacro,233)
nested_formats.texi(mymacro,233) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,233) @item i--tem 1
@@ -225,11 +231,17 @@
nested_formats.texi(mymacro,237) in verbatim
nested_formats.texi(mymacro,237) @end verbatim
nested_formats.texi(mymacro,237)
+nested_formats.texi(mymacro,237) @xml
+nested_formats.texi(mymacro,237) <para> xml para </xml>
+nested_formats.texi(mymacro,237) @end xml
nested_formats.texi(mymacro,237)
nested_formats.texi(mymacro,237) @html
nested_formats.texi(mymacro,237) html
nested_formats.texi(mymacro,237) @end html
nested_formats.texi(mymacro,237)
+nested_formats.texi(mymacro,237) @tex
+nested_formats.texi(mymacro,237) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,237) @end tex
nested_formats.texi(mymacro,237)
nested_formats.texi(mymacro,237) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,237) @item i--tem 1
@@ -381,11 +393,17 @@
nested_formats.texi(mymacro,242) in verbatim
nested_formats.texi(mymacro,242) @end verbatim
nested_formats.texi(mymacro,242)
+nested_formats.texi(mymacro,242) @xml
+nested_formats.texi(mymacro,242) <para> xml para </xml>
+nested_formats.texi(mymacro,242) @end xml
nested_formats.texi(mymacro,242)
nested_formats.texi(mymacro,242) @html
nested_formats.texi(mymacro,242) html
nested_formats.texi(mymacro,242) @end html
nested_formats.texi(mymacro,242)
+nested_formats.texi(mymacro,242) @tex
+nested_formats.texi(mymacro,242) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,242) @end tex
nested_formats.texi(mymacro,242)
nested_formats.texi(mymacro,242) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,242) @item i--tem 1
@@ -535,11 +553,17 @@
nested_formats.texi(mymacro,245) in verbatim
nested_formats.texi(mymacro,245) @end verbatim
nested_formats.texi(mymacro,245)
+nested_formats.texi(mymacro,245) @xml
+nested_formats.texi(mymacro,245) <para> xml para </xml>
+nested_formats.texi(mymacro,245) @end xml
nested_formats.texi(mymacro,245)
nested_formats.texi(mymacro,245) @html
nested_formats.texi(mymacro,245) html
nested_formats.texi(mymacro,245) @end html
nested_formats.texi(mymacro,245)
+nested_formats.texi(mymacro,245) @tex
+nested_formats.texi(mymacro,245) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,245) @end tex
nested_formats.texi(mymacro,245)
nested_formats.texi(mymacro,245) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,245) @item i--tem 1
@@ -690,11 +714,17 @@
nested_formats.texi(mymacro,249) in verbatim
nested_formats.texi(mymacro,249) @end verbatim
nested_formats.texi(mymacro,249)
+nested_formats.texi(mymacro,249) @xml
+nested_formats.texi(mymacro,249) <para> xml para </xml>
+nested_formats.texi(mymacro,249) @end xml
nested_formats.texi(mymacro,249)
nested_formats.texi(mymacro,249) @html
nested_formats.texi(mymacro,249) html
nested_formats.texi(mymacro,249) @end html
nested_formats.texi(mymacro,249)
+nested_formats.texi(mymacro,249) @tex
+nested_formats.texi(mymacro,249) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,249) @end tex
nested_formats.texi(mymacro,249)
nested_formats.texi(mymacro,249) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,249) @item i--tem 1
@@ -847,11 +877,17 @@
nested_formats.texi(mymacro,255) in verbatim
nested_formats.texi(mymacro,255) @end verbatim
nested_formats.texi(mymacro,255)
+nested_formats.texi(mymacro,255) @xml
+nested_formats.texi(mymacro,255) <para> xml para </xml>
+nested_formats.texi(mymacro,255) @end xml
nested_formats.texi(mymacro,255)
nested_formats.texi(mymacro,255) @html
nested_formats.texi(mymacro,255) html
nested_formats.texi(mymacro,255) @end html
nested_formats.texi(mymacro,255)
+nested_formats.texi(mymacro,255) @tex
+nested_formats.texi(mymacro,255) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,255) @end tex
nested_formats.texi(mymacro,255)
nested_formats.texi(mymacro,255) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,255) @item i--tem 1
@@ -1001,11 +1037,17 @@
nested_formats.texi(mymacro,258) in verbatim
nested_formats.texi(mymacro,258) @end verbatim
nested_formats.texi(mymacro,258)
+nested_formats.texi(mymacro,258) @xml
+nested_formats.texi(mymacro,258) <para> xml para </xml>
+nested_formats.texi(mymacro,258) @end xml
nested_formats.texi(mymacro,258)
nested_formats.texi(mymacro,258) @html
nested_formats.texi(mymacro,258) html
nested_formats.texi(mymacro,258) @end html
nested_formats.texi(mymacro,258)
+nested_formats.texi(mymacro,258) @tex
+nested_formats.texi(mymacro,258) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,258) @end tex
nested_formats.texi(mymacro,258)
nested_formats.texi(mymacro,258) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,258) @item i--tem 1
@@ -1156,11 +1198,17 @@
nested_formats.texi(mymacro,262) in verbatim
nested_formats.texi(mymacro,262) @end verbatim
nested_formats.texi(mymacro,262)
+nested_formats.texi(mymacro,262) @xml
+nested_formats.texi(mymacro,262) <para> xml para </xml>
+nested_formats.texi(mymacro,262) @end xml
nested_formats.texi(mymacro,262)
nested_formats.texi(mymacro,262) @html
nested_formats.texi(mymacro,262) html
nested_formats.texi(mymacro,262) @end html
nested_formats.texi(mymacro,262)
+nested_formats.texi(mymacro,262) @tex
+nested_formats.texi(mymacro,262) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,262) @end tex
nested_formats.texi(mymacro,262)
nested_formats.texi(mymacro,262) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,262) @item i--tem 1
@@ -1315,11 +1363,17 @@
nested_formats.texi(mymacro,270) in verbatim
nested_formats.texi(mymacro,270) @end verbatim
nested_formats.texi(mymacro,270)
+nested_formats.texi(mymacro,270) @xml
+nested_formats.texi(mymacro,270) <para> xml para </xml>
+nested_formats.texi(mymacro,270) @end xml
nested_formats.texi(mymacro,270)
nested_formats.texi(mymacro,270) @html
nested_formats.texi(mymacro,270) html
nested_formats.texi(mymacro,270) @end html
nested_formats.texi(mymacro,270)
+nested_formats.texi(mymacro,270) @tex
+nested_formats.texi(mymacro,270) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,270) @end tex
nested_formats.texi(mymacro,270)
nested_formats.texi(mymacro,270) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,270) @item i--tem 1
@@ -1469,11 +1523,17 @@
nested_formats.texi(mymacro,273) in verbatim
nested_formats.texi(mymacro,273) @end verbatim
nested_formats.texi(mymacro,273)
+nested_formats.texi(mymacro,273) @xml
+nested_formats.texi(mymacro,273) <para> xml para </xml>
+nested_formats.texi(mymacro,273) @end xml
nested_formats.texi(mymacro,273)
nested_formats.texi(mymacro,273) @html
nested_formats.texi(mymacro,273) html
nested_formats.texi(mymacro,273) @end html
nested_formats.texi(mymacro,273)
+nested_formats.texi(mymacro,273) @tex
+nested_formats.texi(mymacro,273) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,273) @end tex
nested_formats.texi(mymacro,273)
nested_formats.texi(mymacro,273) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,273) @item i--tem 1
@@ -1624,11 +1684,17 @@
nested_formats.texi(mymacro,277) in verbatim
nested_formats.texi(mymacro,277) @end verbatim
nested_formats.texi(mymacro,277)
+nested_formats.texi(mymacro,277) @xml
+nested_formats.texi(mymacro,277) <para> xml para </xml>
+nested_formats.texi(mymacro,277) @end xml
nested_formats.texi(mymacro,277)
nested_formats.texi(mymacro,277) @html
nested_formats.texi(mymacro,277) html
nested_formats.texi(mymacro,277) @end html
nested_formats.texi(mymacro,277)
+nested_formats.texi(mymacro,277) @tex
+nested_formats.texi(mymacro,277) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,277) @end tex
nested_formats.texi(mymacro,277)
nested_formats.texi(mymacro,277) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,277) @item i--tem 1
@@ -1781,11 +1847,17 @@
nested_formats.texi(mymacro,283) in verbatim
nested_formats.texi(mymacro,283) @end verbatim
nested_formats.texi(mymacro,283)
+nested_formats.texi(mymacro,283) @xml
+nested_formats.texi(mymacro,283) <para> xml para </xml>
+nested_formats.texi(mymacro,283) @end xml
nested_formats.texi(mymacro,283)
nested_formats.texi(mymacro,283) @html
nested_formats.texi(mymacro,283) html
nested_formats.texi(mymacro,283) @end html
nested_formats.texi(mymacro,283)
+nested_formats.texi(mymacro,283) @tex
+nested_formats.texi(mymacro,283) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,283) @end tex
nested_formats.texi(mymacro,283)
nested_formats.texi(mymacro,283) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,283) @item i--tem 1
@@ -1936,11 +2008,17 @@
nested_formats.texi(mymacro,287) in verbatim
nested_formats.texi(mymacro,287) @end verbatim
nested_formats.texi(mymacro,287)
+nested_formats.texi(mymacro,287) @xml
+nested_formats.texi(mymacro,287) <para> xml para </xml>
+nested_formats.texi(mymacro,287) @end xml
nested_formats.texi(mymacro,287)
nested_formats.texi(mymacro,287) @html
nested_formats.texi(mymacro,287) html
nested_formats.texi(mymacro,287) @end html
nested_formats.texi(mymacro,287)
+nested_formats.texi(mymacro,287) @tex
+nested_formats.texi(mymacro,287) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,287) @end tex
nested_formats.texi(mymacro,287)
nested_formats.texi(mymacro,287) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,287) @item i--tem 1
@@ -2092,11 +2170,17 @@
nested_formats.texi(mymacro,292) in verbatim
nested_formats.texi(mymacro,292) @end verbatim
nested_formats.texi(mymacro,292)
+nested_formats.texi(mymacro,292) @xml
+nested_formats.texi(mymacro,292) <para> xml para </xml>
+nested_formats.texi(mymacro,292) @end xml
nested_formats.texi(mymacro,292)
nested_formats.texi(mymacro,292) @html
nested_formats.texi(mymacro,292) html
nested_formats.texi(mymacro,292) @end html
nested_formats.texi(mymacro,292)
+nested_formats.texi(mymacro,292) @tex
+nested_formats.texi(mymacro,292) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,292) @end tex
nested_formats.texi(mymacro,292)
nested_formats.texi(mymacro,292) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,292) @item i--tem 1
@@ -2254,11 +2338,17 @@
nested_formats.texi(mymacro,303) in verbatim
nested_formats.texi(mymacro,303) @end verbatim
nested_formats.texi(mymacro,303)
+nested_formats.texi(mymacro,303) @xml
+nested_formats.texi(mymacro,303) <para> xml para </xml>
+nested_formats.texi(mymacro,303) @end xml
nested_formats.texi(mymacro,303)
nested_formats.texi(mymacro,303) @html
nested_formats.texi(mymacro,303) html
nested_formats.texi(mymacro,303) @end html
nested_formats.texi(mymacro,303)
+nested_formats.texi(mymacro,303) @tex
+nested_formats.texi(mymacro,303) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,303) @end tex
nested_formats.texi(mymacro,303)
nested_formats.texi(mymacro,303) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,303) @item i--tem 1
@@ -2409,11 +2499,17 @@
nested_formats.texi(mymacro,307) in verbatim
nested_formats.texi(mymacro,307) @end verbatim
nested_formats.texi(mymacro,307)
+nested_formats.texi(mymacro,307) @xml
+nested_formats.texi(mymacro,307) <para> xml para </xml>
+nested_formats.texi(mymacro,307) @end xml
nested_formats.texi(mymacro,307)
nested_formats.texi(mymacro,307) @html
nested_formats.texi(mymacro,307) html
nested_formats.texi(mymacro,307) @end html
nested_formats.texi(mymacro,307)
+nested_formats.texi(mymacro,307) @tex
+nested_formats.texi(mymacro,307) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,307) @end tex
nested_formats.texi(mymacro,307)
nested_formats.texi(mymacro,307) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,307) @item i--tem 1
@@ -2566,11 +2662,17 @@
nested_formats.texi(mymacro,313) in verbatim
nested_formats.texi(mymacro,313) @end verbatim
nested_formats.texi(mymacro,313)
+nested_formats.texi(mymacro,313) @xml
+nested_formats.texi(mymacro,313) <para> xml para </xml>
+nested_formats.texi(mymacro,313) @end xml
nested_formats.texi(mymacro,313)
nested_formats.texi(mymacro,313) @html
nested_formats.texi(mymacro,313) html
nested_formats.texi(mymacro,313) @end html
nested_formats.texi(mymacro,313)
+nested_formats.texi(mymacro,313) @tex
+nested_formats.texi(mymacro,313) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,313) @end tex
nested_formats.texi(mymacro,313)
nested_formats.texi(mymacro,313) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,313) @item i--tem 1
@@ -2723,11 +2825,17 @@
nested_formats.texi(mymacro,319) in verbatim
nested_formats.texi(mymacro,319) @end verbatim
nested_formats.texi(mymacro,319)
+nested_formats.texi(mymacro,319) @xml
+nested_formats.texi(mymacro,319) <para> xml para </xml>
+nested_formats.texi(mymacro,319) @end xml
nested_formats.texi(mymacro,319)
nested_formats.texi(mymacro,319) @html
nested_formats.texi(mymacro,319) html
nested_formats.texi(mymacro,319) @end html
nested_formats.texi(mymacro,319)
+nested_formats.texi(mymacro,319) @tex
+nested_formats.texi(mymacro,319) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,319) @end tex
nested_formats.texi(mymacro,319)
nested_formats.texi(mymacro,319) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,319) @item i--tem 1
@@ -2877,11 +2985,17 @@
nested_formats.texi(mymacro,322) in verbatim
nested_formats.texi(mymacro,322) @end verbatim
nested_formats.texi(mymacro,322)
+nested_formats.texi(mymacro,322) @xml
+nested_formats.texi(mymacro,322) <para> xml para </xml>
+nested_formats.texi(mymacro,322) @end xml
nested_formats.texi(mymacro,322)
nested_formats.texi(mymacro,322) @html
nested_formats.texi(mymacro,322) html
nested_formats.texi(mymacro,322) @end html
nested_formats.texi(mymacro,322)
+nested_formats.texi(mymacro,322) @tex
+nested_formats.texi(mymacro,322) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,322) @end tex
nested_formats.texi(mymacro,322)
nested_formats.texi(mymacro,322) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,322) @item i--tem 1
@@ -3032,11 +3146,17 @@
nested_formats.texi(mymacro,326) in verbatim
nested_formats.texi(mymacro,326) @end verbatim
nested_formats.texi(mymacro,326)
+nested_formats.texi(mymacro,326) @xml
+nested_formats.texi(mymacro,326) <para> xml para </xml>
+nested_formats.texi(mymacro,326) @end xml
nested_formats.texi(mymacro,326)
nested_formats.texi(mymacro,326) @html
nested_formats.texi(mymacro,326) html
nested_formats.texi(mymacro,326) @end html
nested_formats.texi(mymacro,326)
+nested_formats.texi(mymacro,326) @tex
+nested_formats.texi(mymacro,326) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,326) @end tex
nested_formats.texi(mymacro,326)
nested_formats.texi(mymacro,326) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,326) @item i--tem 1
@@ -3189,11 +3309,17 @@
nested_formats.texi(mymacro,332) in verbatim
nested_formats.texi(mymacro,332) @end verbatim
nested_formats.texi(mymacro,332)
+nested_formats.texi(mymacro,332) @xml
+nested_formats.texi(mymacro,332) <para> xml para </xml>
+nested_formats.texi(mymacro,332) @end xml
nested_formats.texi(mymacro,332)
nested_formats.texi(mymacro,332) @html
nested_formats.texi(mymacro,332) html
nested_formats.texi(mymacro,332) @end html
nested_formats.texi(mymacro,332)
+nested_formats.texi(mymacro,332) @tex
+nested_formats.texi(mymacro,332) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,332) @end tex
nested_formats.texi(mymacro,332)
nested_formats.texi(mymacro,332) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,332) @item i--tem 1
@@ -3343,11 +3469,17 @@
nested_formats.texi(mymacro,335) in verbatim
nested_formats.texi(mymacro,335) @end verbatim
nested_formats.texi(mymacro,335)
+nested_formats.texi(mymacro,335) @xml
+nested_formats.texi(mymacro,335) <para> xml para </xml>
+nested_formats.texi(mymacro,335) @end xml
nested_formats.texi(mymacro,335)
nested_formats.texi(mymacro,335) @html
nested_formats.texi(mymacro,335) html
nested_formats.texi(mymacro,335) @end html
nested_formats.texi(mymacro,335)
+nested_formats.texi(mymacro,335) @tex
+nested_formats.texi(mymacro,335) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,335) @end tex
nested_formats.texi(mymacro,335)
nested_formats.texi(mymacro,335) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,335) @item i--tem 1
@@ -3497,11 +3629,17 @@
nested_formats.texi(mymacro,338) in verbatim
nested_formats.texi(mymacro,338) @end verbatim
nested_formats.texi(mymacro,338)
+nested_formats.texi(mymacro,338) @xml
+nested_formats.texi(mymacro,338) <para> xml para </xml>
+nested_formats.texi(mymacro,338) @end xml
nested_formats.texi(mymacro,338)
nested_formats.texi(mymacro,338) @html
nested_formats.texi(mymacro,338) html
nested_formats.texi(mymacro,338) @end html
nested_formats.texi(mymacro,338)
+nested_formats.texi(mymacro,338) @tex
+nested_formats.texi(mymacro,338) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,338) @end tex
nested_formats.texi(mymacro,338)
nested_formats.texi(mymacro,338) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,338) @item i--tem 1
@@ -3654,11 +3792,17 @@
nested_formats.texi(mymacro,344) in verbatim
nested_formats.texi(mymacro,344) @end verbatim
nested_formats.texi(mymacro,344)
+nested_formats.texi(mymacro,344) @xml
+nested_formats.texi(mymacro,344) <para> xml para </xml>
+nested_formats.texi(mymacro,344) @end xml
nested_formats.texi(mymacro,344)
nested_formats.texi(mymacro,344) @html
nested_formats.texi(mymacro,344) html
nested_formats.texi(mymacro,344) @end html
nested_formats.texi(mymacro,344)
+nested_formats.texi(mymacro,344) @tex
+nested_formats.texi(mymacro,344) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,344) @end tex
nested_formats.texi(mymacro,344)
nested_formats.texi(mymacro,344) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,344) @item i--tem 1
@@ -3809,11 +3953,17 @@
nested_formats.texi(mymacro,348) in verbatim
nested_formats.texi(mymacro,348) @end verbatim
nested_formats.texi(mymacro,348)
+nested_formats.texi(mymacro,348) @xml
+nested_formats.texi(mymacro,348) <para> xml para </xml>
+nested_formats.texi(mymacro,348) @end xml
nested_formats.texi(mymacro,348)
nested_formats.texi(mymacro,348) @html
nested_formats.texi(mymacro,348) html
nested_formats.texi(mymacro,348) @end html
nested_formats.texi(mymacro,348)
+nested_formats.texi(mymacro,348) @tex
+nested_formats.texi(mymacro,348) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,348) @end tex
nested_formats.texi(mymacro,348)
nested_formats.texi(mymacro,348) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,348) @item i--tem 1
@@ -3965,11 +4115,17 @@
nested_formats.texi(mymacro,353) in verbatim
nested_formats.texi(mymacro,353) @end verbatim
nested_formats.texi(mymacro,353)
+nested_formats.texi(mymacro,353) @xml
+nested_formats.texi(mymacro,353) <para> xml para </xml>
+nested_formats.texi(mymacro,353) @end xml
nested_formats.texi(mymacro,353)
nested_formats.texi(mymacro,353) @html
nested_formats.texi(mymacro,353) html
nested_formats.texi(mymacro,353) @end html
nested_formats.texi(mymacro,353)
+nested_formats.texi(mymacro,353) @tex
+nested_formats.texi(mymacro,353) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,353) @end tex
nested_formats.texi(mymacro,353)
nested_formats.texi(mymacro,353) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,353) @item i--tem 1
@@ -4119,11 +4275,17 @@
nested_formats.texi(mymacro,356) in verbatim
nested_formats.texi(mymacro,356) @end verbatim
nested_formats.texi(mymacro,356)
+nested_formats.texi(mymacro,356) @xml
+nested_formats.texi(mymacro,356) <para> xml para </xml>
+nested_formats.texi(mymacro,356) @end xml
nested_formats.texi(mymacro,356)
nested_formats.texi(mymacro,356) @html
nested_formats.texi(mymacro,356) html
nested_formats.texi(mymacro,356) @end html
nested_formats.texi(mymacro,356)
+nested_formats.texi(mymacro,356) @tex
+nested_formats.texi(mymacro,356) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,356) @end tex
nested_formats.texi(mymacro,356)
nested_formats.texi(mymacro,356) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,356) @item i--tem 1
Index: test/nested_formats/res/texi_nested_formats/nested_formats.passtexi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/nested_formats/res/texi_nested_formats/nested_formats.passtexi,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/nested_formats/res/texi_nested_formats/nested_formats.passtexi 18 Aug
2008 18:06:05 -0000 1.1
+++ test/nested_formats/res/texi_nested_formats/nested_formats.passtexi 9 Jan
2009 21:20:54 -0000 1.2
@@ -70,11 +70,17 @@
nested_formats.texi(mymacro,233) in verbatim
nested_formats.texi(mymacro,233) @end verbatim
nested_formats.texi(mymacro,233)
+nested_formats.texi(mymacro,233) @xml
+nested_formats.texi(mymacro,233) <para> xml para </xml>
+nested_formats.texi(mymacro,233) @end xml
nested_formats.texi(mymacro,233)
nested_formats.texi(mymacro,233) @html
nested_formats.texi(mymacro,233) html
nested_formats.texi(mymacro,233) @end html
nested_formats.texi(mymacro,233)
+nested_formats.texi(mymacro,233) @tex
+nested_formats.texi(mymacro,233) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,233) @end tex
nested_formats.texi(mymacro,233)
nested_formats.texi(mymacro,233) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,233) @item i--tem 1
@@ -225,11 +231,17 @@
nested_formats.texi(mymacro,237) in verbatim
nested_formats.texi(mymacro,237) @end verbatim
nested_formats.texi(mymacro,237)
+nested_formats.texi(mymacro,237) @xml
+nested_formats.texi(mymacro,237) <para> xml para </xml>
+nested_formats.texi(mymacro,237) @end xml
nested_formats.texi(mymacro,237)
nested_formats.texi(mymacro,237) @html
nested_formats.texi(mymacro,237) html
nested_formats.texi(mymacro,237) @end html
nested_formats.texi(mymacro,237)
+nested_formats.texi(mymacro,237) @tex
+nested_formats.texi(mymacro,237) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,237) @end tex
nested_formats.texi(mymacro,237)
nested_formats.texi(mymacro,237) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,237) @item i--tem 1
@@ -381,11 +393,17 @@
nested_formats.texi(mymacro,242) in verbatim
nested_formats.texi(mymacro,242) @end verbatim
nested_formats.texi(mymacro,242)
+nested_formats.texi(mymacro,242) @xml
+nested_formats.texi(mymacro,242) <para> xml para </xml>
+nested_formats.texi(mymacro,242) @end xml
nested_formats.texi(mymacro,242)
nested_formats.texi(mymacro,242) @html
nested_formats.texi(mymacro,242) html
nested_formats.texi(mymacro,242) @end html
nested_formats.texi(mymacro,242)
+nested_formats.texi(mymacro,242) @tex
+nested_formats.texi(mymacro,242) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,242) @end tex
nested_formats.texi(mymacro,242)
nested_formats.texi(mymacro,242) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,242) @item i--tem 1
@@ -535,11 +553,17 @@
nested_formats.texi(mymacro,245) in verbatim
nested_formats.texi(mymacro,245) @end verbatim
nested_formats.texi(mymacro,245)
+nested_formats.texi(mymacro,245) @xml
+nested_formats.texi(mymacro,245) <para> xml para </xml>
+nested_formats.texi(mymacro,245) @end xml
nested_formats.texi(mymacro,245)
nested_formats.texi(mymacro,245) @html
nested_formats.texi(mymacro,245) html
nested_formats.texi(mymacro,245) @end html
nested_formats.texi(mymacro,245)
+nested_formats.texi(mymacro,245) @tex
+nested_formats.texi(mymacro,245) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,245) @end tex
nested_formats.texi(mymacro,245)
nested_formats.texi(mymacro,245) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,245) @item i--tem 1
@@ -690,11 +714,17 @@
nested_formats.texi(mymacro,249) in verbatim
nested_formats.texi(mymacro,249) @end verbatim
nested_formats.texi(mymacro,249)
+nested_formats.texi(mymacro,249) @xml
+nested_formats.texi(mymacro,249) <para> xml para </xml>
+nested_formats.texi(mymacro,249) @end xml
nested_formats.texi(mymacro,249)
nested_formats.texi(mymacro,249) @html
nested_formats.texi(mymacro,249) html
nested_formats.texi(mymacro,249) @end html
nested_formats.texi(mymacro,249)
+nested_formats.texi(mymacro,249) @tex
+nested_formats.texi(mymacro,249) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,249) @end tex
nested_formats.texi(mymacro,249)
nested_formats.texi(mymacro,249) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,249) @item i--tem 1
@@ -847,11 +877,17 @@
nested_formats.texi(mymacro,255) in verbatim
nested_formats.texi(mymacro,255) @end verbatim
nested_formats.texi(mymacro,255)
+nested_formats.texi(mymacro,255) @xml
+nested_formats.texi(mymacro,255) <para> xml para </xml>
+nested_formats.texi(mymacro,255) @end xml
nested_formats.texi(mymacro,255)
nested_formats.texi(mymacro,255) @html
nested_formats.texi(mymacro,255) html
nested_formats.texi(mymacro,255) @end html
nested_formats.texi(mymacro,255)
+nested_formats.texi(mymacro,255) @tex
+nested_formats.texi(mymacro,255) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,255) @end tex
nested_formats.texi(mymacro,255)
nested_formats.texi(mymacro,255) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,255) @item i--tem 1
@@ -1001,11 +1037,17 @@
nested_formats.texi(mymacro,258) in verbatim
nested_formats.texi(mymacro,258) @end verbatim
nested_formats.texi(mymacro,258)
+nested_formats.texi(mymacro,258) @xml
+nested_formats.texi(mymacro,258) <para> xml para </xml>
+nested_formats.texi(mymacro,258) @end xml
nested_formats.texi(mymacro,258)
nested_formats.texi(mymacro,258) @html
nested_formats.texi(mymacro,258) html
nested_formats.texi(mymacro,258) @end html
nested_formats.texi(mymacro,258)
+nested_formats.texi(mymacro,258) @tex
+nested_formats.texi(mymacro,258) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,258) @end tex
nested_formats.texi(mymacro,258)
nested_formats.texi(mymacro,258) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,258) @item i--tem 1
@@ -1156,11 +1198,17 @@
nested_formats.texi(mymacro,262) in verbatim
nested_formats.texi(mymacro,262) @end verbatim
nested_formats.texi(mymacro,262)
+nested_formats.texi(mymacro,262) @xml
+nested_formats.texi(mymacro,262) <para> xml para </xml>
+nested_formats.texi(mymacro,262) @end xml
nested_formats.texi(mymacro,262)
nested_formats.texi(mymacro,262) @html
nested_formats.texi(mymacro,262) html
nested_formats.texi(mymacro,262) @end html
nested_formats.texi(mymacro,262)
+nested_formats.texi(mymacro,262) @tex
+nested_formats.texi(mymacro,262) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,262) @end tex
nested_formats.texi(mymacro,262)
nested_formats.texi(mymacro,262) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,262) @item i--tem 1
@@ -1315,11 +1363,17 @@
nested_formats.texi(mymacro,270) in verbatim
nested_formats.texi(mymacro,270) @end verbatim
nested_formats.texi(mymacro,270)
+nested_formats.texi(mymacro,270) @xml
+nested_formats.texi(mymacro,270) <para> xml para </xml>
+nested_formats.texi(mymacro,270) @end xml
nested_formats.texi(mymacro,270)
nested_formats.texi(mymacro,270) @html
nested_formats.texi(mymacro,270) html
nested_formats.texi(mymacro,270) @end html
nested_formats.texi(mymacro,270)
+nested_formats.texi(mymacro,270) @tex
+nested_formats.texi(mymacro,270) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,270) @end tex
nested_formats.texi(mymacro,270)
nested_formats.texi(mymacro,270) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,270) @item i--tem 1
@@ -1469,11 +1523,17 @@
nested_formats.texi(mymacro,273) in verbatim
nested_formats.texi(mymacro,273) @end verbatim
nested_formats.texi(mymacro,273)
+nested_formats.texi(mymacro,273) @xml
+nested_formats.texi(mymacro,273) <para> xml para </xml>
+nested_formats.texi(mymacro,273) @end xml
nested_formats.texi(mymacro,273)
nested_formats.texi(mymacro,273) @html
nested_formats.texi(mymacro,273) html
nested_formats.texi(mymacro,273) @end html
nested_formats.texi(mymacro,273)
+nested_formats.texi(mymacro,273) @tex
+nested_formats.texi(mymacro,273) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,273) @end tex
nested_formats.texi(mymacro,273)
nested_formats.texi(mymacro,273) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,273) @item i--tem 1
@@ -1624,11 +1684,17 @@
nested_formats.texi(mymacro,277) in verbatim
nested_formats.texi(mymacro,277) @end verbatim
nested_formats.texi(mymacro,277)
+nested_formats.texi(mymacro,277) @xml
+nested_formats.texi(mymacro,277) <para> xml para </xml>
+nested_formats.texi(mymacro,277) @end xml
nested_formats.texi(mymacro,277)
nested_formats.texi(mymacro,277) @html
nested_formats.texi(mymacro,277) html
nested_formats.texi(mymacro,277) @end html
nested_formats.texi(mymacro,277)
+nested_formats.texi(mymacro,277) @tex
+nested_formats.texi(mymacro,277) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,277) @end tex
nested_formats.texi(mymacro,277)
nested_formats.texi(mymacro,277) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,277) @item i--tem 1
@@ -1781,11 +1847,17 @@
nested_formats.texi(mymacro,283) in verbatim
nested_formats.texi(mymacro,283) @end verbatim
nested_formats.texi(mymacro,283)
+nested_formats.texi(mymacro,283) @xml
+nested_formats.texi(mymacro,283) <para> xml para </xml>
+nested_formats.texi(mymacro,283) @end xml
nested_formats.texi(mymacro,283)
nested_formats.texi(mymacro,283) @html
nested_formats.texi(mymacro,283) html
nested_formats.texi(mymacro,283) @end html
nested_formats.texi(mymacro,283)
+nested_formats.texi(mymacro,283) @tex
+nested_formats.texi(mymacro,283) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,283) @end tex
nested_formats.texi(mymacro,283)
nested_formats.texi(mymacro,283) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,283) @item i--tem 1
@@ -1936,11 +2008,17 @@
nested_formats.texi(mymacro,287) in verbatim
nested_formats.texi(mymacro,287) @end verbatim
nested_formats.texi(mymacro,287)
+nested_formats.texi(mymacro,287) @xml
+nested_formats.texi(mymacro,287) <para> xml para </xml>
+nested_formats.texi(mymacro,287) @end xml
nested_formats.texi(mymacro,287)
nested_formats.texi(mymacro,287) @html
nested_formats.texi(mymacro,287) html
nested_formats.texi(mymacro,287) @end html
nested_formats.texi(mymacro,287)
+nested_formats.texi(mymacro,287) @tex
+nested_formats.texi(mymacro,287) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,287) @end tex
nested_formats.texi(mymacro,287)
nested_formats.texi(mymacro,287) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,287) @item i--tem 1
@@ -2092,11 +2170,17 @@
nested_formats.texi(mymacro,292) in verbatim
nested_formats.texi(mymacro,292) @end verbatim
nested_formats.texi(mymacro,292)
+nested_formats.texi(mymacro,292) @xml
+nested_formats.texi(mymacro,292) <para> xml para </xml>
+nested_formats.texi(mymacro,292) @end xml
nested_formats.texi(mymacro,292)
nested_formats.texi(mymacro,292) @html
nested_formats.texi(mymacro,292) html
nested_formats.texi(mymacro,292) @end html
nested_formats.texi(mymacro,292)
+nested_formats.texi(mymacro,292) @tex
+nested_formats.texi(mymacro,292) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,292) @end tex
nested_formats.texi(mymacro,292)
nested_formats.texi(mymacro,292) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,292) @item i--tem 1
@@ -2254,11 +2338,17 @@
nested_formats.texi(mymacro,303) in verbatim
nested_formats.texi(mymacro,303) @end verbatim
nested_formats.texi(mymacro,303)
+nested_formats.texi(mymacro,303) @xml
+nested_formats.texi(mymacro,303) <para> xml para </xml>
+nested_formats.texi(mymacro,303) @end xml
nested_formats.texi(mymacro,303)
nested_formats.texi(mymacro,303) @html
nested_formats.texi(mymacro,303) html
nested_formats.texi(mymacro,303) @end html
nested_formats.texi(mymacro,303)
+nested_formats.texi(mymacro,303) @tex
+nested_formats.texi(mymacro,303) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,303) @end tex
nested_formats.texi(mymacro,303)
nested_formats.texi(mymacro,303) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,303) @item i--tem 1
@@ -2409,11 +2499,17 @@
nested_formats.texi(mymacro,307) in verbatim
nested_formats.texi(mymacro,307) @end verbatim
nested_formats.texi(mymacro,307)
+nested_formats.texi(mymacro,307) @xml
+nested_formats.texi(mymacro,307) <para> xml para </xml>
+nested_formats.texi(mymacro,307) @end xml
nested_formats.texi(mymacro,307)
nested_formats.texi(mymacro,307) @html
nested_formats.texi(mymacro,307) html
nested_formats.texi(mymacro,307) @end html
nested_formats.texi(mymacro,307)
+nested_formats.texi(mymacro,307) @tex
+nested_formats.texi(mymacro,307) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,307) @end tex
nested_formats.texi(mymacro,307)
nested_formats.texi(mymacro,307) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,307) @item i--tem 1
@@ -2566,11 +2662,17 @@
nested_formats.texi(mymacro,313) in verbatim
nested_formats.texi(mymacro,313) @end verbatim
nested_formats.texi(mymacro,313)
+nested_formats.texi(mymacro,313) @xml
+nested_formats.texi(mymacro,313) <para> xml para </xml>
+nested_formats.texi(mymacro,313) @end xml
nested_formats.texi(mymacro,313)
nested_formats.texi(mymacro,313) @html
nested_formats.texi(mymacro,313) html
nested_formats.texi(mymacro,313) @end html
nested_formats.texi(mymacro,313)
+nested_formats.texi(mymacro,313) @tex
+nested_formats.texi(mymacro,313) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,313) @end tex
nested_formats.texi(mymacro,313)
nested_formats.texi(mymacro,313) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,313) @item i--tem 1
@@ -2723,11 +2825,17 @@
nested_formats.texi(mymacro,319) in verbatim
nested_formats.texi(mymacro,319) @end verbatim
nested_formats.texi(mymacro,319)
+nested_formats.texi(mymacro,319) @xml
+nested_formats.texi(mymacro,319) <para> xml para </xml>
+nested_formats.texi(mymacro,319) @end xml
nested_formats.texi(mymacro,319)
nested_formats.texi(mymacro,319) @html
nested_formats.texi(mymacro,319) html
nested_formats.texi(mymacro,319) @end html
nested_formats.texi(mymacro,319)
+nested_formats.texi(mymacro,319) @tex
+nested_formats.texi(mymacro,319) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,319) @end tex
nested_formats.texi(mymacro,319)
nested_formats.texi(mymacro,319) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,319) @item i--tem 1
@@ -2877,11 +2985,17 @@
nested_formats.texi(mymacro,322) in verbatim
nested_formats.texi(mymacro,322) @end verbatim
nested_formats.texi(mymacro,322)
+nested_formats.texi(mymacro,322) @xml
+nested_formats.texi(mymacro,322) <para> xml para </xml>
+nested_formats.texi(mymacro,322) @end xml
nested_formats.texi(mymacro,322)
nested_formats.texi(mymacro,322) @html
nested_formats.texi(mymacro,322) html
nested_formats.texi(mymacro,322) @end html
nested_formats.texi(mymacro,322)
+nested_formats.texi(mymacro,322) @tex
+nested_formats.texi(mymacro,322) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,322) @end tex
nested_formats.texi(mymacro,322)
nested_formats.texi(mymacro,322) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,322) @item i--tem 1
@@ -3032,11 +3146,17 @@
nested_formats.texi(mymacro,326) in verbatim
nested_formats.texi(mymacro,326) @end verbatim
nested_formats.texi(mymacro,326)
+nested_formats.texi(mymacro,326) @xml
+nested_formats.texi(mymacro,326) <para> xml para </xml>
+nested_formats.texi(mymacro,326) @end xml
nested_formats.texi(mymacro,326)
nested_formats.texi(mymacro,326) @html
nested_formats.texi(mymacro,326) html
nested_formats.texi(mymacro,326) @end html
nested_formats.texi(mymacro,326)
+nested_formats.texi(mymacro,326) @tex
+nested_formats.texi(mymacro,326) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,326) @end tex
nested_formats.texi(mymacro,326)
nested_formats.texi(mymacro,326) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,326) @item i--tem 1
@@ -3189,11 +3309,17 @@
nested_formats.texi(mymacro,332) in verbatim
nested_formats.texi(mymacro,332) @end verbatim
nested_formats.texi(mymacro,332)
+nested_formats.texi(mymacro,332) @xml
+nested_formats.texi(mymacro,332) <para> xml para </xml>
+nested_formats.texi(mymacro,332) @end xml
nested_formats.texi(mymacro,332)
nested_formats.texi(mymacro,332) @html
nested_formats.texi(mymacro,332) html
nested_formats.texi(mymacro,332) @end html
nested_formats.texi(mymacro,332)
+nested_formats.texi(mymacro,332) @tex
+nested_formats.texi(mymacro,332) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,332) @end tex
nested_formats.texi(mymacro,332)
nested_formats.texi(mymacro,332) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,332) @item i--tem 1
@@ -3343,11 +3469,17 @@
nested_formats.texi(mymacro,335) in verbatim
nested_formats.texi(mymacro,335) @end verbatim
nested_formats.texi(mymacro,335)
+nested_formats.texi(mymacro,335) @xml
+nested_formats.texi(mymacro,335) <para> xml para </xml>
+nested_formats.texi(mymacro,335) @end xml
nested_formats.texi(mymacro,335)
nested_formats.texi(mymacro,335) @html
nested_formats.texi(mymacro,335) html
nested_formats.texi(mymacro,335) @end html
nested_formats.texi(mymacro,335)
+nested_formats.texi(mymacro,335) @tex
+nested_formats.texi(mymacro,335) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,335) @end tex
nested_formats.texi(mymacro,335)
nested_formats.texi(mymacro,335) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,335) @item i--tem 1
@@ -3497,11 +3629,17 @@
nested_formats.texi(mymacro,338) in verbatim
nested_formats.texi(mymacro,338) @end verbatim
nested_formats.texi(mymacro,338)
+nested_formats.texi(mymacro,338) @xml
+nested_formats.texi(mymacro,338) <para> xml para </xml>
+nested_formats.texi(mymacro,338) @end xml
nested_formats.texi(mymacro,338)
nested_formats.texi(mymacro,338) @html
nested_formats.texi(mymacro,338) html
nested_formats.texi(mymacro,338) @end html
nested_formats.texi(mymacro,338)
+nested_formats.texi(mymacro,338) @tex
+nested_formats.texi(mymacro,338) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,338) @end tex
nested_formats.texi(mymacro,338)
nested_formats.texi(mymacro,338) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,338) @item i--tem 1
@@ -3654,11 +3792,17 @@
nested_formats.texi(mymacro,344) in verbatim
nested_formats.texi(mymacro,344) @end verbatim
nested_formats.texi(mymacro,344)
+nested_formats.texi(mymacro,344) @xml
+nested_formats.texi(mymacro,344) <para> xml para </xml>
+nested_formats.texi(mymacro,344) @end xml
nested_formats.texi(mymacro,344)
nested_formats.texi(mymacro,344) @html
nested_formats.texi(mymacro,344) html
nested_formats.texi(mymacro,344) @end html
nested_formats.texi(mymacro,344)
+nested_formats.texi(mymacro,344) @tex
+nested_formats.texi(mymacro,344) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,344) @end tex
nested_formats.texi(mymacro,344)
nested_formats.texi(mymacro,344) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,344) @item i--tem 1
@@ -3809,11 +3953,17 @@
nested_formats.texi(mymacro,348) in verbatim
nested_formats.texi(mymacro,348) @end verbatim
nested_formats.texi(mymacro,348)
+nested_formats.texi(mymacro,348) @xml
+nested_formats.texi(mymacro,348) <para> xml para </xml>
+nested_formats.texi(mymacro,348) @end xml
nested_formats.texi(mymacro,348)
nested_formats.texi(mymacro,348) @html
nested_formats.texi(mymacro,348) html
nested_formats.texi(mymacro,348) @end html
nested_formats.texi(mymacro,348)
+nested_formats.texi(mymacro,348) @tex
+nested_formats.texi(mymacro,348) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,348) @end tex
nested_formats.texi(mymacro,348)
nested_formats.texi(mymacro,348) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,348) @item i--tem 1
@@ -3965,11 +4115,17 @@
nested_formats.texi(mymacro,353) in verbatim
nested_formats.texi(mymacro,353) @end verbatim
nested_formats.texi(mymacro,353)
+nested_formats.texi(mymacro,353) @xml
+nested_formats.texi(mymacro,353) <para> xml para </xml>
+nested_formats.texi(mymacro,353) @end xml
nested_formats.texi(mymacro,353)
nested_formats.texi(mymacro,353) @html
nested_formats.texi(mymacro,353) html
nested_formats.texi(mymacro,353) @end html
nested_formats.texi(mymacro,353)
+nested_formats.texi(mymacro,353) @tex
+nested_formats.texi(mymacro,353) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,353) @end tex
nested_formats.texi(mymacro,353)
nested_formats.texi(mymacro,353) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,353) @item i--tem 1
@@ -4119,11 +4275,17 @@
nested_formats.texi(mymacro,356) in verbatim
nested_formats.texi(mymacro,356) @end verbatim
nested_formats.texi(mymacro,356)
+nested_formats.texi(mymacro,356) @xml
+nested_formats.texi(mymacro,356) <para> xml para </xml>
+nested_formats.texi(mymacro,356) @end xml
nested_formats.texi(mymacro,356)
nested_formats.texi(mymacro,356) @html
nested_formats.texi(mymacro,356) html
nested_formats.texi(mymacro,356) @end html
nested_formats.texi(mymacro,356)
+nested_formats.texi(mymacro,356) @tex
+nested_formats.texi(mymacro,356) $$\partial_t \eta (t) =
g(\eta(t),\varphi(t))$$
+nested_formats.texi(mymacro,356) @end tex
nested_formats.texi(mymacro,356)
nested_formats.texi(mymacro,356) @itemize @bullet{} a--n itemize line
nested_formats.texi(mymacro,356) @item i--tem 1
Index: test/nested_formats/res/texi_nested_formats/nested_formats.texi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/nested_formats/res/texi_nested_formats/nested_formats.texi,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -b -r1.1 -r1.2
--- test/nested_formats/res/texi_nested_formats/nested_formats.texi 18 Aug
2008 18:06:05 -0000 1.1
+++ test/nested_formats/res/texi_nested_formats/nested_formats.texi 9 Jan
2009 21:20:55 -0000 1.2
@@ -71,11 +71,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -226,11 +232,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -382,11 +394,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -536,11 +554,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -691,11 +715,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -848,11 +878,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -1002,11 +1038,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -1157,11 +1199,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -1316,11 +1364,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -1470,11 +1524,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -1625,11 +1685,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -1782,11 +1848,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -1937,11 +2009,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -2093,11 +2171,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -2255,11 +2339,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -2410,11 +2500,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -2567,11 +2663,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -2724,11 +2826,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -2878,11 +2986,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -3033,11 +3147,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -3190,11 +3310,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -3344,11 +3470,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -3498,11 +3630,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -3655,11 +3793,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -3810,11 +3954,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -3966,11 +4116,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
@@ -4120,11 +4276,17 @@
in verbatim
@end verbatim
address@hidden
+<para> xml para </xml>
address@hidden xml
@html
html
@end html
address@hidden
+$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
address@hidden tex
@itemize @bullet{} a--n itemize line
@item i--tem 1
Index: test/singular_manual/res/texi_singular/singular.passfirst
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/singular_manual/res/texi_singular/singular.passfirst,v
retrieving revision 1.2
retrieving revision 1.3
diff -u -b -r1.2 -r1.3
--- test/singular_manual/res/texi_singular/singular.passfirst 19 Aug 2008
16:53:00 -0000 1.2
+++ test/singular_manual/res/texi_singular/singular.passfirst 9 Jan 2009
21:20:57 -0000 1.3
@@ -309,6 +309,9 @@
start.tex(,78) @sc{Singular}'s development started in 1984 with an
implementation of
start.tex(,79) Mora's Tangent Cone algorithm in Modula-2 on an Atari computer
(K.P.
start.tex(,80) Neuendorf, G. Pfister,
+start.tex(,84) @tex
+start.tex(,85) H.\ Sch\"onemann; Humboldt-Universit\"at
+start.tex(,86) @end tex
start.tex(,87) zu Berlin). The need for a new system arose from the
investigation of
start.tex(,88) mathematical problems coming from singularity theory which none
of the
start.tex(,89) existing systems was able to compute.
@@ -551,6 +554,9 @@
start.tex(,351) @noindent This shows the text of @ref{intmat}, in the printed
manual.
start.tex(,356)
start.tex(,357) Next, we define a
+start.tex(,358) @tex
+start.tex(,359) $3 \times 3$
+start.tex(,360) @end tex
start.tex(,364) matrix of integers and initialize it with some values, row by
row
start.tex(,365) from left to right:
start.tex(,366)
@@ -627,6 +633,9 @@
start.tex(,442) ring variables, and the third part determines the monomial
ordering to
start.tex(,443) be used. So the example above declares a polynomial ring
called @code{r}
start.tex(,444) with a ground field of characteristic
+start.tex(,448) @tex
+start.tex(,449) $0$
+start.tex(,450) @end tex
start.tex(,451) (i.e., the rational
start.tex(,452) numbers) and ring variables called @code{x}, @code{y}, and
@code{z}. The
start.tex(,453) @code{dp} at the end means that the degree reverse
lexicographical
@@ -649,7 +658,13 @@
start.tex(,470)
start.tex(,471) @item ring r4=(0,a),(mu,nu),lp;
start.tex(,472) transcendental extension of
+start.tex(,476) @tex
+start.tex(,477) $Q$
+start.tex(,478) @end tex
start.tex(,479) by
+start.tex(,483) @tex
+start.tex(,484) $a$
+start.tex(,485) @end tex
start.tex(,486) , variable names
start.tex(,487) @code{mu} and @code{nu}.
start.tex(,488)
@@ -678,6 +693,9 @@
start.tex(,511) @c
start.tex(,512) Typing the name of a ring prints its definition. The example
below
start.tex(,513) shows that the default ring in @sc{Singular} is
+start.tex(,517) @tex
+start.tex(,518) $Z/32003[x,y,z]$
+start.tex(,519) @end tex
start.tex(,520)
start.tex(,521) with degree reverse lexicographical ordering:
start.tex(,522)
@@ -707,6 +725,9 @@
start.tex(,551) @end smallexample
start.tex(,552)
start.tex(,553) Once a ring is active, we can define polynomials. A monomial,
say
+start.tex(,554) @tex
+start.tex(,555) $x^3$
+start.tex(,556) @end tex
start.tex(,560) may be entered in two ways: either using the power operator
@code{^},
start.tex(,561) saying @code{x^3}, or in short-hand notation without operator,
saying
start.tex(,562) @code{x3}. Note that the short-hand notation is forbidden if
the name
@@ -825,6 +846,9 @@
start.tex(,677) @end smallexample
start.tex(,678)
start.tex(,679) @noindent gives the desired vector space dimension
+start.tex(,680) @tex
+start.tex(,681) $K[x,y,z]/\hbox{\rm jacob}(f)$.
+start.tex(,682) @end tex
start.tex(,686) As in @sc{Singular} the functions may take the input directly
from
start.tex(,687) earlier calculations, the whole sequence of commands may be
written
start.tex(,688) in one single statement.
@@ -998,6 +1022,9 @@
start.tex(,876)
start.tex(,877) This shows that @code{f} has outside the origin in affine
3-space
start.tex(,878) singularities with local Milnor number adding up to
+start.tex(,879) @tex
+start.tex(,880) $12-4=8$.
+start.tex(,881) @end tex
start.tex(,885) Using global and local orderings as above is a convenient way
to check
start.tex(,886) whether a variety has singularities outside the origin.
start.tex(,887)
@@ -1044,6 +1071,9 @@
start.tex(,928) The algorithm of the standard basis computations may be
start.tex(,929) affected by the command @code{option}. For example, a reduced
standard
start.tex(,930) basis of the ideal generated by the
+start.tex(,931) @tex
+start.tex(,932) $1 \times 1$-minors
+start.tex(,933) @end tex
start.tex(,937) of H is obtained in the following way:
start.tex(,938) @smallexample
start.tex(,939) option(redSB);
@@ -1052,6 +1082,9 @@
start.tex(,942) @end smallexample
start.tex(,943)
start.tex(,944) This shows that 1 is contained in the ideal of the
+start.tex(,945) @tex
+start.tex(,946) $1 \times 1$-minors,
+start.tex(,947) @end tex
start.tex(,951) hence the corresponding variety is empty.
start.tex(,952) @c Coming back to some mathematical considerations, we study
the problem how
start.tex(,953) @c to calculate some ....
@@ -1107,13 +1140,22 @@
start.tex(,1008) @end smallexample
start.tex(,1009)
start.tex(,1010) However the submodule
+start.tex(,1014) @tex
+start.tex(,1015) $MD$
+start.tex(,1016) @end tex
start.tex(,1017) may also be considered as the module
start.tex(,1018) of relations of the factor module
+start.tex(,1019) @tex
+start.tex(,1020) $r^3/MD$.
+start.tex(,1021) @end tex
start.tex(,1025) In this way, @sc{Singular} can treat arbitrary finitely
generated modules
start.tex(,1026) over the
start.tex(,1028) basering (@pxref{Representation of mathematical objects}).
start.tex(,1033)
start.tex(,1034) In order to get the module of relations of
+start.tex(,1038) @tex
+start.tex(,1039) $MD$
+start.tex(,1040) @end tex
start.tex(,1041) ,
start.tex(,1042) we use the command @code{syz}.
start.tex(,1043)
@@ -1124,15 +1166,30 @@
start.tex(,1048)
start.tex(,1049) We want to calculate, as an application, the annihilator of a
given module.
start.tex(,1050) Let
+start.tex(,1051) @tex
+start.tex(,1052) $M = r^3/U$,
+start.tex(,1053) @end tex
start.tex(,1057) where U is our defining module of relations for the module
+start.tex(,1058) @tex
+start.tex(,1059) $M$.
+start.tex(,1060) @end tex
start.tex(,1064)
start.tex(,1065) @smallexample
start.tex(,1066) module U =
[z3,xy2,x3],[yz2,1,xy5z+z3],[y2z,0,x3],[xyz+x2,y2,0],[xyz,x2y,1];
start.tex(,1067) @end smallexample
start.tex(,1068)
start.tex(,1069) Then, by definition, the annihilator of M is the ideal
+start.tex(,1070) @tex
+start.tex(,1071) $\hbox{ann}(M) = \{a \mid aM = 0 \}$
+start.tex(,1072) @end tex
start.tex(,1076) which is by the description of M the same as
+start.tex(,1077) @tex
+start.tex(,1078) $\{ a \mid ar^3 \in U \}$.
+start.tex(,1079) @end tex
start.tex(,1083) Hence we have to calculate the quotient
+start.tex(,1084) @tex
+start.tex(,1085) $U \colon r^3 $.
+start.tex(,1086) @end tex
start.tex(,1090) The rank of the free module is determined by the choice of U
and is the
start.tex(,1091) number of rows of the corresponding matrix. This may be
determined by
start.tex(,1092) the function @code{nrows}. All we have to do now is the
following:
@@ -1152,7 +1209,13 @@
start.tex(,1111) The most general command is @code{res(... ,n)} which
determines heuristically
start.tex(,1112) what method to use for the given problem. It computes the
free resolution
start.tex(,1113) up to the length
+start.tex(,1117) @tex
+start.tex(,1118) $n$
+start.tex(,1119) @end tex
start.tex(,1120) , where
+start.tex(,1124) @tex
+start.tex(,1125) $n=0$
+start.tex(,1126) @end tex
start.tex(,1127) corresponds to the full resolution.
start.tex(,1128)
start.tex(,1129) Here we use the possibility to inspect the calculation
process using the
@@ -1224,7 +1287,13 @@
start.tex(,1195)
start.tex(,1196) In this case, the output is to be interpreted as follows: the
3rd syzygy
start.tex(,1197) module of R/I, @code{rs[3]}, is the rank-2-submodule of
+start.tex(,1198) @tex
+start.tex(,1199) $R^5$
+start.tex(,1200) @end tex
start.tex(,1204) generated by the vectors
+start.tex(,1205) @tex
+start.tex(,1206) $(z^3,0,-y+4z,x+2z,0)$ and
$(-xyz-y^2z-4xz^2+16z^3,-y^2,48z,48z,x+y-z)$.
+start.tex(,1207) @end tex
start.tex(,1211)
singular.texi(,128) @c
----------------------------------------------------------------------------
singular.texi(,129) @node General concepts, Data types, Introduction, Top
@@ -2639,16 +2708,37 @@
general.tex(,1435) @enumerate
general.tex(,1436) @item
general.tex(,1437) the field of rational numbers
+general.tex(,1441) @tex
+general.tex(,1442) $Q$
+general.tex(,1443) @end tex
general.tex(,1444) ,
general.tex(,1445) @item
+general.tex(,1446) @tex
+general.tex(,1447) finite fields $Z/p$, $p$ a prime $\le 2147483629$,
+general.tex(,1448) @end tex
general.tex(,1452) @item
+general.tex(,1453) @tex
+general.tex(,1454) finite fields $\hbox{GF}(p^n)$ with $p^n$ elements, $p$ a
prime, $p^n \le 2^{15}$,
+general.tex(,1455) @end tex
general.tex(,1459) @item
general.tex(,1460) transcendental extension of
+general.tex(,1464) @tex
+general.tex(,1465) $Q$
+general.tex(,1466) @end tex
general.tex(,1467) or
+general.tex(,1471) @tex
+general.tex(,1472) $Z/p$
+general.tex(,1473) @end tex
general.tex(,1474) ,
general.tex(,1475) @item
general.tex(,1476) simple algebraic extension of
+general.tex(,1480) @tex
+general.tex(,1481) $Q$
+general.tex(,1482) @end tex
general.tex(,1483) or
+general.tex(,1487) @tex
+general.tex(,1488) $Z/p$
+general.tex(,1489) @end tex
general.tex(,1490) ,
general.tex(,1491) @item
general.tex(,1492) the field of real numbers represented by floating point
@@ -2699,6 +2789,9 @@
general.tex(,1537) @itemize @bullet
general.tex(,1538) @item
general.tex(,1539) the ring
+general.tex(,1543) @tex
+general.tex(,1544) $Z/32003[x,y,z]$
+general.tex(,1545) @end tex
general.tex(,1546) with degree reverse lexicographical
general.tex(,1547) ordering. The exact ring declaration may be omitted in the
first
general.tex(,1548) example since this is the default ring:
@@ -2710,6 +2803,9 @@
general.tex(,1554)
general.tex(,1555) @item
general.tex(,1556) the ring
+general.tex(,1560) @tex
+general.tex(,1561) $Q[a,b,c,d]$
+general.tex(,1562) @end tex
general.tex(,1563) with lexicographical ordering:
general.tex(,1564)
general.tex(,1565) @smallexample
@@ -2718,6 +2814,9 @@
general.tex(,1568)
general.tex(,1569) @item
general.tex(,1570) the ring
+general.tex(,1574) @tex
+general.tex(,1575) $Z/7[x,y,z]$
+general.tex(,1576) @end tex
general.tex(,1577) with local degree reverse lexicographical
general.tex(,1578) ordering. The non-prime 10 is converted to the next lower
prime in the
general.tex(,1579) second example:
@@ -2729,8 +2828,17 @@
general.tex(,1585)
general.tex(,1586) @item
general.tex(,1587) the ring
+general.tex(,1588) @tex
+general.tex(,1589) $Z/7[x_1,\ldots,x_6]$
+general.tex(,1590) @end tex
general.tex(,1594) with lexicographical ordering for
+general.tex(,1595) @tex
+general.tex(,1596) $x_1,x_2,x_3$
+general.tex(,1597) @end tex
general.tex(,1601) and degree reverse lexicographical ordering for
+general.tex(,1602) @tex
+general.tex(,1603) $x_4,x_5,x_6$:
+general.tex(,1604) @end tex
general.tex(,1608)
general.tex(,1609) @smallexample
general.tex(,1610) ring r = 7,(x(1..6)),(lp(3),dp);
@@ -2738,8 +2846,14 @@
general.tex(,1612)
general.tex(,1613) @item
general.tex(,1614) the localization of
+general.tex(,1618) @tex
+general.tex(,1619) $(Q[a,b,c])[x,y,z]$
+general.tex(,1620) @end tex
general.tex(,1621) at the maximal ideal
general.tex(,1622)
+general.tex(,1626) @tex
+general.tex(,1627) $(x,y,z)$
+general.tex(,1628) @end tex
general.tex(,1629) :
general.tex(,1630)
general.tex(,1631) @smallexample
@@ -2748,10 +2862,22 @@
general.tex(,1634)
general.tex(,1635) @item
general.tex(,1636) the ring
+general.tex(,1640) @tex
+general.tex(,1641) $Q[x,y,z]$
+general.tex(,1642) @end tex
general.tex(,1643) with weighted reverse lexicographical ordering.
general.tex(,1644) The variables
+general.tex(,1648) @tex
+general.tex(,1649) $x$
+general.tex(,1650) @end tex
general.tex(,1651) ,
+general.tex(,1655) @tex
+general.tex(,1656) $y$
+general.tex(,1657) @end tex
general.tex(,1658) , and
+general.tex(,1662) @tex
+general.tex(,1663) $z$
+general.tex(,1664) @end tex
general.tex(,1665) have the weights 2, 1,
general.tex(,1666) and 3, respectively, and vectors are first ordered by
components (in
general.tex(,1667) descending order) and then by monomials:
@@ -2763,12 +2889,30 @@
general.tex(,1673)
general.tex(,1674) @item
general.tex(,1675) the ring
+general.tex(,1679) @tex
+general.tex(,1680) $K[x,y,z]$
+general.tex(,1681) @end tex
general.tex(,1682) , where
+general.tex(,1686) @tex
+general.tex(,1687) $K=Z/7(a,b,c)$
+general.tex(,1688) @end tex
general.tex(,1689) denotes the transcendental
general.tex(,1690) extension of
+general.tex(,1694) @tex
+general.tex(,1695) $Z/7$
+general.tex(,1696) @end tex
general.tex(,1697) by
+general.tex(,1701) @tex
+general.tex(,1702) $a$
+general.tex(,1703) @end tex
general.tex(,1704) ,
+general.tex(,1708) @tex
+general.tex(,1709) $b$
+general.tex(,1710) @end tex
general.tex(,1711) and
+general.tex(,1715) @tex
+general.tex(,1716) $c$
+general.tex(,1717) @end tex
general.tex(,1718) with degree
general.tex(,1719) lexicographical ordering:
general.tex(,1720)
@@ -2778,19 +2922,49 @@
general.tex(,1724)
general.tex(,1725) @item
general.tex(,1726) the ring
+general.tex(,1730) @tex
+general.tex(,1731) $K[x,y,z]$
+general.tex(,1732) @end tex
general.tex(,1733) , where
+general.tex(,1737) @tex
+general.tex(,1738) $K=Z/7[a]$
+general.tex(,1739) @end tex
general.tex(,1740) denotes the algebraic extension of
general.tex(,1741) degree 2 of
+general.tex(,1745) @tex
+general.tex(,1746) $Z/7$
+general.tex(,1747) @end tex
general.tex(,1748) by
+general.tex(,1752) @tex
+general.tex(,1753) $a.$
+general.tex(,1754) @end tex
general.tex(,1755) In other words,
+general.tex(,1759) @tex
+general.tex(,1760) $K$
+general.tex(,1761) @end tex
general.tex(,1762) is the finite field with
general.tex(,1763) 49 elements. In the first case,
+general.tex(,1767) @tex
+general.tex(,1768) $a$
+general.tex(,1769) @end tex
general.tex(,1770) denotes an algebraic
general.tex(,1771) element over
+general.tex(,1775) @tex
+general.tex(,1776) $Z/7$
+general.tex(,1777) @end tex
general.tex(,1778) with minimal polynomial
+general.tex(,1779) @tex
+general.tex(,1780) $\mu_a=a^2+a+3$,
+general.tex(,1781) @end tex
general.tex(,1785) in the second case,
+general.tex(,1789) @tex
+general.tex(,1790) $a$
+general.tex(,1791) @end tex
general.tex(,1792)
general.tex(,1793) refers to some generator of the cyclic group of units of
+general.tex(,1797) @tex
+general.tex(,1798) $K$
+general.tex(,1799) @end tex
general.tex(,1800) :
general.tex(,1801)
general.tex(,1802) @smallexample
@@ -2800,7 +2974,13 @@
general.tex(,1806)
general.tex(,1807) @item
general.tex(,1808) the ring
+general.tex(,1812) @tex
+general.tex(,1813) $R[x,y,z]$
+general.tex(,1814) @end tex
general.tex(,1815) , where
+general.tex(,1819) @tex
+general.tex(,1820) $R$
+general.tex(,1821) @end tex
general.tex(,1822) denotes the field of real
general.tex(,1823) numbers represented by simple precision floating point
numbers. This is
general.tex(,1824) a special case:
@@ -2811,7 +2991,13 @@
general.tex(,1829)
general.tex(,1830) @item
general.tex(,1831) the ring
+general.tex(,1835) @tex
+general.tex(,1836) $R[x,y,z]$
+general.tex(,1837) @end tex
general.tex(,1838) , where
+general.tex(,1842) @tex
+general.tex(,1843) $R$
+general.tex(,1844) @end tex
general.tex(,1845) denotes the field of real
general.tex(,1846) numbers represented by floating point numbers of 50 valid
decimal digits
general.tex(,1847) and the same number of digits for the rest:
@@ -2822,7 +3008,13 @@
general.tex(,1852)
general.tex(,1853) @item
general.tex(,1854) the ring
+general.tex(,1858) @tex
+general.tex(,1859) $R[x,y,z]$
+general.tex(,1860) @end tex
general.tex(,1861) , where
+general.tex(,1865) @tex
+general.tex(,1866) $R$
+general.tex(,1867) @end tex
general.tex(,1868) denotes the field of real
general.tex(,1869) numbers represented by floating point numbers of 10 valid
decimal digits
general.tex(,1870) and with 50 digits for the rest:
@@ -2833,10 +3025,19 @@
general.tex(,1875)
general.tex(,1876) @item
general.tex(,1877) the ring
+general.tex(,1881) @tex
+general.tex(,1882) $R(j)[x,y,z]$
+general.tex(,1883) @end tex
general.tex(,1884) , where
+general.tex(,1888) @tex
+general.tex(,1889) $R$
+general.tex(,1890) @end tex
general.tex(,1891) denotes the field of real
general.tex(,1892) numbers represented by floating point numbers of 30 valid
decimal digits
general.tex(,1893) and the same number for the rest.
+general.tex(,1897) @tex
+general.tex(,1898) $j$
+general.tex(,1899) @end tex
general.tex(,1900) denotes the imaginary unit.
general.tex(,1901)
general.tex(,1902) @smallexample
@@ -2845,10 +3046,19 @@
general.tex(,1905)
general.tex(,1906) @item
general.tex(,1907) the ring
+general.tex(,1911) @tex
+general.tex(,1912) $R(i)[x,y,z]$
+general.tex(,1913) @end tex
general.tex(,1914) , where
+general.tex(,1918) @tex
+general.tex(,1919) $R$
+general.tex(,1920) @end tex
general.tex(,1921) denotes the field of real
general.tex(,1922) numbers represented by floating point numbers of 6 valid
decimal digits
general.tex(,1923) and the same number for the rest.
+general.tex(,1927) @tex
+general.tex(,1928) $i$
+general.tex(,1929) @end tex
general.tex(,1930) is the default for the imaginary unit.
general.tex(,1931)
general.tex(,1932) @smallexample
@@ -2857,8 +3067,14 @@
general.tex(,1935)
general.tex(,1936) @item
general.tex(,1937) the quotient ring
+general.tex(,1941) @tex
+general.tex(,1942) $Z/7[x,y,z]$
+general.tex(,1943) @end tex
general.tex(,1944) modulo the square of the maximal
general.tex(,1945) ideal
+general.tex(,1949) @tex
+general.tex(,1950) $(x,y,z)$
+general.tex(,1951) @end tex
general.tex(,1952) :
general.tex(,1953)
general.tex(,1954) @smallexample
@@ -2911,7 +3127,13 @@
general.tex(,2001) an expression_list of an int_expression and a name.
general.tex(,2002) @* The int_expression has to be a prime number p to the
power of a
general.tex(,2003) positive integer n. This defines the Galois field
+general.tex(,2004) @tex
+general.tex(,2005) $\hbox{GF}(p^n)$ with $p^n$ elements, where $p^n$ has to be
smaller or equal $2^{15}$.
+general.tex(,2006) @end tex
general.tex(,2010) The given name refers to a primitive element of
+general.tex(,2011) @tex
+general.tex(,2012) $\hbox{GF}(p^n)$
+general.tex(,2013) @end tex
general.tex(,2017) generating the multiplicative group. Due to a different
internal
general.tex(,2018) representation, the arithmetic operations in these
coefficient fields
general.tex(,2019) are faster than arithmetic operations in algebraic
extensions as
@@ -3037,7 +3259,13 @@
general.tex(,2139)
general.tex(,2140) @strong{Remark:} The novice user should generally use the
ordering
general.tex(,2141) @code{dp} for computations in the polynomial ring
+general.tex(,2142) @tex
+general.tex(,2143) $K[x_1,\ldots,x_n]$,
+general.tex(,2144) @end tex
general.tex(,2148) resp.@: @code{ds} for computations in the localization
+general.tex(,2149) @tex
+general.tex(,2150) $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n])$.
+general.tex(,2151) @end tex
general.tex(,2155) For more details, see @ref{Polynomial data}.
general.tex(,2156)
general.tex(,2157) In a ring declaration, @sc{Singular} offers the following
orderings:
@@ -3063,8 +3291,14 @@
general.tex(,2177) @end table
general.tex(,2178)
general.tex(,2179) Global orderings are well-orderings, i.e.,
+general.tex(,2183) @tex
+general.tex(,2184) $1 < x$
+general.tex(,2185) @end tex
general.tex(,2186) for each ring
general.tex(,2187) variable
+general.tex(,2191) @tex
+general.tex(,2192) $x$
+general.tex(,2193) @end tex
general.tex(,2194) . They are denoted by a @code{p} as the second
general.tex(,2195) character in their name.
general.tex(,2196)
@@ -3957,6 +4191,9 @@
general.tex(,3083) by the size of expression.
general.tex(,3084) @* But @code{matrix(} expression @code{,} m @code{,} n
@code{)} may also be
general.tex(,3085) used - the result is a
+general.tex(,3086) @tex
+general.tex(,3087) $ m \times n $
+general.tex(,3088) @end tex
general.tex(,3092) matrix (@pxref{matrix type cast})
general.tex(,3093) @item
general.tex(,3094) @ @tab @code{module} @tab expression lists of
@code{int}, @code{number},
@@ -4270,11 +4507,17 @@
general.tex(,3402) @*["help_text"]
general.tex(,3403) @address@hidden@{}
general.tex(,3404) @*
+general.tex(,3405) @tex
+general.tex(,3406) \quad
+general.tex(,3407) @end tex
general.tex(,3408) procedure_body
general.tex(,3409) @address@hidden@}}
general.tex(,3410) @address@hidden
general.tex(,3411) @address@hidden@{}
general.tex(,3412) @*
+general.tex(,3413) @tex
+general.tex(,3414) \quad
+general.tex(,3415) @end tex
general.tex(,3416) sequence_of_commands;
general.tex(,3417) @address@hidden@}}]
general.tex(,3418) @item Purpose:
@@ -5106,6 +5349,9 @@
general.tex(,4210) @code{@@address@hidden@}}
general.tex(,4211) @address@hidden
general.tex(,4212) @*
+general.tex(,4216) @tex
+general.tex(,4217) $\alpha$
+general.tex(,4218) @end tex
general.tex(,4219)
general.tex(,4220) @item Note:
general.tex(,4221) Mathematical expressions inside @code{@@address@hidden@}}
may
@@ -5205,6 +5451,9 @@
general.tex(,4315) @address@hidden
general.tex(,4316) @*Among others, within a texinfo environment one can use
the tex environment
general.tex(,4317) to typeset more complex mathematical like
+general.tex(,4318) @tex
+general.tex(,4319) $ i_{1,1} $
+general.tex(,4320) @end tex
general.tex(,4321) @end table
general.tex(,4322)
general.tex(,4323) @end table
@@ -5499,12 +5748,18 @@
template_lib.tex(,107)
template_lib.tex(,108) @item @strong{Return:}
template_lib.tex(,109) int:
+template_lib.tex(,113) @tex
+template_lib.tex(,114) $i+i+i$
+template_lib.tex(,115) @end tex
template_lib.tex(,116)
template_lib.tex(,117) @item @strong{Note:}
template_lib.tex(,118) Help is in pure Texinfo
template_lib.tex(,119) @*This help string is written in texinfo, which enables
you to use,
template_lib.tex(,120) among others, the @@math command for mathematical
typesetting (like
template_lib.tex(,121)
+template_lib.tex(,125) @tex
+template_lib.tex(,126) $\alpha, \beta$
+template_lib.tex(,127) @end tex
template_lib.tex(,128) ).
template_lib.tex(,129) @*It also gives more control over the layout, but is,
admittingly,
template_lib.tex(,130) more cumbersome to write.
@@ -5545,6 +5800,9 @@
template_lib.tex(,179) @* Use a @@ref constructs for references (like
@pxref{mtripple})
template_lib.tex(,180) @* Use @@code for typewriter font (like @code{i_1})
template_lib.tex(,181) @* Use @@math for simple math mode typesetting (like
+template_lib.tex(,185) @tex
+template_lib.tex(,186) $i_1$
+template_lib.tex(,187) @end tex
template_lib.tex(,188) ).
template_lib.tex(,189) @* Note: No parenthesis like @} are allowed inside
@@math and @@code
template_lib.tex(,190) @* Use @@example for indented preformatted text typeset
in typewriter
@@ -5559,6 +5817,9 @@
template_lib.tex(,199) Use @@texinfo for text in pure texinfo
template_lib.tex(,200)
template_lib.tex(,201) @expansion{}
+template_lib.tex(,202) @tex
+template_lib.tex(,203) $i_{1,1}$
+template_lib.tex(,204) @end tex
template_lib.tex(,205)
template_lib.tex(,206)
template_lib.tex(,207) Notice that
@@ -6208,6 +6469,9 @@
types.tex(,416) set of minors of a matrix (see @ref{minor})
types.tex(,417) @item modulo
types.tex(,418) represents
+types.tex(,419) @tex
+types.tex(,420) $(h1+h2)/h1 \cong h2/(h1 \cap h2)$
+types.tex(,421) @end tex
types.tex(,425) (see @ref{modulo})
types.tex(,426) @item mres
types.tex(,427) minimal free resolution of an ideal resp.@: module w.r.t. a
minimal set of generators of the given ideal resp.@: module
@@ -7959,25 +8223,62 @@
types.tex(,2236) Canonically realized are
types.tex(,2237) @itemize @bullet
types.tex(,2238) @item
+types.tex(,2239) @tex
+types.tex(,2240) $Q \rightarrow Q(a, \ldots)$
+types.tex(,2241) @end tex
types.tex(,2245)
types.tex(,2246) @item
+types.tex(,2247) @tex
+types.tex(,2248) $Q \rightarrow R$
+types.tex(,2249) @end tex
types.tex(,2253)
types.tex(,2254) @item
+types.tex(,2255) @tex
+types.tex(,2256) $Q \rightarrow C$
+types.tex(,2257) @end tex
types.tex(,2261)
types.tex(,2262) @item
+types.tex(,2263) @tex
+types.tex(,2264) $Z/p \rightarrow (Z/p)(a, \ldots)$
+types.tex(,2265) @end tex
types.tex(,2269)
types.tex(,2270) @item
+types.tex(,2271) @tex
+types.tex(,2272) $Z/p \rightarrow GF(p^n)$
+types.tex(,2273) @end tex
types.tex(,2277)
types.tex(,2278) @item
+types.tex(,2279) @tex
+types.tex(,2280) $Z/p \rightarrow R$
+types.tex(,2281) @end tex
types.tex(,2285)
types.tex(,2286) @item
+types.tex(,2287) @tex
+types.tex(,2288) $R \rightarrow C$
+types.tex(,2289) @end tex
types.tex(,2293) @end itemize
types.tex(,2294)
types.tex(,2295) Possible are furthermore
types.tex(,2296) @itemize @bullet
types.tex(,2297) @item
+types.tex(,2298) @tex
+types.tex(,2299) % This is quite a hack, but for now it works.
+types.tex(,2300) $Z/p \rightarrow Q,
+types.tex(,2301) \quad
+types.tex(,2302) [i]_p \mapsto i \in [-p/2, \, p/2]
+types.tex(,2303) \subseteq Z$
+types.tex(,2304) @end tex
types.tex(,2308) @item
+types.tex(,2309) @tex
+types.tex(,2310) $Z/p \rightarrow Z/p^\prime,
+types.tex(,2311) \quad
+types.tex(,2312) [i]_p \mapsto i \in [-p/2, \, p/2] \subseteq Z, \;
+types.tex(,2313) i \mapsto [i]_{p^\prime} \in Z/p^\prime$
+types.tex(,2314) @end tex
types.tex(,2318) @item
+types.tex(,2319) @tex
+types.tex(,2320) $C \rightarrow R, \quad$ the real part
+types.tex(,2321) @end tex
types.tex(,2325) @end itemize
types.tex(,2326)
types.tex(,2327) Finally, in Singular we allow the mapping from rings
@@ -7986,8 +8287,14 @@
types.tex(,2330)
types.tex(,2331) @itemize @bullet
types.tex(,2332) @item
+types.tex(,2333) @tex
+types.tex(,2334) $Q \rightarrow Z/p$
+types.tex(,2335) @end tex
types.tex(,2339)
types.tex(,2340) @item
+types.tex(,2341) @tex
+types.tex(,2342) $Q \rightarrow (Z/p)(a, \ldots)$
+types.tex(,2343) @end tex
types.tex(,2347) @end itemize
types.tex(,2348) In these cases the denominator and the numerator
types.tex(,2349) of a number are mapped separately by the usual
@@ -8431,18 +8738,45 @@
types.tex(,2822) Like vectors they
types.tex(,2823) can only be defined or accessed with respect to a basering.
types.tex(,2824) If
+types.tex(,2828) @tex
+types.tex(,2829) $M$
+types.tex(,2830) @end tex
types.tex(,2831) is a submodule of
+types.tex(,2835) @tex
+types.tex(,2836) $R^n$,
+types.tex(,2837) @end tex
types.tex(,2838)
+types.tex(,2842) @tex
+types.tex(,2843) $R$
+types.tex(,2844) @end tex
types.tex(,2845) the basering, generated by vectors
+types.tex(,2849) @tex
+types.tex(,2850) $v_1, \ldots, v_k$, then $v_1, \ldots, v_k$
+types.tex(,2851) @end tex
types.tex(,2852) may be considered as the generators of relations of
+types.tex(,2856) @tex
+types.tex(,2857) $R^n/M$
+types.tex(,2858) @end tex
types.tex(,2859) between the canonical generators
@code{gen(1)},@dots{},@code{gen(n)}.
types.tex(,2860) Hence any finitely generated
+types.tex(,2864) @tex
+types.tex(,2865) $R$
+types.tex(,2866) @end tex
types.tex(,2867) -module can be represented in @sc{Singular}
types.tex(,2868) by its module of relations. The assignments
types.tex(,2869) @code{module M=v1,...,vk; matrix A=M;}
types.tex(,2870) create the presentation matrix of size
+types.tex(,2874) @tex
+types.tex(,2875) n$\times$k
+types.tex(,2876) @end tex
types.tex(,2877) for
+types.tex(,2881) @tex
+types.tex(,2882) R$^n$/M,
+types.tex(,2883) @end tex
types.tex(,2884) i.e., the columns of A are the vectors
+types.tex(,2888) @tex
+types.tex(,2889) $v_1, \ldots, v_k$
+types.tex(,2890) @end tex
types.tex(,2891) which generate M (cf. @ref{Representation of mathematical
objects}).
types.tex(,2892)
types.tex(,2893) @menu
@@ -8597,6 +8931,9 @@
types.tex(,3058) over a local ring
types.tex(,3059) @item modulo
types.tex(,3060) represents
+types.tex(,3061) @tex
+types.tex(,3062) $(h1+h2)/h1=h2/(h1 \cap h2)$
+types.tex(,3063) @end tex
types.tex(,3067) (see @ref{modulo})
types.tex(,3068) @item mres
types.tex(,3069) minimal free resolution of an ideal resp.@: module w.r.t. a
minimal set of generators of the given module
@@ -9206,11 +9543,17 @@
types.tex(,3796) @*["help_text"]
types.tex(,3797) @address@hidden@{}
types.tex(,3798) @*
+types.tex(,3799) @tex
+types.tex(,3800) \quad
+types.tex(,3801) @end tex
types.tex(,3802) procedure_body
types.tex(,3803) @address@hidden@}}
types.tex(,3804) @address@hidden
types.tex(,3805) @address@hidden@{}
types.tex(,3806) @*
+types.tex(,3807) @tex
+types.tex(,3808) \quad
+types.tex(,3809) @end tex
types.tex(,3810) sequence_of_commands;
types.tex(,3811) @address@hidden@}}]
types.tex(,3812) @address@hidden proc_name @code{=} proc_name @code{;}
@@ -9525,31 +9868,82 @@
types.tex(,4145) @table @asis
types.tex(,4146) @item @code{+}
types.tex(,4147) construct a new ring
+types.tex(,4151) @tex
+types.tex(,4152) $k[X,Y]$
+types.tex(,4153) @end tex
types.tex(,4154) from
+types.tex(,4158) @tex
+types.tex(,4159) $k_1[X]$
+types.tex(,4160) @end tex
types.tex(,4161) and
+types.tex(,4165) @tex
+types.tex(,4166) $k_2[Y]$
+types.tex(,4167) @end tex
types.tex(,4168) .
types.tex(,4169) @end table
types.tex(,4170)
types.tex(,4171) Concerning the ground fields
+types.tex(,4175) @tex
+types.tex(,4176) $k_1$
+types.tex(,4177) @end tex
types.tex(,4178) and
+types.tex(,4182) @tex
+types.tex(,4183) $k_2$
+types.tex(,4184) @end tex
types.tex(,4185) take the
types.tex(,4186) following guide lines into consideration:
types.tex(,4187) @itemize @bullet
types.tex(,4188) @item Neither
+types.tex(,4192) @tex
+types.tex(,4193) $k_1$
+types.tex(,4194) @end tex
types.tex(,4195) nor
+types.tex(,4199) @tex
+types.tex(,4200) $k_2$
+types.tex(,4201) @end tex
types.tex(,4202) may be
+types.tex(,4206) @tex
+types.tex(,4207) $R$
+types.tex(,4208) @end tex
types.tex(,4209) or
+types.tex(,4213) @tex
+types.tex(,4214) $C$
+types.tex(,4215) @end tex
types.tex(,4216) .
types.tex(,4217) @item If the characteristic of
+types.tex(,4221) @tex
+types.tex(,4222) $k_1$
+types.tex(,4223) @end tex
types.tex(,4224) and
+types.tex(,4228) @tex
+types.tex(,4229) $k_2$
+types.tex(,4230) @end tex
types.tex(,4231) differs, then one of them must be
+types.tex(,4235) @tex
+types.tex(,4236) $Q$
+types.tex(,4237) @end tex
types.tex(,4238) .
types.tex(,4239) @item At most one of
+types.tex(,4243) @tex
+types.tex(,4244) $k_1$
+types.tex(,4245) @end tex
types.tex(,4246) and
+types.tex(,4250) @tex
+types.tex(,4251) $k_2$
+types.tex(,4252) @end tex
types.tex(,4253) may be have parameters.
types.tex(,4254) @item If one of
+types.tex(,4258) @tex
+types.tex(,4259) $k_1$
+types.tex(,4260) @end tex
types.tex(,4261) and
+types.tex(,4265) @tex
+types.tex(,4266) $k_2$
+types.tex(,4267) @end tex
types.tex(,4268) is an algebraic extension of
+types.tex(,4272) @tex
+types.tex(,4273) $Z/p$
+types.tex(,4274) @end tex
types.tex(,4275) it may not be defined by a @code{charstr} of type
@code{(p^n,a)}.
types.tex(,4276) @end itemize
types.tex(,4277)
@@ -10445,6 +10839,18 @@
reference.tex(,418) intmat
reference.tex(,419) @item @strong{Purpose:}
reference.tex(,420) with 1 argument: computes the graded Betti numbers of a
minimal resolution of
+reference.tex(,421) @tex
+reference.tex(,422) $R^n/M$, if $R$ denotes the basering and
+reference.tex(,423) $M$ a homogeneous submodule of $R^n$ and the argument
represents a
+reference.tex(,424) resolution of
+reference.tex(,425) $R^n/M$.
+reference.tex(,426) @end tex
+reference.tex(,430) @tex
+reference.tex(,431) The entry d of the intmat at place (i,j) is the minimal
number of
+reference.tex(,432) generators in degree i+j of the j-th syzygy module (=
module of
+reference.tex(,433) relations) of $R^n/M$ (the 0th (resp.\ 1st) syzygy module
of $R^n/M$ is
+reference.tex(,434) $R^n$ (resp.\ $M$)).
+reference.tex(,435) @end tex
reference.tex(,445) The argument is considered to be the result of a
res/sres/mres/nres/lres
reference.tex(,446) command. This implies that a zero is only allowed (and
counted) as a
reference.tex(,447) generator in the first module.
@@ -10512,6 +10918,15 @@
reference.tex(,509) where the generators are the columns of the
reference.tex(,510) displayed matrix and degrees are assigned such that the
corresponding maps
reference.tex(,511) have degree 0:
+reference.tex(,512) @tex
+reference.tex(,513) $$
+reference.tex(,514) 0 \longleftarrow r/j \longleftarrow r(1)
+reference.tex(,515) \buildrel{T[1]}\over{\longleftarrow} r(2) \oplus r^3(3)
+reference.tex(,516) \buildrel{T[2]}\over{\longleftarrow} r^4(4)
+reference.tex(,517) \buildrel{T[3]}\over{\longleftarrow} r(5)
+reference.tex(,518) \longleftarrow 0 \quad .
+reference.tex(,519) $$
+reference.tex(,520) @end tex
reference.tex(,525)
reference.tex(,526) @c inserted refs from reference.doc:455
reference.tex(,551) @c end inserted refs from reference.doc:455
@@ -10825,12 +11240,28 @@
reference.tex(,919) @end format
reference.tex(,920) If J is a vector or a module this procedure is repeated
for each
reference.tex(,921) component and the resulting matrices are address@hidden
+reference.tex(,926) @tex
+reference.tex(,927) The third argument is used to return the matrix T of
coefficients
+reference.tex(,928) such that {\tt matrix}(J) = T*M.
+reference.tex(,929) @end tex
reference.tex(,930) @item @strong{Note:}
reference.tex(,931) @code{coeffs} returns the coefficient 0 at the appropriate
place if a monomial
reference.tex(,932) is not present, while @code{coef} considers only monomials
which really occur
reference.tex(,933) in the given expression. @*
reference.tex(,934) If
+reference.tex(,935) @tex
+reference.tex(,936) $M=(m_{ij})$
+reference.tex(,937) @end tex
reference.tex(,941) then the j-th generator of an ideal J is equal to
+reference.tex(,942) @tex
+reference.tex(,943) $$J_j = z^0 \cdot m_{1j} + z^1 \cdot m_{2j} + ... +
z^{d-1} \cdot m_{dj},$$
+reference.tex(,944) while for a module J the i-th component of the j-th
generator is
+reference.tex(,945) equal to the entry [i,j] of {\tt matrix}(J), and we get
+reference.tex(,946) @end tex
+reference.tex(,956) @tex
+reference.tex(,957) $$ J_{i,j} = z^0 \cdot m_{(i-1)d+1,j} + z^1 \cdot
m_{(i-1)d+2,j} + ... +
+reference.tex(,958) z^{d-1} \cdot m_{id,j}.$$
+reference.tex(,959) @end tex
reference.tex(,968)
reference.tex(,969) @item @strong{Example:}
reference.tex(,970) @smallexample
@@ -10909,7 +11340,14 @@
reference.tex(,1055) producing a m x n matrix.
reference.tex(,1056) @*Contraction is defined on monomials by:
reference.tex(,1057) @*
+reference.tex(,1064) @tex
+reference.tex(,1065) $${\rm contract}(x^A , x^B) := \cases{ x^{(B-A)}, &if
$B\ge A$
+reference.tex(,1066) componentwise\cr 0,&otherwise.\cr}$$
+reference.tex(,1067) @end tex
reference.tex(,1068) where A and B are the multiexponents of the ring
variables represented by
+reference.tex(,1069) @tex
+reference.tex(,1070) $x$.
+reference.tex(,1071) @end tex
reference.tex(,1075) @code{contract} is extended bilinearly to all polynomials.
reference.tex(,1076) @item @strong{Example:}
reference.tex(,1077) @smallexample
@@ -12342,13 +12780,24 @@
reference.tex(,2950) @code{highcorner(I)} returns 0 iff @code{dim(I)>0} or
@code{dim(I)=-1}.
reference.tex(,2951) Otherwise it returns the smallest monomial m not in I
which has the following
reference.tex(,2952) properties (with
+reference.tex(,2956) @tex
+reference.tex(,2957) $x_i$
+reference.tex(,2958) @end tex
reference.tex(,2959) the variables of the basering):
reference.tex(,2960) @itemize @bullet
reference.tex(,2961) @item
reference.tex(,2962) if
+reference.tex(,2966) @tex
+reference.tex(,2967) $x_i>1$ then $x_i$
+reference.tex(,2968) @end tex
reference.tex(,2969) does not divide m (e.g., m=1 if the ordering is global)
reference.tex(,2970) @item
reference.tex(,2971) given any set of generators
+reference.tex(,2977) @tex
+reference.tex(,2978) $f_1,\dots,f_k$ of I, let $f'_i$ be obtained from
+reference.tex(,2979) $f_i$ by deleting the terms divisible by $x_i\cdot m$ for
all i with $x_i<1$.
+reference.tex(,2980) Then $f'_1,\dots,f'_k$ generate I.
+reference.tex(,2981) @end tex
reference.tex(,2982) @end itemize
reference.tex(,2983) @item @strong{Example:}
reference.tex(,2984) @smallexample
@@ -12487,11 +12936,22 @@
reference.tex(,3167)
reference.tex(,3168) More precisely, let R be the basering and I be the given
ideal.
reference.tex(,3169) Then @code{hres} computes a minimal free resolution of R/I
+reference.tex(,3176) @tex
+reference.tex(,3177) $$...\longrightarrow F_2
\buildrel{A_2}\over{\longrightarrow} F_1
+reference.tex(,3178) \buildrel{A_1}\over{\longrightarrow} R\longrightarrow R/I
+reference.tex(,3179) \longrightarrow 0.$$
+reference.tex(,3180) @end tex
reference.tex(,3181) If the int_expression k is not zero then the computation
stops after
reference.tex(,3182) k steps and returns a list of modules
+reference.tex(,3183) @tex
+reference.tex(,3184) $M_i={\tt module} (A_i)$, i=1..k.
+reference.tex(,3185) @end tex
reference.tex(,3189)
reference.tex(,3190) @code{list L=hres(I,0);} returns a list L of n modules
(where n is the
reference.tex(,3191) number of variables of the basering) such that
+reference.tex(,3192) @tex
+reference.tex(,3193) ${\tt L[i]}=M_i$
+reference.tex(,3194) @end tex
reference.tex(,3198) in the above notation.
reference.tex(,3199) @item @strong{Note:}
reference.tex(,3200) The ideal_expression has to be homogeneous.
@@ -12607,6 +13067,9 @@
reference.tex(,3364)
reference.tex(,3365) @item @strong{Note:}
reference.tex(,3366) U is a set of independent variables for I if and only if
+reference.tex(,3367) @tex
+reference.tex(,3368) $I \cap K[U]=(0)$,
+reference.tex(,3369) @end tex
reference.tex(,3373) i.e., eliminating the remaining variables gives (0).
reference.tex(,3374) U is maximal if dim(I)=#U.
reference.tex(,3375) @item @strong{Syntax:}
@@ -12704,19 +13167,47 @@
reference.tex(,3491) @item @strong{Purpose:}
reference.tex(,3492) interreduces a set of polynomials/vectors.
reference.tex(,3493) @*
+reference.tex(,3497) @tex
+reference.tex(,3498) input: $f_1,\dots,f_n$
+reference.tex(,3499) @end tex
reference.tex(,3500) @*
+reference.tex(,3506) @tex
+reference.tex(,3507) output: $g_1,\dots,g_s$ with $s \leq n$ and the properties
+reference.tex(,3508) @end tex
reference.tex(,3509) @itemize @bullet
reference.tex(,3510) @item
+reference.tex(,3514) @tex
+reference.tex(,3515) $(f_1,\dots,f_n) = (g_1,\dots,g_s)$
+reference.tex(,3516) @end tex
reference.tex(,3517) @item
+reference.tex(,3521) @tex
+reference.tex(,3522) $L(g_i)\neq L(g_j)$ for all $i\neq j$
+reference.tex(,3523) @end tex
reference.tex(,3524) @item
reference.tex(,3525) in the case of a global ordering (polynomial ring):
reference.tex(,3526) @*
+reference.tex(,3530) @tex
+reference.tex(,3531) $L(g_i)$
+reference.tex(,3532) @end tex
reference.tex(,3533) does not divide m for all monomials m of
+reference.tex(,3537) @tex
+reference.tex(,3538) $\{g_1,\dots,g_{i-1},g_{i+1},\dots,g_s\}$
+reference.tex(,3539) @end tex
reference.tex(,3540) @item
reference.tex(,3541) in the case of a local or mixed ordering (localization of
polynomial ring):
reference.tex(,3542) @* if
+reference.tex(,3546) @tex
+reference.tex(,3547) $L(g_i) | L(g_j)$ for any $i \neq j$,
+reference.tex(,3548) @end tex
reference.tex(,3549) then
+reference.tex(,3553) @tex
+reference.tex(,3554) $ecart(g_i) > ecart(g_j)$
+reference.tex(,3555) @end tex
reference.tex(,3556) @end itemize
+reference.tex(,3557) @tex
+reference.tex(,3558) Here, $L(g)$ denotes the leading term of $g$ and
+reference.tex(,3559) $ecart(g):=deg(g)-deg(L(g))$.
+reference.tex(,3560) @end tex
reference.tex(,3566) @item @strong{Example:}
reference.tex(,3567) @smallexample
reference.tex(,3568) @c reused example interred reference.doc:2557
@@ -13446,11 +13937,22 @@
reference.tex(,4704)
reference.tex(,4705) More precisely, let R be the basering and I be the given
ideal.
reference.tex(,4706) Then @code{lres} computes a minimal free resolution of R/I
+reference.tex(,4713) @tex
+reference.tex(,4714) $$...\longrightarrow F_2
\buildrel{A_2}\over{\longrightarrow} F_1
+reference.tex(,4715) \buildrel{A_1}\over{\longrightarrow} R\longrightarrow R/I
+reference.tex(,4716) \longrightarrow 0.$$
+reference.tex(,4717) @end tex
reference.tex(,4718) If the int_expression k is not zero then the computation
stops after
reference.tex(,4719) k steps and returns a list of modules
+reference.tex(,4720) @tex
+reference.tex(,4721) $M_i={\tt module}(A_i)$, i=1..k.
+reference.tex(,4722) @end tex
reference.tex(,4726)
reference.tex(,4727) @code{list L=lres(I,0);} returns a list L of n modules
(where n is the
reference.tex(,4728) number of variables of the basering) such that
+reference.tex(,4729) @tex
+reference.tex(,4730) ${\tt L[i]}=M_i$
+reference.tex(,4731) @end tex
reference.tex(,4735) in the above notation.
reference.tex(,4736) @item @strong{Note:}
reference.tex(,4737) The ideal_expression has to be homogeneous.
@@ -13704,9 +14206,25 @@
reference.tex(,5069) module
reference.tex(,5070) @item @strong{Purpose:}
reference.tex(,5071) @code{modulo(h1,h2)}
+reference.tex(,5075) @tex
+reference.tex(,5076) represents $h_1/(h_1 \cap h_2) \cong (h_1+h_2)/h_2$
+reference.tex(,5077) @end tex
reference.tex(,5078) where
+reference.tex(,5079) @tex
+reference.tex(,5080) $h_1$ and $h_2$
+reference.tex(,5081) @end tex
reference.tex(,5085) are considered as submodules of the same free module
+reference.tex(,5086) @tex
+reference.tex(,5087) $R^l$
+reference.tex(,5088) @end tex
reference.tex(,5092) (l=1 for ideals). Let
+reference.tex(,5093) @tex
+reference.tex(,5094) $H_1$, resp.\ $H_2$,
+reference.tex(,5095) @end tex
+reference.tex(,5100) @tex
+reference.tex(,5101) be the matrices of size $l \times k$, resp.\ $l \times
m$, having the
+reference.tex(,5102) generators of $h_1$, resp.\ $h_2$,
+reference.tex(,5103) @end tex
reference.tex(,5107) as columns.
reference.tex(,5108) @c @*
reference.tex(,5109) @c @tex
@@ -13720,7 +14238,14 @@
reference.tex(,5117) @c @end smallexample
reference.tex(,5118) @c @end ifinfo
reference.tex(,5119) Then
+reference.tex(,5120) @tex
+reference.tex(,5121) $h_1/(h_1 \cap h_2) \cong R^k / ker(\overline{H_1})$
+reference.tex(,5122) @end tex
reference.tex(,5131) where
+reference.tex(,5132) @tex
+reference.tex(,5133) $\overline{H_1}: R^k \rightarrow R^l/Im(H_2)=R^l/h_2$
+reference.tex(,5134) is the induced map.
+reference.tex(,5135) @end tex
reference.tex(,5144) @address@hidden(h1,h2)} returns generators of
reference.tex(,5145) the kernel of this induced map.
reference.tex(,5146) @item @strong{Example:}
@@ -13821,17 +14346,32 @@
reference.tex(,5261) computes a minimal free resolution of an ideal or module
M with the
reference.tex(,5262) standard basis method. More precisely, let
address@hidden(M), then @code{mres}
reference.tex(,5263) computes a free resolution of
+reference.tex(,5271) @tex
+reference.tex(,5272) $coker(A)=F_0/M$
+reference.tex(,5273) $$...\longrightarrow F_2
\buildrel{A_2}\over{\longrightarrow} F_1
+reference.tex(,5274) \buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow
F_0/M
+reference.tex(,5275) \longrightarrow 0,$$
+reference.tex(,5276) @end tex
reference.tex(,5277) where the columns of the matrix
+reference.tex(,5278) @tex
+reference.tex(,5279) $A_1$
+reference.tex(,5280) @end tex
reference.tex(,5284) are a minimal set of generators
reference.tex(,5285) of M if the basering is local or if M is homogeneous.
reference.tex(,5286) If the int expression k is not zero then the computation
stops after k steps
reference.tex(,5287) and returns a list of modules
+reference.tex(,5288) @tex
+reference.tex(,5289) $M_i={\tt module}(A_i)$, i=1...k.
+reference.tex(,5290) @end tex
reference.tex(,5294) @address@hidden(M,0)} returns a resolution consisting of
at most n+2 modules,
reference.tex(,5295) where n is the number of variables of the basering.
reference.tex(,5296) Let @code{list L=mres(M,0);}
reference.tex(,5297) then @code{L[1]} consists of a minimal set of generators
of the input, @code{L[2]}
reference.tex(,5298) consists of a minimal set of generators for the first
syzygy module of
reference.tex(,5299) @code{L[1]}, etc., until @code{L[p+1]}, such that
+reference.tex(,5303) @tex
+reference.tex(,5304) ${\tt L[i]}\neq 0$ for $i \le p$,
+reference.tex(,5305) @end tex
reference.tex(,5306) but @code{L[p+1]}, the first syzygy module of
@code{L[p]},
reference.tex(,5307) is 0 (if the basering is not a qring).
reference.tex(,5308) @item @strong{Note:}
@@ -14122,16 +14662,32 @@
reference.tex(,5781) the second module on (by the standard basis method).
reference.tex(,5782)
reference.tex(,5783) More precisely, let
+reference.tex(,5784) @tex
+reference.tex(,5785) $A_1$=matrix(M),
+reference.tex(,5786) @end tex
reference.tex(,5790) then @code{nres} computes a free resolution of
+reference.tex(,5798) @tex
+reference.tex(,5799) $coker(A_1)=F_0/M$
+reference.tex(,5800) $$...\longrightarrow F_2
\buildrel{A_2}\over{\longrightarrow} F_1 \buildrel{A_1}\over{\longrightarrow}
F_0\longrightarrow F_0/M\longrightarrow 0,$$
+reference.tex(,5801) @end tex
reference.tex(,5802) @*where the columns of the matrix
+reference.tex(,5803) @tex
+reference.tex(,5804) $A_1$
+reference.tex(,5805) @end tex
reference.tex(,5809) are the given set of generators of M.
reference.tex(,5810) If the int expression k is not zero then the computation
stops after k steps
reference.tex(,5811) and returns a list of modules
+reference.tex(,5812) @tex
+reference.tex(,5813) $M_i={\tt module}(A_i)$, i=1..k.
+reference.tex(,5814) @end tex
reference.tex(,5818) @address@hidden(M,0)} returns a list of n modules where n
is the number of
reference.tex(,5819) variables of the basering.
reference.tex(,5820) Let @code{list L=nres(M,0);} then @code{L[1]=M} is
identical to the input,
reference.tex(,5821) @code{L[2]} is a minimal set of generators for the first
syzygy
reference.tex(,5822) module of @code{L[1]}, etc.
+reference.tex(,5826) @tex
+reference.tex(,5827) (${\tt L[i]}=M_i$
+reference.tex(,5828) @end tex
reference.tex(,5829) in the notations from above).
reference.tex(,5830) @item @strong{Example:}
reference.tex(,5831) @smallexample
@@ -14750,9 +15306,18 @@
reference.tex(,6638) @table @code
reference.tex(,6639) @item "betti"
reference.tex(,6640) The Betti numbers are printed in a matrix-like format
where the entry
+reference.tex(,6641) @tex
+reference.tex(,6642) $d$ in row $i$ and column $j$
+reference.tex(,6643) @end tex
reference.tex(,6647) is the minimal number of generators in
reference.tex(,6648) degree
+reference.tex(,6649) @tex
+reference.tex(,6650) $i+j$ of the $j$-th
+reference.tex(,6651) @end tex
reference.tex(,6655) syzygy module of
+reference.tex(,6656) @tex
+reference.tex(,6657) $R^n/M$ (the 0th and 1st syzygy module of $R^n/M$ is
$R^n$ and $M$, resp.).
+reference.tex(,6658) @end tex
reference.tex(,6662) @item "%s"
reference.tex(,6663) returns @code{string(} expression @code{)}
reference.tex(,6664) @item "%2s"
@@ -15137,12 +15702,21 @@
reference.tex(,7133) @item @strong{Purpose:}
reference.tex(,7134) computes the ideal quotient, resp.@: module quotient. Let
@code{R} be the
reference.tex(,7135) basering, @code{I,J} ideals and @code{M} a module in
+reference.tex(,7139) @tex
+reference.tex(,7140) ${\tt R}^n$.
+reference.tex(,7141) @end tex
reference.tex(,7142) Then
reference.tex(,7143) @itemize
reference.tex(,7144) @item
reference.tex(,7145) @code{quotient(I,J)}=
+reference.tex(,7149) @tex
+reference.tex(,7150) $\{a \in R \mid aJ \subset I\}$,
+reference.tex(,7151) @end tex
reference.tex(,7152) @item
reference.tex(,7153) @code{quotient(M,J)}=
+reference.tex(,7157) @tex
+reference.tex(,7158) $\{b \in R^n \mid bJ \subset M\}$.
+reference.tex(,7159) @end tex
reference.tex(,7160) @end itemize
reference.tex(,7161) @item @strong{Example:}
reference.tex(,7162) @smallexample
@@ -15332,6 +15906,15 @@
reference.tex(,7410) computes the regularity of a homogeneous ideal, resp.@:
module, from a
reference.tex(,7411) minimal resolution given by the list expression.
reference.tex(,7412) @*
+reference.tex(,7422) @tex
+reference.tex(,7423) \noindent
+reference.tex(,7424) Let $0 \rightarrow\ \bigoplus_a K[x]e_{a,n}\ \rightarrow\
\dots
+reference.tex(,7425) \rightarrow\ \bigoplus_a K[x]e_{a,0}\ \rightarrow\
+reference.tex(,7426) I\ \rightarrow\ 0$
+reference.tex(,7427) be a minimal resolution of I considered with homogeneous
maps of degree 0.
+reference.tex(,7428) The regularity is the smallest number $s$ with the
property deg($e_{a,i})
+reference.tex(,7429) \leq s+i$ for all $i$.
+reference.tex(,7430) @end tex
reference.tex(,7431) @item @strong{Note:}
reference.tex(,7432) If applied to a non minimal resolution only an upper
bound is returned.
reference.tex(,7433) @*If the input to the commands @code{res} and @code{mres}
is homogeneous
@@ -15898,6 +16481,12 @@
reference.tex(,8160) @item @strong{Type:}
reference.tex(,8161) intvec
reference.tex(,8162) @item @strong{Purpose:}
+reference.tex(,8163) @tex
+reference.tex(,8164) computes the permutation {\tt v}
+reference.tex(,8165) which orders the ideal, resp.\ module, {\tt I} by its
initial terms,
+reference.tex(,8166) starting with the smallest, that is, {\tt I(v[i]) <
I(v[i+1])} for all
+reference.tex(,8167) {\tt i}.
+reference.tex(,8168) @end tex
reference.tex(,8175) @item @strong{Example:}
reference.tex(,8176) @smallexample
reference.tex(,8177) @c reused example sortvec reference.doc:5565
@@ -16026,10 +16615,20 @@
reference.tex(,8326) computes a free resolution of an ideal or module with
Schreyer's
reference.tex(,8327) method. The ideal, resp.@: module, has to be a standard
basis.
reference.tex(,8328) More precisely, let M be given by a standard basis and
+reference.tex(,8329) @tex
+reference.tex(,8330) $A_1={\tt matrix}(M)$.
+reference.tex(,8331) @end tex
reference.tex(,8335) Then @code{sres}
reference.tex(,8336) computes a free resolution of
+reference.tex(,8344) @tex
+reference.tex(,8345) $coker(A_1)=F_0/M$
+reference.tex(,8346) $$...\longrightarrow F_2
\buildrel{A_2}\over{\longrightarrow} F_1 \buildrel{A_1}\over{\longrightarrow}
F_0\longrightarrow F_0/M\longrightarrow 0.$$
+reference.tex(,8347) @end tex
reference.tex(,8348) If the int expression k is not zero then the computation
stops after k steps
reference.tex(,8349) and returns a list of modules (given by standard bases)
+reference.tex(,8350) @tex
+reference.tex(,8351) $M_i={\tt module}(A_i)$, i=1..k.
+reference.tex(,8352) @end tex
reference.tex(,8356) @address@hidden(M,0)}
reference.tex(,8357) returns a list of n modules where n is the number of
variables of the basering.
reference.tex(,8358)
@@ -16040,6 +16639,9 @@
reference.tex(,8363) @code{L[2]} is a standard basis with respect to the
Schreyer ordering of
reference.tex(,8364) the first syzygy
reference.tex(,8365) module of @code{L[1]}, etc.
+reference.tex(,8369) @tex
+reference.tex(,8370) (${\tt L[i]}=M_i$
+reference.tex(,8371) @end tex
reference.tex(,8372) in the notations from above.)
reference.tex(,8373) @item @strong{Note:}
reference.tex(,8374) Accessing single elements of a resolution may require
that some partial
@@ -16752,7 +17354,29 @@
reference.tex(,9287) @item @strong{Type:}
reference.tex(,9288) poly
reference.tex(,9289) @item @strong{Purpose:}
+reference.tex(,9303) @tex
+reference.tex(,9304) {\tt vandermonde(p,v,d)} computes the (unique) polynomial
of degree
+reference.tex(,9305) @code{d} with prescribed values {\tt v[1],...,v[N]} at
the points
+reference.tex(,9306) {\tt p}$^0,\dots,$ {\tt p}$^{N-1}$, {\tt N=(d+1)}$^n$,
$n$ the
+reference.tex(,9307) number of ring variables.
+reference.tex(,9308)
+reference.tex(,9309) The returned polynomial is $\sum
+reference.tex(,9310) c_{\alpha_1\ldots\alpha_n}\cdot x_1^{\alpha_1} \cdot
\dots \cdot
+reference.tex(,9311) x_n^{\alpha_n}$, where the coefficients
+reference.tex(,9312) $c_{\alpha_1\ldots\alpha_n}$ are the solution of the
(transposed)
+reference.tex(,9313) Vandermonde system of linear equations
+reference.tex(,9314) $$ \sum_{\alpha_1+\ldots+\alpha_n\leq d}
c_{\alpha_1\ldots\alpha_n} \cdot
+reference.tex(,9315) {\tt p}_1^{(k-1)\alpha_1}\cdot\dots\cdot {\tt
p}_n^{(k-1)\alpha_n} =
+reference.tex(,9316) {\tt v}[k], \quad k=1,\dots,{\tt N}.$$
+reference.tex(,9317) @end tex
reference.tex(,9318) @item @strong{Note:}
+reference.tex(,9326) @tex
+reference.tex(,9327) the ground field has to be the field of rational
+reference.tex(,9328) numbers. Moreover, {\tt ncols(p)==}$n$, the number of
variables in the
+reference.tex(,9329) basering, and all the given generators have to be numbers
different from
+reference.tex(,9330) 0,1 or -1. Finally, {\tt ncols(v)==(d+1)$^n$}, and all
given generators have
+reference.tex(,9331) to be numbers.
+reference.tex(,9332) @end tex
reference.tex(,9333) @item @strong{Example:}
reference.tex(,9334) @smallexample
reference.tex(,9335) @c reused example vandermonde reference.doc:6304
@@ -18398,7 +19022,20 @@
examples.tex(,100)
examples.tex(,101) The Milnor number, resp.@: the Tjurina number, of a power
examples.tex(,102) series f in
+examples.tex(,103) @tex
+examples.tex(,104) $K[[x_1,\ldots,x_n]]$
+examples.tex(,105) @end tex
examples.tex(,109) is
+examples.tex(,116) @tex
+examples.tex(,117) $$
+examples.tex(,118) \hbox{milnor}(f) =
\hbox{dim}_K(K[[x_1,\ldots,x_n]]/\hbox{jacob}(f)),
+examples.tex(,119) $$
+examples.tex(,120) respectively
+examples.tex(,121) $$
+examples.tex(,122) \hbox{tjurina}(f) =
\hbox{dim}_K(K[[x_1,\ldots,x_n]]/((f)+\hbox{jacob}(f)))
+examples.tex(,123) $$
+examples.tex(,124) where
+examples.tex(,125) @end tex
examples.tex(,126) @code{jacob(f)} is the ideal generated by the partials
examples.tex(,127) of @code{f}. @code{tjurina(f)} is finite, if and only if
@code{f} has an
examples.tex(,128) isolated singularity. The same holds for @code{milnor(f)} if
@@ -18407,8 +19044,17 @@
examples.tex(,131)
examples.tex(,132) @sc{Singular} cannot compute with infinite power series.
But it can
examples.tex(,133) work in
+examples.tex(,134) @tex
+examples.tex(,135) $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$,
+examples.tex(,136) @end tex
examples.tex(,140) the localization of
+examples.tex(,141) @tex
+examples.tex(,142) $K[x_1,\ldots,x_n]$
+examples.tex(,143) @end tex
examples.tex(,147) at the maximal ideal
+examples.tex(,148) @tex
+examples.tex(,149) $(x_1,\ldots,x_n)$.
+examples.tex(,150) @end tex
examples.tex(,154) To do this one has to define an
examples.tex(,155) s-ordering like ds, Ds, ls, ws, Ws or an appropriate matrix
examples.tex(,156) ordering (look at the manual to get information about the
possible
@@ -18605,7 +19251,13 @@
examples.tex(,349)
examples.tex(,350) The same computation which computes the Milnor, resp.@: the
Tjurina,
examples.tex(,351) number, but with ordering @code{dp} instead of @code{ds}
(i.e., in
+examples.tex(,352) @tex
+examples.tex(,353) $K[x_1,\ldots,x_n]$
+examples.tex(,354) @end tex
examples.tex(,358) instead of
+examples.tex(,359) @tex
+examples.tex(,360) $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n])$
+examples.tex(,361) @end tex
examples.tex(,365) gives:
examples.tex(,366) @itemize @bullet
examples.tex(,367) @item
@@ -18643,11 +19295,23 @@
examples.tex(,399) @item
examples.tex(,400) The result of the computation here (together with the
previous one in
examples.tex(,401) @ref{Milnor and Tjurina}) shows that (for @code{t}=0)
+examples.tex(,402) @tex
+examples.tex(,403) $\hbox{dim}_K(\hbox{Loc}_{(x,y,z)}K[x,y,z]/\hbox{jacob}(f))$
+examples.tex(,404) @end tex
examples.tex(,408) = 250 (previously computed) while
+examples.tex(,409) @tex
+examples.tex(,410) $\hbox{dim}_K(K[x,y,z]/\hbox{jacob}(f))$
+examples.tex(,411) @end tex
examples.tex(,415) = 536. Hence @code{f} has 286 critical points,
examples.tex(,416) counted with multiplicity, outside the origin.
examples.tex(,417) Moreover, since
+examples.tex(,418) @tex
+examples.tex(,419)
$\hbox{dim}_K(\hbox{Loc}_{(x,y,z)}K[x,y,z]/(\hbox{jacob}(f)+(f)))$
+examples.tex(,420) @end tex
examples.tex(,424) = 195 =
+examples.tex(,425) @tex
+examples.tex(,426) $\hbox{dim}_K(K[x,y,z]/(\hbox{jacob}(f)+(f)))$,
+examples.tex(,427) @end tex
examples.tex(,431) the affine surface @code{f}=0 is smooth outside the origin.
examples.tex(,432) @end itemize
examples.tex(,433)
@@ -18676,27 +19340,72 @@
examples.tex(,461) @cindex Saturation
examples.tex(,462)
examples.tex(,463) Since in the example above, the ideal
+examples.tex(,467) @tex
+examples.tex(,468) $j+(f)$
+examples.tex(,469) @end tex
examples.tex(,470) has the same @code{vdim}
examples.tex(,471) in the polynomial ring and in the localization at 0 (each
195),
examples.tex(,472)
+examples.tex(,476) @tex
+examples.tex(,477) $f=0$
+examples.tex(,478) @end tex
examples.tex(,479) is smooth outside 0.
examples.tex(,480) Hence
+examples.tex(,484) @tex
+examples.tex(,485) $j+(f)$
+examples.tex(,486) @end tex
examples.tex(,487) contains some power of the maximal ideal
+examples.tex(,491) @tex
+examples.tex(,492) $m$
+examples.tex(,493) @end tex
examples.tex(,494) . We shall
examples.tex(,495) check this in a different manner:
examples.tex(,496) For any two ideals
+examples.tex(,500) @tex
+examples.tex(,501) $i, j$
+examples.tex(,502) @end tex
examples.tex(,503) in the basering
+examples.tex(,507) @tex
+examples.tex(,508) $R$
+examples.tex(,509) @end tex
examples.tex(,510) let
+examples.tex(,511) @tex
+examples.tex(,512) $$
+examples.tex(,513) \hbox{sat}(i,j)=\{x\in R\;|\; \exists\;n\hbox{ s.t. }
+examples.tex(,514) x\cdot(j^n)\subseteq i\}
+examples.tex(,515) = \bigcup_{n=1}^\infty i:j^n$$
+examples.tex(,516) @end tex
examples.tex(,521) @*denote the saturation of
+examples.tex(,525) @tex
+examples.tex(,526) $i$
+examples.tex(,527) @end tex
examples.tex(,528) with respect to
+examples.tex(,532) @tex
+examples.tex(,533) $j$
+examples.tex(,534) @end tex
examples.tex(,535) . This defines,
examples.tex(,536) geometrically, the closure of the complement of V(
+examples.tex(,540) @tex
+examples.tex(,541) $j$
+examples.tex(,542) @end tex
examples.tex(,543) ) in V(
+examples.tex(,547) @tex
+examples.tex(,548) $i$
+examples.tex(,549) @end tex
examples.tex(,550) )
examples.tex(,551) (V(
+examples.tex(,555) @tex
+examples.tex(,556) $i$
+examples.tex(,557) @end tex
examples.tex(,558) ) denotes the variety defined by
+examples.tex(,562) @tex
+examples.tex(,563) $i$
+examples.tex(,564) @end tex
examples.tex(,565) ).
examples.tex(,566) In our case,
+examples.tex(,570) @tex
+examples.tex(,571) $sat(j+(f),m)$
+examples.tex(,572) @end tex
examples.tex(,573) must be the whole ring, hence
examples.tex(,574) generated by 1.
examples.tex(,575)
@@ -18836,13 +19545,25 @@
examples.tex(,717) and compute over the ground field Q(t).
examples.tex(,718) We compute the dimension at the generic point,
examples.tex(,725) i.e.,
+examples.tex(,726) @tex
+examples.tex(,727) $dim_{Q(t)}Q(t)[x,y]/j$.
+examples.tex(,728) @end tex
examples.tex(,733) (This gives the
examples.tex(,734) same result as for the deformed ideal above. Hence, the
above small
examples.tex(,735) deformation was "generic".)
examples.tex(,737)
examples.tex(,738) For almost all
+examples.tex(,739) @tex
+examples.tex(,740) $a \in Q$
+examples.tex(,741) @end tex
examples.tex(,745) this is the same as
+examples.tex(,746) @tex
+examples.tex(,747) $dim_Q Q[x,y]/j_0$,
+examples.tex(,748) @end tex
examples.tex(,752) where
+examples.tex(,753) @tex
+examples.tex(,754) $j_0=j|_{t=a}$.
+examples.tex(,755) @end tex
examples.tex(,759)
examples.tex(,760) @smallexample
examples.tex(,761) @c computed example Parameters examples.doc:579
@@ -18876,8 +19597,17 @@
examples.tex(,790) @cindex T2
examples.tex(,791)
examples.tex(,792)
+examples.tex(,796) @tex
+examples.tex(,797) $T^1$
+examples.tex(,798) @end tex
examples.tex(,799) , resp.@:
+examples.tex(,803) @tex
+examples.tex(,804) $T^2$
+examples.tex(,805) @end tex
examples.tex(,806) , of an ideal
+examples.tex(,810) @tex
+examples.tex(,811) $j$
+examples.tex(,812) @end tex
examples.tex(,813) usually denote the modules of
examples.tex(,814) infinitesimal deformations, resp.@: of obstructions.
examples.tex(,815) In @sc{Singular} there are procedures @code{T_1} and
@code{T_2} in
@@ -19047,7 +19777,16 @@
examples.tex(,985) singularity.
examples.tex(,986) @item
examples.tex(,987) The procedure @code{deform} in @code{sing.lib} returns a
matrix whose columns
+examples.tex(,991) @tex
+examples.tex(,992) $h_1,\ldots,h_r$
+examples.tex(,993) @end tex
examples.tex(,994) represent all 1st order deformations. More precisely, if
+examples.tex(,1000) @tex
+examples.tex(,1001) $I \subset R$ is the ideal generated by $f_1,...,f_s$,
then any infinitesimal
+examples.tex(,1002) deformation of $R/I$ over $K[\varepsilon]/(\varepsilon^2)$
is given
+examples.tex(,1003) by $f+\varepsilon g$,
+examples.tex(,1004) where $f=(f_1,...,f_s)$, $g$ a $K$-linear combination of
the $h_i$.
+examples.tex(,1005) @end tex
examples.tex(,1006)
examples.tex(,1007) @item
examples.tex(,1008) The procedure @code{versal} in @code{deform.lib} computes
a formal
@@ -19167,12 +19906,24 @@
examples.tex(,1130) @cindex Finite fields
examples.tex(,1131)
examples.tex(,1132) We define a variety in
+examples.tex(,1136) @tex
+examples.tex(,1137) $n$
+examples.tex(,1138) @end tex
examples.tex(,1139) -space of codimension 2 defined by
examples.tex(,1140) polynomials of degree
+examples.tex(,1144) @tex
+examples.tex(,1145) $d$
+examples.tex(,1146) @end tex
examples.tex(,1147) with generic coefficients over the prime
examples.tex(,1148) field
+examples.tex(,1152) @tex
+examples.tex(,1153) $Z/p$
+examples.tex(,1154) @end tex
examples.tex(,1155) and look for zeros on the torus. First over the prime
examples.tex(,1156) field and then in the finite extension field with
+examples.tex(,1157) @tex
+examples.tex(,1158) $p^k$
+examples.tex(,1159) @end tex
examples.tex(,1163) elements.
examples.tex(,1164) In general there will be many more solutions in the second
case.
examples.tex(,1165) (Since the @sc{Singular} language is interpreted, the
evaluation of many
@@ -19349,9 +20100,24 @@
examples.tex(,1342)
examples.tex(,1343) Elimination is the algebraic counterpart of the geometric
concept of
examples.tex(,1344) projection. If
+examples.tex(,1345) @tex
+examples.tex(,1346) $f=(f_1,\ldots,f_n):k^r\rightarrow k^n$
+examples.tex(,1347) @end tex
examples.tex(,1351) is a polynomial map,
examples.tex(,1352) the Zariski-closure of the image is the zero-set of the
ideal
+examples.tex(,1353) @tex
+examples.tex(,1354) $$
+examples.tex(,1355) \displaylines{
+examples.tex(,1356) j=J \cap k[x_1,\ldots,x_n], \;\quad\hbox{\rm where}\cr
+examples.tex(,1357)
J=(x_1-f_1(t_1,\ldots,t_r),\ldots,x_n-f_n(t_1,\ldots,t_r))\subseteq
+examples.tex(,1358) k[t_1,\ldots,t_r,x_1,\ldots,x_n]
+examples.tex(,1359) }
+examples.tex(,1360) $$
+examples.tex(,1361) @end tex
examples.tex(,1370) i.e, of the ideal j obtained from J by eliminating the
variables
+examples.tex(,1371) @tex
+examples.tex(,1372) $t_1,\ldots,t_r$.
+examples.tex(,1373) @end tex
examples.tex(,1377) This can be done by computing a standard basis of J with
respect to a product
examples.tex(,1378) ordering where the block of t-variables precedes the block
of
examples.tex(,1379) x-variables and then selecting those polynomials which do
not contain
@@ -19363,13 +20129,23 @@
examples.tex(,1385)
examples.tex(,1386) @strong{WARNING:} In the case of a local or a mixed
ordering, elimination needs special
examples.tex(,1387) care. f may be considered as a map of germs
+examples.tex(,1388) @tex
+examples.tex(,1389) $f:(k^r,0)\rightarrow(k^n,0)$,
+examples.tex(,1390) @end tex
examples.tex(,1394) but even
examples.tex(,1395) if this map germ is finite, we are in general not able to
compute the image
examples.tex(,1396) germ because for this we would need an implementation of
the Weierstrass
examples.tex(,1397) preparation theorem. What we can compute, and what
@code{eliminate} actually does,
examples.tex(,1398) is the following: let V(J) be the zero-set of J in
+examples.tex(,1399) @tex
+examples.tex(,1400) $k^r\times(k^n,0)$,
+examples.tex(,1401) @end tex
examples.tex(,1405) then the
examples.tex(,1406) closure of the image of V(J) under the projection
+examples.tex(,1407) @tex
+examples.tex(,1408) $$\hbox{pr}:k^r\times(k^n,0)\rightarrow(k^n,0)$$
+examples.tex(,1409) can be computed.
+examples.tex(,1410) @end tex
examples.tex(,1415) Note that this germ contains also those components
examples.tex(,1416) of V(J) which meet the fiber of pr outside the origin.
examples.tex(,1417) This is achieved by an ordering with the block of
t-variables having a
@@ -19392,6 +20168,9 @@
examples.tex(,1434) @enumerate
examples.tex(,1435) @item
examples.tex(,1436) First we compute the equations of the simple space curve
+examples.tex(,1437) @tex
+examples.tex(,1438) $\hbox{T}[7]^\prime$
+examples.tex(,1439) @end tex
examples.tex(,1443) consisting of two tangential cusps given in parametric
form.
examples.tex(,1444) @item
examples.tex(,1445) We compute weights for the equations such that the
@@ -19399,6 +20178,9 @@
examples.tex(,1447) @item
examples.tex(,1448) Then we compute the tangent developable of the rational
examples.tex(,1449) normal curve in
+examples.tex(,1450) @tex
+examples.tex(,1451) $P^4$.
+examples.tex(,1452) @end tex
examples.tex(,1456) @end enumerate
examples.tex(,1457)
examples.tex(,1458) @smallexample
@@ -19548,11 +20330,20 @@
examples.tex(,1621)
examples.tex(,1622) Now let's look at an example which uses resolutions: The
Hilbert-Burch
examples.tex(,1623) theorem says that the ideal i of a reduced curve in
+examples.tex(,1624) @tex
+examples.tex(,1625) $K^3$
+examples.tex(,1626) @end tex
examples.tex(,1630) has a free resolution of length 2 and that i is given by
the 2x2 minors
examples.tex(,1631) of the 2nd matrix in the resolution.
examples.tex(,1632) We test this for two transversal cusps in
+examples.tex(,1633) @tex
+examples.tex(,1634) $K^3$.
+examples.tex(,1635) @end tex
examples.tex(,1639) Afterwards we compute the resolution of the ideal j of the
tangent developable
examples.tex(,1640) of the rational normal curve in
+examples.tex(,1641) @tex
+examples.tex(,1642) $P^4$
+examples.tex(,1643) @end tex
examples.tex(,1647) from above.
examples.tex(,1648) Finally we demonstrate the use of the type
@code{resolution} in connection with
examples.tex(,1649) the @code{lres} command.
@@ -19673,24 +20464,45 @@
examples.tex(,1765) @cindex Ext
examples.tex(,1766)
examples.tex(,1767) We start by showing how to calculate the
+examples.tex(,1771) @tex
+examples.tex(,1772) $n$
+examples.tex(,1773) @end tex
examples.tex(,1774) -th Ext group of an
examples.tex(,1775) ideal. The ingredients to do this are by the definition of
Ext the
examples.tex(,1776) following: calculate a (minimal) resolution at least up to
length
examples.tex(,1777)
+examples.tex(,1781) @tex
+examples.tex(,1782) $n$
+examples.tex(,1783) @end tex
examples.tex(,1784) , apply the Hom-functor, and calculate the
+examples.tex(,1788) @tex
+examples.tex(,1789) $n$
+examples.tex(,1790) @end tex
examples.tex(,1791) -th homology
examples.tex(,1792) group, that is form the quotient
+examples.tex(,1793) @tex
+examples.tex(,1794) $\hbox{\rm ker} / \hbox{\rm Im}$
+examples.tex(,1795) @end tex
examples.tex(,1799) in the resolution sequence.
examples.tex(,1800)
examples.tex(,1801) The Hom functor is given simply by transposing (hence
dualizing) the
examples.tex(,1802) module or the corresponding matrix with the command
@code{transpose}.
examples.tex(,1803) The image of the
+examples.tex(,1807) @tex
+examples.tex(,1808) $(n-1)$
+examples.tex(,1809) @end tex
examples.tex(,1810) -st map is generated by the columns of the
examples.tex(,1811) corresponding matrix. To calculate the kernel apply the
command
examples.tex(,1812) @code{syz} at the
+examples.tex(,1816) @tex
+examples.tex(,1817) $(n-1)$
+examples.tex(,1818) @end tex
examples.tex(,1819) -st transposed entry of the resolution.
examples.tex(,1820) Finally, the quotient is obtained by the command
@code{modulo}, which
examples.tex(,1821) gives for two modules A = ker, B = Im the module of
relations of
+examples.tex(,1822) @tex
+examples.tex(,1823) $A/(A \cap B)$
+examples.tex(,1824) @end tex
examples.tex(,1828) in the usual way. As we have a chain complex this is
obviously the same
examples.tex(,1829) as ker/Im.
examples.tex(,1830)
@@ -19729,17 +20541,44 @@
examples.tex(,1863) example.
examples.tex(,1864)
examples.tex(,1865) If
+examples.tex(,1869) @tex
+examples.tex(,1870) $M$
+examples.tex(,1871) @end tex
examples.tex(,1872) is a module, then
+examples.tex(,1873) @tex
+examples.tex(,1874) $\hbox{Ext}^1(M,M)$, resp.\ $\hbox{Ext}^2(M,M)$,
+examples.tex(,1875) @end tex
examples.tex(,1879) are the modules of infinitesimal deformations, resp.@: of
obstructions, of
examples.tex(,1880)
+examples.tex(,1884) @tex
+examples.tex(,1885) $M$
+examples.tex(,1886) @end tex
examples.tex(,1887) (like T1 and T2 for a singularity). Similar to the
treatment
examples.tex(,1888) for singularities, the semiuniversal deformation of
+examples.tex(,1892) @tex
+examples.tex(,1893) $M$
+examples.tex(,1894) @end tex
examples.tex(,1895) can be
examples.tex(,1896) computed (if
+examples.tex(,1897) @tex
+examples.tex(,1898) $\hbox{Ext}^1$
+examples.tex(,1899) @end tex
examples.tex(,1903) is finite dimensional) with the help of
+examples.tex(,1904) @tex
+examples.tex(,1905) $\hbox{Ext}^1$, $\hbox{Ext}^2$
+examples.tex(,1906) @end tex
examples.tex(,1910) and the cup product. There is an extra procedure for
+examples.tex(,1911) @tex
+examples.tex(,1912) $\hbox{Ext}^k(R/J,R)$
+examples.tex(,1913) @end tex
examples.tex(,1917) if
+examples.tex(,1921) @tex
+examples.tex(,1922) $J$
+examples.tex(,1923) @end tex
examples.tex(,1924) is an ideal in
+examples.tex(,1928) @tex
+examples.tex(,1929) $R$
+examples.tex(,1930) @end tex
examples.tex(,1931) since this is faster than the
examples.tex(,1932) general Ext.
examples.tex(,1933)
@@ -19747,15 +20586,42 @@
examples.tex(,1935) @itemize @bullet
examples.tex(,1936) @item
examples.tex(,1937) the infinitesimal deformations
+examples.tex(,1938) @tex
+examples.tex(,1939) ($=\hbox{Ext}^1(K,K)$)
+examples.tex(,1940) @end tex
examples.tex(,1944) and obstructions
+examples.tex(,1945) @tex
+examples.tex(,1946) ($=\hbox{Ext}^2(K,K)$)
+examples.tex(,1947) @end tex
examples.tex(,1951) of the residue field
+examples.tex(,1955) @tex
+examples.tex(,1956) $K=R/m$
+examples.tex(,1957) @end tex
examples.tex(,1958) of an ordinary cusp,
+examples.tex(,1959) @tex
+examples.tex(,1960) $R=Loc_m K[x,y]/(x^2-y^3)$, $m=(x,y)$.
+examples.tex(,1961) @end tex
examples.tex(,1965) To compute
+examples.tex(,1966) @tex
+examples.tex(,1967) $\hbox{Ext}^1(m,m)$
+examples.tex(,1968) @end tex
examples.tex(,1972) we have to apply @code{Ext(1,syz(m),syz(m))} with
examples.tex(,1973) @code{syz(m)} the first syzygy module of
+examples.tex(,1977) @tex
+examples.tex(,1978) $m$
+examples.tex(,1979) @end tex
examples.tex(,1980) , which is isomorphic to
+examples.tex(,1981) @tex
+examples.tex(,1982) $\hbox{Ext}^2(K,K)$.
+examples.tex(,1983) @end tex
examples.tex(,1987) @item
+examples.tex(,1988) @tex
+examples.tex(,1989) $\hbox{Ext}^k(R/i,R)$
+examples.tex(,1990) @end tex
examples.tex(,1994) for some ideal
+examples.tex(,1998) @tex
+examples.tex(,1999) $i$
+examples.tex(,2000) @end tex
examples.tex(,2001) and with an extra option.
examples.tex(,2002) @end itemize
examples.tex(,2003)
@@ -19851,18 +20717,45 @@
examples.tex(,2095) @cindex Polar curves
examples.tex(,2096)
examples.tex(,2097) The polar curve of a hypersurface given by a polynomial
+examples.tex(,2098) @tex
+examples.tex(,2099) $f\in k[x_1,\ldots,x_n,t]$
+examples.tex(,2100) @end tex
examples.tex(,2104) with respect to
+examples.tex(,2108) @tex
+examples.tex(,2109) $t$
+examples.tex(,2110) @end tex
examples.tex(,2111) (we may consider
+examples.tex(,2115) @tex
+examples.tex(,2116) $f=0$
+examples.tex(,2117) @end tex
examples.tex(,2118) as a family of
examples.tex(,2119) hypersurfaces parametrized by
+examples.tex(,2123) @tex
+examples.tex(,2124) $t$
+examples.tex(,2125) @end tex
examples.tex(,2126) ) is defined as the Zariski
examples.tex(,2127) closure of
+examples.tex(,2128) @tex
+examples.tex(,2129) $V(\partial f/\partial x_1,\ldots,\partial f/\partial x_n)
\setminus V(f)$
+examples.tex(,2130) @end tex
examples.tex(,2134) if this happens to be a curve. Some authors consider
+examples.tex(,2135) @tex
+examples.tex(,2136) $V(\partial f/\partial x_1,\ldots,\partial f/\partial x_n)$
+examples.tex(,2137) @end tex
examples.tex(,2141) itself as polar curve.
examples.tex(,2142)
examples.tex(,2143) We may consider projective hypersurfaces
+examples.tex(,2144) @tex
+examples.tex(,2145) (in $P^n$),
+examples.tex(,2146) @end tex
examples.tex(,2150) affine hypersurfaces
+examples.tex(,2151) @tex
+examples.tex(,2152) (in $k^n$)
+examples.tex(,2153) @end tex
examples.tex(,2157) or germs of hypersurfaces
+examples.tex(,2158) @tex
+examples.tex(,2159) (in $(k^n,0)$),
+examples.tex(,2160) @end tex
examples.tex(,2164) getting in this way
examples.tex(,2165) projective, affine or local polar curves.
examples.tex(,2166)
@@ -19975,12 +20868,24 @@
examples.tex(,2275) @cindex Depth
examples.tex(,2276)
examples.tex(,2277) We compute the depth of the module of Kaehler differentials
+examples.tex(,2278) @tex
+examples.tex(,2279) D$_k$(R)
+examples.tex(,2280) @end tex
examples.tex(,2284) of the variety defined by the
+examples.tex(,2288) @tex
+examples.tex(,2289) $(m+1)$
+examples.tex(,2290) @end tex
examples.tex(,2291) -minors of a generic symmetric
+examples.tex(,2292) @tex
+examples.tex(,2293) $(n \times n)$-matrix.
+examples.tex(,2294) @end tex
examples.tex(,2298) We do this by computing the resolution over the polynomial
examples.tex(,2299) ring. Then, by the Auslander-Buchsbaum formula, the depth
is equal to
examples.tex(,2300) the number of variables minus the length of a minimal
resolution. This
examples.tex(,2301) example was suggested by U.@: Vetter in order to check
whether his bound
+examples.tex(,2302) @tex
+examples.tex(,2303) $\hbox{depth}(\hbox{D}_k(R))\geq m(m+1)/2 + m-1$
+examples.tex(,2304) @end tex
examples.tex(,2308) could be improved.
examples.tex(,2309)
examples.tex(,2310) @smallexample
@@ -20149,6 +21054,9 @@
examples.tex(,2482)
examples.tex(,2483) We work in characteristic 0 and use the Lie algebra
generated by one
examples.tex(,2484) vector field of the form
+examples.tex(,2485) @tex
+examples.tex(,2486) $\sum x_i \partial /\partial x_{i+1}$.
+examples.tex(,2487) @end tex
examples.tex(,2491) @smallexample
examples.tex(,2492) @c computed example G_a_-Invariants examples.doc:1783
examples.tex(,2493) LIB "ainvar.lib";
@@ -20314,6 +21222,9 @@
examples.tex(,2662)
examples.tex(,2663) We compute the Hamburger-Noether expansion of a plane curve
examples.tex(,2664) singularity given by a polynomial
+examples.tex(,2668) @tex
+examples.tex(,2669) $f$
+examples.tex(,2670) @end tex
examples.tex(,2671) in two variables. This is a
examples.tex(,2672) matrix which allows to compute the parametrization (up to
a given order)
examples.tex(,2673) and all numerical invariants like the
@@ -20331,7 +21242,13 @@
examples.tex(,2685) @end itemize
examples.tex(,2686) Besides this, the library contains procedures to compute
the Newton
examples.tex(,2687) polygon of
+examples.tex(,2691) @tex
+examples.tex(,2692) $f$
+examples.tex(,2693) @end tex
examples.tex(,2694) , the squarefree part of
+examples.tex(,2698) @tex
+examples.tex(,2699) $f$
+examples.tex(,2700) @end tex
examples.tex(,2701) and a procedure to
examples.tex(,2702) convert one set of invariants to another.
examples.tex(,2703)
@@ -20556,9 +21473,15 @@
examples.tex(,2926) @section Normalization
examples.tex(,2927) @cindex Normalization
examples.tex(,2928) The normalization will be computed for a reduced ring
+examples.tex(,2932) @tex
+examples.tex(,2933) $R/I$
+examples.tex(,2934) @end tex
examples.tex(,2935) . The
examples.tex(,2936) result is a list of rings; ideals are always called
@code{norid} in the
examples.tex(,2937) rings of this list. The normalization of
+examples.tex(,2941) @tex
+examples.tex(,2942) $R/I$
+examples.tex(,2943) @end tex
examples.tex(,2944) is the product of
examples.tex(,2945) the factor rings of the rings in the list divided out by
the ideals
examples.tex(,2946) @code{norid}.
@@ -20762,12 +21685,41 @@
examples.tex(,3152) @section Kernel of module homomorphisms
examples.tex(,3153) @cindex Kernel of module homomorphisms
examples.tex(,3154) Let
+examples.tex(,3158) @tex
+examples.tex(,3159) $A$
+examples.tex(,3160) @end tex
examples.tex(,3161) ,
+examples.tex(,3165) @tex
+examples.tex(,3166) $B$
+examples.tex(,3167) @end tex
examples.tex(,3168) be two matrices of size
+examples.tex(,3169) @tex
+examples.tex(,3170) $m\times r$ and $m\times s$
+examples.tex(,3171) @end tex
examples.tex(,3175) over the ring
+examples.tex(,3179) @tex
+examples.tex(,3180) $R$
+examples.tex(,3181) @end tex
examples.tex(,3182) and consider the corresponding maps
+examples.tex(,3183) @tex
+examples.tex(,3184) $$
+examples.tex(,3185) R^r \buildrel{A}\over{\longrightarrow}
+examples.tex(,3186) R^m \buildrel{B}\over{\longleftarrow} R^s\;.
+examples.tex(,3187) $$
+examples.tex(,3188) @end tex
examples.tex(,3202) We want to compute the kernel of the map
+examples.tex(,3203) @tex
+examples.tex(,3204) $R^r \buildrel{A}\over{\longrightarrow}
+examples.tex(,3205) R^m\longrightarrow
+examples.tex(,3206) R^m/\hbox{Im}(B) \;.$
+examples.tex(,3207) @end tex
examples.tex(,3216) This can be done using the @code{modulo} command:
+examples.tex(,3217) @tex
+examples.tex(,3218) $$
+examples.tex(,3219) \hbox{\tt modulo}(A,B)=\hbox{ker}(R^r
+examples.tex(,3220) \buildrel{A}\over{\longrightarrow}R^m/\hbox{Im}(B)) \; .
+examples.tex(,3221) $$
+examples.tex(,3222) @end tex
examples.tex(,3231)
examples.tex(,3232) @smallexample
examples.tex(,3233) @c computed example Kernel_of_module_homomorphisms
examples.doc:2196
@@ -20785,22 +21737,56 @@
examples.tex(,3250) @section Algebraic dependence
examples.tex(,3251) @cindex Algebraic dependence
examples.tex(,3252) Let
+examples.tex(,3253) @tex
+examples.tex(,3254) $g$, $f_1$, \dots, $f_r\in K[x_1,\ldots,x_n]$.
+examples.tex(,3255) @end tex
examples.tex(,3259) We want to check whether
examples.tex(,3260) @enumerate
examples.tex(,3261) @item
+examples.tex(,3262) @tex
+examples.tex(,3263) $f_1$, \dots, $f_r$
+examples.tex(,3264) @end tex
examples.tex(,3268) are algebraically dependent.
examples.tex(,3269)
examples.tex(,3270) Let
+examples.tex(,3271) @tex
+examples.tex(,3272) $I=\langle Y_1-f_1,\ldots,Y_r-f_r \rangle \subseteq
+examples.tex(,3273) K[x_1,\ldots,x_n,Y_1,\ldots,Y_r]$.
+examples.tex(,3274) @end tex
examples.tex(,3282) Then
+examples.tex(,3283) @tex
+examples.tex(,3284) $I \cap K[Y_1,\ldots,Y_r]$
+examples.tex(,3285) @end tex
examples.tex(,3289) are the algebraic relations between
+examples.tex(,3290) @tex
+examples.tex(,3291) $f_1$, \dots, $f_r$.
+examples.tex(,3292) @end tex
examples.tex(,3296)
examples.tex(,3297) @item
+examples.tex(,3298) @tex
+examples.tex(,3299) $g \in K [f_1,\ldots,f_r]$.
+examples.tex(,3300) @end tex
examples.tex(,3304)
+examples.tex(,3305) @tex
+examples.tex(,3306) $g \in K[f_1,\ldots,f_r]$
+examples.tex(,3307) @end tex
examples.tex(,3311) if and only if the normal form of
+examples.tex(,3315) @tex
+examples.tex(,3316) $g$
+examples.tex(,3317) @end tex
examples.tex(,3318) with respect to
+examples.tex(,3322) @tex
+examples.tex(,3323) $I$
+examples.tex(,3324) @end tex
examples.tex(,3325) and a
examples.tex(,3326) block ordering with respect to
+examples.tex(,3327) @tex
+examples.tex(,3328) $X=(x_1,\ldots,x_n)$ and $Y=(Y_1,\ldots,Y_r)$ with $X>Y$
+examples.tex(,3329) @end tex
examples.tex(,3333) is in
+examples.tex(,3337) @tex
+examples.tex(,3338) $K[Y]$
+examples.tex(,3339) @end tex
examples.tex(,3340) .
examples.tex(,3341) @end enumerate
examples.tex(,3342)
@@ -21141,35 +22127,83 @@
pdata.tex(,53) A vector in @sc{Singular} is always an element of a free
module over the
pdata.tex(,54) basering. It is given as a list of polynomials in one of the
following
pdata.tex(,55) formats
+pdata.tex(,56) @tex
+pdata.tex(,57) $[f_1,...,f_n]$ or $f_1*gen(1)+...+f_n*gen(n)$, where $gen(i)$
+pdata.tex(,58) @end tex
pdata.tex(,62) denotes the i-th canonical generator of a free module (with 1
at place i and
pdata.tex(,63) 0 everywhere else).
pdata.tex(,64) Both forms are equivalent. A vector is internally represented in
pdata.tex(,65) the second form with the
+pdata.tex(,66) @tex
+pdata.tex(,67) $gen(i)$
+pdata.tex(,68) @end tex
pdata.tex(,72) being "special" ring variables, ordered accordingly to the
monomial ordering.
pdata.tex(,73) Therefore, the form
+pdata.tex(,74) @tex
+pdata.tex(,75) $[f_1,...,f_n]$
+pdata.tex(,76) @end tex
pdata.tex(,80) is given as output only if the monomial ordering gives priority
to the
pdata.tex(,81) component, i.e@:., is of the form @code{(c,...)} (see
@ref{Module
pdata.tex(,82) orderings}). However, in any case the procedure @code{show}
from the
pdata.tex(,83) library @code{inout.lib} displays the bracket format.
pdata.tex(,84)
pdata.tex(,85) A vector
+pdata.tex(,86) @tex
+pdata.tex(,87) $v=[f_1,...,f_n]$
+pdata.tex(,88) @end tex
pdata.tex(,92) should always be considered as a column vector in a free module
pdata.tex(,93) of rank equal to
+pdata.tex(,94) @tex
+pdata.tex(,95) nrows($v$)
+pdata.tex(,96) @end tex
pdata.tex(,100) where
+pdata.tex(,101) @tex
+pdata.tex(,102) nrows($v$)
+pdata.tex(,103) @end tex
pdata.tex(,107) is equal to the maximal index
+pdata.tex(,108) @tex
+pdata.tex(,109) $r$
+pdata.tex(,110) @end tex
pdata.tex(,114) such that
+pdata.tex(,115) @tex
+pdata.tex(,116) $f_r \not= 0$.
+pdata.tex(,117) @end tex
pdata.tex(,121) This is due to the fact, that internally
+pdata.tex(,122) @tex
+pdata.tex(,123) $v$
+pdata.tex(,124) @end tex
pdata.tex(,128) is a polynomial in a sparse representation, i.e.,
+pdata.tex(,129) @tex
+pdata.tex(,130) $f_i*gen(i)$
+pdata.tex(,131) @end tex
pdata.tex(,135) is not stored if
+pdata.tex(,136) @tex
+pdata.tex(,137) $f_i=0$
+pdata.tex(,138) @end tex
pdata.tex(,142) (for reasons of efficiency), hence the last 0-entries of
+pdata.tex(,143) @tex
+pdata.tex(,144) $v$
+pdata.tex(,145) @end tex
pdata.tex(,149) are lost.
pdata.tex(,150) Only more complex structures are able to keep the rank.
pdata.tex(,151)
pdata.tex(,152) A module
+pdata.tex(,153) @tex
+pdata.tex(,154) $M$
+pdata.tex(,155) @end tex
pdata.tex(,159) in @sc{Singular} is given by a list of vectors
+pdata.tex(,160) @tex
+pdata.tex(,161) $v_1,...,v_k$
+pdata.tex(,162) @end tex
pdata.tex(,166) which generate the module as a submodule of the free module of
rank
pdata.tex(,167) equal to
+pdata.tex(,168) @tex
+pdata.tex(,169) nrows($M$)
+pdata.tex(,170) @end tex
pdata.tex(,174) which is the maximum of
+pdata.tex(,175) @tex
+pdata.tex(,176) nrows($v_i$).
+pdata.tex(,177) @end tex
pdata.tex(,181)
pdata.tex(,182) If one wants to create a module with a larger rank than given
by its
pdata.tex(,183) generators, one has to use the command
@code{attrib(M,"rank",r)} (see
@@ -21184,33 +22218,84 @@
pdata.tex(,192) By the above remarks it might appear that @sc{Singular} is
only able to handle
pdata.tex(,193) submodules of a free module. However, this is not true.
@sc{Singular}
pdata.tex(,194) can compute with any finitely generated module over the
basering
+pdata.tex(,195) @tex
+pdata.tex(,196) $R$.
+pdata.tex(,197) @end tex
pdata.tex(,201) Such a module, say
+pdata.tex(,202) @tex
+pdata.tex(,203) $N$,
+pdata.tex(,204) @end tex
pdata.tex(,208) is not represented by its generators but by its
pdata.tex(,209) (generators and) relations. This means that
+pdata.tex(,210) @tex
+pdata.tex(,211) $N = R^n/M$ where $n$
+pdata.tex(,212) @end tex
pdata.tex(,216) is the number of generators of
+pdata.tex(,217) @tex
+pdata.tex(,218) $N$ and $M \subseteq R^n$
+pdata.tex(,219) @end tex
pdata.tex(,223) is the module of relations.
pdata.tex(,224) In other words, defining a module
+pdata.tex(,225) @tex
+pdata.tex(,226) $M$
+pdata.tex(,227) @end tex
pdata.tex(,231) as a submodule of a free module
+pdata.tex(,232) @tex
+pdata.tex(,233) $R^n$
+pdata.tex(,234) @end tex
pdata.tex(,238) can also be considered as the definition of
+pdata.tex(,239) @tex
+pdata.tex(,240) $N = R^n/M$.
+pdata.tex(,241) @end tex
pdata.tex(,245)
pdata.tex(,246) Note that most functions, when applied to a module
+pdata.tex(,247) @tex
+pdata.tex(,248) $M$,
+pdata.tex(,249) @end tex
pdata.tex(,253) really deal with
+pdata.tex(,254) @tex
+pdata.tex(,255) $M$.
+pdata.tex(,256) @end tex
pdata.tex(,260) However, there are some functions which deal with
+pdata.tex(,261) @tex
+pdata.tex(,262) $N = R^n/M$ instead of $M$.
+pdata.tex(,263) @end tex
pdata.tex(,267)
pdata.tex(,268) For example, @code{std(M)} computes a standard basis of
+pdata.tex(,269) @tex
+pdata.tex(,270) $M$
+pdata.tex(,271) @end tex
pdata.tex(,275) (and thus gives another representation of
+pdata.tex(,276) @tex
+pdata.tex(,277) $N$ as $N = R^n/$std($M$)).
+pdata.tex(,278) @end tex
pdata.tex(,282) However, @code{dim(M)}, resp.@: @code{vdim(M)}, returns
+pdata.tex(,283) @tex
+pdata.tex(,284) dim$(R^n/M)$, resp.@: dim$_k(R^n/M)$
+pdata.tex(,285) @end tex
pdata.tex(,289) (if M is given by a standard basis).
pdata.tex(,290)
pdata.tex(,291) The function @code{syz(M)} returns the first syzygy module of
+pdata.tex(,292) @tex
+pdata.tex(,293) $M$,
+pdata.tex(,294) @end tex
pdata.tex(,298) i.e@:., the module
pdata.tex(,299) of relations of the given generators of
+pdata.tex(,300) @tex
+pdata.tex(,301) $M$
+pdata.tex(,302) @end tex
pdata.tex(,306) which is equal to the second syzygy module of
+pdata.tex(,307) @tex
+pdata.tex(,308) $N$.
+pdata.tex(,309) @end tex
pdata.tex(,313) Refer to the description of each function in
pdata.tex(,314) @ref{Functions} to get information which module the function
deals with.
pdata.tex(,315)
pdata.tex(,316) The numbering in @code{res} and other commands for computing
resolutions
pdata.tex(,317) refers to a resolution of
+pdata.tex(,318) @tex
+pdata.tex(,319) $N = R^n/M$
+pdata.tex(,320) @end tex
pdata.tex(,324) (see @ref{res}; @ref{Syzygies and resolutions}).
pdata.tex(,325)
pdata.tex(,326) It is possible to compute in any field which is a valid ground
field in
@@ -21254,13 +22339,28 @@
pdata.tex(,364) flexibility might also be confusing for the novice user.
Therefore, we
pdata.tex(,365) recommend to those not familiar with monomial orderings to
generally use
pdata.tex(,366) the ordering @code{dp} for computations in the polynomial ring
+pdata.tex(,367) @tex
+pdata.tex(,368) $K[x_1,\ldots,x_n]$,
+pdata.tex(,369) @end tex
pdata.tex(,373) resp.@: @code{ds} for computations in the localization
+pdata.tex(,374) @tex
+pdata.tex(,375) $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$.
+pdata.tex(,376) @end tex
pdata.tex(,380)
pdata.tex(,381) For inhomogeneous input ideals, standard (resp.@: groebner)
bases
pdata.tex(,382) computations are generally faster
pdata.tex(,383) with the orderings
+pdata.tex(,384) @tex
+pdata.tex(,385) $\hbox{Wp}(w_1, \ldots, w_n)$
+pdata.tex(,386) @end tex
pdata.tex(,390) (resp.@:
+pdata.tex(,391) @tex
+pdata.tex(,392) $\hbox{Ws}(w_1, \ldots, w_n)$)
+pdata.tex(,393) @end tex
pdata.tex(,397) if the input is quasihomogeneous w.r.t. the weights
+pdata.tex(,398) @tex
+pdata.tex(,399) $w_1$, $\ldots$, $w_n$ of $x_1$, $\ldots$, $x_n$.
+pdata.tex(,400) @end tex
pdata.tex(,404)
pdata.tex(,405) If the output needs to be "triangular" (resp.@:
"block-triangular"), the
pdata.tex(,406) lexicographical ordering @code{lp} (resp.@: lexicographical
@@ -21275,12 +22375,39 @@
pdata.tex(,415) @cindex term orderings
pdata.tex(,416) @cindex monomial orderings
pdata.tex(,417)
+pdata.tex(,418) @tex
+pdata.tex(,419) A monomial ordering (term ordering) on $K[x_1, \ldots, x_n]$ is
+pdata.tex(,420) a total ordering $<$ on the
+pdata.tex(,421) set of monomials (power products) $\{x^\alpha \mid \alpha \in
\bf{N}^n\}$
+pdata.tex(,422) which is compatible with the
+pdata.tex(,423) natural semigroup structure, i.e., $x^\alpha < x^\beta$
implies $x^\gamma
+pdata.tex(,424) x^\alpha < x^\gamma x^\beta$ for any $\gamma \in \bf{N}^n$.
+pdata.tex(,425) We do not require
+pdata.tex(,426) $<$ to be a well ordering.
+pdata.tex(,427) @end tex
pdata.tex(,439) See the literature cited in @ref{References}.
pdata.tex(,441)
pdata.tex(,442) It is known that any monomial ordering can be represented by a
matrix
+pdata.tex(,443) @tex
+pdata.tex(,444) $M$ in $GL(n,R)$,
+pdata.tex(,445) @end tex
pdata.tex(,449) but, of course, only integer coefficients are of relevance in
pdata.tex(,450) practice.
pdata.tex(,451)
+pdata.tex(,452) @tex
+pdata.tex(,453) Global orderings are well orderings (i.e., \hbox{$1 < x_i$}
for each variable
+pdata.tex(,454) $x_i$), local orderings satisfy $1 > x_i$ for each variable.
If some variables are ordered globally and others locally we
+pdata.tex(,455) call it a mixed ordering. Local or mixed orderings are not
well orderings.
+pdata.tex(,456)
+pdata.tex(,457) Let $K$ be the ground field, \hbox{$x = (x_1, \ldots, x_n)$}
the
+pdata.tex(,458) variables and $<$ a monomial ordering, then Loc $K[x]$ denotes
the
+pdata.tex(,459) localization of $K[x]$ with respect to the multiplicatively
closed set $$\{1 +
+pdata.tex(,460) g \mid g = 0 \hbox{ or } g \in K[x]\backslash \{0\} \hbox{ and
}L(g) <
+pdata.tex(,461) 1\}.$$ Here, $L(g)$
+pdata.tex(,462) denotes the leading monomial of $g$, i.e., the biggest
monomial of $g$ with
+pdata.tex(,463) respect to $<$. The result of any computation which uses
standard basis
+pdata.tex(,464) computations has to be interpreted in Loc $K[x]$.
+pdata.tex(,465) @end tex
pdata.tex(,480)
pdata.tex(,481) Note that the definition of a ring includes the definition of
its
pdata.tex(,482) monomial ordering (see
@@ -21294,6 +22421,9 @@
pdata.tex(,490) @cindex Global orderings
pdata.tex(,491) @cindex orderings, global
pdata.tex(,492)
+pdata.tex(,493) @tex
+pdata.tex(,494) For all these orderings: Loc $K[x]$ = $K[x]$
+pdata.tex(,495) @end tex
pdata.tex(,499)
pdata.tex(,500) @table @asis
pdata.tex(,501) @item lp:
@@ -21301,35 +22431,81 @@
pdata.tex(,503) @cindex lp, global ordering
pdata.tex(,504) @cindex lexicographical ordering
pdata.tex(,505) @*
+pdata.tex(,510) @tex
+pdata.tex(,511) $x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
+pdata.tex(,512) \alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1},
\alpha_i <
+pdata.tex(,513) \beta_i$.
+pdata.tex(,514) @end tex
pdata.tex(,515) @item rp:
pdata.tex(,516) reverse lexicographical ordering:
pdata.tex(,517) @cindex rp, global ordering
pdata.tex(,518) @cindex reverse lexicographical ordering
pdata.tex(,519) @*
+pdata.tex(,524) @tex
+pdata.tex(,525) $x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
+pdata.tex(,526) \alpha_n = \beta_n,
+pdata.tex(,527) \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
+pdata.tex(,528) @end tex
pdata.tex(,529) @item dp:
pdata.tex(,530) degree reverse lexicographical ordering:
pdata.tex(,531) @cindex degree reverse lexicographical ordering
pdata.tex(,532) @cindex dp, global ordering
pdata.tex(,533) @*
+pdata.tex(,537) @tex
+pdata.tex(,538) let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
+pdata.tex(,539) @end tex
+pdata.tex(,547) @tex
+pdata.tex(,548) $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) <
\deg(x^\beta)$ or
+pdata.tex(,549) @end tex
+pdata.tex(,553) @tex
+pdata.tex(,554) \phantom{$x^\alpha < x^\beta \Leftrightarrow $}$
\deg(x^\alpha) =
+pdata.tex(,555) \deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n =
\beta_n,
+pdata.tex(,556) \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
+pdata.tex(,557) @end tex
pdata.tex(,558) @item Dp:
pdata.tex(,559) degree lexicographical ordering:
pdata.tex(,560) @cindex degree lexicographical ordering
pdata.tex(,561) @cindex Dp, global ordering
pdata.tex(,562) @*
+pdata.tex(,566) @tex
+pdata.tex(,567) let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
+pdata.tex(,568) @end tex
+pdata.tex(,576) @tex
+pdata.tex(,577) $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) <
\deg(x^\beta)$ or
+pdata.tex(,578) @end tex
+pdata.tex(,582) @tex
+pdata.tex(,583) \phantom{ $x^\alpha < x^\beta \Leftrightarrow $}
$\deg(x^\alpha) =
+pdata.tex(,584) \deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 =
\beta_1,
+pdata.tex(,585) \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$
+pdata.tex(,586) @end tex
pdata.tex(,587) @item wp:
pdata.tex(,588) weighted reverse lexicographical ordering:
pdata.tex(,589) @cindex weighted reverse lexicographical ordering
pdata.tex(,590) @cindex wp, global ordering
pdata.tex(,591) @*
+pdata.tex(,595) @tex
+pdata.tex(,596) let $w_1, \ldots, w_n$ be positive integers. Then ${\tt
wp}(w_1, \ldots,
+pdata.tex(,597) w_n)$
+pdata.tex(,598) @end tex
pdata.tex(,599) is defined as @code{dp}
pdata.tex(,600) but with
+pdata.tex(,604) @tex
+pdata.tex(,605) $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
+pdata.tex(,606) @end tex
pdata.tex(,607) @item Wp:
pdata.tex(,608) weighted lexicographical ordering:
pdata.tex(,609) @cindex weighted lexicographical ordering
pdata.tex(,610) @cindex WP, global ordering
pdata.tex(,611) @*
+pdata.tex(,615) @tex
+pdata.tex(,616) let $w_1, \ldots, w_n$ be positive integers. Then ${\tt
Wp}(w_1, \ldots,
+pdata.tex(,617) w_n)$
+pdata.tex(,618) @end tex
pdata.tex(,619) is defined as @code{Dp}
pdata.tex(,620) but with
+pdata.tex(,624) @tex
+pdata.tex(,625) $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
+pdata.tex(,626) @end tex
pdata.tex(,627) @end table
pdata.tex(,628) @c
--------------------------------------------------------------------------
pdata.tex(,629) @node Local orderings, Module orderings, Global orderings,
Monomial orderings
@@ -21339,8 +22515,17 @@
pdata.tex(,633)
pdata.tex(,634) For ls, ds, Ds and, if the weights are positive integers, also
for ws and
pdata.tex(,635) Ws, we have
+pdata.tex(,639) @tex
+pdata.tex(,640) Loc $K[x]$ = $K[x]_{(x)}$,
+pdata.tex(,641) @end tex
pdata.tex(,642) the localization of
+pdata.tex(,643) @tex
+pdata.tex(,644) $K[x]$
+pdata.tex(,645) @end tex
pdata.tex(,649) at the maximal ideal
+pdata.tex(,653) @tex
+pdata.tex(,654) \ $(x_1, ..., x_n)$.
+pdata.tex(,655) @end tex
pdata.tex(,656)
pdata.tex(,657) @table @asis
pdata.tex(,658) @item ls:
@@ -21348,36 +22533,81 @@
pdata.tex(,660) @cindex negative lexicographical ordering
pdata.tex(,661) @cindex ls, local ordering
pdata.tex(,662) @*
+pdata.tex(,667) @tex
+pdata.tex(,668) $x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
+pdata.tex(,669) \alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1},
\alpha_i >
+pdata.tex(,670) \beta_i$.
+pdata.tex(,671) @end tex
pdata.tex(,672) @item ds:
pdata.tex(,673) negative degree reverse lexicographical ordering:
pdata.tex(,674) @cindex negative degree reverse lexicographical ordering
pdata.tex(,675) @cindex ds, local ordering
pdata.tex(,676) @*
+pdata.tex(,680) @tex
+pdata.tex(,681) let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
+pdata.tex(,682) @end tex
+pdata.tex(,690) @tex
+pdata.tex(,691) $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) >
\deg(x^\beta)$ or
+pdata.tex(,692) @end tex
+pdata.tex(,696) @tex
+pdata.tex(,697) \phantom{ $x^\alpha < x^\beta \Leftrightarrow$}$
\deg(x^\alpha) =
+pdata.tex(,698) \deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n =
\beta_n,
+pdata.tex(,699) \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
+pdata.tex(,700) @end tex
pdata.tex(,701) @item Ds:
pdata.tex(,702) negative degree lexicographical ordering:
pdata.tex(,703) @cindex negative degree lexicographical ordering
pdata.tex(,704) @cindex Ds, local ordering
pdata.tex(,705) @*
+pdata.tex(,709) @tex
+pdata.tex(,710) let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
+pdata.tex(,711) @end tex
+pdata.tex(,719) @tex
+pdata.tex(,720) $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) >
\deg(x^\beta)$ or
+pdata.tex(,721) @end tex
+pdata.tex(,725) @tex
+pdata.tex(,726) \phantom{ $ x^\alpha < x^\beta \Leftrightarrow$}$
\deg(x^\alpha) =
+pdata.tex(,727) \deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 =
\beta_1,
+pdata.tex(,728) \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$
+pdata.tex(,729) @end tex
pdata.tex(,730) @item ws:
pdata.tex(,731) (general) weighted reverse lexicographical ordering:
pdata.tex(,732) @cindex general weighted reverse lexicographical ordering
pdata.tex(,733) @cindex local weighted reverse lexicographical ordering
pdata.tex(,734) @cindex ws, local ordering
pdata.tex(,735) @*
+pdata.tex(,739) @tex
+pdata.tex(,740) ${\tt ws}(w_1, \ldots, w_n),\; w_1$
+pdata.tex(,741) @end tex
pdata.tex(,742) a nonzero integer,
+pdata.tex(,746) @tex
+pdata.tex(,747) $w_2,\ldots,w_n$
+pdata.tex(,748) @end tex
pdata.tex(,749) any integer (including 0),
pdata.tex(,750) is defined as @code{ds}
pdata.tex(,751) but with
+pdata.tex(,755) @tex
+pdata.tex(,756) $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
+pdata.tex(,757) @end tex
pdata.tex(,758) @item Ws:
pdata.tex(,759) (general) weighted lexicographical ordering:
pdata.tex(,760) @cindex general weighted lexicographical ordering
pdata.tex(,761) @cindex local weighted lexicographical ordering
pdata.tex(,762) @cindex Ws, local ordering
pdata.tex(,763) @*
+pdata.tex(,767) @tex
+pdata.tex(,768) ${\tt Ws}(w_1, \ldots, w_n),\; w_1$
+pdata.tex(,769) @end tex
pdata.tex(,770) a nonzero integer,
+pdata.tex(,774) @tex
+pdata.tex(,775) $w_2,\ldots,w_n$
+pdata.tex(,776) @end tex
pdata.tex(,777) any integer (including 0),
pdata.tex(,778) is defined as @code{Ds}
pdata.tex(,779) but with
+pdata.tex(,783) @tex
+pdata.tex(,784) $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
+pdata.tex(,785) @end tex
pdata.tex(,786) @end table
pdata.tex(,787)
pdata.tex(,788) @c
--------------------------------------------------------------------------
@@ -21386,22 +22616,42 @@
pdata.tex(,791) @cindex Module orderings
pdata.tex(,792)
pdata.tex(,793) @sc{Singular} offers also orderings on the set of ``monomials''
+pdata.tex(,799) @tex
+pdata.tex(,800) $\{ x^a e_i \mid a \in N^n, 1 \leq i \leq r \}$ in Loc
$K[x]^r$ = Loc
+pdata.tex(,801) $K[x]e_1
+pdata.tex(,802) + \ldots +$Loc $K[x]e_r$, where $e_1, \ldots, e_r$ denote the
canonical
+pdata.tex(,803) generators of Loc $K[x]^r$, the r-fold direct sum of Loc
$K[x]$.
+pdata.tex(,804) (The function {\tt gen(i)} yields $e_i$).
+pdata.tex(,805) @end tex
pdata.tex(,806)
pdata.tex(,807) We have two possibilities: either to give priority to the
component of a
pdata.tex(,808) vector in
+pdata.tex(,812) @tex
+pdata.tex(,813) Loc $K[x]^r$
+pdata.tex(,814) @end tex
pdata.tex(,815) or (which is the default in @sc{Singular}) to give priority
pdata.tex(,816) to the coefficients.
pdata.tex(,817) The orderings @code{(<,c)} and @code{(<,C)} give priority to
the
pdata.tex(,818) coefficients; whereas
pdata.tex(,819) @code{(c,<)} and @code{(C,<)} give priority to the components.
pdata.tex(,820) @*Let < be any of the monomial orderings of
+pdata.tex(,821) @tex
+pdata.tex(,822) Loc $K[x]$
+pdata.tex(,823) @end tex
pdata.tex(,827) as above.
pdata.tex(,828)
pdata.tex(,829) @table @asis
pdata.tex(,830) @item (<,C):
pdata.tex(,831) @cindex C, module ordering
pdata.tex(,832) @cindex module ordering C
+pdata.tex(,840) @tex
+pdata.tex(,841) $<_m = (<,C)$ denotes the module ordering (giving priority to
the coefficients):
+pdata.tex(,842) @end tex
pdata.tex(,843) @*
+pdata.tex(,844) @tex
+pdata.tex(,845) \quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow
x^\alpha <
+pdata.tex(,846) x^\beta$ or ($x^\alpha = x^\beta $ and $ i < j$).
+pdata.tex(,847) @end tex
pdata.tex(,848)
pdata.tex(,849) @strong{Example:}
pdata.tex(,850) @smallexample
@@ -21416,6 +22666,13 @@
pdata.tex(,859) @end smallexample
pdata.tex(,860)
pdata.tex(,861) @item (C,<):
+pdata.tex(,870) @tex
+pdata.tex(,871) $<_m = (C, <)$ denotes the module ordering (giving priority to
the component):
+pdata.tex(,872) @end tex
+pdata.tex(,876) @tex
+pdata.tex(,877) \quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow i
< j$ or ($
+pdata.tex(,878) i = j $ and $ x^\alpha < x^\beta $).
+pdata.tex(,879) @end tex
pdata.tex(,880)
pdata.tex(,881) @strong{Example:}
pdata.tex(,882) @smallexample
@@ -21431,6 +22688,13 @@
pdata.tex(,892) @item (<,c):
pdata.tex(,893) @cindex c, module ordering
pdata.tex(,894) @cindex module ordering c
+pdata.tex(,902) @tex
+pdata.tex(,903) $<_m = (<,c)$ denotes the module ordering (giving priority to
the coefficients):
+pdata.tex(,904) @end tex
+pdata.tex(,908) @tex
+pdata.tex(,909) \quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow
x^\alpha <
+pdata.tex(,910) x^\beta$ or ($x^\alpha = x^\beta $ and $ i > j$).
+pdata.tex(,911) @end tex
pdata.tex(,912)
pdata.tex(,913) @strong{Example:}
pdata.tex(,914) @smallexample
@@ -21444,6 +22708,13 @@
pdata.tex(,922) @end smallexample
pdata.tex(,923)
pdata.tex(,924) @item (c,<):
+pdata.tex(,933) @tex
+pdata.tex(,934) $<_m = (c, <)$ denotes the module ordering (giving priority to
the component):
+pdata.tex(,935) @end tex
+pdata.tex(,939) @tex
+pdata.tex(,940) \quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow i
> j$ or ($
+pdata.tex(,941) i = j $ and $ x^\alpha < x^\beta $).
+pdata.tex(,942) @end tex
pdata.tex(,943)
pdata.tex(,944) @strong{Example:}
pdata.tex(,945) @smallexample
@@ -21457,7 +22728,14 @@
pdata.tex(,953) @end smallexample
pdata.tex(,954) @end table
pdata.tex(,955)
+pdata.tex(,960) @tex
+pdata.tex(,961) The output of a vector $v$ in $K[x]^r$ with components $v_1,
+pdata.tex(,962) \ldots, v_r$ has the format $v_1 * gen(1) + \ldots + v_r *
gen(r)$
+pdata.tex(,963) @end tex
pdata.tex(,964) (up to permutation) unless the ordering starts with @code{c}.
+pdata.tex(,968) @tex
+pdata.tex(,969) In this case a vector is written as $[v_1, \ldots, v_r]$.
+pdata.tex(,970) @end tex
pdata.tex(,971) In all cases @sc{Singular} can read input in both formats.
pdata.tex(,972)
pdata.tex(,973) @c
--------------------------------------------------------------------------
@@ -21468,25 +22746,152 @@
pdata.tex(,978) @cindex M, ordering
pdata.tex(,979)
pdata.tex(,980) Let
+pdata.tex(,981) @tex
+pdata.tex(,982) $M$
+pdata.tex(,983) @end tex
pdata.tex(,987) be an invertible
+pdata.tex(,988) @tex
+pdata.tex(,989) $(n \times n)$-matrix
+pdata.tex(,990) @end tex
pdata.tex(,994) with integer coefficients and
+pdata.tex(,998) @tex
+pdata.tex(,999) $M_1, \ldots, M_n$ the rows of $M$.
+pdata.tex(,1000) @end tex
pdata.tex(,1001)
pdata.tex(,1002) The M-ordering < is defined as follows:
pdata.tex(,1003) @*
+pdata.tex(,1008) @tex
+pdata.tex(,1009) \quad \quad $x^a < x^b \Leftrightarrow \exists\ 1 \leq i
\leq n :
+pdata.tex(,1010) M_1 a = \; M_1 b, \ldots, M_{i-1} a = \; M_{i-1} b$ and $M_i
a < \; M_i b$.
+pdata.tex(,1011) @end tex
pdata.tex(,1012)
pdata.tex(,1013) Thus,
+pdata.tex(,1018) @tex
+pdata.tex(,1019) $x^a < x^b$
+pdata.tex(,1020) if and only if $M a$ is smaller than $M b$
+pdata.tex(,1021) @end tex
pdata.tex(,1022) with respect to the lexicographical ordering.
pdata.tex(,1023)
pdata.tex(,1024) The following matrices represent (for 3 variables) the global
and
pdata.tex(,1025) local orderings defined above (note that the matrix is not
uniquely determined
pdata.tex(,1026) by the ordering):
pdata.tex(,1027)
+pdata.tex(,1072) @tex
+pdata.tex(,1073)
+pdata.tex(,1074) $\quad$ lp:
+pdata.tex(,1075) $\left(\matrix{
+pdata.tex(,1076) 1 & 0 & 0 \cr
+pdata.tex(,1077) 0 & 1 & 0 \cr
+pdata.tex(,1078) 0 & 0 & 1 \cr
+pdata.tex(,1079) }\right)$
+pdata.tex(,1080) \quad dp:
+pdata.tex(,1081) $\left(\matrix{
+pdata.tex(,1082) 1 & 1 & 1 \cr
+pdata.tex(,1083) 0 & 0 &-1 \cr
+pdata.tex(,1084) 0 &-1 & 0 \cr
+pdata.tex(,1085) }\right)$
+pdata.tex(,1086) \quad Dp:
+pdata.tex(,1087) $\left(\matrix{
+pdata.tex(,1088) 1 & 1 & 1 \cr
+pdata.tex(,1089) 1 & 0 & 0 \cr
+pdata.tex(,1090) 0 & 1 & 0 \cr
+pdata.tex(,1091) }\right)$
+pdata.tex(,1092)
+pdata.tex(,1093) $\quad$ wp(1,2,3):
+pdata.tex(,1094) $\left(\matrix{
+pdata.tex(,1095) 1 & 2 & 3 \cr
+pdata.tex(,1096) 0 & 0 &-1 \cr
+pdata.tex(,1097) 0 &-1 & 0 \cr
+pdata.tex(,1098) }\right)$
+pdata.tex(,1099) \quad Wp(1,2,3):
+pdata.tex(,1100) $\left(\matrix{
+pdata.tex(,1101) 1 & 2 & 3 \cr
+pdata.tex(,1102) 1 & 0 & 0 \cr
+pdata.tex(,1103) 0 & 1 & 0 \cr
+pdata.tex(,1104) }\right)$
+pdata.tex(,1105)
+pdata.tex(,1106) $\quad$ ls:
+pdata.tex(,1107) $\left(\matrix{
+pdata.tex(,1108) -1 & 0 & 0 \cr
+pdata.tex(,1109) 0 &-1 & 0 \cr
+pdata.tex(,1110) 0 & 0 &-1 \cr
+pdata.tex(,1111) }\right)$
+pdata.tex(,1112) \quad ds:
+pdata.tex(,1113) $\left(\matrix{
+pdata.tex(,1114) -1 &-1 &-1 \cr
+pdata.tex(,1115) 0 & 0 &-1 \cr
+pdata.tex(,1116) 0 &-1 & 0 \cr
+pdata.tex(,1117) }\right)$
+pdata.tex(,1118) \quad Ds:
+pdata.tex(,1119) $\left(\matrix{
+pdata.tex(,1120) -1 &-1 &-1 \cr
+pdata.tex(,1121) 1 & 0 & 0 \cr
+pdata.tex(,1122) 0 & 1 & 0 \cr
+pdata.tex(,1123) }\right)$
+pdata.tex(,1124)
+pdata.tex(,1125) $\quad$ ws(1,2,3):
+pdata.tex(,1126) $\left(\matrix{
+pdata.tex(,1127) -1 &-2 &-3 \cr
+pdata.tex(,1128) 0 & 0 &-1 \cr
+pdata.tex(,1129) 0 &-1 & 0 \cr
+pdata.tex(,1130) }\right)$
+pdata.tex(,1131) \quad Ws(1,2,3):
+pdata.tex(,1132) $\left(\matrix{
+pdata.tex(,1133) -1 &-2 &-3 \cr
+pdata.tex(,1134) 1 & 0 & 0 \cr
+pdata.tex(,1135) 0 & 1 & 0 \cr
+pdata.tex(,1136) }\right)$
+pdata.tex(,1137) @end tex
pdata.tex(,1138)
pdata.tex(,1139) Product orderings (see next section) represented by a matrix:
pdata.tex(,1140)
+pdata.tex(,1159) @tex
+pdata.tex(,1160) $\quad$ (dp(3), wp(1,2,3)):
+pdata.tex(,1161) $\left(\matrix{
+pdata.tex(,1162) 1& 1& 1& 0& 0& 0 \cr
+pdata.tex(,1163) 0& 0& -1& 0& 0& 0 \cr
+pdata.tex(,1164) 0& -1& 0& 0& 0& 0 \cr
+pdata.tex(,1165) 0& 0& 0& 1& 2& 3 \cr
+pdata.tex(,1166) 0& 0& 0& 0& 0& -1 \cr
+pdata.tex(,1167) 0& 0& 0& 0& -1& 0 \cr
+pdata.tex(,1168) }\right)$
+pdata.tex(,1169)
+pdata.tex(,1170) $\quad$ (Dp(3), ds(3)):
+pdata.tex(,1171) $\left(\matrix{
+pdata.tex(,1172) 1& 1& 1& 0& 0& 0 \cr
+pdata.tex(,1173) 1& 0& 0& 0& 0& 0 \cr
+pdata.tex(,1174) 0& 1& 0& 0& 0& 0 \cr
+pdata.tex(,1175) 0& 0& 0& -1& -1& -1 \cr
+pdata.tex(,1176) 0& 0& 0& 0& 0& -1 \cr
+pdata.tex(,1177) 0& 0& 0& 0& -1& 0 \cr
+pdata.tex(,1178) }\right)$
+pdata.tex(,1179) @end tex
pdata.tex(,1180)
pdata.tex(,1181) Orderings with extra weight vector (see below) represented by
a matrix:
pdata.tex(,1182)
+pdata.tex(,1203) @tex
+pdata.tex(,1204) $\quad$ (dp(3), a(1,2,3),dp(3)):
+pdata.tex(,1205) $\left(\matrix{
+pdata.tex(,1206) 1& 1& 1& 0& 0& 0 \cr
+pdata.tex(,1207) 0& 0& -1& 0& 0& 0 \cr
+pdata.tex(,1208) 0& -1& 0& 0& 0& 0 \cr
+pdata.tex(,1209) 0& 0& 0& 1& 2& 3 \cr
+pdata.tex(,1210) 0& 0& 0& 1& 1& 1 \cr
+pdata.tex(,1211) 0& 0& 0& 0& 0& -1 \cr
+pdata.tex(,1212) 0& 0& 0& 0& -1& 0 \cr
+pdata.tex(,1213) }\right)$
+pdata.tex(,1214)
+pdata.tex(,1215) $\quad$ (a(1,2,3,4,5),Dp(3), ds(3)):
+pdata.tex(,1216) $\left(\matrix{
+pdata.tex(,1217) 1& 2& 3& 4& 5& 0 \cr
+pdata.tex(,1218) 1& 1& 1& 0& 0& 0 \cr
+pdata.tex(,1219) 1& 0& 0& 0& 0& 0 \cr
+pdata.tex(,1220) 0& 1& 0& 0& 0& 0 \cr
+pdata.tex(,1221) 0& 0& 0& -1& -1& -1 \cr
+pdata.tex(,1222) 0& 0& 0& 0& 0 & -1 \cr
+pdata.tex(,1223) 0& 0& 0& 0& -1& 0 \cr
+pdata.tex(,1224) }\right)$
+pdata.tex(,1225) @end tex
pdata.tex(,1226)
pdata.tex(,1227) @address@hidden:
pdata.tex(,1228) @smallexample
@@ -21516,7 +22921,13 @@
pdata.tex(,1252) @end smallexample
pdata.tex(,1253)
pdata.tex(,1254) If the ring has
+pdata.tex(,1255) @tex
+pdata.tex(,1256) $n$
+pdata.tex(,1257) @end tex
pdata.tex(,1261) variables and the matrix contains less than
+pdata.tex(,1262) @tex
+pdata.tex(,1263) $n \times n$
+pdata.tex(,1264) @end tex
pdata.tex(,1268) entries an error message is given, if there are more entries,
pdata.tex(,1269) the last ones are ignored.
pdata.tex(,1270)
@@ -21537,6 +22948,9 @@
pdata.tex(,1285) @cindex orderings, product
pdata.tex(,1286)
pdata.tex(,1287) Let
+pdata.tex(,1292) @tex
+pdata.tex(,1293) $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_m)$
+pdata.tex(,1294) @end tex
pdata.tex(,1295) be two ordered sets of variables,
pdata.tex(,1317)
pdata.tex(,1318) Inductively one defines the product ordering of more than two
monomial
@@ -21558,9 +22972,24 @@
pdata.tex(,1334) @cindex a, ordering
pdata.tex(,1335) @cindex orderings, a
pdata.tex(,1336)
+pdata.tex(,1340) @tex
+pdata.tex(,1341) ${\tt a}(w_1, \ldots, w_n),\; $
+pdata.tex(,1342) @end tex
+pdata.tex(,1346) @tex
+pdata.tex(,1347) $w_1,\ldots,w_n$
+pdata.tex(,1348) @end tex
pdata.tex(,1349) any integers (including 0), defines
+pdata.tex(,1353) @tex
+pdata.tex(,1354) $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n$
+pdata.tex(,1355) @end tex
pdata.tex(,1356) and
pdata.tex(,1357) @*
+pdata.tex(,1361) @tex
+pdata.tex(,1362) $$\deg(x^\alpha) < \deg(x^\beta) \Rightarrow x^\alpha <
x^\beta,$$
+pdata.tex(,1363) @end tex
+pdata.tex(,1368) @tex
+pdata.tex(,1369) $$\deg(x^\alpha) > \deg(x^\beta) \Rightarrow x^\alpha >
x^\beta. $$
+pdata.tex(,1370) @end tex
pdata.tex(,1371) @*An extra weight vector does not define a monomial ordering
by itself:
pdata.tex(,1372) it can only be used in combination with other orderings
pdata.tex(,1373) to insert an extra line of weights into the ordering
@@ -21609,14 +23038,44 @@
math.tex(,42) @cindex Standard bases
math.tex(,43)
math.tex(,44) @subheading Definition
+math.tex(,45) @tex
+math.tex(,46) Let $R = \hbox{Loc}_< K[\underline{x}]$ and let $I$ be a
submodule of $R^r$.
+math.tex(,47) Note that for r=1 this means that $I$ is an ideal in $R$.
+math.tex(,48) Denote by $L(I)$ the submodule of $R^r$ generated by the leading
terms
+math.tex(,49) of elements of $I$, i.e. by $\left\{L(f) \mid f \in I\right\}$.
+math.tex(,50) Then $f_1, \ldots, f_s \in I$ is called a {\bf standard basis}
of $I$
+math.tex(,51) if $L(f_1), \ldots, L(f_s)$ generate $L(I)$.
+math.tex(,52) @end tex
math.tex(,60)
math.tex(,61) @subheading Properties
math.tex(,62) @table @asis
math.tex(,63) @item normal form:
math.tex(,64) @cindex Normal form
+math.tex(,65) @tex
+math.tex(,66) A function $\hbox{NF} : R^r \times \{G \mid G\ \hbox{ a standard
+math.tex(,67) basis}\} \to R^r, (p,G) \mapsto \hbox{NF}(p|G)$, is called a
{\bf normal
+math.tex(,68) form} if for any $p \in R^r$ and any standard basis $G$ the
following
+math.tex(,69) holds: if $\hbox{NF}(p|G) \not= 0$ then $L(g)$ does not divide
+math.tex(,70) $L(\hbox{NF}(p|G))$ for all $g \in G$.
+math.tex(,71)
+math.tex(,72) \noindent
+math.tex(,73) $\hbox{NF}(p|G)$ is called a {\bf normal form of} $p$ {\bf with
+math.tex(,74) respect to} $G$ (note that such a function is not unique).
+math.tex(,75) @end tex
math.tex(,84) @item ideal membership:
math.tex(,85) @cindex Ideal membership
+math.tex(,86) @tex
+math.tex(,87) For a standard basis $G$ of $I$ the following holds:
+math.tex(,88) $f \in I$ if and only if $\hbox{NF}(f,G) = 0$.
+math.tex(,89) @end tex
math.tex(,94) @item Hilbert function:
+math.tex(,95) @tex
+math.tex(,96) Let \hbox{$I \subseteq K[\underline{x}]^r$} be a homogeneous
module, then the Hilbert function
+math.tex(,97) $H_I$ of $I$ (see below)
+math.tex(,98) and the Hilbert function $H_{L(I)}$ of the leading module $L(I)$
+math.tex(,99) coincide, i.e.,
+math.tex(,100) $H_I=H_{L(I)}$.
+math.tex(,101) @end tex
math.tex(,106) @end table
math.tex(,107)
math.tex(,108) @c
---------------------------------------------------------------------------
@@ -21624,8 +23083,32 @@
math.tex(,110) @section Hilbert function
math.tex(,111) @cindex Hilbert function
math.tex(,112) @cindex Hilbert series
+math.tex(,113) @tex
+math.tex(,114) Let M $=\bigoplus_i M_i$ be a graded module over
$K[x_1,..,x_n]$ with
+math.tex(,115) respect to weights $(w_1,..w_n)$.
+math.tex(,116) The {\bf Hilbert function} of $M$, $H_M$, is defined (on the
integers) by
+math.tex(,117) $$H_M(k) :=dim_K M_k.$$
+math.tex(,118) The {\bf Hilbert-Poincare series} of $M$ is the power series
+math.tex(,119) $$\hbox{HP}_M(t) :=\sum_{i=-\infty}^\infty
+math.tex(,120) H_M(i)t^i=\sum_{i=-\infty}^\infty dim_K M_i \cdot t^i.$$
+math.tex(,121) It turns out that $\hbox{HP}_M(t)$ can be written in two useful
ways
+math.tex(,122) for weights $(1,..,1)$:
+math.tex(,123) $$\hbox{HP}_M(t)={Q(t)\over (1-t)^n}={P(t)\over
(1-t)^{dim(M)}}$$
+math.tex(,124) where $Q(t)$ and $P(t)$ are polynomials in ${\bf Z}[t]$.
+math.tex(,125) $Q(t)$ is called the {\bf first Hilbert series},
+math.tex(,126) and $P(t)$ the {\bf second Hilbert series}.
+math.tex(,127) If \hbox{$P(t)=\sum_{k=0}^N a_k t^k$}, and \hbox{$d = dim(M)$},
+math.tex(,128) then \hbox{$H_M(s)=\sum_{k=0}^N a_k$ ${d+s-k-1}\choose{d-1}$}
+math.tex(,129) (the {\bf Hilbert polynomial}) for $s \ge N$.
+math.tex(,130) @end tex
math.tex(,156) @*
math.tex(,157) @*
+math.tex(,158) @tex
+math.tex(,159) Generalizing these to quasihomogeneous modules we get
+math.tex(,160) $$\hbox{HP}_M(t)={Q(t)\over {\Pi_{i=1}^n(1-t^{w_i})}}$$
+math.tex(,161) where $Q(t)$ is a polynomial in ${\bf Z}[t]$.
+math.tex(,162) $Q(t)$ is called the {\bf first (weighted) Hilbert series} of M.
+math.tex(,163) @end tex
math.tex(,172)
math.tex(,173) @c
---------------------------------------------------------------------------
math.tex(,174) @node Syzygies and resolutions, Characteristic sets, Hilbert
function, Mathematical background
@@ -21633,11 +23116,22 @@
math.tex(,176) @cindex Syzygies and resolutions
math.tex(,177)
math.tex(,178) @subheading Syzygies
+math.tex(,179) @tex
+math.tex(,180) Let $R$ be a quotient of $\hbox{Loc}_< K[\underline{x}]$ and
let \hbox{$I=(g_1, ..., g_s)$} be a submodule of $R^r$.
+math.tex(,181) Then the {\bf module of syzygies} (or {\bf 1st syzygy module},
{\bf module of relations}) of $I$, syz($I$), is defined to be the kernel of the
map \hbox{$R^s \rightarrow R^r,\; \sum_{i=1}^s w_ie_i \mapsto \sum_{i=1}^s
w_ig_i$.}
+math.tex(,182) @end tex
math.tex(,192)
math.tex(,193) The @strong{k-th syzygy module} is defined inductively to be
the module
math.tex(,194) of syzygies of the
+math.tex(,195) @tex
+math.tex(,196) $(k-1)$-st
+math.tex(,197) @end tex
math.tex(,201) syzygy module.
math.tex(,202)
+math.tex(,203) @tex
+math.tex(,204) Note, that the syzygy modules of $I$ depend on a choice of
generators $g_1, ..., g_s$.
+math.tex(,205) But one can show that they depend on $I$ uniquely up to direct
summands.
+math.tex(,206) @end tex
math.tex(,211)
math.tex(,212) @table @code
math.tex(,213) @item @strong{Example:}
@@ -21655,10 +23149,26 @@
math.tex(,225) @end table
math.tex(,226)
math.tex(,227) @subheading Free resolutions
+math.tex(,228) @tex
+math.tex(,229) Let $I=(g_1,...,g_s)\subseteq R^r$ and $M= R^r/I$.
+math.tex(,230) A {\bf free resolution of $M$} is a long exact sequence
+math.tex(,231) $$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow}
F_1
+math.tex(,232) \buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow
M\longrightarrow
+math.tex(,233) 0,$$
+math.tex(,234) @end tex
math.tex(,242) @*where the columns of the matrix
+math.tex(,243) @tex
+math.tex(,244) $A_1$
+math.tex(,245) @end tex
math.tex(,249) generate
+math.tex(,253) @tex
+math.tex(,254) $I$
+math.tex(,255) @end tex
math.tex(,256) . Note, that resolutions need not to be finite (i.e., of
math.tex(,257) finite length). The Hilbert Syzygy Theorem states, that for
+math.tex(,258) @tex
+math.tex(,259) $R=\hbox{Loc}_< K[\underline{x}]$
+math.tex(,260) @end tex
math.tex(,264) there exists a ("minimal") resolution of length not exceeding
the number of
math.tex(,265) variables.
math.tex(,266)
@@ -21691,11 +23201,37 @@
math.tex(,293) @subheading Betti numbers and regularity
math.tex(,294) @cindex Betti number
math.tex(,295) @cindex regularity
+math.tex(,296) @tex
+math.tex(,297) Let $R$ be a graded ring (e.g., $R = \hbox{Loc}_<
K[\underline{x}]$) and
+math.tex(,298) let $I \subset R^r$ be a graded submodule. Let
+math.tex(,299) $$
+math.tex(,300) R^r = \bigoplus_a R\cdot e_{a,0} \buildrel A_1 \over
\longleftarrow
+math.tex(,301) \bigoplus_a R\cdot e_{a,1} \longleftarrow \ldots
\longleftarrow
+math.tex(,302) \bigoplus_a R\cdot e_{a,n} \longleftarrow 0
+math.tex(,303) $$
+math.tex(,304) be a minimal free resolution of $R^n/I$ considered with
homogeneous maps
+math.tex(,305) of degree 0. Then the {\bf graded Betti number} $b_{i,j}$ of
$R^r/I$ is
+math.tex(,306) the minimal number of generators $e_{a,j}$ in degree $i+j$ of
the $j$-th
+math.tex(,307) syzygy module of $R^r/I$ (i.e., the $(j-1)$-st syzygy module of
+math.tex(,308) $I$). Note, that by definition the $0$-th syzygy module of
$R^r/I$ is $R^r$
+math.tex(,309) and the 1st syzygy module of $R^r/I$ is $I$.
+math.tex(,310) @end tex
math.tex(,325)
math.tex(,326) The @strong{regularity} of
+math.tex(,330) @tex
+math.tex(,331) $I$
+math.tex(,332) @end tex
math.tex(,333) is the smallest integer
+math.tex(,337) @tex
+math.tex(,338) $s$
+math.tex(,339) @end tex
math.tex(,340)
math.tex(,341) such that
+math.tex(,342) @tex
+math.tex(,343) $$
+math.tex(,344) \hbox{deg}(e_{a,j}) \le s+j-1 \quad \hbox{for all $j$.}
+math.tex(,345) $$
+math.tex(,346) @end tex
math.tex(,352)
math.tex(,353) @table @code
math.tex(,354) @item @strong{Example:}
@@ -21728,6 +23264,43 @@
math.tex(,381) @section Characteristic sets
math.tex(,382) @cindex Characteristic sets
math.tex(,383)
+math.tex(,384) @tex
+math.tex(,385) Let $<$ be the lexicographical ordering on $R=K[x_1,...,x_n]$
with $x_1
+math.tex(,386) < ... < x_n$.
+math.tex(,387) For $f \in R$ let lvar($f$) (the leading variable of $f$) be
the largest
+math.tex(,388) variable in $f$,
+math.tex(,389) i.e., if $f=a_s(x_1,...,x_{k-1})x_k^s+...+a_0(x_1,...,x_{k-1})$
for some
+math.tex(,390) $k \leq n$ then lvar$(f)=x_k$.
+math.tex(,391)
+math.tex(,392) Moreover, let
+math.tex(,393) \hbox{ini}$(f):=a_s(x_1,...,x_{k-1})$. The pseudo remainder
+math.tex(,394) $r=\hbox{prem}(g,f)$ of $g$ with respect to $f$ is
+math.tex(,395) defined by the equality $\hbox{ini}(f)^a\cdot g = qf+r$ with
+math.tex(,396) $\hbox{deg}_{lvar(f)}(r)<\hbox{deg}_{lvar(f)}(f)$ and $a$
+math.tex(,397) minimal.
+math.tex(,398)
+math.tex(,399) A set $T=\{f_1,...,f_r\} \subset R$ is called triangular if
+math.tex(,400) $\hbox{lvar}(f_1)<...<\hbox{lvar}(f_r)$. Moreover, let $ U
\subset T $,
+math.tex(,401) then $(T,U)$ is called a triangular system, if $T$ is a
triangular set
+math.tex(,402) such that $\hbox{ini}(T)$ does not vanish on $V(T) \setminus
V(U)
+math.tex(,403) (=:V(T\setminus U))$.
+math.tex(,404)
+math.tex(,405) $T$ is called irreducible if for every $i$ there are no
+math.tex(,406) $d_i$,$f_i'$,$f_i''$ such that
+math.tex(,407) $$ \hbox{lvar}(d_i)<\hbox{lvar}(f_i) =
+math.tex(,408) \hbox{lvar}(f_i')=\hbox{lvar}(f_i''),$$
+math.tex(,409) $$ 0 \not\in \hbox{prem}(\{ d_i, \hbox{ini}(f_i'),
+math.tex(,410) \hbox{ini}(f_i'')\},\{ f_1,...,f_{i-1}\}),$$
+math.tex(,411) $$\hbox{prem}(d_if_i-f_i'f_i'',\{f_1,...,f_{i-1}\})=0.$$
+math.tex(,412) Furthermore, $(T,U)$ is called irreducible if $T$ is
irreducible.
+math.tex(,413)
+math.tex(,414) The main result on triangular sets is the following:
+math.tex(,415) let $G=\{g_1,...,g_s\} \subset R$ then there are irreducible
triangular sets $T_1,...,T_l$
+math.tex(,416) such that $V(G)=\bigcup_{i=1}^{l}(V(T_i\setminus I_i))$
+math.tex(,417) where $I_i=\{\hbox{ini}(f) \mid f \in T_i \}$. Such a set
+math.tex(,418) $\{T_1,...,T_l\}$ is called an {\bf irreducible characteristic
series} of
+math.tex(,419) the ideal $(G)$.
+math.tex(,420) @end tex
math.tex(,456)
math.tex(,457) @table @code
math.tex(,458) @item @strong{Example:}
@@ -21752,14 +23325,61 @@
math.tex(,477) @c tex and info versions of it. It end just before the
introducing text
math.tex(,478) @c to the first example.
math.tex(,479)
+math.tex(,480) @tex
+math.tex(,481) Let $f\colon(C^{n+1},0)\rightarrow(C,0)$ be a complex isolated
hypersurface singularity given by a polynomial with algebraic coefficients
which we also denote by $f$.
+math.tex(,482) Let $O=C[x_0,\ldots,x_n]_{(x_0,\ldots,x_n)}$ be the local ring
at the origin and $J_f$ the Jacobian ideal of $f$.
+math.tex(,483)
+math.tex(,484) A {\bf Milnor representative} of $f$ defines a differentiable
fibre bundle over the punctured disc with fibres of homotopy type of $\mu$
$n$-spheres.
+math.tex(,485) The $n$-th cohomology bundle is a flat vector bundle of
dimension $n$ and carries a natural flat connection with covariant derivative
$\partial_t$.
+math.tex(,486) The {\bf monodromy operator} is the action of a positively
oriented generator of the fundamental group of the puctured disc on the Milnor
fibre.
+math.tex(,487) Sections in the cohomology bundle of {\bf moderate growth} at
$0$ form a regular $D=C\{t\}[\partial_t]$-module $G$, the {\bf Gauss-Manin
connection}.
+math.tex(,488)
+math.tex(,489) By integrating along flat multivalued families of cycles, one
can consider fibrewise global holomorphic differential forms as elements of $G$.
+math.tex(,490) This factors through an inclusion of the {\bf Brieskorn
lattice} $H'':=\Omega^{n+1}_{C^{n+1},0}/df\wedge d\Omega^{n-1}_{C^{n+1},0}$ in
$G$.
+math.tex(,491)
+math.tex(,492) The $D$-module structure defines the {\bf V-filtration} $V$ on
$G$ by $V^\alpha:=\sum_{\beta\ge\alpha}C\{t\}ker(t\partial_t-\beta)^{n+1}$.
+math.tex(,493) The Brieskorn lattice defines the {\bf Hodge filtration} $F$ on
$G$ by $F_k=\partial_t^kH''$ which comes from the {\bf mixed Hodge structure}
on the Milnor fibre.
+math.tex(,494) Note that $F_{-1}=H'$.
+math.tex(,495)
+math.tex(,496) The induced V-filtration on the Brieskorn lattice determines
the {\bf singularity spectrum} $Sp$ by $Sp(\alpha):=\dim_CGr_V^\alpha Gr^F_0G$.
+math.tex(,497) The spectrum consists of $\mu$ rational numbers
$\alpha_1,\dots,\alpha_\mu$ such that $e^{2\pi i\alpha_1},\dots,e^{2\pi
i\alpha_\mu}$ are the eigenvalues of the monodromy.
+math.tex(,498) These {\bf spectral numbers} lie in the open interval $(-1,n)$,
symmetric about the midpoint $(n-1)/2$.
+math.tex(,499)
+math.tex(,500) The spectrum is constant under $\mu$-constant deformations and
has the following semicontinuity property:
+math.tex(,501) The number of spectral numbers in an interval $(a,a+1]$ of all
singularities of a small deformation of $f$ is greater or equal to that of f in
this interval.
+math.tex(,502) For semiquasihomogeneous singularities, this also holds for
intervals of the form $(a,a+1)$.
+math.tex(,503)
+math.tex(,504) Two given isolated singularities $f$ and $g$ determine two
spectra and from these spectra we get an integer.
+math.tex(,505) This integer is the maximal positive integer $k$ such that the
semicontinuity holds for the spectrum of $f$ and $k$ times the spectrum of $g$.
+math.tex(,506) These numbers give bounds for the maximal number of isolated
singularities of a specific type on a hypersurface $X\subset{P}^n$ of degree
$d$:
+math.tex(,507) such a hypersurface has a smooth hyperplane section, and the
complement is a small deformation of a cone over this hyperplane section.
+math.tex(,508) The cone itself being a $\mu$-constant deformation of
$x_0^d+\dots+x_n^d=0$, the singularities are bounded by the spectrum of
$x_0^d+\dots+x_n^d$.
+math.tex(,509)
+math.tex(,510) Using the library {\tt gaussman.lib} one can compute the {\bf
monodromy}, the V-filtration on $H''/H'$, and the spectrum.
+math.tex(,511) @end tex
math.tex(,512)
math.tex(,545)
math.tex(,546) Let us consider as an example
+math.tex(,550) @tex
+math.tex(,551) $f=x^5+x^2y^2+y^5$
+math.tex(,552) @end tex
math.tex(,553) .
math.tex(,554) First, we compute a matrix
+math.tex(,558) @tex
+math.tex(,559) $M$
+math.tex(,560) @end tex
math.tex(,561) such that
+math.tex(,562) @tex
+math.tex(,563) $\exp(2\pi iM)$
+math.tex(,564) @end tex
math.tex(,568) is a monodromy matrix of
+math.tex(,572) @tex
+math.tex(,573) $f$
+math.tex(,574) @end tex
math.tex(,575) and the Jordan normal form of
+math.tex(,579) @tex
+math.tex(,580) $M$
+math.tex(,581) @end tex
math.tex(,582) :
math.tex(,583) @smallexample
math.tex(,584) @c reused example Gauss-Manin_connection math.doc:505
@@ -21784,6 +23404,9 @@
math.tex(,603) @end smallexample
math.tex(,604)
math.tex(,605) Now, we compute the V-filtration on
+math.tex(,609) @tex
+math.tex(,610) $H''/H'$
+math.tex(,611) @end tex
math.tex(,612) and the spectrum:
math.tex(,613) @smallexample
math.tex(,614) @c reused example Gauss-Manin_connection_1 math.doc:517
@@ -21835,17 +23458,36 @@
math.tex(,660) @c end example Gauss-Manin_connection_1 math.doc:517
math.tex(,661) @end smallexample
math.tex(,662) Here @code{l[1]} contains the spectral numbers, @code{l[2]} the
corresponding multiplicities, @code{l[3]} a
+math.tex(,666) @tex
+math.tex(,667) $C$
+math.tex(,668) @end tex
math.tex(,669) -basis of the V-filtration on
+math.tex(,673) @tex
+math.tex(,674) $H''/H'$
+math.tex(,675) @end tex
math.tex(,676) in terms of the monomial basis of
+math.tex(,677) @tex
+math.tex(,678) $O/J_f\cong H''/H'$
+math.tex(,679) @end tex
math.tex(,683) in @code{l[4]}.
math.tex(,684)
+math.tex(,685) @tex
+math.tex(,686) If the principal part of $f$ is $C$-nondegenerate, one can
compute the spectrum using the library {\tt spectrum.lib}.
+math.tex(,687) In this case, the V-filtration on $H''$ coincides with the
Newton-filtration on $H''$ which allows to compute the spectrum more
efficiently.
+math.tex(,688) @end tex
math.tex(,689)
math.tex(,694)
math.tex(,695) Let us calculate one specific example, the maximal number
math.tex(,696) of triple points of type
+math.tex(,697) @tex
+math.tex(,698) $\tilde{E}_6$ on a surface $X\subset{P}^3$
+math.tex(,699) @end tex
math.tex(,703) of degree seven.
math.tex(,704) This calculation can be done over the rationals.
math.tex(,705) So choose a local ordering on
+math.tex(,709) @tex
+math.tex(,710) $Q[x,y,z]$
+math.tex(,711) @end tex
math.tex(,712) . Here we take the
math.tex(,713) negative degree lexicographical ordering which is denoted
math.tex(,714) @code{ds} in @sc{Singular}:
@@ -21880,21 +23522,44 @@
math.tex(,743) @end smallexample
math.tex(,744)
math.tex(,745) The command @code{spectrumnd(f)} computes the spectrum of
+math.tex(,749) @tex
+math.tex(,750) $f$
+math.tex(,751) @end tex
math.tex(,752) and
math.tex(,753) returns a list with six entries:
math.tex(,754) The Milnor number
+math.tex(,755) @tex
+math.tex(,756) $\mu(f)$, the geometric genus $p_g(f)$
+math.tex(,757) @end tex
math.tex(,761) and the number of different spectrum numbers.
math.tex(,762) The other three entries are of type @code{intvec}.
math.tex(,763) They contain the numerators, denominators and
math.tex(,764) multiplicities of the spectrum numbers. So
+math.tex(,765) @tex
+math.tex(,766) $x^7+y^7+z^7=0$
+math.tex(,767) @end tex
math.tex(,771) has Milnor number 216 and geometrical
math.tex(,772) genus 35. Its spectrum consists of the 16 different rationals
math.tex(,773) @*
+math.tex(,774) @tex
+math.tex(,775) ${3 \over 7}, {4 \over 7}, {5 \over 7}, {6 \over 7}, {1 \over
1},
+math.tex(,776) {8 \over 7}, {9 \over 7}, {10 \over 7}, {11 \over 7}, {12 \over
7},
+math.tex(,777) {13 \over 7}, {2 \over 1}, {15 \over 7}, {16 \over 7}, {17
\over 7},
+math.tex(,778) {18 \over 7}$
+math.tex(,779) @end tex
math.tex(,784) @*appearing with multiplicities
math.tex(,785) @*1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1.
math.tex(,786)
+math.tex(,787) @tex
+math.tex(,788) The singularities of type $\tilde{E}_6$ form a
+math.tex(,789) $\mu$-constant one parameter family given by
+math.tex(,790) $x^3+y^3+z^3+\lambda xyz=0,\quad \lambda^3\neq-27$.
+math.tex(,791) @end tex
math.tex(,797) Therefore they have all the same spectrum, which we compute
math.tex(,798) for
+math.tex(,799) @tex
+math.tex(,800) $x^3+y^3+z^3$.
+math.tex(,801) @end tex
math.tex(,805)
math.tex(,806) @smallexample
math.tex(,807) poly g=x^3+y^3+z^3;
@@ -21920,6 +23585,9 @@
math.tex(,827) @end smallexample
math.tex(,828)
math.tex(,829) This tells us that there are at most 18 singularities of type
+math.tex(,830) @tex
+math.tex(,831) $\tilde{E}_6$ on a septic in $P^3$. But $x^7+y^7+z^7$
+math.tex(,832) @end tex
math.tex(,836) is semiquasihomogeneous (sqh), so we can also apply the stronger
math.tex(,837) form of semicontinuity:
math.tex(,838)
@@ -21929,12 +23597,21 @@
math.tex(,842) @end smallexample
math.tex(,843)
math.tex(,844) So in fact a septic has at most 17 triple points of type
+math.tex(,845) @tex
+math.tex(,846) $\tilde{E}_6$.
+math.tex(,847) @end tex
math.tex(,851)
math.tex(,852) Note that @code{spectrumnd(f)} works only if
+math.tex(,856) @tex
+math.tex(,857) $f$
+math.tex(,858) @end tex
math.tex(,859) has nondegenerate
math.tex(,860) principal part. In fact @code{spectrumnd} will detect a
degenerate
math.tex(,861) principal part in many cases and print out an error message.
math.tex(,862) However if it is known in advance that
+math.tex(,866) @tex
+math.tex(,867) $f$
+math.tex(,868) @end tex
math.tex(,869) has nondegenerate
math.tex(,870) principal part, then the spectrum may be computed much faster
math.tex(,871) using @code{spectrumnd(f,1)}.
@@ -21958,10 +23635,33 @@
ti_ip.tex(,12) @comment DO NOT EDIT DIRECTLY, BUT EDIT ti_ip.doc INSTEAD
ti_ip.tex(,13) @cindex ideal, toric
ti_ip.tex(,14)
+ti_ip.tex(,15) @tex
+ti_ip.tex(,16) Let $A$ denote an $m\times n$ matrix with integral
coefficients. For $u
+ti_ip.tex(,17) \in Z\!\!\! Z^n$, we define $u^+,u^-$ to be the uniquely
determined
+ti_ip.tex(,18) vectors with nonnegative coefficients and disjoint support
(i.e.,
+ti_ip.tex(,19) $u_i^+=0$ or $u_i^-=0$ for each component $i$) such that
+ti_ip.tex(,20) $u=u^+-u^-$. For $u\geq 0$ component-wise, let $x^u$ denote the
monomial
+ti_ip.tex(,21) $x_1^{u_1}\cdot\ldots\cdot x_n^{u_n}\in K[x_1,\ldots,x_n]$.
+ti_ip.tex(,22)
+ti_ip.tex(,23) The ideal
+ti_ip.tex(,24) $$ I_A:=<x^{u^+}-x^{u^-} | u\in\ker(A)\cap Z\!\!\! Z^n>\ \subset
+ti_ip.tex(,25) K[x_1,\ldots,x_n] $$
+ti_ip.tex(,26) is called a \bf toric ideal. \rm
+ti_ip.tex(,27)
+ti_ip.tex(,28) The first problem in computing toric ideals is to find a finite
+ti_ip.tex(,29) generating set: Let $v_1,\ldots,v_r$ be a lattice basis of
$\ker(A)\cap
+ti_ip.tex(,30) Z\!\!\! Z^n$ (i.e, a basis of the $Z\!\!\! Z$-module). Then
+ti_ip.tex(,31) $$ I_A:=I:(x_1\cdot\ldots\cdot x_n)^\infty $$
+ti_ip.tex(,32) where
+ti_ip.tex(,33) $$ I=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
+ti_ip.tex(,34) @end tex
ti_ip.tex(,35)
ti_ip.tex(,61)
ti_ip.tex(,62) The required lattice basis can be computed using the
LLL-algorithm (@pxref{[Coh93]}). For the computation of the saturation, there
are various
ti_ip.tex(,63) possibilities described in the
+ti_ip.tex(,64) @tex
+ti_ip.tex(,65) section Algorithms.
+ti_ip.tex(,66) @end tex
ti_ip.tex(,70)
ti_ip.tex(,71) @menu
ti_ip.tex(,72) * Algorithms:: Various algorithms for computing
toric ideals.
@@ -21989,6 +23689,23 @@
ti_ip.tex(,94)
ti_ip.tex(,95)
ti_ip.tex(,96) The algorithm of Conti and Traverso (@pxref{[CoTr91]})
+ti_ip.tex(,97) @tex
+ti_ip.tex(,98) computes $I_A$ via the
+ti_ip.tex(,99) extended matrix $B=(I_m|A)$,
+ti_ip.tex(,100) where $I_m$ is the $m\times m$ unity matrix. A lattice basis
of $B$ is
+ti_ip.tex(,101) given by the set of vectors $(a^j,-e_j)\in Z\!\!\! Z^{m+n}$,
where $a^j$
+ti_ip.tex(,102) is the $j$-th row of $A$ and $e_j$ the $j$-th coordinate
vector. We
+ti_ip.tex(,103) look at the ideal in $K[y_1,\ldots,y_m,x_1,\ldots,x_n]$
corresponding to
+ti_ip.tex(,104) these vectors, namely
+ti_ip.tex(,105) $$ I_1=<y^{a_j^+}- x_j y^{a_j^-} | j=1,\ldots, n>.$$
+ti_ip.tex(,106) We introduce a further variable $t$ and adjoin the binomial
$t\cdot
+ti_ip.tex(,107) y_1\cdot\ldots\cdot y_m -1$ to the generating set of $I_1$,
obtaining
+ti_ip.tex(,108) an ideal $I_2$ in the polynomial ring $K[t,
+ti_ip.tex(,109) y_1,\ldots,y_m,x_1,\ldots,x_n]$. $I_2$ is saturated w.r.t. all
+ti_ip.tex(,110) variables because all variables are invertible modulo $I_2$.
Now $I_A$
+ti_ip.tex(,111) can be computed from $I_2$ by eliminating the variables
+ti_ip.tex(,112) $t,y_1,\ldots,y_m$.
+ti_ip.tex(,113) @end tex
ti_ip.tex(,131)
ti_ip.tex(,132) Because of the big number of auxiliary variables needed to
compute a
ti_ip.tex(,133) toric ideal, this algorithm is rather slow in practice.
However, it has
@@ -22003,6 +23720,16 @@
ti_ip.tex(,142)
ti_ip.tex(,143)
ti_ip.tex(,144) The algorithm of Pottier (@pxref{[Pot94]}) starts by computing
a lattice
+ti_ip.tex(,145) @tex
+ti_ip.tex(,146) basis $v_1,\ldots,v_r$ for the integer kernel of $A$ using the
+ti_ip.tex(,147) LLL-algorithm. The ideal corresponding to the lattice basis
vectors
+ti_ip.tex(,148) $$ I_1=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
+ti_ip.tex(,149) is saturated -- as in the algorithm of Conti and Traverso -- by
+ti_ip.tex(,150) inversion of all variables: One adds an auxiliary variable $t$
and the
+ti_ip.tex(,151) generator $t\cdot x_1\cdot\ldots\cdot x_n -1$ to obtain an
ideal $I_2$
+ti_ip.tex(,152) in $K[t,x_1,\ldots,x_n]$ from which one computes $I_A$ by
elimination of
+ti_ip.tex(,153) $t$.
+ti_ip.tex(,154) @end tex
ti_ip.tex(,167)
ti_ip.tex(,168)
ti_ip.tex(,169) @node Hosten and Sturmfels, Di Biase and Urbanke, Pottier,
Algorithms
@@ -22013,6 +23740,33 @@
ti_ip.tex(,174)
ti_ip.tex(,175)
ti_ip.tex(,176) The algorithm of Hosten and Sturmfels (@pxref{[HoSt95]})
allows to
+ti_ip.tex(,177) @tex
+ti_ip.tex(,178) compute $I_A$ without any auxiliary variables, provided that
$A$ contains a vector $w$
+ti_ip.tex(,179) with positive coefficients in its row space. This is a real
restriction,
+ti_ip.tex(,180) i.e., the algorithm will not necessarily work in the general
case.
+ti_ip.tex(,181)
+ti_ip.tex(,182) A lattice basis $v_1,\ldots,v_r$ is again computed via the
+ti_ip.tex(,183) LLL-algorithm. The saturation step is performed in the
following way:
+ti_ip.tex(,184) First note that $w$ induces a positive grading w.r.t. which
the ideal
+ti_ip.tex(,185) $$ I=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
+ti_ip.tex(,186) corresponding to our lattice basis is homogeneous. We use the
following
+ti_ip.tex(,187) lemma:
+ti_ip.tex(,188)
+ti_ip.tex(,189) Let $I$ be a homogeneous ideal w.r.t. the weighted reverse
+ti_ip.tex(,190) lexicographical ordering with weight vector $w$ and variable
order $x_1
+ti_ip.tex(,191) > x_2 > \ldots > x_n$. Let $G$ denote a Groebner basis of $I$
w.r.t. to
+ti_ip.tex(,192) this ordering. Then a Groebner basis of $(I:x_n^\infty)$ is
obtained by
+ti_ip.tex(,193) dividing each element of $G$ by the highest possible power of
$x_n$.
+ti_ip.tex(,194)
+ti_ip.tex(,195) From this fact, we can successively compute
+ti_ip.tex(,196) $$ I_A= I:(x_1\cdot\ldots\cdot x_n)^\infty
+ti_ip.tex(,197) =(((I:x_1^\infty):x_2^\infty):\ldots :x_n^\infty); $$
+ti_ip.tex(,198) in the $i$-th step we take $x_i$ as the cheapest variable and
apply the
+ti_ip.tex(,199) lemma with $x_i$ instead of $x_n$.
+ti_ip.tex(,200)
+ti_ip.tex(,201) This procedure involves $n$ Groebner basis computations.
Actually, this
+ti_ip.tex(,202) number can be reduced to at most $n/2$
+ti_ip.tex(,203) @end tex
ti_ip.tex(,235) (@pxref{[HoSh98]}), and the single
ti_ip.tex(,236) computations -- except from the first one -- show to be easy
and fast in
ti_ip.tex(,237) practice.
@@ -22025,6 +23779,38 @@
ti_ip.tex(,244)
ti_ip.tex(,245) Like the algorithm of Hosten and Sturmfels, the algorithm of
Di Biase
ti_ip.tex(,246) and Urbanke (@pxref{[DBUr95]}) performs up
+ti_ip.tex(,247) @tex
+ti_ip.tex(,248) to $n/2$ Groebner basis
+ti_ip.tex(,249) computations. It needs no auxiliary variables, but a
supplementary
+ti_ip.tex(,250) precondition; namely, the existence of a vector without zero
components
+ti_ip.tex(,251) in the kernel of $A$.
+ti_ip.tex(,252)
+ti_ip.tex(,253) The main idea comes from the following observation:
+ti_ip.tex(,254)
+ti_ip.tex(,255) Let $B$ be an integer matrix, $u_1,\ldots,u_r$ a lattice basis
of the
+ti_ip.tex(,256) integer kernel of $B$. Assume that all components of $u_1$ are
+ti_ip.tex(,257) positive. Then
+ti_ip.tex(,258) $$ I_B=<x^{u_i^+}-x^{u_i^-}|i=1,\ldots,r>, $$
+ti_ip.tex(,259) i.e., the ideal on the right is already saturated w.r.t. all
variables.
+ti_ip.tex(,260)
+ti_ip.tex(,261) The algorithm starts by finding a lattice basis
$v_1,\ldots,v_r$ of the
+ti_ip.tex(,262) kernel of $A$ such that $v_1$ has no zero component. Let
+ti_ip.tex(,263) $\{i_1,\ldots,i_l\}$ be the set of indices $i$ with
+ti_ip.tex(,264) $v_{1,i}<0$. Multiplying the components $i_1,\ldots,i_l$ of
+ti_ip.tex(,265) $v_1,\ldots,v_r$ and the columns $i_1,\ldots,i_l$ of $A$ by
$-1$ yields
+ti_ip.tex(,266) a matrix $B$ and a lattice basis $u_1,\ldots,u_r$ of the
kernel of $B$
+ti_ip.tex(,267) that fulfill the assumption of the observation above. We are
then able
+ti_ip.tex(,268) to compute a generating set of $I_A$ by applying the following
+ti_ip.tex(,269) ``variable flip'' successively to $i=i_1,\ldots,i_l$:
+ti_ip.tex(,270)
+ti_ip.tex(,271) Let $>$ be an elimination ordering for $x_i$. Let $A_i$ be the
matrix
+ti_ip.tex(,272) obtained by multiplying the $i$-th column of $A$ with $-1$. Let
+ti_ip.tex(,273) $$\{x_i^{r_j} x^{a_j} - x^{b_j} | j\in J \}$$
+ti_ip.tex(,274) be a Groebner basis of $I_{A_i}$ w.r.t. $>$ (where $x_i$ is
neither
+ti_ip.tex(,275) involved in $x^{a_j}$ nor in $x^{b_j}$). Then
+ti_ip.tex(,276) $$\{x^{a_j} - x_i^{r_j} x^{b_j} | j\in J \}$$
+ti_ip.tex(,277) is a generating set for $I_A$.
+ti_ip.tex(,278) @end tex
ti_ip.tex(,316)
ti_ip.tex(,317) @node Bigatti and La Scala and Robbiano, , Di Biase and
Urbanke, Algorithms
ti_ip.tex(,318)
@@ -22035,6 +23821,12 @@
ti_ip.tex(,323) The algorithm of Bigatti, La Scala and Robbiano
(@pxref{[BLR98]}) combines the ideas of
ti_ip.tex(,324) the algorithms of Pottier and of Hosten and Sturmfels. The
ti_ip.tex(,325) computations are performed on a graded ideal with one auxiliary
+ti_ip.tex(,326) @tex
+ti_ip.tex(,327) variable $u$ and one supplementary generator
$x_1\cdot\ldots\cdot x_n -
+ti_ip.tex(,328) u$ (instead of the generator $t\cdot x_1\cdot\ldots\cdot x_n
-1$ in
+ti_ip.tex(,329) the algorithm of Pottier). The algorithm uses a quite unusual
technique to
+ti_ip.tex(,330) get rid of the variable $u$ again.
+ti_ip.tex(,331) @end tex
ti_ip.tex(,338)
ti_ip.tex(,339) There is another algorithm of the authors which tries to
parallelize
ti_ip.tex(,340) the computations (but which is not implemented in this
library).
@@ -22059,6 +23851,25 @@
ti_ip.tex(,359) @subsection Integer programming
ti_ip.tex(,360) @cindex integer programming
ti_ip.tex(,361)
+ti_ip.tex(,362) @tex
+ti_ip.tex(,363) Let $A$ be an $m\times n$ matrix with integral coefficients,
$b\in
+ti_ip.tex(,364) Z\!\!\! Z^m$ and $c\in Z\!\!\! Z^n$. The problem
+ti_ip.tex(,365) $$ \min\{c^T x | x\in Z\!\!\! Z^n, Ax=b, x\geq 0\hbox{
+ti_ip.tex(,366) component-wise}\} $$
+ti_ip.tex(,367) is called an instance of the \bf integer programming problem
\rm or
+ti_ip.tex(,368) \bf IP problem. \rm
+ti_ip.tex(,369)
+ti_ip.tex(,370) The IP problem is very hard; namely, it is NP-complete.
+ti_ip.tex(,371)
+ti_ip.tex(,372) For the following discussion let $c\geq 0$ (component-wise). We
+ti_ip.tex(,373) consider $c$ as a weight vector; because of its
non-negativity, $c$ can
+ti_ip.tex(,374) be refined into a monomial ordering $>_c$. It turns out that
we can
+ti_ip.tex(,375) solve such an IP instance with the help of toric ideals:
+ti_ip.tex(,376)
+ti_ip.tex(,377) First we assume that an initial solution $v$ (i.e., $v\in
Z\!\!\!
+ti_ip.tex(,378) Z^n, v\geq 0, Av=b$) is already known. We obtain the optimal
solution
+ti_ip.tex(,379) $v_0$ (i.e., with $c^T v_0$ minimal) by the following
procedure:
+ti_ip.tex(,380) @end tex
ti_ip.tex(,381) @c \begin{itemize}
ti_ip.tex(,382) @c \item (1) Compute the toric ideal $I_A$ using one of the
algorithms in the
ti_ip.tex(,383) @c previous section.
@@ -22073,11 +23884,23 @@
ti_ip.tex(,412) @itemize @bullet
ti_ip.tex(,413) @item (1) Compute the toric ideal I(A) using one of the
algorithms in the previous section.
ti_ip.tex(,414) @item (2) Compute the reduced Groebner basis G(c) of I(A)
w.r.t.@:
+ti_ip.tex(,418) @tex
+ti_ip.tex(,419) $>_c$
+ti_ip.tex(,420) @end tex
ti_ip.tex(,421) .
ti_ip.tex(,422) @item (3) Reduce
+ti_ip.tex(,426) @tex
+ti_ip.tex(,427) $x^v$
+ti_ip.tex(,428) @end tex
ti_ip.tex(,429) modulo G(c) using the Hironaka division algorithm.
ti_ip.tex(,430) If the result of this reduction is
+ti_ip.tex(,434) @tex
+ti_ip.tex(,435) $x^(v_0)$
+ti_ip.tex(,436) @end tex
ti_ip.tex(,437) , then
+ti_ip.tex(,441) @tex
+ti_ip.tex(,442) $v_0$
+ti_ip.tex(,443) @end tex
ti_ip.tex(,444) is an optimal
ti_ip.tex(,445) solution of the given instance.
ti_ip.tex(,446) @end itemize
@@ -22097,6 +23920,9 @@
ti_ip.tex(,460) methods seem to be faster in general than the methods using
toric
ti_ip.tex(,461) ideals. But the latter have one great advantage: If one wants
to solve
ti_ip.tex(,462) various instances that differ only by the vector
+ti_ip.tex(,466) @tex
+ti_ip.tex(,467) $b$
+ti_ip.tex(,468) @end tex
ti_ip.tex(,469) , one has to
ti_ip.tex(,470) perform steps (1) and (2) above only once. As the running time
of step (3)
ti_ip.tex(,471) is very short, solving all the instances is not much harder
than
@@ -22239,6 +24065,9 @@
math.tex(,959) Symbolic Computation
math.tex(,960)
math.tex(,961) @item
+math.tex(,962) @tex
+math.tex(,963) Faug\`ere,
+math.tex(,964) @end tex
math.tex(,968) J. C.; Gianni, P.; Lazard, D.; Mora, T.: Efficient computation
math.tex(,969) of zero-dimensional
math.tex(,970) Gr@"obner bases by change of ordering. Journal of Symbolic
Computation, 1989
@@ -47789,6 +49618,9 @@
brnoeth_lib.tex(,685)
brnoeth_lib.tex(,686) @item @strong{Warnings:}
brnoeth_lib.tex(,687) G should satisfy
+brnoeth_lib.tex(,691) @tex
+brnoeth_lib.tex(,692) $ 2*genus-2 < deg(G) < size(D) $
+brnoeth_lib.tex(,693) @end tex
brnoeth_lib.tex(,694) , which is
brnoeth_lib.tex(,695) not checked by the algorithm.
brnoeth_lib.tex(,696) @*G and D should have disjoint supports (checked by the
algorithm).
@@ -47853,10 +49685,16 @@
brnoeth_lib.tex(,771) for more details)address@hidden
brnoeth_lib.tex(,772) The code computes the residues of a vector space basis of
brnoeth_lib.tex(,773)
+brnoeth_lib.tex(,777) @tex
+brnoeth_lib.tex(,778) $\Omega(G-D)$
+brnoeth_lib.tex(,779) @end tex
brnoeth_lib.tex(,780) at the rational places given by D.
brnoeth_lib.tex(,781)
brnoeth_lib.tex(,782) @item @strong{Warnings:}
brnoeth_lib.tex(,783) G should satisfy
+brnoeth_lib.tex(,787) @tex
+brnoeth_lib.tex(,788) $ 2*genus-2 < deg(G) < size(D) $
+brnoeth_lib.tex(,789) @end tex
brnoeth_lib.tex(,790) , which is
brnoeth_lib.tex(,791) not checked by the algorithm.
brnoeth_lib.tex(,792) @*G and D should have disjoint supports (checked by the
algorithm).
@@ -47913,8 +49751,14 @@
brnoeth_lib.tex(,859) E[2] ... E[n+2]: matrices used in the procedure
decodeSV
brnoeth_lib.tex(,860) E[n+3]: intvec with
brnoeth_lib.tex(,861) E[n+3][1]: correction capacity
+brnoeth_lib.tex(,865) @tex
+brnoeth_lib.tex(,866) $epsilon$
+brnoeth_lib.tex(,867) @end tex
brnoeth_lib.tex(,868) of the algorithm
brnoeth_lib.tex(,869) E[n+3][2]: designed Goppa distance
+brnoeth_lib.tex(,873) @tex
+brnoeth_lib.tex(,874) $delta$
+brnoeth_lib.tex(,875) @end tex
brnoeth_lib.tex(,876) of the current AG code
brnoeth_lib.tex(,877) @end format
brnoeth_lib.tex(,878)
@@ -47930,6 +49774,9 @@
brnoeth_lib.tex(,888) The current AG code is
@code{AGcode_Omega(G,D,EC)address@hidden
brnoeth_lib.tex(,889) If you know the exact minimum distance d and you want to
use it in
brnoeth_lib.tex(,890) @code{decodeSV} instead of
+brnoeth_lib.tex(,894) @tex
+brnoeth_lib.tex(,895) $delta$
+brnoeth_lib.tex(,896) @end tex
brnoeth_lib.tex(,897) , you can change the value
brnoeth_lib.tex(,898) of E[n+3][2] to d before applying decodeSV.
brnoeth_lib.tex(,899) @*If you have a systematic encoding for the current code
and want to
@@ -47940,10 +49787,19 @@
brnoeth_lib.tex(,904) @item @strong{Warnings:}
brnoeth_lib.tex(,905) F must be a divisor with support disjoint from the
support of D and
brnoeth_lib.tex(,906) with degree
+brnoeth_lib.tex(,910) @tex
+brnoeth_lib.tex(,911) $epsilon + genus$
+brnoeth_lib.tex(,912) @end tex
brnoeth_lib.tex(,913) , where
brnoeth_lib.tex(,914)
+brnoeth_lib.tex(,918) @tex
+brnoeth_lib.tex(,919) $epsilon:=[(deg(G)-3*genus+1)/2]$
+brnoeth_lib.tex(,920) @end tex
brnoeth_lib.tex(,921) address@hidden
brnoeth_lib.tex(,922) G should satisfy
+brnoeth_lib.tex(,926) @tex
+brnoeth_lib.tex(,927) $ 2*genus-2 < deg(G) < size(D) $
+brnoeth_lib.tex(,928) @end tex
brnoeth_lib.tex(,929) , which is
brnoeth_lib.tex(,930) not checked by the algorithm.
brnoeth_lib.tex(,931) @*G and D should also have disjoint supports (checked by
the
Index: test/singular_manual/res/texi_singular/singular.passtexi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/singular_manual/res/texi_singular/singular.passtexi,v
retrieving revision 1.3
retrieving revision 1.4
diff -u -b -r1.3 -r1.4
--- test/singular_manual/res/texi_singular/singular.passtexi 29 Aug 2008
15:06:51 -0000 1.3
+++ test/singular_manual/res/texi_singular/singular.passtexi 9 Jan 2009
21:20:58 -0000 1.4
@@ -332,6 +332,9 @@
start.tex(,78) @sc{Singular}'s development started in 1984 with an
implementation of
start.tex(,79) Mora's Tangent Cone algorithm in Modula-2 on an Atari computer
(K.P.
start.tex(,80) Neuendorf, G. Pfister,
+start.tex(,84) @tex
+start.tex(,85) H.\ Sch\"onemann; Humboldt-Universit\"at
+start.tex(,86) @end tex
start.tex(,87) zu Berlin). The need for a new system arose from the
investigation of
start.tex(,88) mathematical problems coming from singularity theory which none
of the
start.tex(,89) existing systems was able to compute.
@@ -574,6 +577,9 @@
start.tex(,351) @noindent This shows the text of @ref{intmat}, in the printed
manual.
start.tex(,356)
start.tex(,357) Next, we define a
+start.tex(,358) @tex
+start.tex(,359) $3 \times 3$
+start.tex(,360) @end tex
start.tex(,364) matrix of integers and initialize it with some values, row by
row
start.tex(,365) from left to right:
start.tex(,366)
@@ -650,6 +656,9 @@
start.tex(,442) ring variables, and the third part determines the monomial
ordering to
start.tex(,443) be used. So the example above declares a polynomial ring
called @code{r}
start.tex(,444) with a ground field of characteristic
+start.tex(,448) @tex
+start.tex(,449) $0$
+start.tex(,450) @end tex
start.tex(,451) (i.e., the rational
start.tex(,452) numbers) and ring variables called @code{x}, @code{y}, and
@code{z}. The
start.tex(,453) @code{dp} at the end means that the degree reverse
lexicographical
@@ -672,7 +681,13 @@
start.tex(,470)
start.tex(,471) @item ring r4=(0,a),(mu,nu),lp;
start.tex(,472) transcendental extension of
+start.tex(,476) @tex
+start.tex(,477) $Q$
+start.tex(,478) @end tex
start.tex(,479) by
+start.tex(,483) @tex
+start.tex(,484) $a$
+start.tex(,485) @end tex
start.tex(,486) , variable names
start.tex(,487) @code{mu} and @code{nu}.
start.tex(,488)
@@ -701,6 +716,9 @@
start.tex(,511) @c
start.tex(,512) Typing the name of a ring prints its definition. The example
below
start.tex(,513) shows that the default ring in @sc{Singular} is
+start.tex(,517) @tex
+start.tex(,518) $Z/32003[x,y,z]$
+start.tex(,519) @end tex
start.tex(,520)
start.tex(,521) with degree reverse lexicographical ordering:
start.tex(,522)
@@ -730,6 +748,9 @@
start.tex(,551) @end smallexample
start.tex(,552)
start.tex(,553) Once a ring is active, we can define polynomials. A monomial,
say
+start.tex(,554) @tex
+start.tex(,555) $x^3$
+start.tex(,556) @end tex
start.tex(,560) may be entered in two ways: either using the power operator
@code{^},
start.tex(,561) saying @code{x^3}, or in short-hand notation without operator,
saying
start.tex(,562) @code{x3}. Note that the short-hand notation is forbidden if
the name
@@ -848,6 +869,9 @@
start.tex(,677) @end smallexample
start.tex(,678)
start.tex(,679) @noindent gives the desired vector space dimension
+start.tex(,680) @tex
+start.tex(,681) $K[x,y,z]/\hbox{\rm jacob}(f)$.
+start.tex(,682) @end tex
start.tex(,686) As in @sc{Singular} the functions may take the input directly
from
start.tex(,687) earlier calculations, the whole sequence of commands may be
written
start.tex(,688) in one single statement.
@@ -1021,6 +1045,9 @@
start.tex(,876)
start.tex(,877) This shows that @code{f} has outside the origin in affine
3-space
start.tex(,878) singularities with local Milnor number adding up to
+start.tex(,879) @tex
+start.tex(,880) $12-4=8$.
+start.tex(,881) @end tex
start.tex(,885) Using global and local orderings as above is a convenient way
to check
start.tex(,886) whether a variety has singularities outside the origin.
start.tex(,887)
@@ -1067,6 +1094,9 @@
start.tex(,928) The algorithm of the standard basis computations may be
start.tex(,929) affected by the command @code{option}. For example, a reduced
standard
start.tex(,930) basis of the ideal generated by the
+start.tex(,931) @tex
+start.tex(,932) $1 \times 1$-minors
+start.tex(,933) @end tex
start.tex(,937) of H is obtained in the following way:
start.tex(,938) @smallexample
start.tex(,939) option(redSB);
@@ -1075,6 +1105,9 @@
start.tex(,942) @end smallexample
start.tex(,943)
start.tex(,944) This shows that 1 is contained in the ideal of the
+start.tex(,945) @tex
+start.tex(,946) $1 \times 1$-minors,
+start.tex(,947) @end tex
start.tex(,951) hence the corresponding variety is empty.
start.tex(,952) @c Coming back to some mathematical considerations, we study
the problem how
start.tex(,953) @c to calculate some ....
@@ -1130,13 +1163,22 @@
start.tex(,1008) @end smallexample
start.tex(,1009)
start.tex(,1010) However the submodule
+start.tex(,1014) @tex
+start.tex(,1015) $MD$
+start.tex(,1016) @end tex
start.tex(,1017) may also be considered as the module
start.tex(,1018) of relations of the factor module
+start.tex(,1019) @tex
+start.tex(,1020) $r^3/MD$.
+start.tex(,1021) @end tex
start.tex(,1025) In this way, @sc{Singular} can treat arbitrary finitely
generated modules
start.tex(,1026) over the
start.tex(,1028) basering (@pxref{Representation of mathematical objects}).
start.tex(,1033)
start.tex(,1034) In order to get the module of relations of
+start.tex(,1038) @tex
+start.tex(,1039) $MD$
+start.tex(,1040) @end tex
start.tex(,1041) ,
start.tex(,1042) we use the command @code{syz}.
start.tex(,1043)
@@ -1147,15 +1189,30 @@
start.tex(,1048)
start.tex(,1049) We want to calculate, as an application, the annihilator of a
given module.
start.tex(,1050) Let
+start.tex(,1051) @tex
+start.tex(,1052) $M = r^3/U$,
+start.tex(,1053) @end tex
start.tex(,1057) where U is our defining module of relations for the module
+start.tex(,1058) @tex
+start.tex(,1059) $M$.
+start.tex(,1060) @end tex
start.tex(,1064)
start.tex(,1065) @smallexample
start.tex(,1066) module U =
[z3,xy2,x3],[yz2,1,xy5z+z3],[y2z,0,x3],[xyz+x2,y2,0],[xyz,x2y,1];
start.tex(,1067) @end smallexample
start.tex(,1068)
start.tex(,1069) Then, by definition, the annihilator of M is the ideal
+start.tex(,1070) @tex
+start.tex(,1071) $\hbox{ann}(M) = \{a \mid aM = 0 \}$
+start.tex(,1072) @end tex
start.tex(,1076) which is by the description of M the same as
+start.tex(,1077) @tex
+start.tex(,1078) $\{ a \mid ar^3 \in U \}$.
+start.tex(,1079) @end tex
start.tex(,1083) Hence we have to calculate the quotient
+start.tex(,1084) @tex
+start.tex(,1085) $U \colon r^3 $.
+start.tex(,1086) @end tex
start.tex(,1090) The rank of the free module is determined by the choice of U
and is the
start.tex(,1091) number of rows of the corresponding matrix. This may be
determined by
start.tex(,1092) the function @code{nrows}. All we have to do now is the
following:
@@ -1175,7 +1232,13 @@
start.tex(,1111) The most general command is @code{res(... ,n)} which
determines heuristically
start.tex(,1112) what method to use for the given problem. It computes the
free resolution
start.tex(,1113) up to the length
+start.tex(,1117) @tex
+start.tex(,1118) $n$
+start.tex(,1119) @end tex
start.tex(,1120) , where
+start.tex(,1124) @tex
+start.tex(,1125) $n=0$
+start.tex(,1126) @end tex
start.tex(,1127) corresponds to the full resolution.
start.tex(,1128)
start.tex(,1129) Here we use the possibility to inspect the calculation
process using the
@@ -1247,7 +1310,13 @@
start.tex(,1195)
start.tex(,1196) In this case, the output is to be interpreted as follows: the
3rd syzygy
start.tex(,1197) module of R/I, @code{rs[3]}, is the rank-2-submodule of
+start.tex(,1198) @tex
+start.tex(,1199) $R^5$
+start.tex(,1200) @end tex
start.tex(,1204) generated by the vectors
+start.tex(,1205) @tex
+start.tex(,1206) $(z^3,0,-y+4z,x+2z,0)$ and
$(-xyz-y^2z-4xz^2+16z^3,-y^2,48z,48z,x+y-z)$.
+start.tex(,1207) @end tex
start.tex(,1211)
singular.texi(,128) @c
----------------------------------------------------------------------------
singular.texi(,129) @node General concepts, Data types, Introduction, Top
@@ -2662,16 +2731,37 @@
general.tex(,1435) @enumerate
general.tex(,1436) @item
general.tex(,1437) the field of rational numbers
+general.tex(,1441) @tex
+general.tex(,1442) $Q$
+general.tex(,1443) @end tex
general.tex(,1444) ,
general.tex(,1445) @item
+general.tex(,1446) @tex
+general.tex(,1447) finite fields $Z/p$, $p$ a prime $\le 2147483629$,
+general.tex(,1448) @end tex
general.tex(,1452) @item
+general.tex(,1453) @tex
+general.tex(,1454) finite fields $\hbox{GF}(p^n)$ with $p^n$ elements, $p$ a
prime, $p^n \le 2^{15}$,
+general.tex(,1455) @end tex
general.tex(,1459) @item
general.tex(,1460) transcendental extension of
+general.tex(,1464) @tex
+general.tex(,1465) $Q$
+general.tex(,1466) @end tex
general.tex(,1467) or
+general.tex(,1471) @tex
+general.tex(,1472) $Z/p$
+general.tex(,1473) @end tex
general.tex(,1474) ,
general.tex(,1475) @item
general.tex(,1476) simple algebraic extension of
+general.tex(,1480) @tex
+general.tex(,1481) $Q$
+general.tex(,1482) @end tex
general.tex(,1483) or
+general.tex(,1487) @tex
+general.tex(,1488) $Z/p$
+general.tex(,1489) @end tex
general.tex(,1490) ,
general.tex(,1491) @item
general.tex(,1492) the field of real numbers represented by floating point
@@ -2722,6 +2812,9 @@
general.tex(,1537) @itemize @bullet
general.tex(,1538) @item
general.tex(,1539) the ring
+general.tex(,1543) @tex
+general.tex(,1544) $Z/32003[x,y,z]$
+general.tex(,1545) @end tex
general.tex(,1546) with degree reverse lexicographical
general.tex(,1547) ordering. The exact ring declaration may be omitted in the
first
general.tex(,1548) example since this is the default ring:
@@ -2733,6 +2826,9 @@
general.tex(,1554)
general.tex(,1555) @item
general.tex(,1556) the ring
+general.tex(,1560) @tex
+general.tex(,1561) $Q[a,b,c,d]$
+general.tex(,1562) @end tex
general.tex(,1563) with lexicographical ordering:
general.tex(,1564)
general.tex(,1565) @smallexample
@@ -2741,6 +2837,9 @@
general.tex(,1568)
general.tex(,1569) @item
general.tex(,1570) the ring
+general.tex(,1574) @tex
+general.tex(,1575) $Z/7[x,y,z]$
+general.tex(,1576) @end tex
general.tex(,1577) with local degree reverse lexicographical
general.tex(,1578) ordering. The non-prime 10 is converted to the next lower
prime in the
general.tex(,1579) second example:
@@ -2752,8 +2851,17 @@
general.tex(,1585)
general.tex(,1586) @item
general.tex(,1587) the ring
+general.tex(,1588) @tex
+general.tex(,1589) $Z/7[x_1,\ldots,x_6]$
+general.tex(,1590) @end tex
general.tex(,1594) with lexicographical ordering for
+general.tex(,1595) @tex
+general.tex(,1596) $x_1,x_2,x_3$
+general.tex(,1597) @end tex
general.tex(,1601) and degree reverse lexicographical ordering for
+general.tex(,1602) @tex
+general.tex(,1603) $x_4,x_5,x_6$:
+general.tex(,1604) @end tex
general.tex(,1608)
general.tex(,1609) @smallexample
general.tex(,1610) ring r = 7,(x(1..6)),(lp(3),dp);
@@ -2761,8 +2869,14 @@
general.tex(,1612)
general.tex(,1613) @item
general.tex(,1614) the localization of
+general.tex(,1618) @tex
+general.tex(,1619) $(Q[a,b,c])[x,y,z]$
+general.tex(,1620) @end tex
general.tex(,1621) at the maximal ideal
general.tex(,1622)
+general.tex(,1626) @tex
+general.tex(,1627) $(x,y,z)$
+general.tex(,1628) @end tex
general.tex(,1629) :
general.tex(,1630)
general.tex(,1631) @smallexample
@@ -2771,10 +2885,22 @@
general.tex(,1634)
general.tex(,1635) @item
general.tex(,1636) the ring
+general.tex(,1640) @tex
+general.tex(,1641) $Q[x,y,z]$
+general.tex(,1642) @end tex
general.tex(,1643) with weighted reverse lexicographical ordering.
general.tex(,1644) The variables
+general.tex(,1648) @tex
+general.tex(,1649) $x$
+general.tex(,1650) @end tex
general.tex(,1651) ,
+general.tex(,1655) @tex
+general.tex(,1656) $y$
+general.tex(,1657) @end tex
general.tex(,1658) , and
+general.tex(,1662) @tex
+general.tex(,1663) $z$
+general.tex(,1664) @end tex
general.tex(,1665) have the weights 2, 1,
general.tex(,1666) and 3, respectively, and vectors are first ordered by
components (in
general.tex(,1667) descending order) and then by monomials:
@@ -2786,12 +2912,30 @@
general.tex(,1673)
general.tex(,1674) @item
general.tex(,1675) the ring
+general.tex(,1679) @tex
+general.tex(,1680) $K[x,y,z]$
+general.tex(,1681) @end tex
general.tex(,1682) , where
+general.tex(,1686) @tex
+general.tex(,1687) $K=Z/7(a,b,c)$
+general.tex(,1688) @end tex
general.tex(,1689) denotes the transcendental
general.tex(,1690) extension of
+general.tex(,1694) @tex
+general.tex(,1695) $Z/7$
+general.tex(,1696) @end tex
general.tex(,1697) by
+general.tex(,1701) @tex
+general.tex(,1702) $a$
+general.tex(,1703) @end tex
general.tex(,1704) ,
+general.tex(,1708) @tex
+general.tex(,1709) $b$
+general.tex(,1710) @end tex
general.tex(,1711) and
+general.tex(,1715) @tex
+general.tex(,1716) $c$
+general.tex(,1717) @end tex
general.tex(,1718) with degree
general.tex(,1719) lexicographical ordering:
general.tex(,1720)
@@ -2801,19 +2945,49 @@
general.tex(,1724)
general.tex(,1725) @item
general.tex(,1726) the ring
+general.tex(,1730) @tex
+general.tex(,1731) $K[x,y,z]$
+general.tex(,1732) @end tex
general.tex(,1733) , where
+general.tex(,1737) @tex
+general.tex(,1738) $K=Z/7[a]$
+general.tex(,1739) @end tex
general.tex(,1740) denotes the algebraic extension of
general.tex(,1741) degree 2 of
+general.tex(,1745) @tex
+general.tex(,1746) $Z/7$
+general.tex(,1747) @end tex
general.tex(,1748) by
+general.tex(,1752) @tex
+general.tex(,1753) $a.$
+general.tex(,1754) @end tex
general.tex(,1755) In other words,
+general.tex(,1759) @tex
+general.tex(,1760) $K$
+general.tex(,1761) @end tex
general.tex(,1762) is the finite field with
general.tex(,1763) 49 elements. In the first case,
+general.tex(,1767) @tex
+general.tex(,1768) $a$
+general.tex(,1769) @end tex
general.tex(,1770) denotes an algebraic
general.tex(,1771) element over
+general.tex(,1775) @tex
+general.tex(,1776) $Z/7$
+general.tex(,1777) @end tex
general.tex(,1778) with minimal polynomial
+general.tex(,1779) @tex
+general.tex(,1780) $\mu_a=a^2+a+3$,
+general.tex(,1781) @end tex
general.tex(,1785) in the second case,
+general.tex(,1789) @tex
+general.tex(,1790) $a$
+general.tex(,1791) @end tex
general.tex(,1792)
general.tex(,1793) refers to some generator of the cyclic group of units of
+general.tex(,1797) @tex
+general.tex(,1798) $K$
+general.tex(,1799) @end tex
general.tex(,1800) :
general.tex(,1801)
general.tex(,1802) @smallexample
@@ -2823,7 +2997,13 @@
general.tex(,1806)
general.tex(,1807) @item
general.tex(,1808) the ring
+general.tex(,1812) @tex
+general.tex(,1813) $R[x,y,z]$
+general.tex(,1814) @end tex
general.tex(,1815) , where
+general.tex(,1819) @tex
+general.tex(,1820) $R$
+general.tex(,1821) @end tex
general.tex(,1822) denotes the field of real
general.tex(,1823) numbers represented by simple precision floating point
numbers. This is
general.tex(,1824) a special case:
@@ -2834,7 +3014,13 @@
general.tex(,1829)
general.tex(,1830) @item
general.tex(,1831) the ring
+general.tex(,1835) @tex
+general.tex(,1836) $R[x,y,z]$
+general.tex(,1837) @end tex
general.tex(,1838) , where
+general.tex(,1842) @tex
+general.tex(,1843) $R$
+general.tex(,1844) @end tex
general.tex(,1845) denotes the field of real
general.tex(,1846) numbers represented by floating point numbers of 50 valid
decimal digits
general.tex(,1847) and the same number of digits for the rest:
@@ -2845,7 +3031,13 @@
general.tex(,1852)
general.tex(,1853) @item
general.tex(,1854) the ring
+general.tex(,1858) @tex
+general.tex(,1859) $R[x,y,z]$
+general.tex(,1860) @end tex
general.tex(,1861) , where
+general.tex(,1865) @tex
+general.tex(,1866) $R$
+general.tex(,1867) @end tex
general.tex(,1868) denotes the field of real
general.tex(,1869) numbers represented by floating point numbers of 10 valid
decimal digits
general.tex(,1870) and with 50 digits for the rest:
@@ -2856,10 +3048,19 @@
general.tex(,1875)
general.tex(,1876) @item
general.tex(,1877) the ring
+general.tex(,1881) @tex
+general.tex(,1882) $R(j)[x,y,z]$
+general.tex(,1883) @end tex
general.tex(,1884) , where
+general.tex(,1888) @tex
+general.tex(,1889) $R$
+general.tex(,1890) @end tex
general.tex(,1891) denotes the field of real
general.tex(,1892) numbers represented by floating point numbers of 30 valid
decimal digits
general.tex(,1893) and the same number for the rest.
+general.tex(,1897) @tex
+general.tex(,1898) $j$
+general.tex(,1899) @end tex
general.tex(,1900) denotes the imaginary unit.
general.tex(,1901)
general.tex(,1902) @smallexample
@@ -2868,10 +3069,19 @@
general.tex(,1905)
general.tex(,1906) @item
general.tex(,1907) the ring
+general.tex(,1911) @tex
+general.tex(,1912) $R(i)[x,y,z]$
+general.tex(,1913) @end tex
general.tex(,1914) , where
+general.tex(,1918) @tex
+general.tex(,1919) $R$
+general.tex(,1920) @end tex
general.tex(,1921) denotes the field of real
general.tex(,1922) numbers represented by floating point numbers of 6 valid
decimal digits
general.tex(,1923) and the same number for the rest.
+general.tex(,1927) @tex
+general.tex(,1928) $i$
+general.tex(,1929) @end tex
general.tex(,1930) is the default for the imaginary unit.
general.tex(,1931)
general.tex(,1932) @smallexample
@@ -2880,8 +3090,14 @@
general.tex(,1935)
general.tex(,1936) @item
general.tex(,1937) the quotient ring
+general.tex(,1941) @tex
+general.tex(,1942) $Z/7[x,y,z]$
+general.tex(,1943) @end tex
general.tex(,1944) modulo the square of the maximal
general.tex(,1945) ideal
+general.tex(,1949) @tex
+general.tex(,1950) $(x,y,z)$
+general.tex(,1951) @end tex
general.tex(,1952) :
general.tex(,1953)
general.tex(,1954) @smallexample
@@ -2934,7 +3150,13 @@
general.tex(,2001) an expression_list of an int_expression and a name.
general.tex(,2002) @* The int_expression has to be a prime number p to the
power of a
general.tex(,2003) positive integer n. This defines the Galois field
+general.tex(,2004) @tex
+general.tex(,2005) $\hbox{GF}(p^n)$ with $p^n$ elements, where $p^n$ has to be
smaller or equal $2^{15}$.
+general.tex(,2006) @end tex
general.tex(,2010) The given name refers to a primitive element of
+general.tex(,2011) @tex
+general.tex(,2012) $\hbox{GF}(p^n)$
+general.tex(,2013) @end tex
general.tex(,2017) generating the multiplicative group. Due to a different
internal
general.tex(,2018) representation, the arithmetic operations in these
coefficient fields
general.tex(,2019) are faster than arithmetic operations in algebraic
extensions as
@@ -3060,7 +3282,13 @@
general.tex(,2139)
general.tex(,2140) @strong{Remark:} The novice user should generally use the
ordering
general.tex(,2141) @code{dp} for computations in the polynomial ring
+general.tex(,2142) @tex
+general.tex(,2143) $K[x_1,\ldots,x_n]$,
+general.tex(,2144) @end tex
general.tex(,2148) resp.@: @code{ds} for computations in the localization
+general.tex(,2149) @tex
+general.tex(,2150) $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n])$.
+general.tex(,2151) @end tex
general.tex(,2155) For more details, see @ref{Polynomial data}.
general.tex(,2156)
general.tex(,2157) In a ring declaration, @sc{Singular} offers the following
orderings:
@@ -3086,8 +3314,14 @@
general.tex(,2177) @end table
general.tex(,2178)
general.tex(,2179) Global orderings are well-orderings, i.e.,
+general.tex(,2183) @tex
+general.tex(,2184) $1 < x$
+general.tex(,2185) @end tex
general.tex(,2186) for each ring
general.tex(,2187) variable
+general.tex(,2191) @tex
+general.tex(,2192) $x$
+general.tex(,2193) @end tex
general.tex(,2194) . They are denoted by a @code{p} as the second
general.tex(,2195) character in their name.
general.tex(,2196)
@@ -3980,6 +4214,9 @@
general.tex(,3083) by the size of expression.
general.tex(,3084) @* But @code{matrix(} expression @code{,} m @code{,} n
@code{)} may also be
general.tex(,3085) used - the result is a
+general.tex(,3086) @tex
+general.tex(,3087) $ m \times n $
+general.tex(,3088) @end tex
general.tex(,3092) matrix (@pxref{matrix type cast})
general.tex(,3093) @item
general.tex(,3094) @ @tab @code{module} @tab expression lists of
@code{int}, @code{number},
@@ -4293,11 +4530,17 @@
general.tex(,3402) @*["help_text"]
general.tex(,3403) @address@hidden@{}
general.tex(,3404) @*
+general.tex(,3405) @tex
+general.tex(,3406) \quad
+general.tex(,3407) @end tex
general.tex(,3408) procedure_body
general.tex(,3409) @address@hidden@}}
general.tex(,3410) @address@hidden
general.tex(,3411) @address@hidden@{}
general.tex(,3412) @*
+general.tex(,3413) @tex
+general.tex(,3414) \quad
+general.tex(,3415) @end tex
general.tex(,3416) sequence_of_commands;
general.tex(,3417) @address@hidden@}}]
general.tex(,3418) @item Purpose:
@@ -5129,6 +5372,9 @@
general.tex(,4210) @code{@@address@hidden@}}
general.tex(,4211) @address@hidden
general.tex(,4212) @*
+general.tex(,4216) @tex
+general.tex(,4217) $\alpha$
+general.tex(,4218) @end tex
general.tex(,4219)
general.tex(,4220) @item Note:
general.tex(,4221) Mathematical expressions inside @code{@@address@hidden@}}
may
@@ -5228,6 +5474,9 @@
general.tex(,4315) @address@hidden
general.tex(,4316) @*Among others, within a texinfo environment one can use
the tex environment
general.tex(,4317) to typeset more complex mathematical like
+general.tex(,4318) @tex
+general.tex(,4319) $ i_{1,1} $
+general.tex(,4320) @end tex
general.tex(,4321) @end table
general.tex(,4322)
general.tex(,4323) @end table
@@ -5522,12 +5771,18 @@
template_lib.tex(,107)
template_lib.tex(,108) @item @strong{Return:}
template_lib.tex(,109) int:
+template_lib.tex(,113) @tex
+template_lib.tex(,114) $i+i+i$
+template_lib.tex(,115) @end tex
template_lib.tex(,116)
template_lib.tex(,117) @item @strong{Note:}
template_lib.tex(,118) Help is in pure Texinfo
template_lib.tex(,119) @*This help string is written in texinfo, which enables
you to use,
template_lib.tex(,120) among others, the @@math command for mathematical
typesetting (like
template_lib.tex(,121)
+template_lib.tex(,125) @tex
+template_lib.tex(,126) $\alpha, \beta$
+template_lib.tex(,127) @end tex
template_lib.tex(,128) ).
template_lib.tex(,129) @*It also gives more control over the layout, but is,
admittingly,
template_lib.tex(,130) more cumbersome to write.
@@ -5568,6 +5823,9 @@
template_lib.tex(,179) @* Use a @@ref constructs for references (like
@pxref{mtripple})
template_lib.tex(,180) @* Use @@code for typewriter font (like @code{i_1})
template_lib.tex(,181) @* Use @@math for simple math mode typesetting (like
+template_lib.tex(,185) @tex
+template_lib.tex(,186) $i_1$
+template_lib.tex(,187) @end tex
template_lib.tex(,188) ).
template_lib.tex(,189) @* Note: No parenthesis like @} are allowed inside
@@math and @@code
template_lib.tex(,190) @* Use @@example for indented preformatted text typeset
in typewriter
@@ -5582,6 +5840,9 @@
template_lib.tex(,199) Use @@texinfo for text in pure texinfo
template_lib.tex(,200)
template_lib.tex(,201) @expansion{}
+template_lib.tex(,202) @tex
+template_lib.tex(,203) $i_{1,1}$
+template_lib.tex(,204) @end tex
template_lib.tex(,205)
template_lib.tex(,206)
template_lib.tex(,207) Notice that
@@ -6231,6 +6492,9 @@
types.tex(,416) set of minors of a matrix (see @ref{minor})
types.tex(,417) @item modulo
types.tex(,418) represents
+types.tex(,419) @tex
+types.tex(,420) $(h1+h2)/h1 \cong h2/(h1 \cap h2)$
+types.tex(,421) @end tex
types.tex(,425) (see @ref{modulo})
types.tex(,426) @item mres
types.tex(,427) minimal free resolution of an ideal resp.@: module w.r.t. a
minimal set of generators of the given ideal resp.@: module
@@ -7982,25 +8246,62 @@
types.tex(,2236) Canonically realized are
types.tex(,2237) @itemize @bullet
types.tex(,2238) @item
+types.tex(,2239) @tex
+types.tex(,2240) $Q \rightarrow Q(a, \ldots)$
+types.tex(,2241) @end tex
types.tex(,2245)
types.tex(,2246) @item
+types.tex(,2247) @tex
+types.tex(,2248) $Q \rightarrow R$
+types.tex(,2249) @end tex
types.tex(,2253)
types.tex(,2254) @item
+types.tex(,2255) @tex
+types.tex(,2256) $Q \rightarrow C$
+types.tex(,2257) @end tex
types.tex(,2261)
types.tex(,2262) @item
+types.tex(,2263) @tex
+types.tex(,2264) $Z/p \rightarrow (Z/p)(a, \ldots)$
+types.tex(,2265) @end tex
types.tex(,2269)
types.tex(,2270) @item
+types.tex(,2271) @tex
+types.tex(,2272) $Z/p \rightarrow GF(p^n)$
+types.tex(,2273) @end tex
types.tex(,2277)
types.tex(,2278) @item
+types.tex(,2279) @tex
+types.tex(,2280) $Z/p \rightarrow R$
+types.tex(,2281) @end tex
types.tex(,2285)
types.tex(,2286) @item
+types.tex(,2287) @tex
+types.tex(,2288) $R \rightarrow C$
+types.tex(,2289) @end tex
types.tex(,2293) @end itemize
types.tex(,2294)
types.tex(,2295) Possible are furthermore
types.tex(,2296) @itemize @bullet
types.tex(,2297) @item
+types.tex(,2298) @tex
+types.tex(,2299) % This is quite a hack, but for now it works.
+types.tex(,2300) $Z/p \rightarrow Q,
+types.tex(,2301) \quad
+types.tex(,2302) [i]_p \mapsto i \in [-p/2, \, p/2]
+types.tex(,2303) \subseteq Z$
+types.tex(,2304) @end tex
types.tex(,2308) @item
+types.tex(,2309) @tex
+types.tex(,2310) $Z/p \rightarrow Z/p^\prime,
+types.tex(,2311) \quad
+types.tex(,2312) [i]_p \mapsto i \in [-p/2, \, p/2] \subseteq Z, \;
+types.tex(,2313) i \mapsto [i]_{p^\prime} \in Z/p^\prime$
+types.tex(,2314) @end tex
types.tex(,2318) @item
+types.tex(,2319) @tex
+types.tex(,2320) $C \rightarrow R, \quad$ the real part
+types.tex(,2321) @end tex
types.tex(,2325) @end itemize
types.tex(,2326)
types.tex(,2327) Finally, in Singular we allow the mapping from rings
@@ -8009,8 +8310,14 @@
types.tex(,2330)
types.tex(,2331) @itemize @bullet
types.tex(,2332) @item
+types.tex(,2333) @tex
+types.tex(,2334) $Q \rightarrow Z/p$
+types.tex(,2335) @end tex
types.tex(,2339)
types.tex(,2340) @item
+types.tex(,2341) @tex
+types.tex(,2342) $Q \rightarrow (Z/p)(a, \ldots)$
+types.tex(,2343) @end tex
types.tex(,2347) @end itemize
types.tex(,2348) In these cases the denominator and the numerator
types.tex(,2349) of a number are mapped separately by the usual
@@ -8454,18 +8761,45 @@
types.tex(,2822) Like vectors they
types.tex(,2823) can only be defined or accessed with respect to a basering.
types.tex(,2824) If
+types.tex(,2828) @tex
+types.tex(,2829) $M$
+types.tex(,2830) @end tex
types.tex(,2831) is a submodule of
+types.tex(,2835) @tex
+types.tex(,2836) $R^n$,
+types.tex(,2837) @end tex
types.tex(,2838)
+types.tex(,2842) @tex
+types.tex(,2843) $R$
+types.tex(,2844) @end tex
types.tex(,2845) the basering, generated by vectors
+types.tex(,2849) @tex
+types.tex(,2850) $v_1, \ldots, v_k$, then $v_1, \ldots, v_k$
+types.tex(,2851) @end tex
types.tex(,2852) may be considered as the generators of relations of
+types.tex(,2856) @tex
+types.tex(,2857) $R^n/M$
+types.tex(,2858) @end tex
types.tex(,2859) between the canonical generators
@code{gen(1)},@dots{},@code{gen(n)}.
types.tex(,2860) Hence any finitely generated
+types.tex(,2864) @tex
+types.tex(,2865) $R$
+types.tex(,2866) @end tex
types.tex(,2867) -module can be represented in @sc{Singular}
types.tex(,2868) by its module of relations. The assignments
types.tex(,2869) @code{module M=v1,...,vk; matrix A=M;}
types.tex(,2870) create the presentation matrix of size
+types.tex(,2874) @tex
+types.tex(,2875) n$\times$k
+types.tex(,2876) @end tex
types.tex(,2877) for
+types.tex(,2881) @tex
+types.tex(,2882) R$^n$/M,
+types.tex(,2883) @end tex
types.tex(,2884) i.e., the columns of A are the vectors
+types.tex(,2888) @tex
+types.tex(,2889) $v_1, \ldots, v_k$
+types.tex(,2890) @end tex
types.tex(,2891) which generate M (cf. @ref{Representation of mathematical
objects}).
types.tex(,2892)
types.tex(,2893) @menu
@@ -8620,6 +8954,9 @@
types.tex(,3058) over a local ring
types.tex(,3059) @item modulo
types.tex(,3060) represents
+types.tex(,3061) @tex
+types.tex(,3062) $(h1+h2)/h1=h2/(h1 \cap h2)$
+types.tex(,3063) @end tex
types.tex(,3067) (see @ref{modulo})
types.tex(,3068) @item mres
types.tex(,3069) minimal free resolution of an ideal resp.@: module w.r.t. a
minimal set of generators of the given module
@@ -9229,11 +9566,17 @@
types.tex(,3796) @*["help_text"]
types.tex(,3797) @address@hidden@{}
types.tex(,3798) @*
+types.tex(,3799) @tex
+types.tex(,3800) \quad
+types.tex(,3801) @end tex
types.tex(,3802) procedure_body
types.tex(,3803) @address@hidden@}}
types.tex(,3804) @address@hidden
types.tex(,3805) @address@hidden@{}
types.tex(,3806) @*
+types.tex(,3807) @tex
+types.tex(,3808) \quad
+types.tex(,3809) @end tex
types.tex(,3810) sequence_of_commands;
types.tex(,3811) @address@hidden@}}]
types.tex(,3812) @address@hidden proc_name @code{=} proc_name @code{;}
@@ -9548,31 +9891,82 @@
types.tex(,4145) @table @asis
types.tex(,4146) @item @code{+}
types.tex(,4147) construct a new ring
+types.tex(,4151) @tex
+types.tex(,4152) $k[X,Y]$
+types.tex(,4153) @end tex
types.tex(,4154) from
+types.tex(,4158) @tex
+types.tex(,4159) $k_1[X]$
+types.tex(,4160) @end tex
types.tex(,4161) and
+types.tex(,4165) @tex
+types.tex(,4166) $k_2[Y]$
+types.tex(,4167) @end tex
types.tex(,4168) .
types.tex(,4169) @end table
types.tex(,4170)
types.tex(,4171) Concerning the ground fields
+types.tex(,4175) @tex
+types.tex(,4176) $k_1$
+types.tex(,4177) @end tex
types.tex(,4178) and
+types.tex(,4182) @tex
+types.tex(,4183) $k_2$
+types.tex(,4184) @end tex
types.tex(,4185) take the
types.tex(,4186) following guide lines into consideration:
types.tex(,4187) @itemize @bullet
types.tex(,4188) @item Neither
+types.tex(,4192) @tex
+types.tex(,4193) $k_1$
+types.tex(,4194) @end tex
types.tex(,4195) nor
+types.tex(,4199) @tex
+types.tex(,4200) $k_2$
+types.tex(,4201) @end tex
types.tex(,4202) may be
+types.tex(,4206) @tex
+types.tex(,4207) $R$
+types.tex(,4208) @end tex
types.tex(,4209) or
+types.tex(,4213) @tex
+types.tex(,4214) $C$
+types.tex(,4215) @end tex
types.tex(,4216) .
types.tex(,4217) @item If the characteristic of
+types.tex(,4221) @tex
+types.tex(,4222) $k_1$
+types.tex(,4223) @end tex
types.tex(,4224) and
+types.tex(,4228) @tex
+types.tex(,4229) $k_2$
+types.tex(,4230) @end tex
types.tex(,4231) differs, then one of them must be
+types.tex(,4235) @tex
+types.tex(,4236) $Q$
+types.tex(,4237) @end tex
types.tex(,4238) .
types.tex(,4239) @item At most one of
+types.tex(,4243) @tex
+types.tex(,4244) $k_1$
+types.tex(,4245) @end tex
types.tex(,4246) and
+types.tex(,4250) @tex
+types.tex(,4251) $k_2$
+types.tex(,4252) @end tex
types.tex(,4253) may be have parameters.
types.tex(,4254) @item If one of
+types.tex(,4258) @tex
+types.tex(,4259) $k_1$
+types.tex(,4260) @end tex
types.tex(,4261) and
+types.tex(,4265) @tex
+types.tex(,4266) $k_2$
+types.tex(,4267) @end tex
types.tex(,4268) is an algebraic extension of
+types.tex(,4272) @tex
+types.tex(,4273) $Z/p$
+types.tex(,4274) @end tex
types.tex(,4275) it may not be defined by a @code{charstr} of type
@code{(p^n,a)}.
types.tex(,4276) @end itemize
types.tex(,4277)
@@ -10468,6 +10862,18 @@
reference.tex(,418) intmat
reference.tex(,419) @item @strong{Purpose:}
reference.tex(,420) with 1 argument: computes the graded Betti numbers of a
minimal resolution of
+reference.tex(,421) @tex
+reference.tex(,422) $R^n/M$, if $R$ denotes the basering and
+reference.tex(,423) $M$ a homogeneous submodule of $R^n$ and the argument
represents a
+reference.tex(,424) resolution of
+reference.tex(,425) $R^n/M$.
+reference.tex(,426) @end tex
+reference.tex(,430) @tex
+reference.tex(,431) The entry d of the intmat at place (i,j) is the minimal
number of
+reference.tex(,432) generators in degree i+j of the j-th syzygy module (=
module of
+reference.tex(,433) relations) of $R^n/M$ (the 0th (resp.\ 1st) syzygy module
of $R^n/M$ is
+reference.tex(,434) $R^n$ (resp.\ $M$)).
+reference.tex(,435) @end tex
reference.tex(,445) The argument is considered to be the result of a
res/sres/mres/nres/lres
reference.tex(,446) command. This implies that a zero is only allowed (and
counted) as a
reference.tex(,447) generator in the first module.
@@ -10535,6 +10941,15 @@
reference.tex(,509) where the generators are the columns of the
reference.tex(,510) displayed matrix and degrees are assigned such that the
corresponding maps
reference.tex(,511) have degree 0:
+reference.tex(,512) @tex
+reference.tex(,513) $$
+reference.tex(,514) 0 \longleftarrow r/j \longleftarrow r(1)
+reference.tex(,515) \buildrel{T[1]}\over{\longleftarrow} r(2) \oplus r^3(3)
+reference.tex(,516) \buildrel{T[2]}\over{\longleftarrow} r^4(4)
+reference.tex(,517) \buildrel{T[3]}\over{\longleftarrow} r(5)
+reference.tex(,518) \longleftarrow 0 \quad .
+reference.tex(,519) $$
+reference.tex(,520) @end tex
reference.tex(,525)
reference.tex(,526) @c inserted refs from reference.doc:455
reference.tex(,551) @c end inserted refs from reference.doc:455
@@ -10848,12 +11263,28 @@
reference.tex(,919) @end format
reference.tex(,920) If J is a vector or a module this procedure is repeated
for each
reference.tex(,921) component and the resulting matrices are address@hidden
+reference.tex(,926) @tex
+reference.tex(,927) The third argument is used to return the matrix T of
coefficients
+reference.tex(,928) such that {\tt matrix}(J) = T*M.
+reference.tex(,929) @end tex
reference.tex(,930) @item @strong{Note:}
reference.tex(,931) @code{coeffs} returns the coefficient 0 at the appropriate
place if a monomial
reference.tex(,932) is not present, while @code{coef} considers only monomials
which really occur
reference.tex(,933) in the given expression. @*
reference.tex(,934) If
+reference.tex(,935) @tex
+reference.tex(,936) $M=(m_{ij})$
+reference.tex(,937) @end tex
reference.tex(,941) then the j-th generator of an ideal J is equal to
+reference.tex(,942) @tex
+reference.tex(,943) $$J_j = z^0 \cdot m_{1j} + z^1 \cdot m_{2j} + ... +
z^{d-1} \cdot m_{dj},$$
+reference.tex(,944) while for a module J the i-th component of the j-th
generator is
+reference.tex(,945) equal to the entry [i,j] of {\tt matrix}(J), and we get
+reference.tex(,946) @end tex
+reference.tex(,956) @tex
+reference.tex(,957) $$ J_{i,j} = z^0 \cdot m_{(i-1)d+1,j} + z^1 \cdot
m_{(i-1)d+2,j} + ... +
+reference.tex(,958) z^{d-1} \cdot m_{id,j}.$$
+reference.tex(,959) @end tex
reference.tex(,968)
reference.tex(,969) @item @strong{Example:}
reference.tex(,970) @smallexample
@@ -10932,7 +11363,14 @@
reference.tex(,1055) producing a m x n matrix.
reference.tex(,1056) @*Contraction is defined on monomials by:
reference.tex(,1057) @*
+reference.tex(,1064) @tex
+reference.tex(,1065) $${\rm contract}(x^A , x^B) := \cases{ x^{(B-A)}, &if
$B\ge A$
+reference.tex(,1066) componentwise\cr 0,&otherwise.\cr}$$
+reference.tex(,1067) @end tex
reference.tex(,1068) where A and B are the multiexponents of the ring
variables represented by
+reference.tex(,1069) @tex
+reference.tex(,1070) $x$.
+reference.tex(,1071) @end tex
reference.tex(,1075) @code{contract} is extended bilinearly to all polynomials.
reference.tex(,1076) @item @strong{Example:}
reference.tex(,1077) @smallexample
@@ -12365,13 +12803,24 @@
reference.tex(,2950) @code{highcorner(I)} returns 0 iff @code{dim(I)>0} or
@code{dim(I)=-1}.
reference.tex(,2951) Otherwise it returns the smallest monomial m not in I
which has the following
reference.tex(,2952) properties (with
+reference.tex(,2956) @tex
+reference.tex(,2957) $x_i$
+reference.tex(,2958) @end tex
reference.tex(,2959) the variables of the basering):
reference.tex(,2960) @itemize @bullet
reference.tex(,2961) @item
reference.tex(,2962) if
+reference.tex(,2966) @tex
+reference.tex(,2967) $x_i>1$ then $x_i$
+reference.tex(,2968) @end tex
reference.tex(,2969) does not divide m (e.g., m=1 if the ordering is global)
reference.tex(,2970) @item
reference.tex(,2971) given any set of generators
+reference.tex(,2977) @tex
+reference.tex(,2978) $f_1,\dots,f_k$ of I, let $f'_i$ be obtained from
+reference.tex(,2979) $f_i$ by deleting the terms divisible by $x_i\cdot m$ for
all i with $x_i<1$.
+reference.tex(,2980) Then $f'_1,\dots,f'_k$ generate I.
+reference.tex(,2981) @end tex
reference.tex(,2982) @end itemize
reference.tex(,2983) @item @strong{Example:}
reference.tex(,2984) @smallexample
@@ -12510,11 +12959,22 @@
reference.tex(,3167)
reference.tex(,3168) More precisely, let R be the basering and I be the given
ideal.
reference.tex(,3169) Then @code{hres} computes a minimal free resolution of R/I
+reference.tex(,3176) @tex
+reference.tex(,3177) $$...\longrightarrow F_2
\buildrel{A_2}\over{\longrightarrow} F_1
+reference.tex(,3178) \buildrel{A_1}\over{\longrightarrow} R\longrightarrow R/I
+reference.tex(,3179) \longrightarrow 0.$$
+reference.tex(,3180) @end tex
reference.tex(,3181) If the int_expression k is not zero then the computation
stops after
reference.tex(,3182) k steps and returns a list of modules
+reference.tex(,3183) @tex
+reference.tex(,3184) $M_i={\tt module} (A_i)$, i=1..k.
+reference.tex(,3185) @end tex
reference.tex(,3189)
reference.tex(,3190) @code{list L=hres(I,0);} returns a list L of n modules
(where n is the
reference.tex(,3191) number of variables of the basering) such that
+reference.tex(,3192) @tex
+reference.tex(,3193) ${\tt L[i]}=M_i$
+reference.tex(,3194) @end tex
reference.tex(,3198) in the above notation.
reference.tex(,3199) @item @strong{Note:}
reference.tex(,3200) The ideal_expression has to be homogeneous.
@@ -12630,6 +13090,9 @@
reference.tex(,3364)
reference.tex(,3365) @item @strong{Note:}
reference.tex(,3366) U is a set of independent variables for I if and only if
+reference.tex(,3367) @tex
+reference.tex(,3368) $I \cap K[U]=(0)$,
+reference.tex(,3369) @end tex
reference.tex(,3373) i.e., eliminating the remaining variables gives (0).
reference.tex(,3374) U is maximal if dim(I)=#U.
reference.tex(,3375) @item @strong{Syntax:}
@@ -12727,19 +13190,47 @@
reference.tex(,3491) @item @strong{Purpose:}
reference.tex(,3492) interreduces a set of polynomials/vectors.
reference.tex(,3493) @*
+reference.tex(,3497) @tex
+reference.tex(,3498) input: $f_1,\dots,f_n$
+reference.tex(,3499) @end tex
reference.tex(,3500) @*
+reference.tex(,3506) @tex
+reference.tex(,3507) output: $g_1,\dots,g_s$ with $s \leq n$ and the properties
+reference.tex(,3508) @end tex
reference.tex(,3509) @itemize @bullet
reference.tex(,3510) @item
+reference.tex(,3514) @tex
+reference.tex(,3515) $(f_1,\dots,f_n) = (g_1,\dots,g_s)$
+reference.tex(,3516) @end tex
reference.tex(,3517) @item
+reference.tex(,3521) @tex
+reference.tex(,3522) $L(g_i)\neq L(g_j)$ for all $i\neq j$
+reference.tex(,3523) @end tex
reference.tex(,3524) @item
reference.tex(,3525) in the case of a global ordering (polynomial ring):
reference.tex(,3526) @*
+reference.tex(,3530) @tex
+reference.tex(,3531) $L(g_i)$
+reference.tex(,3532) @end tex
reference.tex(,3533) does not divide m for all monomials m of
+reference.tex(,3537) @tex
+reference.tex(,3538) $\{g_1,\dots,g_{i-1},g_{i+1},\dots,g_s\}$
+reference.tex(,3539) @end tex
reference.tex(,3540) @item
reference.tex(,3541) in the case of a local or mixed ordering (localization of
polynomial ring):
reference.tex(,3542) @* if
+reference.tex(,3546) @tex
+reference.tex(,3547) $L(g_i) | L(g_j)$ for any $i \neq j$,
+reference.tex(,3548) @end tex
reference.tex(,3549) then
+reference.tex(,3553) @tex
+reference.tex(,3554) $ecart(g_i) > ecart(g_j)$
+reference.tex(,3555) @end tex
reference.tex(,3556) @end itemize
+reference.tex(,3557) @tex
+reference.tex(,3558) Here, $L(g)$ denotes the leading term of $g$ and
+reference.tex(,3559) $ecart(g):=deg(g)-deg(L(g))$.
+reference.tex(,3560) @end tex
reference.tex(,3566) @item @strong{Example:}
reference.tex(,3567) @smallexample
reference.tex(,3568) @c reused example interred reference.doc:2557
@@ -13469,11 +13960,22 @@
reference.tex(,4704)
reference.tex(,4705) More precisely, let R be the basering and I be the given
ideal.
reference.tex(,4706) Then @code{lres} computes a minimal free resolution of R/I
+reference.tex(,4713) @tex
+reference.tex(,4714) $$...\longrightarrow F_2
\buildrel{A_2}\over{\longrightarrow} F_1
+reference.tex(,4715) \buildrel{A_1}\over{\longrightarrow} R\longrightarrow R/I
+reference.tex(,4716) \longrightarrow 0.$$
+reference.tex(,4717) @end tex
reference.tex(,4718) If the int_expression k is not zero then the computation
stops after
reference.tex(,4719) k steps and returns a list of modules
+reference.tex(,4720) @tex
+reference.tex(,4721) $M_i={\tt module}(A_i)$, i=1..k.
+reference.tex(,4722) @end tex
reference.tex(,4726)
reference.tex(,4727) @code{list L=lres(I,0);} returns a list L of n modules
(where n is the
reference.tex(,4728) number of variables of the basering) such that
+reference.tex(,4729) @tex
+reference.tex(,4730) ${\tt L[i]}=M_i$
+reference.tex(,4731) @end tex
reference.tex(,4735) in the above notation.
reference.tex(,4736) @item @strong{Note:}
reference.tex(,4737) The ideal_expression has to be homogeneous.
@@ -13727,9 +14229,25 @@
reference.tex(,5069) module
reference.tex(,5070) @item @strong{Purpose:}
reference.tex(,5071) @code{modulo(h1,h2)}
+reference.tex(,5075) @tex
+reference.tex(,5076) represents $h_1/(h_1 \cap h_2) \cong (h_1+h_2)/h_2$
+reference.tex(,5077) @end tex
reference.tex(,5078) where
+reference.tex(,5079) @tex
+reference.tex(,5080) $h_1$ and $h_2$
+reference.tex(,5081) @end tex
reference.tex(,5085) are considered as submodules of the same free module
+reference.tex(,5086) @tex
+reference.tex(,5087) $R^l$
+reference.tex(,5088) @end tex
reference.tex(,5092) (l=1 for ideals). Let
+reference.tex(,5093) @tex
+reference.tex(,5094) $H_1$, resp.\ $H_2$,
+reference.tex(,5095) @end tex
+reference.tex(,5100) @tex
+reference.tex(,5101) be the matrices of size $l \times k$, resp.\ $l \times
m$, having the
+reference.tex(,5102) generators of $h_1$, resp.\ $h_2$,
+reference.tex(,5103) @end tex
reference.tex(,5107) as columns.
reference.tex(,5108) @c @*
reference.tex(,5109) @c @tex
@@ -13743,7 +14261,14 @@
reference.tex(,5117) @c @end smallexample
reference.tex(,5118) @c @end ifinfo
reference.tex(,5119) Then
+reference.tex(,5120) @tex
+reference.tex(,5121) $h_1/(h_1 \cap h_2) \cong R^k / ker(\overline{H_1})$
+reference.tex(,5122) @end tex
reference.tex(,5131) where
+reference.tex(,5132) @tex
+reference.tex(,5133) $\overline{H_1}: R^k \rightarrow R^l/Im(H_2)=R^l/h_2$
+reference.tex(,5134) is the induced map.
+reference.tex(,5135) @end tex
reference.tex(,5144) @address@hidden(h1,h2)} returns generators of
reference.tex(,5145) the kernel of this induced map.
reference.tex(,5146) @item @strong{Example:}
@@ -13844,17 +14369,32 @@
reference.tex(,5261) computes a minimal free resolution of an ideal or module
M with the
reference.tex(,5262) standard basis method. More precisely, let
address@hidden(M), then @code{mres}
reference.tex(,5263) computes a free resolution of
+reference.tex(,5271) @tex
+reference.tex(,5272) $coker(A)=F_0/M$
+reference.tex(,5273) $$...\longrightarrow F_2
\buildrel{A_2}\over{\longrightarrow} F_1
+reference.tex(,5274) \buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow
F_0/M
+reference.tex(,5275) \longrightarrow 0,$$
+reference.tex(,5276) @end tex
reference.tex(,5277) where the columns of the matrix
+reference.tex(,5278) @tex
+reference.tex(,5279) $A_1$
+reference.tex(,5280) @end tex
reference.tex(,5284) are a minimal set of generators
reference.tex(,5285) of M if the basering is local or if M is homogeneous.
reference.tex(,5286) If the int expression k is not zero then the computation
stops after k steps
reference.tex(,5287) and returns a list of modules
+reference.tex(,5288) @tex
+reference.tex(,5289) $M_i={\tt module}(A_i)$, i=1...k.
+reference.tex(,5290) @end tex
reference.tex(,5294) @address@hidden(M,0)} returns a resolution consisting of
at most n+2 modules,
reference.tex(,5295) where n is the number of variables of the basering.
reference.tex(,5296) Let @code{list L=mres(M,0);}
reference.tex(,5297) then @code{L[1]} consists of a minimal set of generators
of the input, @code{L[2]}
reference.tex(,5298) consists of a minimal set of generators for the first
syzygy module of
reference.tex(,5299) @code{L[1]}, etc., until @code{L[p+1]}, such that
+reference.tex(,5303) @tex
+reference.tex(,5304) ${\tt L[i]}\neq 0$ for $i \le p$,
+reference.tex(,5305) @end tex
reference.tex(,5306) but @code{L[p+1]}, the first syzygy module of
@code{L[p]},
reference.tex(,5307) is 0 (if the basering is not a qring).
reference.tex(,5308) @item @strong{Note:}
@@ -14145,16 +14685,32 @@
reference.tex(,5781) the second module on (by the standard basis method).
reference.tex(,5782)
reference.tex(,5783) More precisely, let
+reference.tex(,5784) @tex
+reference.tex(,5785) $A_1$=matrix(M),
+reference.tex(,5786) @end tex
reference.tex(,5790) then @code{nres} computes a free resolution of
+reference.tex(,5798) @tex
+reference.tex(,5799) $coker(A_1)=F_0/M$
+reference.tex(,5800) $$...\longrightarrow F_2
\buildrel{A_2}\over{\longrightarrow} F_1 \buildrel{A_1}\over{\longrightarrow}
F_0\longrightarrow F_0/M\longrightarrow 0,$$
+reference.tex(,5801) @end tex
reference.tex(,5802) @*where the columns of the matrix
+reference.tex(,5803) @tex
+reference.tex(,5804) $A_1$
+reference.tex(,5805) @end tex
reference.tex(,5809) are the given set of generators of M.
reference.tex(,5810) If the int expression k is not zero then the computation
stops after k steps
reference.tex(,5811) and returns a list of modules
+reference.tex(,5812) @tex
+reference.tex(,5813) $M_i={\tt module}(A_i)$, i=1..k.
+reference.tex(,5814) @end tex
reference.tex(,5818) @address@hidden(M,0)} returns a list of n modules where n
is the number of
reference.tex(,5819) variables of the basering.
reference.tex(,5820) Let @code{list L=nres(M,0);} then @code{L[1]=M} is
identical to the input,
reference.tex(,5821) @code{L[2]} is a minimal set of generators for the first
syzygy
reference.tex(,5822) module of @code{L[1]}, etc.
+reference.tex(,5826) @tex
+reference.tex(,5827) (${\tt L[i]}=M_i$
+reference.tex(,5828) @end tex
reference.tex(,5829) in the notations from above).
reference.tex(,5830) @item @strong{Example:}
reference.tex(,5831) @smallexample
@@ -14773,9 +15329,18 @@
reference.tex(,6638) @table @code
reference.tex(,6639) @item "betti"
reference.tex(,6640) The Betti numbers are printed in a matrix-like format
where the entry
+reference.tex(,6641) @tex
+reference.tex(,6642) $d$ in row $i$ and column $j$
+reference.tex(,6643) @end tex
reference.tex(,6647) is the minimal number of generators in
reference.tex(,6648) degree
+reference.tex(,6649) @tex
+reference.tex(,6650) $i+j$ of the $j$-th
+reference.tex(,6651) @end tex
reference.tex(,6655) syzygy module of
+reference.tex(,6656) @tex
+reference.tex(,6657) $R^n/M$ (the 0th and 1st syzygy module of $R^n/M$ is
$R^n$ and $M$, resp.).
+reference.tex(,6658) @end tex
reference.tex(,6662) @item "%s"
reference.tex(,6663) returns @code{string(} expression @code{)}
reference.tex(,6664) @item "%2s"
@@ -15160,12 +15725,21 @@
reference.tex(,7133) @item @strong{Purpose:}
reference.tex(,7134) computes the ideal quotient, resp.@: module quotient. Let
@code{R} be the
reference.tex(,7135) basering, @code{I,J} ideals and @code{M} a module in
+reference.tex(,7139) @tex
+reference.tex(,7140) ${\tt R}^n$.
+reference.tex(,7141) @end tex
reference.tex(,7142) Then
reference.tex(,7143) @itemize
reference.tex(,7144) @item
reference.tex(,7145) @code{quotient(I,J)}=
+reference.tex(,7149) @tex
+reference.tex(,7150) $\{a \in R \mid aJ \subset I\}$,
+reference.tex(,7151) @end tex
reference.tex(,7152) @item
reference.tex(,7153) @code{quotient(M,J)}=
+reference.tex(,7157) @tex
+reference.tex(,7158) $\{b \in R^n \mid bJ \subset M\}$.
+reference.tex(,7159) @end tex
reference.tex(,7160) @end itemize
reference.tex(,7161) @item @strong{Example:}
reference.tex(,7162) @smallexample
@@ -15355,6 +15929,15 @@
reference.tex(,7410) computes the regularity of a homogeneous ideal, resp.@:
module, from a
reference.tex(,7411) minimal resolution given by the list expression.
reference.tex(,7412) @*
+reference.tex(,7422) @tex
+reference.tex(,7423) \noindent
+reference.tex(,7424) Let $0 \rightarrow\ \bigoplus_a K[x]e_{a,n}\ \rightarrow\
\dots
+reference.tex(,7425) \rightarrow\ \bigoplus_a K[x]e_{a,0}\ \rightarrow\
+reference.tex(,7426) I\ \rightarrow\ 0$
+reference.tex(,7427) be a minimal resolution of I considered with homogeneous
maps of degree 0.
+reference.tex(,7428) The regularity is the smallest number $s$ with the
property deg($e_{a,i})
+reference.tex(,7429) \leq s+i$ for all $i$.
+reference.tex(,7430) @end tex
reference.tex(,7431) @item @strong{Note:}
reference.tex(,7432) If applied to a non minimal resolution only an upper
bound is returned.
reference.tex(,7433) @*If the input to the commands @code{res} and @code{mres}
is homogeneous
@@ -15921,6 +16504,12 @@
reference.tex(,8160) @item @strong{Type:}
reference.tex(,8161) intvec
reference.tex(,8162) @item @strong{Purpose:}
+reference.tex(,8163) @tex
+reference.tex(,8164) computes the permutation {\tt v}
+reference.tex(,8165) which orders the ideal, resp.\ module, {\tt I} by its
initial terms,
+reference.tex(,8166) starting with the smallest, that is, {\tt I(v[i]) <
I(v[i+1])} for all
+reference.tex(,8167) {\tt i}.
+reference.tex(,8168) @end tex
reference.tex(,8175) @item @strong{Example:}
reference.tex(,8176) @smallexample
reference.tex(,8177) @c reused example sortvec reference.doc:5565
@@ -16049,10 +16638,20 @@
reference.tex(,8326) computes a free resolution of an ideal or module with
Schreyer's
reference.tex(,8327) method. The ideal, resp.@: module, has to be a standard
basis.
reference.tex(,8328) More precisely, let M be given by a standard basis and
+reference.tex(,8329) @tex
+reference.tex(,8330) $A_1={\tt matrix}(M)$.
+reference.tex(,8331) @end tex
reference.tex(,8335) Then @code{sres}
reference.tex(,8336) computes a free resolution of
+reference.tex(,8344) @tex
+reference.tex(,8345) $coker(A_1)=F_0/M$
+reference.tex(,8346) $$...\longrightarrow F_2
\buildrel{A_2}\over{\longrightarrow} F_1 \buildrel{A_1}\over{\longrightarrow}
F_0\longrightarrow F_0/M\longrightarrow 0.$$
+reference.tex(,8347) @end tex
reference.tex(,8348) If the int expression k is not zero then the computation
stops after k steps
reference.tex(,8349) and returns a list of modules (given by standard bases)
+reference.tex(,8350) @tex
+reference.tex(,8351) $M_i={\tt module}(A_i)$, i=1..k.
+reference.tex(,8352) @end tex
reference.tex(,8356) @address@hidden(M,0)}
reference.tex(,8357) returns a list of n modules where n is the number of
variables of the basering.
reference.tex(,8358)
@@ -16063,6 +16662,9 @@
reference.tex(,8363) @code{L[2]} is a standard basis with respect to the
Schreyer ordering of
reference.tex(,8364) the first syzygy
reference.tex(,8365) module of @code{L[1]}, etc.
+reference.tex(,8369) @tex
+reference.tex(,8370) (${\tt L[i]}=M_i$
+reference.tex(,8371) @end tex
reference.tex(,8372) in the notations from above.)
reference.tex(,8373) @item @strong{Note:}
reference.tex(,8374) Accessing single elements of a resolution may require
that some partial
@@ -16775,7 +17377,29 @@
reference.tex(,9287) @item @strong{Type:}
reference.tex(,9288) poly
reference.tex(,9289) @item @strong{Purpose:}
+reference.tex(,9303) @tex
+reference.tex(,9304) {\tt vandermonde(p,v,d)} computes the (unique) polynomial
of degree
+reference.tex(,9305) @code{d} with prescribed values {\tt v[1],...,v[N]} at
the points
+reference.tex(,9306) {\tt p}$^0,\dots,$ {\tt p}$^{N-1}$, {\tt N=(d+1)}$^n$,
$n$ the
+reference.tex(,9307) number of ring variables.
+reference.tex(,9308)
+reference.tex(,9309) The returned polynomial is $\sum
+reference.tex(,9310) c_{\alpha_1\ldots\alpha_n}\cdot x_1^{\alpha_1} \cdot
\dots \cdot
+reference.tex(,9311) x_n^{\alpha_n}$, where the coefficients
+reference.tex(,9312) $c_{\alpha_1\ldots\alpha_n}$ are the solution of the
(transposed)
+reference.tex(,9313) Vandermonde system of linear equations
+reference.tex(,9314) $$ \sum_{\alpha_1+\ldots+\alpha_n\leq d}
c_{\alpha_1\ldots\alpha_n} \cdot
+reference.tex(,9315) {\tt p}_1^{(k-1)\alpha_1}\cdot\dots\cdot {\tt
p}_n^{(k-1)\alpha_n} =
+reference.tex(,9316) {\tt v}[k], \quad k=1,\dots,{\tt N}.$$
+reference.tex(,9317) @end tex
reference.tex(,9318) @item @strong{Note:}
+reference.tex(,9326) @tex
+reference.tex(,9327) the ground field has to be the field of rational
+reference.tex(,9328) numbers. Moreover, {\tt ncols(p)==}$n$, the number of
variables in the
+reference.tex(,9329) basering, and all the given generators have to be numbers
different from
+reference.tex(,9330) 0,1 or -1. Finally, {\tt ncols(v)==(d+1)$^n$}, and all
given generators have
+reference.tex(,9331) to be numbers.
+reference.tex(,9332) @end tex
reference.tex(,9333) @item @strong{Example:}
reference.tex(,9334) @smallexample
reference.tex(,9335) @c reused example vandermonde reference.doc:6304
@@ -18421,7 +19045,20 @@
examples.tex(,100)
examples.tex(,101) The Milnor number, resp.@: the Tjurina number, of a power
examples.tex(,102) series f in
+examples.tex(,103) @tex
+examples.tex(,104) $K[[x_1,\ldots,x_n]]$
+examples.tex(,105) @end tex
examples.tex(,109) is
+examples.tex(,116) @tex
+examples.tex(,117) $$
+examples.tex(,118) \hbox{milnor}(f) =
\hbox{dim}_K(K[[x_1,\ldots,x_n]]/\hbox{jacob}(f)),
+examples.tex(,119) $$
+examples.tex(,120) respectively
+examples.tex(,121) $$
+examples.tex(,122) \hbox{tjurina}(f) =
\hbox{dim}_K(K[[x_1,\ldots,x_n]]/((f)+\hbox{jacob}(f)))
+examples.tex(,123) $$
+examples.tex(,124) where
+examples.tex(,125) @end tex
examples.tex(,126) @code{jacob(f)} is the ideal generated by the partials
examples.tex(,127) of @code{f}. @code{tjurina(f)} is finite, if and only if
@code{f} has an
examples.tex(,128) isolated singularity. The same holds for @code{milnor(f)} if
@@ -18430,8 +19067,17 @@
examples.tex(,131)
examples.tex(,132) @sc{Singular} cannot compute with infinite power series.
But it can
examples.tex(,133) work in
+examples.tex(,134) @tex
+examples.tex(,135) $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$,
+examples.tex(,136) @end tex
examples.tex(,140) the localization of
+examples.tex(,141) @tex
+examples.tex(,142) $K[x_1,\ldots,x_n]$
+examples.tex(,143) @end tex
examples.tex(,147) at the maximal ideal
+examples.tex(,148) @tex
+examples.tex(,149) $(x_1,\ldots,x_n)$.
+examples.tex(,150) @end tex
examples.tex(,154) To do this one has to define an
examples.tex(,155) s-ordering like ds, Ds, ls, ws, Ws or an appropriate matrix
examples.tex(,156) ordering (look at the manual to get information about the
possible
@@ -18628,7 +19274,13 @@
examples.tex(,349)
examples.tex(,350) The same computation which computes the Milnor, resp.@: the
Tjurina,
examples.tex(,351) number, but with ordering @code{dp} instead of @code{ds}
(i.e., in
+examples.tex(,352) @tex
+examples.tex(,353) $K[x_1,\ldots,x_n]$
+examples.tex(,354) @end tex
examples.tex(,358) instead of
+examples.tex(,359) @tex
+examples.tex(,360) $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n])$
+examples.tex(,361) @end tex
examples.tex(,365) gives:
examples.tex(,366) @itemize @bullet
examples.tex(,367) @item
@@ -18666,11 +19318,23 @@
examples.tex(,399) @item
examples.tex(,400) The result of the computation here (together with the
previous one in
examples.tex(,401) @ref{Milnor and Tjurina}) shows that (for @code{t}=0)
+examples.tex(,402) @tex
+examples.tex(,403) $\hbox{dim}_K(\hbox{Loc}_{(x,y,z)}K[x,y,z]/\hbox{jacob}(f))$
+examples.tex(,404) @end tex
examples.tex(,408) = 250 (previously computed) while
+examples.tex(,409) @tex
+examples.tex(,410) $\hbox{dim}_K(K[x,y,z]/\hbox{jacob}(f))$
+examples.tex(,411) @end tex
examples.tex(,415) = 536. Hence @code{f} has 286 critical points,
examples.tex(,416) counted with multiplicity, outside the origin.
examples.tex(,417) Moreover, since
+examples.tex(,418) @tex
+examples.tex(,419)
$\hbox{dim}_K(\hbox{Loc}_{(x,y,z)}K[x,y,z]/(\hbox{jacob}(f)+(f)))$
+examples.tex(,420) @end tex
examples.tex(,424) = 195 =
+examples.tex(,425) @tex
+examples.tex(,426) $\hbox{dim}_K(K[x,y,z]/(\hbox{jacob}(f)+(f)))$,
+examples.tex(,427) @end tex
examples.tex(,431) the affine surface @code{f}=0 is smooth outside the origin.
examples.tex(,432) @end itemize
examples.tex(,433)
@@ -18699,27 +19363,72 @@
examples.tex(,461) @cindex Saturation
examples.tex(,462)
examples.tex(,463) Since in the example above, the ideal
+examples.tex(,467) @tex
+examples.tex(,468) $j+(f)$
+examples.tex(,469) @end tex
examples.tex(,470) has the same @code{vdim}
examples.tex(,471) in the polynomial ring and in the localization at 0 (each
195),
examples.tex(,472)
+examples.tex(,476) @tex
+examples.tex(,477) $f=0$
+examples.tex(,478) @end tex
examples.tex(,479) is smooth outside 0.
examples.tex(,480) Hence
+examples.tex(,484) @tex
+examples.tex(,485) $j+(f)$
+examples.tex(,486) @end tex
examples.tex(,487) contains some power of the maximal ideal
+examples.tex(,491) @tex
+examples.tex(,492) $m$
+examples.tex(,493) @end tex
examples.tex(,494) . We shall
examples.tex(,495) check this in a different manner:
examples.tex(,496) For any two ideals
+examples.tex(,500) @tex
+examples.tex(,501) $i, j$
+examples.tex(,502) @end tex
examples.tex(,503) in the basering
+examples.tex(,507) @tex
+examples.tex(,508) $R$
+examples.tex(,509) @end tex
examples.tex(,510) let
+examples.tex(,511) @tex
+examples.tex(,512) $$
+examples.tex(,513) \hbox{sat}(i,j)=\{x\in R\;|\; \exists\;n\hbox{ s.t. }
+examples.tex(,514) x\cdot(j^n)\subseteq i\}
+examples.tex(,515) = \bigcup_{n=1}^\infty i:j^n$$
+examples.tex(,516) @end tex
examples.tex(,521) @*denote the saturation of
+examples.tex(,525) @tex
+examples.tex(,526) $i$
+examples.tex(,527) @end tex
examples.tex(,528) with respect to
+examples.tex(,532) @tex
+examples.tex(,533) $j$
+examples.tex(,534) @end tex
examples.tex(,535) . This defines,
examples.tex(,536) geometrically, the closure of the complement of V(
+examples.tex(,540) @tex
+examples.tex(,541) $j$
+examples.tex(,542) @end tex
examples.tex(,543) ) in V(
+examples.tex(,547) @tex
+examples.tex(,548) $i$
+examples.tex(,549) @end tex
examples.tex(,550) )
examples.tex(,551) (V(
+examples.tex(,555) @tex
+examples.tex(,556) $i$
+examples.tex(,557) @end tex
examples.tex(,558) ) denotes the variety defined by
+examples.tex(,562) @tex
+examples.tex(,563) $i$
+examples.tex(,564) @end tex
examples.tex(,565) ).
examples.tex(,566) In our case,
+examples.tex(,570) @tex
+examples.tex(,571) $sat(j+(f),m)$
+examples.tex(,572) @end tex
examples.tex(,573) must be the whole ring, hence
examples.tex(,574) generated by 1.
examples.tex(,575)
@@ -18859,13 +19568,25 @@
examples.tex(,717) and compute over the ground field Q(t).
examples.tex(,718) We compute the dimension at the generic point,
examples.tex(,725) i.e.,
+examples.tex(,726) @tex
+examples.tex(,727) $dim_{Q(t)}Q(t)[x,y]/j$.
+examples.tex(,728) @end tex
examples.tex(,733) (This gives the
examples.tex(,734) same result as for the deformed ideal above. Hence, the
above small
examples.tex(,735) deformation was "generic".)
examples.tex(,737)
examples.tex(,738) For almost all
+examples.tex(,739) @tex
+examples.tex(,740) $a \in Q$
+examples.tex(,741) @end tex
examples.tex(,745) this is the same as
+examples.tex(,746) @tex
+examples.tex(,747) $dim_Q Q[x,y]/j_0$,
+examples.tex(,748) @end tex
examples.tex(,752) where
+examples.tex(,753) @tex
+examples.tex(,754) $j_0=j|_{t=a}$.
+examples.tex(,755) @end tex
examples.tex(,759)
examples.tex(,760) @smallexample
examples.tex(,761) @c computed example Parameters examples.doc:579
@@ -18899,8 +19620,17 @@
examples.tex(,790) @cindex T2
examples.tex(,791)
examples.tex(,792)
+examples.tex(,796) @tex
+examples.tex(,797) $T^1$
+examples.tex(,798) @end tex
examples.tex(,799) , resp.@:
+examples.tex(,803) @tex
+examples.tex(,804) $T^2$
+examples.tex(,805) @end tex
examples.tex(,806) , of an ideal
+examples.tex(,810) @tex
+examples.tex(,811) $j$
+examples.tex(,812) @end tex
examples.tex(,813) usually denote the modules of
examples.tex(,814) infinitesimal deformations, resp.@: of obstructions.
examples.tex(,815) In @sc{Singular} there are procedures @code{T_1} and
@code{T_2} in
@@ -19070,7 +19800,16 @@
examples.tex(,985) singularity.
examples.tex(,986) @item
examples.tex(,987) The procedure @code{deform} in @code{sing.lib} returns a
matrix whose columns
+examples.tex(,991) @tex
+examples.tex(,992) $h_1,\ldots,h_r$
+examples.tex(,993) @end tex
examples.tex(,994) represent all 1st order deformations. More precisely, if
+examples.tex(,1000) @tex
+examples.tex(,1001) $I \subset R$ is the ideal generated by $f_1,...,f_s$,
then any infinitesimal
+examples.tex(,1002) deformation of $R/I$ over $K[\varepsilon]/(\varepsilon^2)$
is given
+examples.tex(,1003) by $f+\varepsilon g$,
+examples.tex(,1004) where $f=(f_1,...,f_s)$, $g$ a $K$-linear combination of
the $h_i$.
+examples.tex(,1005) @end tex
examples.tex(,1006)
examples.tex(,1007) @item
examples.tex(,1008) The procedure @code{versal} in @code{deform.lib} computes
a formal
@@ -19190,12 +19929,24 @@
examples.tex(,1130) @cindex Finite fields
examples.tex(,1131)
examples.tex(,1132) We define a variety in
+examples.tex(,1136) @tex
+examples.tex(,1137) $n$
+examples.tex(,1138) @end tex
examples.tex(,1139) -space of codimension 2 defined by
examples.tex(,1140) polynomials of degree
+examples.tex(,1144) @tex
+examples.tex(,1145) $d$
+examples.tex(,1146) @end tex
examples.tex(,1147) with generic coefficients over the prime
examples.tex(,1148) field
+examples.tex(,1152) @tex
+examples.tex(,1153) $Z/p$
+examples.tex(,1154) @end tex
examples.tex(,1155) and look for zeros on the torus. First over the prime
examples.tex(,1156) field and then in the finite extension field with
+examples.tex(,1157) @tex
+examples.tex(,1158) $p^k$
+examples.tex(,1159) @end tex
examples.tex(,1163) elements.
examples.tex(,1164) In general there will be many more solutions in the second
case.
examples.tex(,1165) (Since the @sc{Singular} language is interpreted, the
evaluation of many
@@ -19372,9 +20123,24 @@
examples.tex(,1342)
examples.tex(,1343) Elimination is the algebraic counterpart of the geometric
concept of
examples.tex(,1344) projection. If
+examples.tex(,1345) @tex
+examples.tex(,1346) $f=(f_1,\ldots,f_n):k^r\rightarrow k^n$
+examples.tex(,1347) @end tex
examples.tex(,1351) is a polynomial map,
examples.tex(,1352) the Zariski-closure of the image is the zero-set of the
ideal
+examples.tex(,1353) @tex
+examples.tex(,1354) $$
+examples.tex(,1355) \displaylines{
+examples.tex(,1356) j=J \cap k[x_1,\ldots,x_n], \;\quad\hbox{\rm where}\cr
+examples.tex(,1357)
J=(x_1-f_1(t_1,\ldots,t_r),\ldots,x_n-f_n(t_1,\ldots,t_r))\subseteq
+examples.tex(,1358) k[t_1,\ldots,t_r,x_1,\ldots,x_n]
+examples.tex(,1359) }
+examples.tex(,1360) $$
+examples.tex(,1361) @end tex
examples.tex(,1370) i.e, of the ideal j obtained from J by eliminating the
variables
+examples.tex(,1371) @tex
+examples.tex(,1372) $t_1,\ldots,t_r$.
+examples.tex(,1373) @end tex
examples.tex(,1377) This can be done by computing a standard basis of J with
respect to a product
examples.tex(,1378) ordering where the block of t-variables precedes the block
of
examples.tex(,1379) x-variables and then selecting those polynomials which do
not contain
@@ -19386,13 +20152,23 @@
examples.tex(,1385)
examples.tex(,1386) @strong{WARNING:} In the case of a local or a mixed
ordering, elimination needs special
examples.tex(,1387) care. f may be considered as a map of germs
+examples.tex(,1388) @tex
+examples.tex(,1389) $f:(k^r,0)\rightarrow(k^n,0)$,
+examples.tex(,1390) @end tex
examples.tex(,1394) but even
examples.tex(,1395) if this map germ is finite, we are in general not able to
compute the image
examples.tex(,1396) germ because for this we would need an implementation of
the Weierstrass
examples.tex(,1397) preparation theorem. What we can compute, and what
@code{eliminate} actually does,
examples.tex(,1398) is the following: let V(J) be the zero-set of J in
+examples.tex(,1399) @tex
+examples.tex(,1400) $k^r\times(k^n,0)$,
+examples.tex(,1401) @end tex
examples.tex(,1405) then the
examples.tex(,1406) closure of the image of V(J) under the projection
+examples.tex(,1407) @tex
+examples.tex(,1408) $$\hbox{pr}:k^r\times(k^n,0)\rightarrow(k^n,0)$$
+examples.tex(,1409) can be computed.
+examples.tex(,1410) @end tex
examples.tex(,1415) Note that this germ contains also those components
examples.tex(,1416) of V(J) which meet the fiber of pr outside the origin.
examples.tex(,1417) This is achieved by an ordering with the block of
t-variables having a
@@ -19415,6 +20191,9 @@
examples.tex(,1434) @enumerate
examples.tex(,1435) @item
examples.tex(,1436) First we compute the equations of the simple space curve
+examples.tex(,1437) @tex
+examples.tex(,1438) $\hbox{T}[7]^\prime$
+examples.tex(,1439) @end tex
examples.tex(,1443) consisting of two tangential cusps given in parametric
form.
examples.tex(,1444) @item
examples.tex(,1445) We compute weights for the equations such that the
@@ -19422,6 +20201,9 @@
examples.tex(,1447) @item
examples.tex(,1448) Then we compute the tangent developable of the rational
examples.tex(,1449) normal curve in
+examples.tex(,1450) @tex
+examples.tex(,1451) $P^4$.
+examples.tex(,1452) @end tex
examples.tex(,1456) @end enumerate
examples.tex(,1457)
examples.tex(,1458) @smallexample
@@ -19571,11 +20353,20 @@
examples.tex(,1621)
examples.tex(,1622) Now let's look at an example which uses resolutions: The
Hilbert-Burch
examples.tex(,1623) theorem says that the ideal i of a reduced curve in
+examples.tex(,1624) @tex
+examples.tex(,1625) $K^3$
+examples.tex(,1626) @end tex
examples.tex(,1630) has a free resolution of length 2 and that i is given by
the 2x2 minors
examples.tex(,1631) of the 2nd matrix in the resolution.
examples.tex(,1632) We test this for two transversal cusps in
+examples.tex(,1633) @tex
+examples.tex(,1634) $K^3$.
+examples.tex(,1635) @end tex
examples.tex(,1639) Afterwards we compute the resolution of the ideal j of the
tangent developable
examples.tex(,1640) of the rational normal curve in
+examples.tex(,1641) @tex
+examples.tex(,1642) $P^4$
+examples.tex(,1643) @end tex
examples.tex(,1647) from above.
examples.tex(,1648) Finally we demonstrate the use of the type
@code{resolution} in connection with
examples.tex(,1649) the @code{lres} command.
@@ -19696,24 +20487,45 @@
examples.tex(,1765) @cindex Ext
examples.tex(,1766)
examples.tex(,1767) We start by showing how to calculate the
+examples.tex(,1771) @tex
+examples.tex(,1772) $n$
+examples.tex(,1773) @end tex
examples.tex(,1774) -th Ext group of an
examples.tex(,1775) ideal. The ingredients to do this are by the definition of
Ext the
examples.tex(,1776) following: calculate a (minimal) resolution at least up to
length
examples.tex(,1777)
+examples.tex(,1781) @tex
+examples.tex(,1782) $n$
+examples.tex(,1783) @end tex
examples.tex(,1784) , apply the Hom-functor, and calculate the
+examples.tex(,1788) @tex
+examples.tex(,1789) $n$
+examples.tex(,1790) @end tex
examples.tex(,1791) -th homology
examples.tex(,1792) group, that is form the quotient
+examples.tex(,1793) @tex
+examples.tex(,1794) $\hbox{\rm ker} / \hbox{\rm Im}$
+examples.tex(,1795) @end tex
examples.tex(,1799) in the resolution sequence.
examples.tex(,1800)
examples.tex(,1801) The Hom functor is given simply by transposing (hence
dualizing) the
examples.tex(,1802) module or the corresponding matrix with the command
@code{transpose}.
examples.tex(,1803) The image of the
+examples.tex(,1807) @tex
+examples.tex(,1808) $(n-1)$
+examples.tex(,1809) @end tex
examples.tex(,1810) -st map is generated by the columns of the
examples.tex(,1811) corresponding matrix. To calculate the kernel apply the
command
examples.tex(,1812) @code{syz} at the
+examples.tex(,1816) @tex
+examples.tex(,1817) $(n-1)$
+examples.tex(,1818) @end tex
examples.tex(,1819) -st transposed entry of the resolution.
examples.tex(,1820) Finally, the quotient is obtained by the command
@code{modulo}, which
examples.tex(,1821) gives for two modules A = ker, B = Im the module of
relations of
+examples.tex(,1822) @tex
+examples.tex(,1823) $A/(A \cap B)$
+examples.tex(,1824) @end tex
examples.tex(,1828) in the usual way. As we have a chain complex this is
obviously the same
examples.tex(,1829) as ker/Im.
examples.tex(,1830)
@@ -19752,17 +20564,44 @@
examples.tex(,1863) example.
examples.tex(,1864)
examples.tex(,1865) If
+examples.tex(,1869) @tex
+examples.tex(,1870) $M$
+examples.tex(,1871) @end tex
examples.tex(,1872) is a module, then
+examples.tex(,1873) @tex
+examples.tex(,1874) $\hbox{Ext}^1(M,M)$, resp.\ $\hbox{Ext}^2(M,M)$,
+examples.tex(,1875) @end tex
examples.tex(,1879) are the modules of infinitesimal deformations, resp.@: of
obstructions, of
examples.tex(,1880)
+examples.tex(,1884) @tex
+examples.tex(,1885) $M$
+examples.tex(,1886) @end tex
examples.tex(,1887) (like T1 and T2 for a singularity). Similar to the
treatment
examples.tex(,1888) for singularities, the semiuniversal deformation of
+examples.tex(,1892) @tex
+examples.tex(,1893) $M$
+examples.tex(,1894) @end tex
examples.tex(,1895) can be
examples.tex(,1896) computed (if
+examples.tex(,1897) @tex
+examples.tex(,1898) $\hbox{Ext}^1$
+examples.tex(,1899) @end tex
examples.tex(,1903) is finite dimensional) with the help of
+examples.tex(,1904) @tex
+examples.tex(,1905) $\hbox{Ext}^1$, $\hbox{Ext}^2$
+examples.tex(,1906) @end tex
examples.tex(,1910) and the cup product. There is an extra procedure for
+examples.tex(,1911) @tex
+examples.tex(,1912) $\hbox{Ext}^k(R/J,R)$
+examples.tex(,1913) @end tex
examples.tex(,1917) if
+examples.tex(,1921) @tex
+examples.tex(,1922) $J$
+examples.tex(,1923) @end tex
examples.tex(,1924) is an ideal in
+examples.tex(,1928) @tex
+examples.tex(,1929) $R$
+examples.tex(,1930) @end tex
examples.tex(,1931) since this is faster than the
examples.tex(,1932) general Ext.
examples.tex(,1933)
@@ -19770,15 +20609,42 @@
examples.tex(,1935) @itemize @bullet
examples.tex(,1936) @item
examples.tex(,1937) the infinitesimal deformations
+examples.tex(,1938) @tex
+examples.tex(,1939) ($=\hbox{Ext}^1(K,K)$)
+examples.tex(,1940) @end tex
examples.tex(,1944) and obstructions
+examples.tex(,1945) @tex
+examples.tex(,1946) ($=\hbox{Ext}^2(K,K)$)
+examples.tex(,1947) @end tex
examples.tex(,1951) of the residue field
+examples.tex(,1955) @tex
+examples.tex(,1956) $K=R/m$
+examples.tex(,1957) @end tex
examples.tex(,1958) of an ordinary cusp,
+examples.tex(,1959) @tex
+examples.tex(,1960) $R=Loc_m K[x,y]/(x^2-y^3)$, $m=(x,y)$.
+examples.tex(,1961) @end tex
examples.tex(,1965) To compute
+examples.tex(,1966) @tex
+examples.tex(,1967) $\hbox{Ext}^1(m,m)$
+examples.tex(,1968) @end tex
examples.tex(,1972) we have to apply @code{Ext(1,syz(m),syz(m))} with
examples.tex(,1973) @code{syz(m)} the first syzygy module of
+examples.tex(,1977) @tex
+examples.tex(,1978) $m$
+examples.tex(,1979) @end tex
examples.tex(,1980) , which is isomorphic to
+examples.tex(,1981) @tex
+examples.tex(,1982) $\hbox{Ext}^2(K,K)$.
+examples.tex(,1983) @end tex
examples.tex(,1987) @item
+examples.tex(,1988) @tex
+examples.tex(,1989) $\hbox{Ext}^k(R/i,R)$
+examples.tex(,1990) @end tex
examples.tex(,1994) for some ideal
+examples.tex(,1998) @tex
+examples.tex(,1999) $i$
+examples.tex(,2000) @end tex
examples.tex(,2001) and with an extra option.
examples.tex(,2002) @end itemize
examples.tex(,2003)
@@ -19874,18 +20740,45 @@
examples.tex(,2095) @cindex Polar curves
examples.tex(,2096)
examples.tex(,2097) The polar curve of a hypersurface given by a polynomial
+examples.tex(,2098) @tex
+examples.tex(,2099) $f\in k[x_1,\ldots,x_n,t]$
+examples.tex(,2100) @end tex
examples.tex(,2104) with respect to
+examples.tex(,2108) @tex
+examples.tex(,2109) $t$
+examples.tex(,2110) @end tex
examples.tex(,2111) (we may consider
+examples.tex(,2115) @tex
+examples.tex(,2116) $f=0$
+examples.tex(,2117) @end tex
examples.tex(,2118) as a family of
examples.tex(,2119) hypersurfaces parametrized by
+examples.tex(,2123) @tex
+examples.tex(,2124) $t$
+examples.tex(,2125) @end tex
examples.tex(,2126) ) is defined as the Zariski
examples.tex(,2127) closure of
+examples.tex(,2128) @tex
+examples.tex(,2129) $V(\partial f/\partial x_1,\ldots,\partial f/\partial x_n)
\setminus V(f)$
+examples.tex(,2130) @end tex
examples.tex(,2134) if this happens to be a curve. Some authors consider
+examples.tex(,2135) @tex
+examples.tex(,2136) $V(\partial f/\partial x_1,\ldots,\partial f/\partial x_n)$
+examples.tex(,2137) @end tex
examples.tex(,2141) itself as polar curve.
examples.tex(,2142)
examples.tex(,2143) We may consider projective hypersurfaces
+examples.tex(,2144) @tex
+examples.tex(,2145) (in $P^n$),
+examples.tex(,2146) @end tex
examples.tex(,2150) affine hypersurfaces
+examples.tex(,2151) @tex
+examples.tex(,2152) (in $k^n$)
+examples.tex(,2153) @end tex
examples.tex(,2157) or germs of hypersurfaces
+examples.tex(,2158) @tex
+examples.tex(,2159) (in $(k^n,0)$),
+examples.tex(,2160) @end tex
examples.tex(,2164) getting in this way
examples.tex(,2165) projective, affine or local polar curves.
examples.tex(,2166)
@@ -19998,12 +20891,24 @@
examples.tex(,2275) @cindex Depth
examples.tex(,2276)
examples.tex(,2277) We compute the depth of the module of Kaehler differentials
+examples.tex(,2278) @tex
+examples.tex(,2279) D$_k$(R)
+examples.tex(,2280) @end tex
examples.tex(,2284) of the variety defined by the
+examples.tex(,2288) @tex
+examples.tex(,2289) $(m+1)$
+examples.tex(,2290) @end tex
examples.tex(,2291) -minors of a generic symmetric
+examples.tex(,2292) @tex
+examples.tex(,2293) $(n \times n)$-matrix.
+examples.tex(,2294) @end tex
examples.tex(,2298) We do this by computing the resolution over the polynomial
examples.tex(,2299) ring. Then, by the Auslander-Buchsbaum formula, the depth
is equal to
examples.tex(,2300) the number of variables minus the length of a minimal
resolution. This
examples.tex(,2301) example was suggested by U.@: Vetter in order to check
whether his bound
+examples.tex(,2302) @tex
+examples.tex(,2303) $\hbox{depth}(\hbox{D}_k(R))\geq m(m+1)/2 + m-1$
+examples.tex(,2304) @end tex
examples.tex(,2308) could be improved.
examples.tex(,2309)
examples.tex(,2310) @smallexample
@@ -20172,6 +21077,9 @@
examples.tex(,2482)
examples.tex(,2483) We work in characteristic 0 and use the Lie algebra
generated by one
examples.tex(,2484) vector field of the form
+examples.tex(,2485) @tex
+examples.tex(,2486) $\sum x_i \partial /\partial x_{i+1}$.
+examples.tex(,2487) @end tex
examples.tex(,2491) @smallexample
examples.tex(,2492) @c computed example G_a_-Invariants examples.doc:1783
examples.tex(,2493) LIB "ainvar.lib";
@@ -20337,6 +21245,9 @@
examples.tex(,2662)
examples.tex(,2663) We compute the Hamburger-Noether expansion of a plane curve
examples.tex(,2664) singularity given by a polynomial
+examples.tex(,2668) @tex
+examples.tex(,2669) $f$
+examples.tex(,2670) @end tex
examples.tex(,2671) in two variables. This is a
examples.tex(,2672) matrix which allows to compute the parametrization (up to
a given order)
examples.tex(,2673) and all numerical invariants like the
@@ -20354,7 +21265,13 @@
examples.tex(,2685) @end itemize
examples.tex(,2686) Besides this, the library contains procedures to compute
the Newton
examples.tex(,2687) polygon of
+examples.tex(,2691) @tex
+examples.tex(,2692) $f$
+examples.tex(,2693) @end tex
examples.tex(,2694) , the squarefree part of
+examples.tex(,2698) @tex
+examples.tex(,2699) $f$
+examples.tex(,2700) @end tex
examples.tex(,2701) and a procedure to
examples.tex(,2702) convert one set of invariants to another.
examples.tex(,2703)
@@ -20579,9 +21496,15 @@
examples.tex(,2926) @section Normalization
examples.tex(,2927) @cindex Normalization
examples.tex(,2928) The normalization will be computed for a reduced ring
+examples.tex(,2932) @tex
+examples.tex(,2933) $R/I$
+examples.tex(,2934) @end tex
examples.tex(,2935) . The
examples.tex(,2936) result is a list of rings; ideals are always called
@code{norid} in the
examples.tex(,2937) rings of this list. The normalization of
+examples.tex(,2941) @tex
+examples.tex(,2942) $R/I$
+examples.tex(,2943) @end tex
examples.tex(,2944) is the product of
examples.tex(,2945) the factor rings of the rings in the list divided out by
the ideals
examples.tex(,2946) @code{norid}.
@@ -20785,12 +21708,41 @@
examples.tex(,3152) @section Kernel of module homomorphisms
examples.tex(,3153) @cindex Kernel of module homomorphisms
examples.tex(,3154) Let
+examples.tex(,3158) @tex
+examples.tex(,3159) $A$
+examples.tex(,3160) @end tex
examples.tex(,3161) ,
+examples.tex(,3165) @tex
+examples.tex(,3166) $B$
+examples.tex(,3167) @end tex
examples.tex(,3168) be two matrices of size
+examples.tex(,3169) @tex
+examples.tex(,3170) $m\times r$ and $m\times s$
+examples.tex(,3171) @end tex
examples.tex(,3175) over the ring
+examples.tex(,3179) @tex
+examples.tex(,3180) $R$
+examples.tex(,3181) @end tex
examples.tex(,3182) and consider the corresponding maps
+examples.tex(,3183) @tex
+examples.tex(,3184) $$
+examples.tex(,3185) R^r \buildrel{A}\over{\longrightarrow}
+examples.tex(,3186) R^m \buildrel{B}\over{\longleftarrow} R^s\;.
+examples.tex(,3187) $$
+examples.tex(,3188) @end tex
examples.tex(,3202) We want to compute the kernel of the map
+examples.tex(,3203) @tex
+examples.tex(,3204) $R^r \buildrel{A}\over{\longrightarrow}
+examples.tex(,3205) R^m\longrightarrow
+examples.tex(,3206) R^m/\hbox{Im}(B) \;.$
+examples.tex(,3207) @end tex
examples.tex(,3216) This can be done using the @code{modulo} command:
+examples.tex(,3217) @tex
+examples.tex(,3218) $$
+examples.tex(,3219) \hbox{\tt modulo}(A,B)=\hbox{ker}(R^r
+examples.tex(,3220) \buildrel{A}\over{\longrightarrow}R^m/\hbox{Im}(B)) \; .
+examples.tex(,3221) $$
+examples.tex(,3222) @end tex
examples.tex(,3231)
examples.tex(,3232) @smallexample
examples.tex(,3233) @c computed example Kernel_of_module_homomorphisms
examples.doc:2196
@@ -20808,22 +21760,56 @@
examples.tex(,3250) @section Algebraic dependence
examples.tex(,3251) @cindex Algebraic dependence
examples.tex(,3252) Let
+examples.tex(,3253) @tex
+examples.tex(,3254) $g$, $f_1$, \dots, $f_r\in K[x_1,\ldots,x_n]$.
+examples.tex(,3255) @end tex
examples.tex(,3259) We want to check whether
examples.tex(,3260) @enumerate
examples.tex(,3261) @item
+examples.tex(,3262) @tex
+examples.tex(,3263) $f_1$, \dots, $f_r$
+examples.tex(,3264) @end tex
examples.tex(,3268) are algebraically dependent.
examples.tex(,3269)
examples.tex(,3270) Let
+examples.tex(,3271) @tex
+examples.tex(,3272) $I=\langle Y_1-f_1,\ldots,Y_r-f_r \rangle \subseteq
+examples.tex(,3273) K[x_1,\ldots,x_n,Y_1,\ldots,Y_r]$.
+examples.tex(,3274) @end tex
examples.tex(,3282) Then
+examples.tex(,3283) @tex
+examples.tex(,3284) $I \cap K[Y_1,\ldots,Y_r]$
+examples.tex(,3285) @end tex
examples.tex(,3289) are the algebraic relations between
+examples.tex(,3290) @tex
+examples.tex(,3291) $f_1$, \dots, $f_r$.
+examples.tex(,3292) @end tex
examples.tex(,3296)
examples.tex(,3297) @item
+examples.tex(,3298) @tex
+examples.tex(,3299) $g \in K [f_1,\ldots,f_r]$.
+examples.tex(,3300) @end tex
examples.tex(,3304)
+examples.tex(,3305) @tex
+examples.tex(,3306) $g \in K[f_1,\ldots,f_r]$
+examples.tex(,3307) @end tex
examples.tex(,3311) if and only if the normal form of
+examples.tex(,3315) @tex
+examples.tex(,3316) $g$
+examples.tex(,3317) @end tex
examples.tex(,3318) with respect to
+examples.tex(,3322) @tex
+examples.tex(,3323) $I$
+examples.tex(,3324) @end tex
examples.tex(,3325) and a
examples.tex(,3326) block ordering with respect to
+examples.tex(,3327) @tex
+examples.tex(,3328) $X=(x_1,\ldots,x_n)$ and $Y=(Y_1,\ldots,Y_r)$ with $X>Y$
+examples.tex(,3329) @end tex
examples.tex(,3333) is in
+examples.tex(,3337) @tex
+examples.tex(,3338) $K[Y]$
+examples.tex(,3339) @end tex
examples.tex(,3340) .
examples.tex(,3341) @end enumerate
examples.tex(,3342)
@@ -21164,35 +22150,83 @@
pdata.tex(,53) A vector in @sc{Singular} is always an element of a free
module over the
pdata.tex(,54) basering. It is given as a list of polynomials in one of the
following
pdata.tex(,55) formats
+pdata.tex(,56) @tex
+pdata.tex(,57) $[f_1,...,f_n]$ or $f_1*gen(1)+...+f_n*gen(n)$, where $gen(i)$
+pdata.tex(,58) @end tex
pdata.tex(,62) denotes the i-th canonical generator of a free module (with 1
at place i and
pdata.tex(,63) 0 everywhere else).
pdata.tex(,64) Both forms are equivalent. A vector is internally represented in
pdata.tex(,65) the second form with the
+pdata.tex(,66) @tex
+pdata.tex(,67) $gen(i)$
+pdata.tex(,68) @end tex
pdata.tex(,72) being "special" ring variables, ordered accordingly to the
monomial ordering.
pdata.tex(,73) Therefore, the form
+pdata.tex(,74) @tex
+pdata.tex(,75) $[f_1,...,f_n]$
+pdata.tex(,76) @end tex
pdata.tex(,80) is given as output only if the monomial ordering gives priority
to the
pdata.tex(,81) component, i.e@:., is of the form @code{(c,...)} (see
@ref{Module
pdata.tex(,82) orderings}). However, in any case the procedure @code{show}
from the
pdata.tex(,83) library @code{inout.lib} displays the bracket format.
pdata.tex(,84)
pdata.tex(,85) A vector
+pdata.tex(,86) @tex
+pdata.tex(,87) $v=[f_1,...,f_n]$
+pdata.tex(,88) @end tex
pdata.tex(,92) should always be considered as a column vector in a free module
pdata.tex(,93) of rank equal to
+pdata.tex(,94) @tex
+pdata.tex(,95) nrows($v$)
+pdata.tex(,96) @end tex
pdata.tex(,100) where
+pdata.tex(,101) @tex
+pdata.tex(,102) nrows($v$)
+pdata.tex(,103) @end tex
pdata.tex(,107) is equal to the maximal index
+pdata.tex(,108) @tex
+pdata.tex(,109) $r$
+pdata.tex(,110) @end tex
pdata.tex(,114) such that
+pdata.tex(,115) @tex
+pdata.tex(,116) $f_r \not= 0$.
+pdata.tex(,117) @end tex
pdata.tex(,121) This is due to the fact, that internally
+pdata.tex(,122) @tex
+pdata.tex(,123) $v$
+pdata.tex(,124) @end tex
pdata.tex(,128) is a polynomial in a sparse representation, i.e.,
+pdata.tex(,129) @tex
+pdata.tex(,130) $f_i*gen(i)$
+pdata.tex(,131) @end tex
pdata.tex(,135) is not stored if
+pdata.tex(,136) @tex
+pdata.tex(,137) $f_i=0$
+pdata.tex(,138) @end tex
pdata.tex(,142) (for reasons of efficiency), hence the last 0-entries of
+pdata.tex(,143) @tex
+pdata.tex(,144) $v$
+pdata.tex(,145) @end tex
pdata.tex(,149) are lost.
pdata.tex(,150) Only more complex structures are able to keep the rank.
pdata.tex(,151)
pdata.tex(,152) A module
+pdata.tex(,153) @tex
+pdata.tex(,154) $M$
+pdata.tex(,155) @end tex
pdata.tex(,159) in @sc{Singular} is given by a list of vectors
+pdata.tex(,160) @tex
+pdata.tex(,161) $v_1,...,v_k$
+pdata.tex(,162) @end tex
pdata.tex(,166) which generate the module as a submodule of the free module of
rank
pdata.tex(,167) equal to
+pdata.tex(,168) @tex
+pdata.tex(,169) nrows($M$)
+pdata.tex(,170) @end tex
pdata.tex(,174) which is the maximum of
+pdata.tex(,175) @tex
+pdata.tex(,176) nrows($v_i$).
+pdata.tex(,177) @end tex
pdata.tex(,181)
pdata.tex(,182) If one wants to create a module with a larger rank than given
by its
pdata.tex(,183) generators, one has to use the command
@code{attrib(M,"rank",r)} (see
@@ -21207,33 +22241,84 @@
pdata.tex(,192) By the above remarks it might appear that @sc{Singular} is
only able to handle
pdata.tex(,193) submodules of a free module. However, this is not true.
@sc{Singular}
pdata.tex(,194) can compute with any finitely generated module over the
basering
+pdata.tex(,195) @tex
+pdata.tex(,196) $R$.
+pdata.tex(,197) @end tex
pdata.tex(,201) Such a module, say
+pdata.tex(,202) @tex
+pdata.tex(,203) $N$,
+pdata.tex(,204) @end tex
pdata.tex(,208) is not represented by its generators but by its
pdata.tex(,209) (generators and) relations. This means that
+pdata.tex(,210) @tex
+pdata.tex(,211) $N = R^n/M$ where $n$
+pdata.tex(,212) @end tex
pdata.tex(,216) is the number of generators of
+pdata.tex(,217) @tex
+pdata.tex(,218) $N$ and $M \subseteq R^n$
+pdata.tex(,219) @end tex
pdata.tex(,223) is the module of relations.
pdata.tex(,224) In other words, defining a module
+pdata.tex(,225) @tex
+pdata.tex(,226) $M$
+pdata.tex(,227) @end tex
pdata.tex(,231) as a submodule of a free module
+pdata.tex(,232) @tex
+pdata.tex(,233) $R^n$
+pdata.tex(,234) @end tex
pdata.tex(,238) can also be considered as the definition of
+pdata.tex(,239) @tex
+pdata.tex(,240) $N = R^n/M$.
+pdata.tex(,241) @end tex
pdata.tex(,245)
pdata.tex(,246) Note that most functions, when applied to a module
+pdata.tex(,247) @tex
+pdata.tex(,248) $M$,
+pdata.tex(,249) @end tex
pdata.tex(,253) really deal with
+pdata.tex(,254) @tex
+pdata.tex(,255) $M$.
+pdata.tex(,256) @end tex
pdata.tex(,260) However, there are some functions which deal with
+pdata.tex(,261) @tex
+pdata.tex(,262) $N = R^n/M$ instead of $M$.
+pdata.tex(,263) @end tex
pdata.tex(,267)
pdata.tex(,268) For example, @code{std(M)} computes a standard basis of
+pdata.tex(,269) @tex
+pdata.tex(,270) $M$
+pdata.tex(,271) @end tex
pdata.tex(,275) (and thus gives another representation of
+pdata.tex(,276) @tex
+pdata.tex(,277) $N$ as $N = R^n/$std($M$)).
+pdata.tex(,278) @end tex
pdata.tex(,282) However, @code{dim(M)}, resp.@: @code{vdim(M)}, returns
+pdata.tex(,283) @tex
+pdata.tex(,284) dim$(R^n/M)$, resp.@: dim$_k(R^n/M)$
+pdata.tex(,285) @end tex
pdata.tex(,289) (if M is given by a standard basis).
pdata.tex(,290)
pdata.tex(,291) The function @code{syz(M)} returns the first syzygy module of
+pdata.tex(,292) @tex
+pdata.tex(,293) $M$,
+pdata.tex(,294) @end tex
pdata.tex(,298) i.e@:., the module
pdata.tex(,299) of relations of the given generators of
+pdata.tex(,300) @tex
+pdata.tex(,301) $M$
+pdata.tex(,302) @end tex
pdata.tex(,306) which is equal to the second syzygy module of
+pdata.tex(,307) @tex
+pdata.tex(,308) $N$.
+pdata.tex(,309) @end tex
pdata.tex(,313) Refer to the description of each function in
pdata.tex(,314) @ref{Functions} to get information which module the function
deals with.
pdata.tex(,315)
pdata.tex(,316) The numbering in @code{res} and other commands for computing
resolutions
pdata.tex(,317) refers to a resolution of
+pdata.tex(,318) @tex
+pdata.tex(,319) $N = R^n/M$
+pdata.tex(,320) @end tex
pdata.tex(,324) (see @ref{res}; @ref{Syzygies and resolutions}).
pdata.tex(,325)
pdata.tex(,326) It is possible to compute in any field which is a valid ground
field in
@@ -21277,13 +22362,28 @@
pdata.tex(,364) flexibility might also be confusing for the novice user.
Therefore, we
pdata.tex(,365) recommend to those not familiar with monomial orderings to
generally use
pdata.tex(,366) the ordering @code{dp} for computations in the polynomial ring
+pdata.tex(,367) @tex
+pdata.tex(,368) $K[x_1,\ldots,x_n]$,
+pdata.tex(,369) @end tex
pdata.tex(,373) resp.@: @code{ds} for computations in the localization
+pdata.tex(,374) @tex
+pdata.tex(,375) $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$.
+pdata.tex(,376) @end tex
pdata.tex(,380)
pdata.tex(,381) For inhomogeneous input ideals, standard (resp.@: groebner)
bases
pdata.tex(,382) computations are generally faster
pdata.tex(,383) with the orderings
+pdata.tex(,384) @tex
+pdata.tex(,385) $\hbox{Wp}(w_1, \ldots, w_n)$
+pdata.tex(,386) @end tex
pdata.tex(,390) (resp.@:
+pdata.tex(,391) @tex
+pdata.tex(,392) $\hbox{Ws}(w_1, \ldots, w_n)$)
+pdata.tex(,393) @end tex
pdata.tex(,397) if the input is quasihomogeneous w.r.t. the weights
+pdata.tex(,398) @tex
+pdata.tex(,399) $w_1$, $\ldots$, $w_n$ of $x_1$, $\ldots$, $x_n$.
+pdata.tex(,400) @end tex
pdata.tex(,404)
pdata.tex(,405) If the output needs to be "triangular" (resp.@:
"block-triangular"), the
pdata.tex(,406) lexicographical ordering @code{lp} (resp.@: lexicographical
@@ -21298,12 +22398,39 @@
pdata.tex(,415) @cindex term orderings
pdata.tex(,416) @cindex monomial orderings
pdata.tex(,417)
+pdata.tex(,418) @tex
+pdata.tex(,419) A monomial ordering (term ordering) on $K[x_1, \ldots, x_n]$ is
+pdata.tex(,420) a total ordering $<$ on the
+pdata.tex(,421) set of monomials (power products) $\{x^\alpha \mid \alpha \in
\bf{N}^n\}$
+pdata.tex(,422) which is compatible with the
+pdata.tex(,423) natural semigroup structure, i.e., $x^\alpha < x^\beta$
implies $x^\gamma
+pdata.tex(,424) x^\alpha < x^\gamma x^\beta$ for any $\gamma \in \bf{N}^n$.
+pdata.tex(,425) We do not require
+pdata.tex(,426) $<$ to be a well ordering.
+pdata.tex(,427) @end tex
pdata.tex(,439) See the literature cited in @ref{References}.
pdata.tex(,441)
pdata.tex(,442) It is known that any monomial ordering can be represented by a
matrix
+pdata.tex(,443) @tex
+pdata.tex(,444) $M$ in $GL(n,R)$,
+pdata.tex(,445) @end tex
pdata.tex(,449) but, of course, only integer coefficients are of relevance in
pdata.tex(,450) practice.
pdata.tex(,451)
+pdata.tex(,452) @tex
+pdata.tex(,453) Global orderings are well orderings (i.e., \hbox{$1 < x_i$}
for each variable
+pdata.tex(,454) $x_i$), local orderings satisfy $1 > x_i$ for each variable.
If some variables are ordered globally and others locally we
+pdata.tex(,455) call it a mixed ordering. Local or mixed orderings are not
well orderings.
+pdata.tex(,456)
+pdata.tex(,457) Let $K$ be the ground field, \hbox{$x = (x_1, \ldots, x_n)$}
the
+pdata.tex(,458) variables and $<$ a monomial ordering, then Loc $K[x]$ denotes
the
+pdata.tex(,459) localization of $K[x]$ with respect to the multiplicatively
closed set $$\{1 +
+pdata.tex(,460) g \mid g = 0 \hbox{ or } g \in K[x]\backslash \{0\} \hbox{ and
}L(g) <
+pdata.tex(,461) 1\}.$$ Here, $L(g)$
+pdata.tex(,462) denotes the leading monomial of $g$, i.e., the biggest
monomial of $g$ with
+pdata.tex(,463) respect to $<$. The result of any computation which uses
standard basis
+pdata.tex(,464) computations has to be interpreted in Loc $K[x]$.
+pdata.tex(,465) @end tex
pdata.tex(,480)
pdata.tex(,481) Note that the definition of a ring includes the definition of
its
pdata.tex(,482) monomial ordering (see
@@ -21317,6 +22444,9 @@
pdata.tex(,490) @cindex Global orderings
pdata.tex(,491) @cindex orderings, global
pdata.tex(,492)
+pdata.tex(,493) @tex
+pdata.tex(,494) For all these orderings: Loc $K[x]$ = $K[x]$
+pdata.tex(,495) @end tex
pdata.tex(,499)
pdata.tex(,500) @table @asis
pdata.tex(,501) @item lp:
@@ -21324,35 +22454,81 @@
pdata.tex(,503) @cindex lp, global ordering
pdata.tex(,504) @cindex lexicographical ordering
pdata.tex(,505) @*
+pdata.tex(,510) @tex
+pdata.tex(,511) $x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
+pdata.tex(,512) \alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1},
\alpha_i <
+pdata.tex(,513) \beta_i$.
+pdata.tex(,514) @end tex
pdata.tex(,515) @item rp:
pdata.tex(,516) reverse lexicographical ordering:
pdata.tex(,517) @cindex rp, global ordering
pdata.tex(,518) @cindex reverse lexicographical ordering
pdata.tex(,519) @*
+pdata.tex(,524) @tex
+pdata.tex(,525) $x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
+pdata.tex(,526) \alpha_n = \beta_n,
+pdata.tex(,527) \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
+pdata.tex(,528) @end tex
pdata.tex(,529) @item dp:
pdata.tex(,530) degree reverse lexicographical ordering:
pdata.tex(,531) @cindex degree reverse lexicographical ordering
pdata.tex(,532) @cindex dp, global ordering
pdata.tex(,533) @*
+pdata.tex(,537) @tex
+pdata.tex(,538) let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
+pdata.tex(,539) @end tex
+pdata.tex(,547) @tex
+pdata.tex(,548) $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) <
\deg(x^\beta)$ or
+pdata.tex(,549) @end tex
+pdata.tex(,553) @tex
+pdata.tex(,554) \phantom{$x^\alpha < x^\beta \Leftrightarrow $}$
\deg(x^\alpha) =
+pdata.tex(,555) \deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n =
\beta_n,
+pdata.tex(,556) \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
+pdata.tex(,557) @end tex
pdata.tex(,558) @item Dp:
pdata.tex(,559) degree lexicographical ordering:
pdata.tex(,560) @cindex degree lexicographical ordering
pdata.tex(,561) @cindex Dp, global ordering
pdata.tex(,562) @*
+pdata.tex(,566) @tex
+pdata.tex(,567) let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
+pdata.tex(,568) @end tex
+pdata.tex(,576) @tex
+pdata.tex(,577) $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) <
\deg(x^\beta)$ or
+pdata.tex(,578) @end tex
+pdata.tex(,582) @tex
+pdata.tex(,583) \phantom{ $x^\alpha < x^\beta \Leftrightarrow $}
$\deg(x^\alpha) =
+pdata.tex(,584) \deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 =
\beta_1,
+pdata.tex(,585) \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$
+pdata.tex(,586) @end tex
pdata.tex(,587) @item wp:
pdata.tex(,588) weighted reverse lexicographical ordering:
pdata.tex(,589) @cindex weighted reverse lexicographical ordering
pdata.tex(,590) @cindex wp, global ordering
pdata.tex(,591) @*
+pdata.tex(,595) @tex
+pdata.tex(,596) let $w_1, \ldots, w_n$ be positive integers. Then ${\tt
wp}(w_1, \ldots,
+pdata.tex(,597) w_n)$
+pdata.tex(,598) @end tex
pdata.tex(,599) is defined as @code{dp}
pdata.tex(,600) but with
+pdata.tex(,604) @tex
+pdata.tex(,605) $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
+pdata.tex(,606) @end tex
pdata.tex(,607) @item Wp:
pdata.tex(,608) weighted lexicographical ordering:
pdata.tex(,609) @cindex weighted lexicographical ordering
pdata.tex(,610) @cindex WP, global ordering
pdata.tex(,611) @*
+pdata.tex(,615) @tex
+pdata.tex(,616) let $w_1, \ldots, w_n$ be positive integers. Then ${\tt
Wp}(w_1, \ldots,
+pdata.tex(,617) w_n)$
+pdata.tex(,618) @end tex
pdata.tex(,619) is defined as @code{Dp}
pdata.tex(,620) but with
+pdata.tex(,624) @tex
+pdata.tex(,625) $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
+pdata.tex(,626) @end tex
pdata.tex(,627) @end table
pdata.tex(,628) @c
--------------------------------------------------------------------------
pdata.tex(,629) @node Local orderings, Module orderings, Global orderings,
Monomial orderings
@@ -21362,8 +22538,17 @@
pdata.tex(,633)
pdata.tex(,634) For ls, ds, Ds and, if the weights are positive integers, also
for ws and
pdata.tex(,635) Ws, we have
+pdata.tex(,639) @tex
+pdata.tex(,640) Loc $K[x]$ = $K[x]_{(x)}$,
+pdata.tex(,641) @end tex
pdata.tex(,642) the localization of
+pdata.tex(,643) @tex
+pdata.tex(,644) $K[x]$
+pdata.tex(,645) @end tex
pdata.tex(,649) at the maximal ideal
+pdata.tex(,653) @tex
+pdata.tex(,654) \ $(x_1, ..., x_n)$.
+pdata.tex(,655) @end tex
pdata.tex(,656)
pdata.tex(,657) @table @asis
pdata.tex(,658) @item ls:
@@ -21371,36 +22556,81 @@
pdata.tex(,660) @cindex negative lexicographical ordering
pdata.tex(,661) @cindex ls, local ordering
pdata.tex(,662) @*
+pdata.tex(,667) @tex
+pdata.tex(,668) $x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
+pdata.tex(,669) \alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1},
\alpha_i >
+pdata.tex(,670) \beta_i$.
+pdata.tex(,671) @end tex
pdata.tex(,672) @item ds:
pdata.tex(,673) negative degree reverse lexicographical ordering:
pdata.tex(,674) @cindex negative degree reverse lexicographical ordering
pdata.tex(,675) @cindex ds, local ordering
pdata.tex(,676) @*
+pdata.tex(,680) @tex
+pdata.tex(,681) let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
+pdata.tex(,682) @end tex
+pdata.tex(,690) @tex
+pdata.tex(,691) $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) >
\deg(x^\beta)$ or
+pdata.tex(,692) @end tex
+pdata.tex(,696) @tex
+pdata.tex(,697) \phantom{ $x^\alpha < x^\beta \Leftrightarrow$}$
\deg(x^\alpha) =
+pdata.tex(,698) \deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n =
\beta_n,
+pdata.tex(,699) \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
+pdata.tex(,700) @end tex
pdata.tex(,701) @item Ds:
pdata.tex(,702) negative degree lexicographical ordering:
pdata.tex(,703) @cindex negative degree lexicographical ordering
pdata.tex(,704) @cindex Ds, local ordering
pdata.tex(,705) @*
+pdata.tex(,709) @tex
+pdata.tex(,710) let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
+pdata.tex(,711) @end tex
+pdata.tex(,719) @tex
+pdata.tex(,720) $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) >
\deg(x^\beta)$ or
+pdata.tex(,721) @end tex
+pdata.tex(,725) @tex
+pdata.tex(,726) \phantom{ $ x^\alpha < x^\beta \Leftrightarrow$}$
\deg(x^\alpha) =
+pdata.tex(,727) \deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 =
\beta_1,
+pdata.tex(,728) \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$
+pdata.tex(,729) @end tex
pdata.tex(,730) @item ws:
pdata.tex(,731) (general) weighted reverse lexicographical ordering:
pdata.tex(,732) @cindex general weighted reverse lexicographical ordering
pdata.tex(,733) @cindex local weighted reverse lexicographical ordering
pdata.tex(,734) @cindex ws, local ordering
pdata.tex(,735) @*
+pdata.tex(,739) @tex
+pdata.tex(,740) ${\tt ws}(w_1, \ldots, w_n),\; w_1$
+pdata.tex(,741) @end tex
pdata.tex(,742) a nonzero integer,
+pdata.tex(,746) @tex
+pdata.tex(,747) $w_2,\ldots,w_n$
+pdata.tex(,748) @end tex
pdata.tex(,749) any integer (including 0),
pdata.tex(,750) is defined as @code{ds}
pdata.tex(,751) but with
+pdata.tex(,755) @tex
+pdata.tex(,756) $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
+pdata.tex(,757) @end tex
pdata.tex(,758) @item Ws:
pdata.tex(,759) (general) weighted lexicographical ordering:
pdata.tex(,760) @cindex general weighted lexicographical ordering
pdata.tex(,761) @cindex local weighted lexicographical ordering
pdata.tex(,762) @cindex Ws, local ordering
pdata.tex(,763) @*
+pdata.tex(,767) @tex
+pdata.tex(,768) ${\tt Ws}(w_1, \ldots, w_n),\; w_1$
+pdata.tex(,769) @end tex
pdata.tex(,770) a nonzero integer,
+pdata.tex(,774) @tex
+pdata.tex(,775) $w_2,\ldots,w_n$
+pdata.tex(,776) @end tex
pdata.tex(,777) any integer (including 0),
pdata.tex(,778) is defined as @code{Ds}
pdata.tex(,779) but with
+pdata.tex(,783) @tex
+pdata.tex(,784) $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
+pdata.tex(,785) @end tex
pdata.tex(,786) @end table
pdata.tex(,787)
pdata.tex(,788) @c
--------------------------------------------------------------------------
@@ -21409,22 +22639,42 @@
pdata.tex(,791) @cindex Module orderings
pdata.tex(,792)
pdata.tex(,793) @sc{Singular} offers also orderings on the set of ``monomials''
+pdata.tex(,799) @tex
+pdata.tex(,800) $\{ x^a e_i \mid a \in N^n, 1 \leq i \leq r \}$ in Loc
$K[x]^r$ = Loc
+pdata.tex(,801) $K[x]e_1
+pdata.tex(,802) + \ldots +$Loc $K[x]e_r$, where $e_1, \ldots, e_r$ denote the
canonical
+pdata.tex(,803) generators of Loc $K[x]^r$, the r-fold direct sum of Loc
$K[x]$.
+pdata.tex(,804) (The function {\tt gen(i)} yields $e_i$).
+pdata.tex(,805) @end tex
pdata.tex(,806)
pdata.tex(,807) We have two possibilities: either to give priority to the
component of a
pdata.tex(,808) vector in
+pdata.tex(,812) @tex
+pdata.tex(,813) Loc $K[x]^r$
+pdata.tex(,814) @end tex
pdata.tex(,815) or (which is the default in @sc{Singular}) to give priority
pdata.tex(,816) to the coefficients.
pdata.tex(,817) The orderings @code{(<,c)} and @code{(<,C)} give priority to
the
pdata.tex(,818) coefficients; whereas
pdata.tex(,819) @code{(c,<)} and @code{(C,<)} give priority to the components.
pdata.tex(,820) @*Let < be any of the monomial orderings of
+pdata.tex(,821) @tex
+pdata.tex(,822) Loc $K[x]$
+pdata.tex(,823) @end tex
pdata.tex(,827) as above.
pdata.tex(,828)
pdata.tex(,829) @table @asis
pdata.tex(,830) @item (<,C):
pdata.tex(,831) @cindex C, module ordering
pdata.tex(,832) @cindex module ordering C
+pdata.tex(,840) @tex
+pdata.tex(,841) $<_m = (<,C)$ denotes the module ordering (giving priority to
the coefficients):
+pdata.tex(,842) @end tex
pdata.tex(,843) @*
+pdata.tex(,844) @tex
+pdata.tex(,845) \quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow
x^\alpha <
+pdata.tex(,846) x^\beta$ or ($x^\alpha = x^\beta $ and $ i < j$).
+pdata.tex(,847) @end tex
pdata.tex(,848)
pdata.tex(,849) @strong{Example:}
pdata.tex(,850) @smallexample
@@ -21439,6 +22689,13 @@
pdata.tex(,859) @end smallexample
pdata.tex(,860)
pdata.tex(,861) @item (C,<):
+pdata.tex(,870) @tex
+pdata.tex(,871) $<_m = (C, <)$ denotes the module ordering (giving priority to
the component):
+pdata.tex(,872) @end tex
+pdata.tex(,876) @tex
+pdata.tex(,877) \quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow i
< j$ or ($
+pdata.tex(,878) i = j $ and $ x^\alpha < x^\beta $).
+pdata.tex(,879) @end tex
pdata.tex(,880)
pdata.tex(,881) @strong{Example:}
pdata.tex(,882) @smallexample
@@ -21454,6 +22711,13 @@
pdata.tex(,892) @item (<,c):
pdata.tex(,893) @cindex c, module ordering
pdata.tex(,894) @cindex module ordering c
+pdata.tex(,902) @tex
+pdata.tex(,903) $<_m = (<,c)$ denotes the module ordering (giving priority to
the coefficients):
+pdata.tex(,904) @end tex
+pdata.tex(,908) @tex
+pdata.tex(,909) \quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow
x^\alpha <
+pdata.tex(,910) x^\beta$ or ($x^\alpha = x^\beta $ and $ i > j$).
+pdata.tex(,911) @end tex
pdata.tex(,912)
pdata.tex(,913) @strong{Example:}
pdata.tex(,914) @smallexample
@@ -21467,6 +22731,13 @@
pdata.tex(,922) @end smallexample
pdata.tex(,923)
pdata.tex(,924) @item (c,<):
+pdata.tex(,933) @tex
+pdata.tex(,934) $<_m = (c, <)$ denotes the module ordering (giving priority to
the component):
+pdata.tex(,935) @end tex
+pdata.tex(,939) @tex
+pdata.tex(,940) \quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow i
> j$ or ($
+pdata.tex(,941) i = j $ and $ x^\alpha < x^\beta $).
+pdata.tex(,942) @end tex
pdata.tex(,943)
pdata.tex(,944) @strong{Example:}
pdata.tex(,945) @smallexample
@@ -21480,7 +22751,14 @@
pdata.tex(,953) @end smallexample
pdata.tex(,954) @end table
pdata.tex(,955)
+pdata.tex(,960) @tex
+pdata.tex(,961) The output of a vector $v$ in $K[x]^r$ with components $v_1,
+pdata.tex(,962) \ldots, v_r$ has the format $v_1 * gen(1) + \ldots + v_r *
gen(r)$
+pdata.tex(,963) @end tex
pdata.tex(,964) (up to permutation) unless the ordering starts with @code{c}.
+pdata.tex(,968) @tex
+pdata.tex(,969) In this case a vector is written as $[v_1, \ldots, v_r]$.
+pdata.tex(,970) @end tex
pdata.tex(,971) In all cases @sc{Singular} can read input in both formats.
pdata.tex(,972)
pdata.tex(,973) @c
--------------------------------------------------------------------------
@@ -21491,25 +22769,152 @@
pdata.tex(,978) @cindex M, ordering
pdata.tex(,979)
pdata.tex(,980) Let
+pdata.tex(,981) @tex
+pdata.tex(,982) $M$
+pdata.tex(,983) @end tex
pdata.tex(,987) be an invertible
+pdata.tex(,988) @tex
+pdata.tex(,989) $(n \times n)$-matrix
+pdata.tex(,990) @end tex
pdata.tex(,994) with integer coefficients and
+pdata.tex(,998) @tex
+pdata.tex(,999) $M_1, \ldots, M_n$ the rows of $M$.
+pdata.tex(,1000) @end tex
pdata.tex(,1001)
pdata.tex(,1002) The M-ordering < is defined as follows:
pdata.tex(,1003) @*
+pdata.tex(,1008) @tex
+pdata.tex(,1009) \quad \quad $x^a < x^b \Leftrightarrow \exists\ 1 \leq i
\leq n :
+pdata.tex(,1010) M_1 a = \; M_1 b, \ldots, M_{i-1} a = \; M_{i-1} b$ and $M_i
a < \; M_i b$.
+pdata.tex(,1011) @end tex
pdata.tex(,1012)
pdata.tex(,1013) Thus,
+pdata.tex(,1018) @tex
+pdata.tex(,1019) $x^a < x^b$
+pdata.tex(,1020) if and only if $M a$ is smaller than $M b$
+pdata.tex(,1021) @end tex
pdata.tex(,1022) with respect to the lexicographical ordering.
pdata.tex(,1023)
pdata.tex(,1024) The following matrices represent (for 3 variables) the global
and
pdata.tex(,1025) local orderings defined above (note that the matrix is not
uniquely determined
pdata.tex(,1026) by the ordering):
pdata.tex(,1027)
+pdata.tex(,1072) @tex
+pdata.tex(,1073)
+pdata.tex(,1074) $\quad$ lp:
+pdata.tex(,1075) $\left(\matrix{
+pdata.tex(,1076) 1 & 0 & 0 \cr
+pdata.tex(,1077) 0 & 1 & 0 \cr
+pdata.tex(,1078) 0 & 0 & 1 \cr
+pdata.tex(,1079) }\right)$
+pdata.tex(,1080) \quad dp:
+pdata.tex(,1081) $\left(\matrix{
+pdata.tex(,1082) 1 & 1 & 1 \cr
+pdata.tex(,1083) 0 & 0 &-1 \cr
+pdata.tex(,1084) 0 &-1 & 0 \cr
+pdata.tex(,1085) }\right)$
+pdata.tex(,1086) \quad Dp:
+pdata.tex(,1087) $\left(\matrix{
+pdata.tex(,1088) 1 & 1 & 1 \cr
+pdata.tex(,1089) 1 & 0 & 0 \cr
+pdata.tex(,1090) 0 & 1 & 0 \cr
+pdata.tex(,1091) }\right)$
+pdata.tex(,1092)
+pdata.tex(,1093) $\quad$ wp(1,2,3):
+pdata.tex(,1094) $\left(\matrix{
+pdata.tex(,1095) 1 & 2 & 3 \cr
+pdata.tex(,1096) 0 & 0 &-1 \cr
+pdata.tex(,1097) 0 &-1 & 0 \cr
+pdata.tex(,1098) }\right)$
+pdata.tex(,1099) \quad Wp(1,2,3):
+pdata.tex(,1100) $\left(\matrix{
+pdata.tex(,1101) 1 & 2 & 3 \cr
+pdata.tex(,1102) 1 & 0 & 0 \cr
+pdata.tex(,1103) 0 & 1 & 0 \cr
+pdata.tex(,1104) }\right)$
+pdata.tex(,1105)
+pdata.tex(,1106) $\quad$ ls:
+pdata.tex(,1107) $\left(\matrix{
+pdata.tex(,1108) -1 & 0 & 0 \cr
+pdata.tex(,1109) 0 &-1 & 0 \cr
+pdata.tex(,1110) 0 & 0 &-1 \cr
+pdata.tex(,1111) }\right)$
+pdata.tex(,1112) \quad ds:
+pdata.tex(,1113) $\left(\matrix{
+pdata.tex(,1114) -1 &-1 &-1 \cr
+pdata.tex(,1115) 0 & 0 &-1 \cr
+pdata.tex(,1116) 0 &-1 & 0 \cr
+pdata.tex(,1117) }\right)$
+pdata.tex(,1118) \quad Ds:
+pdata.tex(,1119) $\left(\matrix{
+pdata.tex(,1120) -1 &-1 &-1 \cr
+pdata.tex(,1121) 1 & 0 & 0 \cr
+pdata.tex(,1122) 0 & 1 & 0 \cr
+pdata.tex(,1123) }\right)$
+pdata.tex(,1124)
+pdata.tex(,1125) $\quad$ ws(1,2,3):
+pdata.tex(,1126) $\left(\matrix{
+pdata.tex(,1127) -1 &-2 &-3 \cr
+pdata.tex(,1128) 0 & 0 &-1 \cr
+pdata.tex(,1129) 0 &-1 & 0 \cr
+pdata.tex(,1130) }\right)$
+pdata.tex(,1131) \quad Ws(1,2,3):
+pdata.tex(,1132) $\left(\matrix{
+pdata.tex(,1133) -1 &-2 &-3 \cr
+pdata.tex(,1134) 1 & 0 & 0 \cr
+pdata.tex(,1135) 0 & 1 & 0 \cr
+pdata.tex(,1136) }\right)$
+pdata.tex(,1137) @end tex
pdata.tex(,1138)
pdata.tex(,1139) Product orderings (see next section) represented by a matrix:
pdata.tex(,1140)
+pdata.tex(,1159) @tex
+pdata.tex(,1160) $\quad$ (dp(3), wp(1,2,3)):
+pdata.tex(,1161) $\left(\matrix{
+pdata.tex(,1162) 1& 1& 1& 0& 0& 0 \cr
+pdata.tex(,1163) 0& 0& -1& 0& 0& 0 \cr
+pdata.tex(,1164) 0& -1& 0& 0& 0& 0 \cr
+pdata.tex(,1165) 0& 0& 0& 1& 2& 3 \cr
+pdata.tex(,1166) 0& 0& 0& 0& 0& -1 \cr
+pdata.tex(,1167) 0& 0& 0& 0& -1& 0 \cr
+pdata.tex(,1168) }\right)$
+pdata.tex(,1169)
+pdata.tex(,1170) $\quad$ (Dp(3), ds(3)):
+pdata.tex(,1171) $\left(\matrix{
+pdata.tex(,1172) 1& 1& 1& 0& 0& 0 \cr
+pdata.tex(,1173) 1& 0& 0& 0& 0& 0 \cr
+pdata.tex(,1174) 0& 1& 0& 0& 0& 0 \cr
+pdata.tex(,1175) 0& 0& 0& -1& -1& -1 \cr
+pdata.tex(,1176) 0& 0& 0& 0& 0& -1 \cr
+pdata.tex(,1177) 0& 0& 0& 0& -1& 0 \cr
+pdata.tex(,1178) }\right)$
+pdata.tex(,1179) @end tex
pdata.tex(,1180)
pdata.tex(,1181) Orderings with extra weight vector (see below) represented by
a matrix:
pdata.tex(,1182)
+pdata.tex(,1203) @tex
+pdata.tex(,1204) $\quad$ (dp(3), a(1,2,3),dp(3)):
+pdata.tex(,1205) $\left(\matrix{
+pdata.tex(,1206) 1& 1& 1& 0& 0& 0 \cr
+pdata.tex(,1207) 0& 0& -1& 0& 0& 0 \cr
+pdata.tex(,1208) 0& -1& 0& 0& 0& 0 \cr
+pdata.tex(,1209) 0& 0& 0& 1& 2& 3 \cr
+pdata.tex(,1210) 0& 0& 0& 1& 1& 1 \cr
+pdata.tex(,1211) 0& 0& 0& 0& 0& -1 \cr
+pdata.tex(,1212) 0& 0& 0& 0& -1& 0 \cr
+pdata.tex(,1213) }\right)$
+pdata.tex(,1214)
+pdata.tex(,1215) $\quad$ (a(1,2,3,4,5),Dp(3), ds(3)):
+pdata.tex(,1216) $\left(\matrix{
+pdata.tex(,1217) 1& 2& 3& 4& 5& 0 \cr
+pdata.tex(,1218) 1& 1& 1& 0& 0& 0 \cr
+pdata.tex(,1219) 1& 0& 0& 0& 0& 0 \cr
+pdata.tex(,1220) 0& 1& 0& 0& 0& 0 \cr
+pdata.tex(,1221) 0& 0& 0& -1& -1& -1 \cr
+pdata.tex(,1222) 0& 0& 0& 0& 0 & -1 \cr
+pdata.tex(,1223) 0& 0& 0& 0& -1& 0 \cr
+pdata.tex(,1224) }\right)$
+pdata.tex(,1225) @end tex
pdata.tex(,1226)
pdata.tex(,1227) @address@hidden:
pdata.tex(,1228) @smallexample
@@ -21539,7 +22944,13 @@
pdata.tex(,1252) @end smallexample
pdata.tex(,1253)
pdata.tex(,1254) If the ring has
+pdata.tex(,1255) @tex
+pdata.tex(,1256) $n$
+pdata.tex(,1257) @end tex
pdata.tex(,1261) variables and the matrix contains less than
+pdata.tex(,1262) @tex
+pdata.tex(,1263) $n \times n$
+pdata.tex(,1264) @end tex
pdata.tex(,1268) entries an error message is given, if there are more entries,
pdata.tex(,1269) the last ones are ignored.
pdata.tex(,1270)
@@ -21560,6 +22971,9 @@
pdata.tex(,1285) @cindex orderings, product
pdata.tex(,1286)
pdata.tex(,1287) Let
+pdata.tex(,1292) @tex
+pdata.tex(,1293) $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_m)$
+pdata.tex(,1294) @end tex
pdata.tex(,1295) be two ordered sets of variables,
pdata.tex(,1317)
pdata.tex(,1318) Inductively one defines the product ordering of more than two
monomial
@@ -21581,9 +22995,24 @@
pdata.tex(,1334) @cindex a, ordering
pdata.tex(,1335) @cindex orderings, a
pdata.tex(,1336)
+pdata.tex(,1340) @tex
+pdata.tex(,1341) ${\tt a}(w_1, \ldots, w_n),\; $
+pdata.tex(,1342) @end tex
+pdata.tex(,1346) @tex
+pdata.tex(,1347) $w_1,\ldots,w_n$
+pdata.tex(,1348) @end tex
pdata.tex(,1349) any integers (including 0), defines
+pdata.tex(,1353) @tex
+pdata.tex(,1354) $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n$
+pdata.tex(,1355) @end tex
pdata.tex(,1356) and
pdata.tex(,1357) @*
+pdata.tex(,1361) @tex
+pdata.tex(,1362) $$\deg(x^\alpha) < \deg(x^\beta) \Rightarrow x^\alpha <
x^\beta,$$
+pdata.tex(,1363) @end tex
+pdata.tex(,1368) @tex
+pdata.tex(,1369) $$\deg(x^\alpha) > \deg(x^\beta) \Rightarrow x^\alpha >
x^\beta. $$
+pdata.tex(,1370) @end tex
pdata.tex(,1371) @*An extra weight vector does not define a monomial ordering
by itself:
pdata.tex(,1372) it can only be used in combination with other orderings
pdata.tex(,1373) to insert an extra line of weights into the ordering
@@ -21632,14 +23061,44 @@
math.tex(,42) @cindex Standard bases
math.tex(,43)
math.tex(,44) @subheading Definition
+math.tex(,45) @tex
+math.tex(,46) Let $R = \hbox{Loc}_< K[\underline{x}]$ and let $I$ be a
submodule of $R^r$.
+math.tex(,47) Note that for r=1 this means that $I$ is an ideal in $R$.
+math.tex(,48) Denote by $L(I)$ the submodule of $R^r$ generated by the leading
terms
+math.tex(,49) of elements of $I$, i.e. by $\left\{L(f) \mid f \in I\right\}$.
+math.tex(,50) Then $f_1, \ldots, f_s \in I$ is called a {\bf standard basis}
of $I$
+math.tex(,51) if $L(f_1), \ldots, L(f_s)$ generate $L(I)$.
+math.tex(,52) @end tex
math.tex(,60)
math.tex(,61) @subheading Properties
math.tex(,62) @table @asis
math.tex(,63) @item normal form:
math.tex(,64) @cindex Normal form
+math.tex(,65) @tex
+math.tex(,66) A function $\hbox{NF} : R^r \times \{G \mid G\ \hbox{ a standard
+math.tex(,67) basis}\} \to R^r, (p,G) \mapsto \hbox{NF}(p|G)$, is called a
{\bf normal
+math.tex(,68) form} if for any $p \in R^r$ and any standard basis $G$ the
following
+math.tex(,69) holds: if $\hbox{NF}(p|G) \not= 0$ then $L(g)$ does not divide
+math.tex(,70) $L(\hbox{NF}(p|G))$ for all $g \in G$.
+math.tex(,71)
+math.tex(,72) \noindent
+math.tex(,73) $\hbox{NF}(p|G)$ is called a {\bf normal form of} $p$ {\bf with
+math.tex(,74) respect to} $G$ (note that such a function is not unique).
+math.tex(,75) @end tex
math.tex(,84) @item ideal membership:
math.tex(,85) @cindex Ideal membership
+math.tex(,86) @tex
+math.tex(,87) For a standard basis $G$ of $I$ the following holds:
+math.tex(,88) $f \in I$ if and only if $\hbox{NF}(f,G) = 0$.
+math.tex(,89) @end tex
math.tex(,94) @item Hilbert function:
+math.tex(,95) @tex
+math.tex(,96) Let \hbox{$I \subseteq K[\underline{x}]^r$} be a homogeneous
module, then the Hilbert function
+math.tex(,97) $H_I$ of $I$ (see below)
+math.tex(,98) and the Hilbert function $H_{L(I)}$ of the leading module $L(I)$
+math.tex(,99) coincide, i.e.,
+math.tex(,100) $H_I=H_{L(I)}$.
+math.tex(,101) @end tex
math.tex(,106) @end table
math.tex(,107)
math.tex(,108) @c
---------------------------------------------------------------------------
@@ -21647,8 +23106,32 @@
math.tex(,110) @section Hilbert function
math.tex(,111) @cindex Hilbert function
math.tex(,112) @cindex Hilbert series
+math.tex(,113) @tex
+math.tex(,114) Let M $=\bigoplus_i M_i$ be a graded module over
$K[x_1,..,x_n]$ with
+math.tex(,115) respect to weights $(w_1,..w_n)$.
+math.tex(,116) The {\bf Hilbert function} of $M$, $H_M$, is defined (on the
integers) by
+math.tex(,117) $$H_M(k) :=dim_K M_k.$$
+math.tex(,118) The {\bf Hilbert-Poincare series} of $M$ is the power series
+math.tex(,119) $$\hbox{HP}_M(t) :=\sum_{i=-\infty}^\infty
+math.tex(,120) H_M(i)t^i=\sum_{i=-\infty}^\infty dim_K M_i \cdot t^i.$$
+math.tex(,121) It turns out that $\hbox{HP}_M(t)$ can be written in two useful
ways
+math.tex(,122) for weights $(1,..,1)$:
+math.tex(,123) $$\hbox{HP}_M(t)={Q(t)\over (1-t)^n}={P(t)\over
(1-t)^{dim(M)}}$$
+math.tex(,124) where $Q(t)$ and $P(t)$ are polynomials in ${\bf Z}[t]$.
+math.tex(,125) $Q(t)$ is called the {\bf first Hilbert series},
+math.tex(,126) and $P(t)$ the {\bf second Hilbert series}.
+math.tex(,127) If \hbox{$P(t)=\sum_{k=0}^N a_k t^k$}, and \hbox{$d = dim(M)$},
+math.tex(,128) then \hbox{$H_M(s)=\sum_{k=0}^N a_k$ ${d+s-k-1}\choose{d-1}$}
+math.tex(,129) (the {\bf Hilbert polynomial}) for $s \ge N$.
+math.tex(,130) @end tex
math.tex(,156) @*
math.tex(,157) @*
+math.tex(,158) @tex
+math.tex(,159) Generalizing these to quasihomogeneous modules we get
+math.tex(,160) $$\hbox{HP}_M(t)={Q(t)\over {\Pi_{i=1}^n(1-t^{w_i})}}$$
+math.tex(,161) where $Q(t)$ is a polynomial in ${\bf Z}[t]$.
+math.tex(,162) $Q(t)$ is called the {\bf first (weighted) Hilbert series} of M.
+math.tex(,163) @end tex
math.tex(,172)
math.tex(,173) @c
---------------------------------------------------------------------------
math.tex(,174) @node Syzygies and resolutions, Characteristic sets, Hilbert
function, Mathematical background
@@ -21656,11 +23139,22 @@
math.tex(,176) @cindex Syzygies and resolutions
math.tex(,177)
math.tex(,178) @subheading Syzygies
+math.tex(,179) @tex
+math.tex(,180) Let $R$ be a quotient of $\hbox{Loc}_< K[\underline{x}]$ and
let \hbox{$I=(g_1, ..., g_s)$} be a submodule of $R^r$.
+math.tex(,181) Then the {\bf module of syzygies} (or {\bf 1st syzygy module},
{\bf module of relations}) of $I$, syz($I$), is defined to be the kernel of the
map \hbox{$R^s \rightarrow R^r,\; \sum_{i=1}^s w_ie_i \mapsto \sum_{i=1}^s
w_ig_i$.}
+math.tex(,182) @end tex
math.tex(,192)
math.tex(,193) The @strong{k-th syzygy module} is defined inductively to be
the module
math.tex(,194) of syzygies of the
+math.tex(,195) @tex
+math.tex(,196) $(k-1)$-st
+math.tex(,197) @end tex
math.tex(,201) syzygy module.
math.tex(,202)
+math.tex(,203) @tex
+math.tex(,204) Note, that the syzygy modules of $I$ depend on a choice of
generators $g_1, ..., g_s$.
+math.tex(,205) But one can show that they depend on $I$ uniquely up to direct
summands.
+math.tex(,206) @end tex
math.tex(,211)
math.tex(,212) @table @code
math.tex(,213) @item @strong{Example:}
@@ -21678,10 +23172,26 @@
math.tex(,225) @end table
math.tex(,226)
math.tex(,227) @subheading Free resolutions
+math.tex(,228) @tex
+math.tex(,229) Let $I=(g_1,...,g_s)\subseteq R^r$ and $M= R^r/I$.
+math.tex(,230) A {\bf free resolution of $M$} is a long exact sequence
+math.tex(,231) $$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow}
F_1
+math.tex(,232) \buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow
M\longrightarrow
+math.tex(,233) 0,$$
+math.tex(,234) @end tex
math.tex(,242) @*where the columns of the matrix
+math.tex(,243) @tex
+math.tex(,244) $A_1$
+math.tex(,245) @end tex
math.tex(,249) generate
+math.tex(,253) @tex
+math.tex(,254) $I$
+math.tex(,255) @end tex
math.tex(,256) . Note, that resolutions need not to be finite (i.e., of
math.tex(,257) finite length). The Hilbert Syzygy Theorem states, that for
+math.tex(,258) @tex
+math.tex(,259) $R=\hbox{Loc}_< K[\underline{x}]$
+math.tex(,260) @end tex
math.tex(,264) there exists a ("minimal") resolution of length not exceeding
the number of
math.tex(,265) variables.
math.tex(,266)
@@ -21714,11 +23224,37 @@
math.tex(,293) @subheading Betti numbers and regularity
math.tex(,294) @cindex Betti number
math.tex(,295) @cindex regularity
+math.tex(,296) @tex
+math.tex(,297) Let $R$ be a graded ring (e.g., $R = \hbox{Loc}_<
K[\underline{x}]$) and
+math.tex(,298) let $I \subset R^r$ be a graded submodule. Let
+math.tex(,299) $$
+math.tex(,300) R^r = \bigoplus_a R\cdot e_{a,0} \buildrel A_1 \over
\longleftarrow
+math.tex(,301) \bigoplus_a R\cdot e_{a,1} \longleftarrow \ldots
\longleftarrow
+math.tex(,302) \bigoplus_a R\cdot e_{a,n} \longleftarrow 0
+math.tex(,303) $$
+math.tex(,304) be a minimal free resolution of $R^n/I$ considered with
homogeneous maps
+math.tex(,305) of degree 0. Then the {\bf graded Betti number} $b_{i,j}$ of
$R^r/I$ is
+math.tex(,306) the minimal number of generators $e_{a,j}$ in degree $i+j$ of
the $j$-th
+math.tex(,307) syzygy module of $R^r/I$ (i.e., the $(j-1)$-st syzygy module of
+math.tex(,308) $I$). Note, that by definition the $0$-th syzygy module of
$R^r/I$ is $R^r$
+math.tex(,309) and the 1st syzygy module of $R^r/I$ is $I$.
+math.tex(,310) @end tex
math.tex(,325)
math.tex(,326) The @strong{regularity} of
+math.tex(,330) @tex
+math.tex(,331) $I$
+math.tex(,332) @end tex
math.tex(,333) is the smallest integer
+math.tex(,337) @tex
+math.tex(,338) $s$
+math.tex(,339) @end tex
math.tex(,340)
math.tex(,341) such that
+math.tex(,342) @tex
+math.tex(,343) $$
+math.tex(,344) \hbox{deg}(e_{a,j}) \le s+j-1 \quad \hbox{for all $j$.}
+math.tex(,345) $$
+math.tex(,346) @end tex
math.tex(,352)
math.tex(,353) @table @code
math.tex(,354) @item @strong{Example:}
@@ -21751,6 +23287,43 @@
math.tex(,381) @section Characteristic sets
math.tex(,382) @cindex Characteristic sets
math.tex(,383)
+math.tex(,384) @tex
+math.tex(,385) Let $<$ be the lexicographical ordering on $R=K[x_1,...,x_n]$
with $x_1
+math.tex(,386) < ... < x_n$.
+math.tex(,387) For $f \in R$ let lvar($f$) (the leading variable of $f$) be
the largest
+math.tex(,388) variable in $f$,
+math.tex(,389) i.e., if $f=a_s(x_1,...,x_{k-1})x_k^s+...+a_0(x_1,...,x_{k-1})$
for some
+math.tex(,390) $k \leq n$ then lvar$(f)=x_k$.
+math.tex(,391)
+math.tex(,392) Moreover, let
+math.tex(,393) \hbox{ini}$(f):=a_s(x_1,...,x_{k-1})$. The pseudo remainder
+math.tex(,394) $r=\hbox{prem}(g,f)$ of $g$ with respect to $f$ is
+math.tex(,395) defined by the equality $\hbox{ini}(f)^a\cdot g = qf+r$ with
+math.tex(,396) $\hbox{deg}_{lvar(f)}(r)<\hbox{deg}_{lvar(f)}(f)$ and $a$
+math.tex(,397) minimal.
+math.tex(,398)
+math.tex(,399) A set $T=\{f_1,...,f_r\} \subset R$ is called triangular if
+math.tex(,400) $\hbox{lvar}(f_1)<...<\hbox{lvar}(f_r)$. Moreover, let $ U
\subset T $,
+math.tex(,401) then $(T,U)$ is called a triangular system, if $T$ is a
triangular set
+math.tex(,402) such that $\hbox{ini}(T)$ does not vanish on $V(T) \setminus
V(U)
+math.tex(,403) (=:V(T\setminus U))$.
+math.tex(,404)
+math.tex(,405) $T$ is called irreducible if for every $i$ there are no
+math.tex(,406) $d_i$,$f_i'$,$f_i''$ such that
+math.tex(,407) $$ \hbox{lvar}(d_i)<\hbox{lvar}(f_i) =
+math.tex(,408) \hbox{lvar}(f_i')=\hbox{lvar}(f_i''),$$
+math.tex(,409) $$ 0 \not\in \hbox{prem}(\{ d_i, \hbox{ini}(f_i'),
+math.tex(,410) \hbox{ini}(f_i'')\},\{ f_1,...,f_{i-1}\}),$$
+math.tex(,411) $$\hbox{prem}(d_if_i-f_i'f_i'',\{f_1,...,f_{i-1}\})=0.$$
+math.tex(,412) Furthermore, $(T,U)$ is called irreducible if $T$ is
irreducible.
+math.tex(,413)
+math.tex(,414) The main result on triangular sets is the following:
+math.tex(,415) let $G=\{g_1,...,g_s\} \subset R$ then there are irreducible
triangular sets $T_1,...,T_l$
+math.tex(,416) such that $V(G)=\bigcup_{i=1}^{l}(V(T_i\setminus I_i))$
+math.tex(,417) where $I_i=\{\hbox{ini}(f) \mid f \in T_i \}$. Such a set
+math.tex(,418) $\{T_1,...,T_l\}$ is called an {\bf irreducible characteristic
series} of
+math.tex(,419) the ideal $(G)$.
+math.tex(,420) @end tex
math.tex(,456)
math.tex(,457) @table @code
math.tex(,458) @item @strong{Example:}
@@ -21775,14 +23348,61 @@
math.tex(,477) @c tex and info versions of it. It end just before the
introducing text
math.tex(,478) @c to the first example.
math.tex(,479)
+math.tex(,480) @tex
+math.tex(,481) Let $f\colon(C^{n+1},0)\rightarrow(C,0)$ be a complex isolated
hypersurface singularity given by a polynomial with algebraic coefficients
which we also denote by $f$.
+math.tex(,482) Let $O=C[x_0,\ldots,x_n]_{(x_0,\ldots,x_n)}$ be the local ring
at the origin and $J_f$ the Jacobian ideal of $f$.
+math.tex(,483)
+math.tex(,484) A {\bf Milnor representative} of $f$ defines a differentiable
fibre bundle over the punctured disc with fibres of homotopy type of $\mu$
$n$-spheres.
+math.tex(,485) The $n$-th cohomology bundle is a flat vector bundle of
dimension $n$ and carries a natural flat connection with covariant derivative
$\partial_t$.
+math.tex(,486) The {\bf monodromy operator} is the action of a positively
oriented generator of the fundamental group of the puctured disc on the Milnor
fibre.
+math.tex(,487) Sections in the cohomology bundle of {\bf moderate growth} at
$0$ form a regular $D=C\{t\}[\partial_t]$-module $G$, the {\bf Gauss-Manin
connection}.
+math.tex(,488)
+math.tex(,489) By integrating along flat multivalued families of cycles, one
can consider fibrewise global holomorphic differential forms as elements of $G$.
+math.tex(,490) This factors through an inclusion of the {\bf Brieskorn
lattice} $H'':=\Omega^{n+1}_{C^{n+1},0}/df\wedge d\Omega^{n-1}_{C^{n+1},0}$ in
$G$.
+math.tex(,491)
+math.tex(,492) The $D$-module structure defines the {\bf V-filtration} $V$ on
$G$ by $V^\alpha:=\sum_{\beta\ge\alpha}C\{t\}ker(t\partial_t-\beta)^{n+1}$.
+math.tex(,493) The Brieskorn lattice defines the {\bf Hodge filtration} $F$ on
$G$ by $F_k=\partial_t^kH''$ which comes from the {\bf mixed Hodge structure}
on the Milnor fibre.
+math.tex(,494) Note that $F_{-1}=H'$.
+math.tex(,495)
+math.tex(,496) The induced V-filtration on the Brieskorn lattice determines
the {\bf singularity spectrum} $Sp$ by $Sp(\alpha):=\dim_CGr_V^\alpha Gr^F_0G$.
+math.tex(,497) The spectrum consists of $\mu$ rational numbers
$\alpha_1,\dots,\alpha_\mu$ such that $e^{2\pi i\alpha_1},\dots,e^{2\pi
i\alpha_\mu}$ are the eigenvalues of the monodromy.
+math.tex(,498) These {\bf spectral numbers} lie in the open interval $(-1,n)$,
symmetric about the midpoint $(n-1)/2$.
+math.tex(,499)
+math.tex(,500) The spectrum is constant under $\mu$-constant deformations and
has the following semicontinuity property:
+math.tex(,501) The number of spectral numbers in an interval $(a,a+1]$ of all
singularities of a small deformation of $f$ is greater or equal to that of f in
this interval.
+math.tex(,502) For semiquasihomogeneous singularities, this also holds for
intervals of the form $(a,a+1)$.
+math.tex(,503)
+math.tex(,504) Two given isolated singularities $f$ and $g$ determine two
spectra and from these spectra we get an integer.
+math.tex(,505) This integer is the maximal positive integer $k$ such that the
semicontinuity holds for the spectrum of $f$ and $k$ times the spectrum of $g$.
+math.tex(,506) These numbers give bounds for the maximal number of isolated
singularities of a specific type on a hypersurface $X\subset{P}^n$ of degree
$d$:
+math.tex(,507) such a hypersurface has a smooth hyperplane section, and the
complement is a small deformation of a cone over this hyperplane section.
+math.tex(,508) The cone itself being a $\mu$-constant deformation of
$x_0^d+\dots+x_n^d=0$, the singularities are bounded by the spectrum of
$x_0^d+\dots+x_n^d$.
+math.tex(,509)
+math.tex(,510) Using the library {\tt gaussman.lib} one can compute the {\bf
monodromy}, the V-filtration on $H''/H'$, and the spectrum.
+math.tex(,511) @end tex
math.tex(,512)
math.tex(,545)
math.tex(,546) Let us consider as an example
+math.tex(,550) @tex
+math.tex(,551) $f=x^5+x^2y^2+y^5$
+math.tex(,552) @end tex
math.tex(,553) .
math.tex(,554) First, we compute a matrix
+math.tex(,558) @tex
+math.tex(,559) $M$
+math.tex(,560) @end tex
math.tex(,561) such that
+math.tex(,562) @tex
+math.tex(,563) $\exp(2\pi iM)$
+math.tex(,564) @end tex
math.tex(,568) is a monodromy matrix of
+math.tex(,572) @tex
+math.tex(,573) $f$
+math.tex(,574) @end tex
math.tex(,575) and the Jordan normal form of
+math.tex(,579) @tex
+math.tex(,580) $M$
+math.tex(,581) @end tex
math.tex(,582) :
math.tex(,583) @smallexample
math.tex(,584) @c reused example Gauss-Manin_connection math.doc:505
@@ -21807,6 +23427,9 @@
math.tex(,603) @end smallexample
math.tex(,604)
math.tex(,605) Now, we compute the V-filtration on
+math.tex(,609) @tex
+math.tex(,610) $H''/H'$
+math.tex(,611) @end tex
math.tex(,612) and the spectrum:
math.tex(,613) @smallexample
math.tex(,614) @c reused example Gauss-Manin_connection_1 math.doc:517
@@ -21858,17 +23481,36 @@
math.tex(,660) @c end example Gauss-Manin_connection_1 math.doc:517
math.tex(,661) @end smallexample
math.tex(,662) Here @code{l[1]} contains the spectral numbers, @code{l[2]} the
corresponding multiplicities, @code{l[3]} a
+math.tex(,666) @tex
+math.tex(,667) $C$
+math.tex(,668) @end tex
math.tex(,669) -basis of the V-filtration on
+math.tex(,673) @tex
+math.tex(,674) $H''/H'$
+math.tex(,675) @end tex
math.tex(,676) in terms of the monomial basis of
+math.tex(,677) @tex
+math.tex(,678) $O/J_f\cong H''/H'$
+math.tex(,679) @end tex
math.tex(,683) in @code{l[4]}.
math.tex(,684)
+math.tex(,685) @tex
+math.tex(,686) If the principal part of $f$ is $C$-nondegenerate, one can
compute the spectrum using the library {\tt spectrum.lib}.
+math.tex(,687) In this case, the V-filtration on $H''$ coincides with the
Newton-filtration on $H''$ which allows to compute the spectrum more
efficiently.
+math.tex(,688) @end tex
math.tex(,689)
math.tex(,694)
math.tex(,695) Let us calculate one specific example, the maximal number
math.tex(,696) of triple points of type
+math.tex(,697) @tex
+math.tex(,698) $\tilde{E}_6$ on a surface $X\subset{P}^3$
+math.tex(,699) @end tex
math.tex(,703) of degree seven.
math.tex(,704) This calculation can be done over the rationals.
math.tex(,705) So choose a local ordering on
+math.tex(,709) @tex
+math.tex(,710) $Q[x,y,z]$
+math.tex(,711) @end tex
math.tex(,712) . Here we take the
math.tex(,713) negative degree lexicographical ordering which is denoted
math.tex(,714) @code{ds} in @sc{Singular}:
@@ -21903,21 +23545,44 @@
math.tex(,743) @end smallexample
math.tex(,744)
math.tex(,745) The command @code{spectrumnd(f)} computes the spectrum of
+math.tex(,749) @tex
+math.tex(,750) $f$
+math.tex(,751) @end tex
math.tex(,752) and
math.tex(,753) returns a list with six entries:
math.tex(,754) The Milnor number
+math.tex(,755) @tex
+math.tex(,756) $\mu(f)$, the geometric genus $p_g(f)$
+math.tex(,757) @end tex
math.tex(,761) and the number of different spectrum numbers.
math.tex(,762) The other three entries are of type @code{intvec}.
math.tex(,763) They contain the numerators, denominators and
math.tex(,764) multiplicities of the spectrum numbers. So
+math.tex(,765) @tex
+math.tex(,766) $x^7+y^7+z^7=0$
+math.tex(,767) @end tex
math.tex(,771) has Milnor number 216 and geometrical
math.tex(,772) genus 35. Its spectrum consists of the 16 different rationals
math.tex(,773) @*
+math.tex(,774) @tex
+math.tex(,775) ${3 \over 7}, {4 \over 7}, {5 \over 7}, {6 \over 7}, {1 \over
1},
+math.tex(,776) {8 \over 7}, {9 \over 7}, {10 \over 7}, {11 \over 7}, {12 \over
7},
+math.tex(,777) {13 \over 7}, {2 \over 1}, {15 \over 7}, {16 \over 7}, {17
\over 7},
+math.tex(,778) {18 \over 7}$
+math.tex(,779) @end tex
math.tex(,784) @*appearing with multiplicities
math.tex(,785) @*1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1.
math.tex(,786)
+math.tex(,787) @tex
+math.tex(,788) The singularities of type $\tilde{E}_6$ form a
+math.tex(,789) $\mu$-constant one parameter family given by
+math.tex(,790) $x^3+y^3+z^3+\lambda xyz=0,\quad \lambda^3\neq-27$.
+math.tex(,791) @end tex
math.tex(,797) Therefore they have all the same spectrum, which we compute
math.tex(,798) for
+math.tex(,799) @tex
+math.tex(,800) $x^3+y^3+z^3$.
+math.tex(,801) @end tex
math.tex(,805)
math.tex(,806) @smallexample
math.tex(,807) poly g=x^3+y^3+z^3;
@@ -21943,6 +23608,9 @@
math.tex(,827) @end smallexample
math.tex(,828)
math.tex(,829) This tells us that there are at most 18 singularities of type
+math.tex(,830) @tex
+math.tex(,831) $\tilde{E}_6$ on a septic in $P^3$. But $x^7+y^7+z^7$
+math.tex(,832) @end tex
math.tex(,836) is semiquasihomogeneous (sqh), so we can also apply the stronger
math.tex(,837) form of semicontinuity:
math.tex(,838)
@@ -21952,12 +23620,21 @@
math.tex(,842) @end smallexample
math.tex(,843)
math.tex(,844) So in fact a septic has at most 17 triple points of type
+math.tex(,845) @tex
+math.tex(,846) $\tilde{E}_6$.
+math.tex(,847) @end tex
math.tex(,851)
math.tex(,852) Note that @code{spectrumnd(f)} works only if
+math.tex(,856) @tex
+math.tex(,857) $f$
+math.tex(,858) @end tex
math.tex(,859) has nondegenerate
math.tex(,860) principal part. In fact @code{spectrumnd} will detect a
degenerate
math.tex(,861) principal part in many cases and print out an error message.
math.tex(,862) However if it is known in advance that
+math.tex(,866) @tex
+math.tex(,867) $f$
+math.tex(,868) @end tex
math.tex(,869) has nondegenerate
math.tex(,870) principal part, then the spectrum may be computed much faster
math.tex(,871) using @code{spectrumnd(f,1)}.
@@ -21981,10 +23658,33 @@
ti_ip.tex(,12) @comment DO NOT EDIT DIRECTLY, BUT EDIT ti_ip.doc INSTEAD
ti_ip.tex(,13) @cindex ideal, toric
ti_ip.tex(,14)
+ti_ip.tex(,15) @tex
+ti_ip.tex(,16) Let $A$ denote an $m\times n$ matrix with integral
coefficients. For $u
+ti_ip.tex(,17) \in Z\!\!\! Z^n$, we define $u^+,u^-$ to be the uniquely
determined
+ti_ip.tex(,18) vectors with nonnegative coefficients and disjoint support
(i.e.,
+ti_ip.tex(,19) $u_i^+=0$ or $u_i^-=0$ for each component $i$) such that
+ti_ip.tex(,20) $u=u^+-u^-$. For $u\geq 0$ component-wise, let $x^u$ denote the
monomial
+ti_ip.tex(,21) $x_1^{u_1}\cdot\ldots\cdot x_n^{u_n}\in K[x_1,\ldots,x_n]$.
+ti_ip.tex(,22)
+ti_ip.tex(,23) The ideal
+ti_ip.tex(,24) $$ I_A:=<x^{u^+}-x^{u^-} | u\in\ker(A)\cap Z\!\!\! Z^n>\ \subset
+ti_ip.tex(,25) K[x_1,\ldots,x_n] $$
+ti_ip.tex(,26) is called a \bf toric ideal. \rm
+ti_ip.tex(,27)
+ti_ip.tex(,28) The first problem in computing toric ideals is to find a finite
+ti_ip.tex(,29) generating set: Let $v_1,\ldots,v_r$ be a lattice basis of
$\ker(A)\cap
+ti_ip.tex(,30) Z\!\!\! Z^n$ (i.e, a basis of the $Z\!\!\! Z$-module). Then
+ti_ip.tex(,31) $$ I_A:=I:(x_1\cdot\ldots\cdot x_n)^\infty $$
+ti_ip.tex(,32) where
+ti_ip.tex(,33) $$ I=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
+ti_ip.tex(,34) @end tex
ti_ip.tex(,35)
ti_ip.tex(,61)
ti_ip.tex(,62) The required lattice basis can be computed using the
LLL-algorithm (@pxref{[Coh93]}). For the computation of the saturation, there
are various
ti_ip.tex(,63) possibilities described in the
+ti_ip.tex(,64) @tex
+ti_ip.tex(,65) section Algorithms.
+ti_ip.tex(,66) @end tex
ti_ip.tex(,70)
ti_ip.tex(,71) @menu
ti_ip.tex(,72) * Algorithms:: Various algorithms for computing
toric ideals.
@@ -22012,6 +23712,23 @@
ti_ip.tex(,94)
ti_ip.tex(,95)
ti_ip.tex(,96) The algorithm of Conti and Traverso (@pxref{[CoTr91]})
+ti_ip.tex(,97) @tex
+ti_ip.tex(,98) computes $I_A$ via the
+ti_ip.tex(,99) extended matrix $B=(I_m|A)$,
+ti_ip.tex(,100) where $I_m$ is the $m\times m$ unity matrix. A lattice basis
of $B$ is
+ti_ip.tex(,101) given by the set of vectors $(a^j,-e_j)\in Z\!\!\! Z^{m+n}$,
where $a^j$
+ti_ip.tex(,102) is the $j$-th row of $A$ and $e_j$ the $j$-th coordinate
vector. We
+ti_ip.tex(,103) look at the ideal in $K[y_1,\ldots,y_m,x_1,\ldots,x_n]$
corresponding to
+ti_ip.tex(,104) these vectors, namely
+ti_ip.tex(,105) $$ I_1=<y^{a_j^+}- x_j y^{a_j^-} | j=1,\ldots, n>.$$
+ti_ip.tex(,106) We introduce a further variable $t$ and adjoin the binomial
$t\cdot
+ti_ip.tex(,107) y_1\cdot\ldots\cdot y_m -1$ to the generating set of $I_1$,
obtaining
+ti_ip.tex(,108) an ideal $I_2$ in the polynomial ring $K[t,
+ti_ip.tex(,109) y_1,\ldots,y_m,x_1,\ldots,x_n]$. $I_2$ is saturated w.r.t. all
+ti_ip.tex(,110) variables because all variables are invertible modulo $I_2$.
Now $I_A$
+ti_ip.tex(,111) can be computed from $I_2$ by eliminating the variables
+ti_ip.tex(,112) $t,y_1,\ldots,y_m$.
+ti_ip.tex(,113) @end tex
ti_ip.tex(,131)
ti_ip.tex(,132) Because of the big number of auxiliary variables needed to
compute a
ti_ip.tex(,133) toric ideal, this algorithm is rather slow in practice.
However, it has
@@ -22026,6 +23743,16 @@
ti_ip.tex(,142)
ti_ip.tex(,143)
ti_ip.tex(,144) The algorithm of Pottier (@pxref{[Pot94]}) starts by computing
a lattice
+ti_ip.tex(,145) @tex
+ti_ip.tex(,146) basis $v_1,\ldots,v_r$ for the integer kernel of $A$ using the
+ti_ip.tex(,147) LLL-algorithm. The ideal corresponding to the lattice basis
vectors
+ti_ip.tex(,148) $$ I_1=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
+ti_ip.tex(,149) is saturated -- as in the algorithm of Conti and Traverso -- by
+ti_ip.tex(,150) inversion of all variables: One adds an auxiliary variable $t$
and the
+ti_ip.tex(,151) generator $t\cdot x_1\cdot\ldots\cdot x_n -1$ to obtain an
ideal $I_2$
+ti_ip.tex(,152) in $K[t,x_1,\ldots,x_n]$ from which one computes $I_A$ by
elimination of
+ti_ip.tex(,153) $t$.
+ti_ip.tex(,154) @end tex
ti_ip.tex(,167)
ti_ip.tex(,168)
ti_ip.tex(,169) @node Hosten and Sturmfels, Di Biase and Urbanke, Pottier,
Algorithms
@@ -22036,6 +23763,33 @@
ti_ip.tex(,174)
ti_ip.tex(,175)
ti_ip.tex(,176) The algorithm of Hosten and Sturmfels (@pxref{[HoSt95]})
allows to
+ti_ip.tex(,177) @tex
+ti_ip.tex(,178) compute $I_A$ without any auxiliary variables, provided that
$A$ contains a vector $w$
+ti_ip.tex(,179) with positive coefficients in its row space. This is a real
restriction,
+ti_ip.tex(,180) i.e., the algorithm will not necessarily work in the general
case.
+ti_ip.tex(,181)
+ti_ip.tex(,182) A lattice basis $v_1,\ldots,v_r$ is again computed via the
+ti_ip.tex(,183) LLL-algorithm. The saturation step is performed in the
following way:
+ti_ip.tex(,184) First note that $w$ induces a positive grading w.r.t. which
the ideal
+ti_ip.tex(,185) $$ I=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
+ti_ip.tex(,186) corresponding to our lattice basis is homogeneous. We use the
following
+ti_ip.tex(,187) lemma:
+ti_ip.tex(,188)
+ti_ip.tex(,189) Let $I$ be a homogeneous ideal w.r.t. the weighted reverse
+ti_ip.tex(,190) lexicographical ordering with weight vector $w$ and variable
order $x_1
+ti_ip.tex(,191) > x_2 > \ldots > x_n$. Let $G$ denote a Groebner basis of $I$
w.r.t. to
+ti_ip.tex(,192) this ordering. Then a Groebner basis of $(I:x_n^\infty)$ is
obtained by
+ti_ip.tex(,193) dividing each element of $G$ by the highest possible power of
$x_n$.
+ti_ip.tex(,194)
+ti_ip.tex(,195) From this fact, we can successively compute
+ti_ip.tex(,196) $$ I_A= I:(x_1\cdot\ldots\cdot x_n)^\infty
+ti_ip.tex(,197) =(((I:x_1^\infty):x_2^\infty):\ldots :x_n^\infty); $$
+ti_ip.tex(,198) in the $i$-th step we take $x_i$ as the cheapest variable and
apply the
+ti_ip.tex(,199) lemma with $x_i$ instead of $x_n$.
+ti_ip.tex(,200)
+ti_ip.tex(,201) This procedure involves $n$ Groebner basis computations.
Actually, this
+ti_ip.tex(,202) number can be reduced to at most $n/2$
+ti_ip.tex(,203) @end tex
ti_ip.tex(,235) (@pxref{[HoSh98]}), and the single
ti_ip.tex(,236) computations -- except from the first one -- show to be easy
and fast in
ti_ip.tex(,237) practice.
@@ -22048,6 +23802,38 @@
ti_ip.tex(,244)
ti_ip.tex(,245) Like the algorithm of Hosten and Sturmfels, the algorithm of
Di Biase
ti_ip.tex(,246) and Urbanke (@pxref{[DBUr95]}) performs up
+ti_ip.tex(,247) @tex
+ti_ip.tex(,248) to $n/2$ Groebner basis
+ti_ip.tex(,249) computations. It needs no auxiliary variables, but a
supplementary
+ti_ip.tex(,250) precondition; namely, the existence of a vector without zero
components
+ti_ip.tex(,251) in the kernel of $A$.
+ti_ip.tex(,252)
+ti_ip.tex(,253) The main idea comes from the following observation:
+ti_ip.tex(,254)
+ti_ip.tex(,255) Let $B$ be an integer matrix, $u_1,\ldots,u_r$ a lattice basis
of the
+ti_ip.tex(,256) integer kernel of $B$. Assume that all components of $u_1$ are
+ti_ip.tex(,257) positive. Then
+ti_ip.tex(,258) $$ I_B=<x^{u_i^+}-x^{u_i^-}|i=1,\ldots,r>, $$
+ti_ip.tex(,259) i.e., the ideal on the right is already saturated w.r.t. all
variables.
+ti_ip.tex(,260)
+ti_ip.tex(,261) The algorithm starts by finding a lattice basis
$v_1,\ldots,v_r$ of the
+ti_ip.tex(,262) kernel of $A$ such that $v_1$ has no zero component. Let
+ti_ip.tex(,263) $\{i_1,\ldots,i_l\}$ be the set of indices $i$ with
+ti_ip.tex(,264) $v_{1,i}<0$. Multiplying the components $i_1,\ldots,i_l$ of
+ti_ip.tex(,265) $v_1,\ldots,v_r$ and the columns $i_1,\ldots,i_l$ of $A$ by
$-1$ yields
+ti_ip.tex(,266) a matrix $B$ and a lattice basis $u_1,\ldots,u_r$ of the
kernel of $B$
+ti_ip.tex(,267) that fulfill the assumption of the observation above. We are
then able
+ti_ip.tex(,268) to compute a generating set of $I_A$ by applying the following
+ti_ip.tex(,269) ``variable flip'' successively to $i=i_1,\ldots,i_l$:
+ti_ip.tex(,270)
+ti_ip.tex(,271) Let $>$ be an elimination ordering for $x_i$. Let $A_i$ be the
matrix
+ti_ip.tex(,272) obtained by multiplying the $i$-th column of $A$ with $-1$. Let
+ti_ip.tex(,273) $$\{x_i^{r_j} x^{a_j} - x^{b_j} | j\in J \}$$
+ti_ip.tex(,274) be a Groebner basis of $I_{A_i}$ w.r.t. $>$ (where $x_i$ is
neither
+ti_ip.tex(,275) involved in $x^{a_j}$ nor in $x^{b_j}$). Then
+ti_ip.tex(,276) $$\{x^{a_j} - x_i^{r_j} x^{b_j} | j\in J \}$$
+ti_ip.tex(,277) is a generating set for $I_A$.
+ti_ip.tex(,278) @end tex
ti_ip.tex(,316)
ti_ip.tex(,317) @node Bigatti and La Scala and Robbiano, , Di Biase and
Urbanke, Algorithms
ti_ip.tex(,318)
@@ -22058,6 +23844,12 @@
ti_ip.tex(,323) The algorithm of Bigatti, La Scala and Robbiano
(@pxref{[BLR98]}) combines the ideas of
ti_ip.tex(,324) the algorithms of Pottier and of Hosten and Sturmfels. The
ti_ip.tex(,325) computations are performed on a graded ideal with one auxiliary
+ti_ip.tex(,326) @tex
+ti_ip.tex(,327) variable $u$ and one supplementary generator
$x_1\cdot\ldots\cdot x_n -
+ti_ip.tex(,328) u$ (instead of the generator $t\cdot x_1\cdot\ldots\cdot x_n
-1$ in
+ti_ip.tex(,329) the algorithm of Pottier). The algorithm uses a quite unusual
technique to
+ti_ip.tex(,330) get rid of the variable $u$ again.
+ti_ip.tex(,331) @end tex
ti_ip.tex(,338)
ti_ip.tex(,339) There is another algorithm of the authors which tries to
parallelize
ti_ip.tex(,340) the computations (but which is not implemented in this
library).
@@ -22082,6 +23874,25 @@
ti_ip.tex(,359) @subsection Integer programming
ti_ip.tex(,360) @cindex integer programming
ti_ip.tex(,361)
+ti_ip.tex(,362) @tex
+ti_ip.tex(,363) Let $A$ be an $m\times n$ matrix with integral coefficients,
$b\in
+ti_ip.tex(,364) Z\!\!\! Z^m$ and $c\in Z\!\!\! Z^n$. The problem
+ti_ip.tex(,365) $$ \min\{c^T x | x\in Z\!\!\! Z^n, Ax=b, x\geq 0\hbox{
+ti_ip.tex(,366) component-wise}\} $$
+ti_ip.tex(,367) is called an instance of the \bf integer programming problem
\rm or
+ti_ip.tex(,368) \bf IP problem. \rm
+ti_ip.tex(,369)
+ti_ip.tex(,370) The IP problem is very hard; namely, it is NP-complete.
+ti_ip.tex(,371)
+ti_ip.tex(,372) For the following discussion let $c\geq 0$ (component-wise). We
+ti_ip.tex(,373) consider $c$ as a weight vector; because of its
non-negativity, $c$ can
+ti_ip.tex(,374) be refined into a monomial ordering $>_c$. It turns out that
we can
+ti_ip.tex(,375) solve such an IP instance with the help of toric ideals:
+ti_ip.tex(,376)
+ti_ip.tex(,377) First we assume that an initial solution $v$ (i.e., $v\in
Z\!\!\!
+ti_ip.tex(,378) Z^n, v\geq 0, Av=b$) is already known. We obtain the optimal
solution
+ti_ip.tex(,379) $v_0$ (i.e., with $c^T v_0$ minimal) by the following
procedure:
+ti_ip.tex(,380) @end tex
ti_ip.tex(,381) @c \begin{itemize}
ti_ip.tex(,382) @c \item (1) Compute the toric ideal $I_A$ using one of the
algorithms in the
ti_ip.tex(,383) @c previous section.
@@ -22096,11 +23907,23 @@
ti_ip.tex(,412) @itemize @bullet
ti_ip.tex(,413) @item (1) Compute the toric ideal I(A) using one of the
algorithms in the previous section.
ti_ip.tex(,414) @item (2) Compute the reduced Groebner basis G(c) of I(A)
w.r.t.@:
+ti_ip.tex(,418) @tex
+ti_ip.tex(,419) $>_c$
+ti_ip.tex(,420) @end tex
ti_ip.tex(,421) .
ti_ip.tex(,422) @item (3) Reduce
+ti_ip.tex(,426) @tex
+ti_ip.tex(,427) $x^v$
+ti_ip.tex(,428) @end tex
ti_ip.tex(,429) modulo G(c) using the Hironaka division algorithm.
ti_ip.tex(,430) If the result of this reduction is
+ti_ip.tex(,434) @tex
+ti_ip.tex(,435) $x^(v_0)$
+ti_ip.tex(,436) @end tex
ti_ip.tex(,437) , then
+ti_ip.tex(,441) @tex
+ti_ip.tex(,442) $v_0$
+ti_ip.tex(,443) @end tex
ti_ip.tex(,444) is an optimal
ti_ip.tex(,445) solution of the given instance.
ti_ip.tex(,446) @end itemize
@@ -22120,6 +23943,9 @@
ti_ip.tex(,460) methods seem to be faster in general than the methods using
toric
ti_ip.tex(,461) ideals. But the latter have one great advantage: If one wants
to solve
ti_ip.tex(,462) various instances that differ only by the vector
+ti_ip.tex(,466) @tex
+ti_ip.tex(,467) $b$
+ti_ip.tex(,468) @end tex
ti_ip.tex(,469) , one has to
ti_ip.tex(,470) perform steps (1) and (2) above only once. As the running time
of step (3)
ti_ip.tex(,471) is very short, solving all the instances is not much harder
than
@@ -22262,6 +24088,9 @@
math.tex(,959) Symbolic Computation
math.tex(,960)
math.tex(,961) @item
+math.tex(,962) @tex
+math.tex(,963) Faug\`ere,
+math.tex(,964) @end tex
math.tex(,968) J. C.; Gianni, P.; Lazard, D.; Mora, T.: Efficient computation
math.tex(,969) of zero-dimensional
math.tex(,970) Gr@"obner bases by change of ordering. Journal of Symbolic
Computation, 1989
@@ -47812,6 +49641,9 @@
brnoeth_lib.tex(,685)
brnoeth_lib.tex(,686) @item @strong{Warnings:}
brnoeth_lib.tex(,687) G should satisfy
+brnoeth_lib.tex(,691) @tex
+brnoeth_lib.tex(,692) $ 2*genus-2 < deg(G) < size(D) $
+brnoeth_lib.tex(,693) @end tex
brnoeth_lib.tex(,694) , which is
brnoeth_lib.tex(,695) not checked by the algorithm.
brnoeth_lib.tex(,696) @*G and D should have disjoint supports (checked by the
algorithm).
@@ -47876,10 +49708,16 @@
brnoeth_lib.tex(,771) for more details)address@hidden
brnoeth_lib.tex(,772) The code computes the residues of a vector space basis of
brnoeth_lib.tex(,773)
+brnoeth_lib.tex(,777) @tex
+brnoeth_lib.tex(,778) $\Omega(G-D)$
+brnoeth_lib.tex(,779) @end tex
brnoeth_lib.tex(,780) at the rational places given by D.
brnoeth_lib.tex(,781)
brnoeth_lib.tex(,782) @item @strong{Warnings:}
brnoeth_lib.tex(,783) G should satisfy
+brnoeth_lib.tex(,787) @tex
+brnoeth_lib.tex(,788) $ 2*genus-2 < deg(G) < size(D) $
+brnoeth_lib.tex(,789) @end tex
brnoeth_lib.tex(,790) , which is
brnoeth_lib.tex(,791) not checked by the algorithm.
brnoeth_lib.tex(,792) @*G and D should have disjoint supports (checked by the
algorithm).
@@ -47936,8 +49774,14 @@
brnoeth_lib.tex(,859) E[2] ... E[n+2]: matrices used in the procedure
decodeSV
brnoeth_lib.tex(,860) E[n+3]: intvec with
brnoeth_lib.tex(,861) E[n+3][1]: correction capacity
+brnoeth_lib.tex(,865) @tex
+brnoeth_lib.tex(,866) $epsilon$
+brnoeth_lib.tex(,867) @end tex
brnoeth_lib.tex(,868) of the algorithm
brnoeth_lib.tex(,869) E[n+3][2]: designed Goppa distance
+brnoeth_lib.tex(,873) @tex
+brnoeth_lib.tex(,874) $delta$
+brnoeth_lib.tex(,875) @end tex
brnoeth_lib.tex(,876) of the current AG code
brnoeth_lib.tex(,877) @end format
brnoeth_lib.tex(,878)
@@ -47953,6 +49797,9 @@
brnoeth_lib.tex(,888) The current AG code is
@code{AGcode_Omega(G,D,EC)address@hidden
brnoeth_lib.tex(,889) If you know the exact minimum distance d and you want to
use it in
brnoeth_lib.tex(,890) @code{decodeSV} instead of
+brnoeth_lib.tex(,894) @tex
+brnoeth_lib.tex(,895) $delta$
+brnoeth_lib.tex(,896) @end tex
brnoeth_lib.tex(,897) , you can change the value
brnoeth_lib.tex(,898) of E[n+3][2] to d before applying decodeSV.
brnoeth_lib.tex(,899) @*If you have a systematic encoding for the current code
and want to
@@ -47963,10 +49810,19 @@
brnoeth_lib.tex(,904) @item @strong{Warnings:}
brnoeth_lib.tex(,905) F must be a divisor with support disjoint from the
support of D and
brnoeth_lib.tex(,906) with degree
+brnoeth_lib.tex(,910) @tex
+brnoeth_lib.tex(,911) $epsilon + genus$
+brnoeth_lib.tex(,912) @end tex
brnoeth_lib.tex(,913) , where
brnoeth_lib.tex(,914)
+brnoeth_lib.tex(,918) @tex
+brnoeth_lib.tex(,919) $epsilon:=[(deg(G)-3*genus+1)/2]$
+brnoeth_lib.tex(,920) @end tex
brnoeth_lib.tex(,921) address@hidden
brnoeth_lib.tex(,922) G should satisfy
+brnoeth_lib.tex(,926) @tex
+brnoeth_lib.tex(,927) $ 2*genus-2 < deg(G) < size(D) $
+brnoeth_lib.tex(,928) @end tex
brnoeth_lib.tex(,929) , which is
brnoeth_lib.tex(,930) not checked by the algorithm.
brnoeth_lib.tex(,931) @*G and D should also have disjoint supports (checked by
the
Index: test/singular_manual/res/texi_singular/singular.texi
===================================================================
RCS file:
/cvsroot/texi2html/texi2html/test/singular_manual/res/texi_singular/singular.texi,v
retrieving revision 1.2
retrieving revision 1.3
diff -u -b -r1.2 -r1.3
--- test/singular_manual/res/texi_singular/singular.texi 19 Aug 2008
16:53:01 -0000 1.2
+++ test/singular_manual/res/texi_singular/singular.texi 9 Jan 2009
21:20:59 -0000 1.3
@@ -333,6 +333,9 @@
@sc{Singular}'s development started in 1984 with an implementation of
Mora's Tangent Cone algorithm in Modula-2 on an Atari computer (K.P.
Neuendorf, G. Pfister,
address@hidden
+H.\ Sch\"onemann; Humboldt-Universit\"at
address@hidden tex
zu Berlin). The need for a new system arose from the investigation of
mathematical problems coming from singularity theory which none of the
existing systems was able to compute.
@@ -575,6 +578,9 @@
@noindent This shows the text of @ref{intmat}, in the printed manual.
Next, we define a
address@hidden
+$3 \times 3$
address@hidden tex
matrix of integers and initialize it with some values, row by row
from left to right:
@@ -651,6 +657,9 @@
ring variables, and the third part determines the monomial ordering to
be used. So the example above declares a polynomial ring called @code{r}
with a ground field of characteristic
address@hidden
+$0$
address@hidden tex
(i.e., the rational
numbers) and ring variables called @code{x}, @code{y}, and @code{z}. The
@code{dp} at the end means that the degree reverse lexicographical
@@ -673,7 +682,13 @@
@item ring r4=(0,a),(mu,nu),lp;
transcendental extension of
address@hidden
+$Q$
address@hidden tex
by
address@hidden
+$a$
address@hidden tex
, variable names
@code{mu} and @code{nu}.
@@ -702,6 +717,9 @@
@c
Typing the name of a ring prints its definition. The example below
shows that the default ring in @sc{Singular} is
address@hidden
+$Z/32003[x,y,z]$
address@hidden tex
with degree reverse lexicographical ordering:
@@ -731,6 +749,9 @@
@end smallexample
Once a ring is active, we can define polynomials. A monomial, say
address@hidden
+$x^3$
address@hidden tex
may be entered in two ways: either using the power operator @code{^},
saying @code{x^3}, or in short-hand notation without operator, saying
@code{x3}. Note that the short-hand notation is forbidden if the name
@@ -849,6 +870,9 @@
@end smallexample
@noindent gives the desired vector space dimension
address@hidden
+$K[x,y,z]/\hbox{\rm jacob}(f)$.
address@hidden tex
As in @sc{Singular} the functions may take the input directly from
earlier calculations, the whole sequence of commands may be written
in one single statement.
@@ -1022,6 +1046,9 @@
This shows that @code{f} has outside the origin in affine 3-space
singularities with local Milnor number adding up to
address@hidden
+$12-4=8$.
address@hidden tex
Using global and local orderings as above is a convenient way to check
whether a variety has singularities outside the origin.
@@ -1068,6 +1095,9 @@
The algorithm of the standard basis computations may be
affected by the command @code{option}. For example, a reduced standard
basis of the ideal generated by the
address@hidden
+$1 \times 1$-minors
address@hidden tex
of H is obtained in the following way:
@smallexample
option(redSB);
@@ -1076,6 +1106,9 @@
@end smallexample
This shows that 1 is contained in the ideal of the
address@hidden
+$1 \times 1$-minors,
address@hidden tex
hence the corresponding variety is empty.
@c Coming back to some mathematical considerations, we study the problem how
@c to calculate some ....
@@ -1131,13 +1164,22 @@
@end smallexample
However the submodule
address@hidden
+$MD$
address@hidden tex
may also be considered as the module
of relations of the factor module
address@hidden
+$r^3/MD$.
address@hidden tex
In this way, @sc{Singular} can treat arbitrary finitely generated modules
over the
basering (@pxref{Representation of mathematical objects}).
In order to get the module of relations of
address@hidden
+$MD$
address@hidden tex
,
we use the command @code{syz}.
@@ -1148,15 +1190,30 @@
We want to calculate, as an application, the annihilator of a given module.
Let
address@hidden
+$M = r^3/U$,
address@hidden tex
where U is our defining module of relations for the module
address@hidden
+$M$.
address@hidden tex
@smallexample
module U = [z3,xy2,x3],[yz2,1,xy5z+z3],[y2z,0,x3],[xyz+x2,y2,0],[xyz,x2y,1];
@end smallexample
Then, by definition, the annihilator of M is the ideal
address@hidden
+$\hbox{ann}(M) = \{a \mid aM = 0 \}$
address@hidden tex
which is by the description of M the same as
address@hidden
+$\{ a \mid ar^3 \in U \}$.
address@hidden tex
Hence we have to calculate the quotient
address@hidden
+$U \colon r^3 $.
address@hidden tex
The rank of the free module is determined by the choice of U and is the
number of rows of the corresponding matrix. This may be determined by
the function @code{nrows}. All we have to do now is the following:
@@ -1176,7 +1233,13 @@
The most general command is @code{res(... ,n)} which determines heuristically
what method to use for the given problem. It computes the free resolution
up to the length
address@hidden
+$n$
address@hidden tex
, where
address@hidden
+$n=0$
address@hidden tex
corresponds to the full resolution.
Here we use the possibility to inspect the calculation process using the
@@ -1248,7 +1311,13 @@
In this case, the output is to be interpreted as follows: the 3rd syzygy
module of R/I, @code{rs[3]}, is the rank-2-submodule of
address@hidden
+$R^5$
address@hidden tex
generated by the vectors
address@hidden
+$(z^3,0,-y+4z,x+2z,0)$ and $(-xyz-y^2z-4xz^2+16z^3,-y^2,48z,48z,x+y-z)$.
address@hidden tex
@c ----------------------------------------------------------------------------
@node General concepts, Data types, Introduction, Top
@@ -2663,16 +2732,37 @@
@enumerate
@item
the field of rational numbers
address@hidden
+$Q$
address@hidden tex
,
@item
address@hidden
address@hidden
+finite fields $Z/p$, $p$ a prime $\le 2147483629$,
address@hidden tex
address@hidden
address@hidden
+finite fields $\hbox{GF}(p^n)$ with $p^n$ elements, $p$ a prime, $p^n \le
2^{15}$,
address@hidden tex
@item
transcendental extension of
address@hidden
+$Q$
address@hidden tex
or
address@hidden
+$Z/p$
address@hidden tex
,
@item
simple algebraic extension of
address@hidden
+$Q$
address@hidden tex
or
address@hidden
+$Z/p$
address@hidden tex
,
@item
the field of real numbers represented by floating point
@@ -2723,6 +2813,9 @@
@itemize @bullet
@item
the ring
address@hidden
+$Z/32003[x,y,z]$
address@hidden tex
with degree reverse lexicographical
ordering. The exact ring declaration may be omitted in the first
example since this is the default ring:
@@ -2734,6 +2827,9 @@
@item
the ring
address@hidden
+$Q[a,b,c,d]$
address@hidden tex
with lexicographical ordering:
@smallexample
@@ -2742,6 +2838,9 @@
@item
the ring
address@hidden
+$Z/7[x,y,z]$
address@hidden tex
with local degree reverse lexicographical
ordering. The non-prime 10 is converted to the next lower prime in the
second example:
@@ -2753,8 +2852,17 @@
@item
the ring
address@hidden
+$Z/7[x_1,\ldots,x_6]$
address@hidden tex
with lexicographical ordering for
address@hidden
+$x_1,x_2,x_3$
address@hidden tex
and degree reverse lexicographical ordering for
address@hidden
+$x_4,x_5,x_6$:
address@hidden tex
@smallexample
ring r = 7,(x(1..6)),(lp(3),dp);
@@ -2762,8 +2870,14 @@
@item
the localization of
address@hidden
+$(Q[a,b,c])[x,y,z]$
address@hidden tex
at the maximal ideal
address@hidden
+$(x,y,z)$
address@hidden tex
:
@smallexample
@@ -2772,10 +2886,22 @@
@item
the ring
address@hidden
+$Q[x,y,z]$
address@hidden tex
with weighted reverse lexicographical ordering.
The variables
address@hidden
+$x$
address@hidden tex
,
address@hidden
+$y$
address@hidden tex
, and
address@hidden
+$z$
address@hidden tex
have the weights 2, 1,
and 3, respectively, and vectors are first ordered by components (in
descending order) and then by monomials:
@@ -2787,12 +2913,30 @@
@item
the ring
address@hidden
+$K[x,y,z]$
address@hidden tex
, where
address@hidden
+$K=Z/7(a,b,c)$
address@hidden tex
denotes the transcendental
extension of
address@hidden
+$Z/7$
address@hidden tex
by
address@hidden
+$a$
address@hidden tex
,
address@hidden
+$b$
address@hidden tex
and
address@hidden
+$c$
address@hidden tex
with degree
lexicographical ordering:
@@ -2802,19 +2946,49 @@
@item
the ring
address@hidden
+$K[x,y,z]$
address@hidden tex
, where
address@hidden
+$K=Z/7[a]$
address@hidden tex
denotes the algebraic extension of
degree 2 of
address@hidden
+$Z/7$
address@hidden tex
by
address@hidden
+$a.$
address@hidden tex
In other words,
address@hidden
+$K$
address@hidden tex
is the finite field with
49 elements. In the first case,
address@hidden
+$a$
address@hidden tex
denotes an algebraic
element over
address@hidden
+$Z/7$
address@hidden tex
with minimal polynomial
address@hidden
+$\mu_a=a^2+a+3$,
address@hidden tex
in the second case,
address@hidden
+$a$
address@hidden tex
refers to some generator of the cyclic group of units of
address@hidden
+$K$
address@hidden tex
:
@smallexample
@@ -2824,7 +2998,13 @@
@item
the ring
address@hidden
+$R[x,y,z]$
address@hidden tex
, where
address@hidden
+$R$
address@hidden tex
denotes the field of real
numbers represented by simple precision floating point numbers. This is
a special case:
@@ -2835,7 +3015,13 @@
@item
the ring
address@hidden
+$R[x,y,z]$
address@hidden tex
, where
address@hidden
+$R$
address@hidden tex
denotes the field of real
numbers represented by floating point numbers of 50 valid decimal digits
and the same number of digits for the rest:
@@ -2846,7 +3032,13 @@
@item
the ring
address@hidden
+$R[x,y,z]$
address@hidden tex
, where
address@hidden
+$R$
address@hidden tex
denotes the field of real
numbers represented by floating point numbers of 10 valid decimal digits
and with 50 digits for the rest:
@@ -2857,10 +3049,19 @@
@item
the ring
address@hidden
+$R(j)[x,y,z]$
address@hidden tex
, where
address@hidden
+$R$
address@hidden tex
denotes the field of real
numbers represented by floating point numbers of 30 valid decimal digits
and the same number for the rest.
address@hidden
+$j$
address@hidden tex
denotes the imaginary unit.
@smallexample
@@ -2869,10 +3070,19 @@
@item
the ring
address@hidden
+$R(i)[x,y,z]$
address@hidden tex
, where
address@hidden
+$R$
address@hidden tex
denotes the field of real
numbers represented by floating point numbers of 6 valid decimal digits
and the same number for the rest.
address@hidden
+$i$
address@hidden tex
is the default for the imaginary unit.
@smallexample
@@ -2881,8 +3091,14 @@
@item
the quotient ring
address@hidden
+$Z/7[x,y,z]$
address@hidden tex
modulo the square of the maximal
ideal
address@hidden
+$(x,y,z)$
address@hidden tex
:
@smallexample
@@ -2935,7 +3151,13 @@
an expression_list of an int_expression and a name.
@* The int_expression has to be a prime number p to the power of a
positive integer n. This defines the Galois field
address@hidden
+$\hbox{GF}(p^n)$ with $p^n$ elements, where $p^n$ has to be smaller or equal
$2^{15}$.
address@hidden tex
The given name refers to a primitive element of
address@hidden
+$\hbox{GF}(p^n)$
address@hidden tex
generating the multiplicative group. Due to a different internal
representation, the arithmetic operations in these coefficient fields
are faster than arithmetic operations in algebraic extensions as
@@ -3061,7 +3283,13 @@
@strong{Remark:} The novice user should generally use the ordering
@code{dp} for computations in the polynomial ring
address@hidden
+$K[x_1,\ldots,x_n]$,
address@hidden tex
resp.@: @code{ds} for computations in the localization
address@hidden
+$\hbox{Loc}_{(x)}K[x_1,\ldots,x_n])$.
address@hidden tex
For more details, see @ref{Polynomial data}.
In a ring declaration, @sc{Singular} offers the following orderings:
@@ -3087,8 +3315,14 @@
@end table
Global orderings are well-orderings, i.e.,
address@hidden
+$1 < x$
address@hidden tex
for each ring
variable
address@hidden
+$x$
address@hidden tex
. They are denoted by a @code{p} as the second
character in their name.
@@ -3981,6 +4215,9 @@
by the size of expression.
@* But @code{matrix(} expression @code{,} m @code{,} n @code{)} may also be
used - the result is a
address@hidden
+$ m \times n $
address@hidden tex
matrix (@pxref{matrix type cast})
@item
@ @tab @code{module} @tab expression lists of @code{int}, @code{number},
@@ -4294,11 +4531,17 @@
@*["help_text"]
@address@hidden@{}
@*
address@hidden
+\quad
address@hidden tex
procedure_body
@address@hidden@}}
@address@hidden
@address@hidden@{}
@*
address@hidden
+\quad
address@hidden tex
sequence_of_commands;
@address@hidden@}}]
@item Purpose:
@@ -5130,6 +5373,9 @@
@code{@@address@hidden@}}
@address@hidden
@*
address@hidden
+$\alpha$
address@hidden tex
@item Note:
Mathematical expressions inside @code{@@address@hidden@}} may
@@ -5229,6 +5475,9 @@
@address@hidden
@*Among others, within a texinfo environment one can use the tex environment
to typeset more complex mathematical like
address@hidden
+$ i_{1,1} $
address@hidden tex
@end table
@end table
@@ -5523,12 +5772,18 @@
@item @strong{Return:}
int:
address@hidden
+$i+i+i$
address@hidden tex
@item @strong{Note:}
Help is in pure Texinfo
@*This help string is written in texinfo, which enables you to use,
among others, the @@math command for mathematical typesetting (like
address@hidden
+$\alpha, \beta$
address@hidden tex
).
@*It also gives more control over the layout, but is, admittingly,
more cumbersome to write.
@@ -5569,6 +5824,9 @@
@* Use a @@ref constructs for references (like @pxref{mtripple})
@* Use @@code for typewriter font (like @code{i_1})
@* Use @@math for simple math mode typesetting (like
address@hidden
+$i_1$
address@hidden tex
).
@* Note: No parenthesis like @} are allowed inside @@math and @@code
@* Use @@example for indented preformatted text typeset in typewriter
@@ -5583,6 +5841,9 @@
Use @@texinfo for text in pure texinfo
@expansion{}
address@hidden
+$i_{1,1}$
address@hidden tex
Notice that
@@ -6232,6 +6493,9 @@
set of minors of a matrix (see @ref{minor})
@item modulo
represents
address@hidden
+$(h1+h2)/h1 \cong h2/(h1 \cap h2)$
address@hidden tex
(see @ref{modulo})
@item mres
minimal free resolution of an ideal resp.@: module w.r.t. a minimal set of
generators of the given ideal resp.@: module
@@ -7983,25 +8247,62 @@
Canonically realized are
@itemize @bullet
@item
address@hidden
+$Q \rightarrow Q(a, \ldots)$
address@hidden tex
@item
address@hidden
+$Q \rightarrow R$
address@hidden tex
@item
address@hidden
+$Q \rightarrow C$
address@hidden tex
@item
address@hidden
+$Z/p \rightarrow (Z/p)(a, \ldots)$
address@hidden tex
@item
address@hidden
+$Z/p \rightarrow GF(p^n)$
address@hidden tex
@item
address@hidden
+$Z/p \rightarrow R$
address@hidden tex
@item
address@hidden
+$R \rightarrow C$
address@hidden tex
@end itemize
Possible are furthermore
@itemize @bullet
@item
address@hidden
address@hidden
address@hidden
+% This is quite a hack, but for now it works.
+$Z/p \rightarrow Q,
+\quad
+[i]_p \mapsto i \in [-p/2, \, p/2]
+\subseteq Z$
address@hidden tex
address@hidden
address@hidden
+$Z/p \rightarrow Z/p^\prime,
+\quad
+[i]_p \mapsto i \in [-p/2, \, p/2] \subseteq Z, \;
+i \mapsto [i]_{p^\prime} \in Z/p^\prime$
address@hidden tex
address@hidden
address@hidden
+$C \rightarrow R, \quad$ the real part
address@hidden tex
@end itemize
Finally, in Singular we allow the mapping from rings
@@ -8010,8 +8311,14 @@
@itemize @bullet
@item
address@hidden
+$Q \rightarrow Z/p$
address@hidden tex
@item
address@hidden
+$Q \rightarrow (Z/p)(a, \ldots)$
address@hidden tex
@end itemize
In these cases the denominator and the numerator
of a number are mapped separately by the usual
@@ -8455,18 +8762,45 @@
Like vectors they
can only be defined or accessed with respect to a basering.
If
address@hidden
+$M$
address@hidden tex
is a submodule of
-
address@hidden
+$R^n$,
address@hidden tex
+
address@hidden
+$R$
address@hidden tex
the basering, generated by vectors
address@hidden
+$v_1, \ldots, v_k$, then $v_1, \ldots, v_k$
address@hidden tex
may be considered as the generators of relations of
address@hidden
+$R^n/M$
address@hidden tex
between the canonical generators @code{gen(1)},@dots{},@code{gen(n)}.
Hence any finitely generated
address@hidden
+$R$
address@hidden tex
-module can be represented in @sc{Singular}
by its module of relations. The assignments
@code{module M=v1,...,vk; matrix A=M;}
create the presentation matrix of size
address@hidden
+n$\times$k
address@hidden tex
for
address@hidden
+R$^n$/M,
address@hidden tex
i.e., the columns of A are the vectors
address@hidden
+$v_1, \ldots, v_k$
address@hidden tex
which generate M (cf. @ref{Representation of mathematical objects}).
@menu
@@ -8621,6 +8955,9 @@
over a local ring
@item modulo
represents
address@hidden
+$(h1+h2)/h1=h2/(h1 \cap h2)$
address@hidden tex
(see @ref{modulo})
@item mres
minimal free resolution of an ideal resp.@: module w.r.t. a minimal set of
generators of the given module
@@ -9230,11 +9567,17 @@
@*["help_text"]
@address@hidden@{}
@*
address@hidden
+\quad
address@hidden tex
procedure_body
@address@hidden@}}
@address@hidden
@address@hidden@{}
@*
address@hidden
+\quad
address@hidden tex
sequence_of_commands;
@address@hidden@}}]
@address@hidden proc_name @code{=} proc_name @code{;}
@@ -9549,31 +9892,82 @@
@table @asis
@item @code{+}
construct a new ring
address@hidden
+$k[X,Y]$
address@hidden tex
from
address@hidden
+$k_1[X]$
address@hidden tex
and
address@hidden
+$k_2[Y]$
address@hidden tex
.
@end table
Concerning the ground fields
address@hidden
+$k_1$
address@hidden tex
and
address@hidden
+$k_2$
address@hidden tex
take the
following guide lines into consideration:
@itemize @bullet
@item Neither
address@hidden
+$k_1$
address@hidden tex
nor
address@hidden
+$k_2$
address@hidden tex
may be
address@hidden
+$R$
address@hidden tex
or
address@hidden
+$C$
address@hidden tex
.
@item If the characteristic of
address@hidden
+$k_1$
address@hidden tex
and
address@hidden
+$k_2$
address@hidden tex
differs, then one of them must be
address@hidden
+$Q$
address@hidden tex
.
@item At most one of
address@hidden
+$k_1$
address@hidden tex
and
address@hidden
+$k_2$
address@hidden tex
may be have parameters.
@item If one of
address@hidden
+$k_1$
address@hidden tex
and
address@hidden
+$k_2$
address@hidden tex
is an algebraic extension of
address@hidden
+$Z/p$
address@hidden tex
it may not be defined by a @code{charstr} of type @code{(p^n,a)}.
@end itemize
@@ -10469,6 +10863,18 @@
intmat
@item @strong{Purpose:}
with 1 argument: computes the graded Betti numbers of a minimal resolution of
address@hidden
+$R^n/M$, if $R$ denotes the basering and
+$M$ a homogeneous submodule of $R^n$ and the argument represents a
+resolution of
+$R^n/M$.
address@hidden tex
address@hidden
+The entry d of the intmat at place (i,j) is the minimal number of
+generators in degree i+j of the j-th syzygy module (= module of
+relations) of $R^n/M$ (the 0th (resp.\ 1st) syzygy module of $R^n/M$ is
+$R^n$ (resp.\ $M$)).
address@hidden tex
The argument is considered to be the result of a res/sres/mres/nres/lres
command. This implies that a zero is only allowed (and counted) as a
generator in the first module.
@@ -10536,6 +10942,15 @@
where the generators are the columns of the
displayed matrix and degrees are assigned such that the corresponding maps
have degree 0:
address@hidden
+$$
+0 \longleftarrow r/j \longleftarrow r(1)
+\buildrel{T[1]}\over{\longleftarrow} r(2) \oplus r^3(3)
+\buildrel{T[2]}\over{\longleftarrow} r^4(4)
+\buildrel{T[3]}\over{\longleftarrow} r(5)
+\longleftarrow 0 \quad .
+$$
address@hidden tex
@c inserted refs from reference.doc:455
@c end inserted refs from reference.doc:455
@@ -10849,12 +11264,28 @@
@end format
If J is a vector or a module this procedure is repeated for each
component and the resulting matrices are address@hidden
address@hidden
+The third argument is used to return the matrix T of coefficients
+such that {\tt matrix}(J) = T*M.
address@hidden tex
@item @strong{Note:}
@code{coeffs} returns the coefficient 0 at the appropriate place if a monomial
is not present, while @code{coef} considers only monomials which really occur
in the given expression. @*
If
address@hidden
+$M=(m_{ij})$
address@hidden tex
then the j-th generator of an ideal J is equal to
address@hidden
+$$J_j = z^0 \cdot m_{1j} + z^1 \cdot m_{2j} + ... + z^{d-1} \cdot m_{dj},$$
+while for a module J the i-th component of the j-th generator is
+equal to the entry [i,j] of {\tt matrix}(J), and we get
address@hidden tex
address@hidden
+$$ J_{i,j} = z^0 \cdot m_{(i-1)d+1,j} + z^1 \cdot m_{(i-1)d+2,j} + ... +
+z^{d-1} \cdot m_{id,j}.$$
address@hidden tex
@item @strong{Example:}
@smallexample
@@ -10933,7 +11364,14 @@
producing a m x n matrix.
@*Contraction is defined on monomials by:
@*
address@hidden
+$${\rm contract}(x^A , x^B) := \cases{ x^{(B-A)}, &if $B\ge A$
+componentwise\cr 0,&otherwise.\cr}$$
address@hidden tex
where A and B are the multiexponents of the ring variables represented by
address@hidden
+$x$.
address@hidden tex
@code{contract} is extended bilinearly to all polynomials.
@item @strong{Example:}
@smallexample
@@ -12366,13 +12804,24 @@
@code{highcorner(I)} returns 0 iff @code{dim(I)>0} or @code{dim(I)=-1}.
Otherwise it returns the smallest monomial m not in I which has the following
properties (with
address@hidden
+$x_i$
address@hidden tex
the variables of the basering):
@itemize @bullet
@item
if
address@hidden
+$x_i>1$ then $x_i$
address@hidden tex
does not divide m (e.g., m=1 if the ordering is global)
@item
given any set of generators
address@hidden
+$f_1,\dots,f_k$ of I, let $f'_i$ be obtained from
+$f_i$ by deleting the terms divisible by $x_i\cdot m$ for all i with $x_i<1$.
+Then $f'_1,\dots,f'_k$ generate I.
address@hidden tex
@end itemize
@item @strong{Example:}
@smallexample
@@ -12511,11 +12960,22 @@
More precisely, let R be the basering and I be the given ideal.
Then @code{hres} computes a minimal free resolution of R/I
address@hidden
+$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1
+\buildrel{A_1}\over{\longrightarrow} R\longrightarrow R/I
+\longrightarrow 0.$$
address@hidden tex
If the int_expression k is not zero then the computation stops after
k steps and returns a list of modules
address@hidden
+$M_i={\tt module} (A_i)$, i=1..k.
address@hidden tex
@code{list L=hres(I,0);} returns a list L of n modules (where n is the
number of variables of the basering) such that
address@hidden
+${\tt L[i]}=M_i$
address@hidden tex
in the above notation.
@item @strong{Note:}
The ideal_expression has to be homogeneous.
@@ -12631,6 +13091,9 @@
@item @strong{Note:}
U is a set of independent variables for I if and only if
address@hidden
+$I \cap K[U]=(0)$,
address@hidden tex
i.e., eliminating the remaining variables gives (0).
U is maximal if dim(I)=#U.
@item @strong{Syntax:}
@@ -12728,19 +13191,47 @@
@item @strong{Purpose:}
interreduces a set of polynomials/vectors.
@*
address@hidden
+input: $f_1,\dots,f_n$
address@hidden tex
@*
address@hidden
+output: $g_1,\dots,g_s$ with $s \leq n$ and the properties
address@hidden tex
@itemize @bullet
@item
address@hidden
address@hidden
+$(f_1,\dots,f_n) = (g_1,\dots,g_s)$
address@hidden tex
address@hidden
address@hidden
+$L(g_i)\neq L(g_j)$ for all $i\neq j$
address@hidden tex
@item
in the case of a global ordering (polynomial ring):
@*
address@hidden
+$L(g_i)$
address@hidden tex
does not divide m for all monomials m of
address@hidden
+$\{g_1,\dots,g_{i-1},g_{i+1},\dots,g_s\}$
address@hidden tex
@item
in the case of a local or mixed ordering (localization of polynomial ring):
@* if
address@hidden
+$L(g_i) | L(g_j)$ for any $i \neq j$,
address@hidden tex
then
address@hidden
+$ecart(g_i) > ecart(g_j)$
address@hidden tex
@end itemize
address@hidden
+Here, $L(g)$ denotes the leading term of $g$ and
+$ecart(g):=deg(g)-deg(L(g))$.
address@hidden tex
@item @strong{Example:}
@smallexample
@c reused example interred reference.doc:2557
@@ -13470,11 +13961,22 @@
More precisely, let R be the basering and I be the given ideal.
Then @code{lres} computes a minimal free resolution of R/I
address@hidden
+$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1
+\buildrel{A_1}\over{\longrightarrow} R\longrightarrow R/I
+\longrightarrow 0.$$
address@hidden tex
If the int_expression k is not zero then the computation stops after
k steps and returns a list of modules
address@hidden
+$M_i={\tt module}(A_i)$, i=1..k.
address@hidden tex
@code{list L=lres(I,0);} returns a list L of n modules (where n is the
number of variables of the basering) such that
address@hidden
+${\tt L[i]}=M_i$
address@hidden tex
in the above notation.
@item @strong{Note:}
The ideal_expression has to be homogeneous.
@@ -13728,9 +14230,25 @@
module
@item @strong{Purpose:}
@code{modulo(h1,h2)}
address@hidden
+represents $h_1/(h_1 \cap h_2) \cong (h_1+h_2)/h_2$
address@hidden tex
where
address@hidden
+$h_1$ and $h_2$
address@hidden tex
are considered as submodules of the same free module
address@hidden
+$R^l$
address@hidden tex
(l=1 for ideals). Let
address@hidden
+$H_1$, resp.\ $H_2$,
address@hidden tex
address@hidden
+be the matrices of size $l \times k$, resp.\ $l \times m$, having the
+generators of $h_1$, resp.\ $h_2$,
address@hidden tex
as columns.
@c @*
@c @tex
@@ -13744,7 +14262,14 @@
@c @end smallexample
@c @end ifinfo
Then
address@hidden
+$h_1/(h_1 \cap h_2) \cong R^k / ker(\overline{H_1})$
address@hidden tex
where
address@hidden
+$\overline{H_1}: R^k \rightarrow R^l/Im(H_2)=R^l/h_2$
+is the induced map.
address@hidden tex
@address@hidden(h1,h2)} returns generators of
the kernel of this induced map.
@item @strong{Example:}
@@ -13845,17 +14370,32 @@
computes a minimal free resolution of an ideal or module M with the
standard basis method. More precisely, let address@hidden(M), then @code{mres}
computes a free resolution of
address@hidden
+$coker(A)=F_0/M$
+$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1
+\buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow F_0/M
+\longrightarrow 0,$$
address@hidden tex
where the columns of the matrix
address@hidden
+$A_1$
address@hidden tex
are a minimal set of generators
of M if the basering is local or if M is homogeneous.
If the int expression k is not zero then the computation stops after k steps
and returns a list of modules
address@hidden
+$M_i={\tt module}(A_i)$, i=1...k.
address@hidden tex
@address@hidden(M,0)} returns a resolution consisting of at most n+2 modules,
where n is the number of variables of the basering.
Let @code{list L=mres(M,0);}
then @code{L[1]} consists of a minimal set of generators of the input,
@code{L[2]}
consists of a minimal set of generators for the first syzygy module of
@code{L[1]}, etc., until @code{L[p+1]}, such that
address@hidden
+${\tt L[i]}\neq 0$ for $i \le p$,
address@hidden tex
but @code{L[p+1]}, the first syzygy module of @code{L[p]},
is 0 (if the basering is not a qring).
@item @strong{Note:}
@@ -14146,16 +14686,32 @@
the second module on (by the standard basis method).
More precisely, let
address@hidden
+$A_1$=matrix(M),
address@hidden tex
then @code{nres} computes a free resolution of
address@hidden
+$coker(A_1)=F_0/M$
+$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1
\buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow F_0/M\longrightarrow
0,$$
address@hidden tex
@*where the columns of the matrix
address@hidden
+$A_1$
address@hidden tex
are the given set of generators of M.
If the int expression k is not zero then the computation stops after k steps
and returns a list of modules
address@hidden
+$M_i={\tt module}(A_i)$, i=1..k.
address@hidden tex
@address@hidden(M,0)} returns a list of n modules where n is the number of
variables of the basering.
Let @code{list L=nres(M,0);} then @code{L[1]=M} is identical to the input,
@code{L[2]} is a minimal set of generators for the first syzygy
module of @code{L[1]}, etc.
address@hidden
+(${\tt L[i]}=M_i$
address@hidden tex
in the notations from above).
@item @strong{Example:}
@smallexample
@@ -14774,9 +15330,18 @@
@table @code
@item "betti"
The Betti numbers are printed in a matrix-like format where the entry
address@hidden
+$d$ in row $i$ and column $j$
address@hidden tex
is the minimal number of generators in
degree
address@hidden
+$i+j$ of the $j$-th
address@hidden tex
syzygy module of
address@hidden
+$R^n/M$ (the 0th and 1st syzygy module of $R^n/M$ is $R^n$ and $M$, resp.).
address@hidden tex
@item "%s"
returns @code{string(} expression @code{)}
@item "%2s"
@@ -15161,12 +15726,21 @@
@item @strong{Purpose:}
computes the ideal quotient, resp.@: module quotient. Let @code{R} be the
basering, @code{I,J} ideals and @code{M} a module in
address@hidden
+${\tt R}^n$.
address@hidden tex
Then
@itemize
@item
@code{quotient(I,J)}=
address@hidden
+$\{a \in R \mid aJ \subset I\}$,
address@hidden tex
@item
@code{quotient(M,J)}=
address@hidden
+$\{b \in R^n \mid bJ \subset M\}$.
address@hidden tex
@end itemize
@item @strong{Example:}
@smallexample
@@ -15356,6 +15930,15 @@
computes the regularity of a homogeneous ideal, resp.@: module, from a
minimal resolution given by the list expression.
@*
address@hidden
+\noindent
+Let $0 \rightarrow\ \bigoplus_a K[x]e_{a,n}\ \rightarrow\ \dots
+ \rightarrow\ \bigoplus_a K[x]e_{a,0}\ \rightarrow\
+ I\ \rightarrow\ 0$
+be a minimal resolution of I considered with homogeneous maps of degree 0.
+The regularity is the smallest number $s$ with the property deg($e_{a,i})
+ \leq s+i$ for all $i$.
address@hidden tex
@item @strong{Note:}
If applied to a non minimal resolution only an upper bound is returned.
@*If the input to the commands @code{res} and @code{mres} is homogeneous
@@ -15922,6 +16505,12 @@
@item @strong{Type:}
intvec
@item @strong{Purpose:}
address@hidden
+computes the permutation {\tt v}
+which orders the ideal, resp.\ module, {\tt I} by its initial terms,
+starting with the smallest, that is, {\tt I(v[i]) < I(v[i+1])} for all
+{\tt i}.
address@hidden tex
@item @strong{Example:}
@smallexample
@c reused example sortvec reference.doc:5565
@@ -16050,10 +16639,20 @@
computes a free resolution of an ideal or module with Schreyer's
method. The ideal, resp.@: module, has to be a standard basis.
More precisely, let M be given by a standard basis and
address@hidden
+$A_1={\tt matrix}(M)$.
address@hidden tex
Then @code{sres}
computes a free resolution of
address@hidden
+$coker(A_1)=F_0/M$
+$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1
\buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow F_0/M\longrightarrow
0.$$
address@hidden tex
If the int expression k is not zero then the computation stops after k steps
and returns a list of modules (given by standard bases)
address@hidden
+$M_i={\tt module}(A_i)$, i=1..k.
address@hidden tex
@address@hidden(M,0)}
returns a list of n modules where n is the number of variables of the basering.
@@ -16064,6 +16663,9 @@
@code{L[2]} is a standard basis with respect to the Schreyer ordering of
the first syzygy
module of @code{L[1]}, etc.
address@hidden
+(${\tt L[i]}=M_i$
address@hidden tex
in the notations from above.)
@item @strong{Note:}
Accessing single elements of a resolution may require that some partial
@@ -16776,7 +17378,29 @@
@item @strong{Type:}
poly
@item @strong{Purpose:}
address@hidden @strong{Note:}
address@hidden
+{\tt vandermonde(p,v,d)} computes the (unique) polynomial of degree
address@hidden with prescribed values {\tt v[1],...,v[N]} at the points
+{\tt p}$^0,\dots,$ {\tt p}$^{N-1}$, {\tt N=(d+1)}$^n$, $n$ the
+number of ring variables.
+
+The returned polynomial is $\sum
+c_{\alpha_1\ldots\alpha_n}\cdot x_1^{\alpha_1} \cdot \dots \cdot
+x_n^{\alpha_n}$, where the coefficients
+$c_{\alpha_1\ldots\alpha_n}$ are the solution of the (transposed)
+Vandermonde system of linear equations
+$$ \sum_{\alpha_1+\ldots+\alpha_n\leq d} c_{\alpha_1\ldots\alpha_n} \cdot
+{\tt p}_1^{(k-1)\alpha_1}\cdot\dots\cdot {\tt p}_n^{(k-1)\alpha_n} =
+{\tt v}[k], \quad k=1,\dots,{\tt N}.$$
address@hidden tex
address@hidden @strong{Note:}
address@hidden
+the ground field has to be the field of rational
+numbers. Moreover, {\tt ncols(p)==}$n$, the number of variables in the
+basering, and all the given generators have to be numbers different from
+0,1 or -1. Finally, {\tt ncols(v)==(d+1)$^n$}, and all given generators have
+to be numbers.
address@hidden tex
@item @strong{Example:}
@smallexample
@c reused example vandermonde reference.doc:6304
@@ -18422,7 +19046,20 @@
The Milnor number, resp.@: the Tjurina number, of a power
series f in
address@hidden
+$K[[x_1,\ldots,x_n]]$
address@hidden tex
is
address@hidden
+$$
+\hbox{milnor}(f) = \hbox{dim}_K(K[[x_1,\ldots,x_n]]/\hbox{jacob}(f)),
+$$
+respectively
+$$
+\hbox{tjurina}(f) = \hbox{dim}_K(K[[x_1,\ldots,x_n]]/((f)+\hbox{jacob}(f)))
+$$
+where
address@hidden tex
@code{jacob(f)} is the ideal generated by the partials
of @code{f}. @code{tjurina(f)} is finite, if and only if @code{f} has an
isolated singularity. The same holds for @code{milnor(f)} if
@@ -18431,8 +19068,17 @@
@sc{Singular} cannot compute with infinite power series. But it can
work in
address@hidden
+$\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$,
address@hidden tex
the localization of
address@hidden
+$K[x_1,\ldots,x_n]$
address@hidden tex
at the maximal ideal
address@hidden
+$(x_1,\ldots,x_n)$.
address@hidden tex
To do this one has to define an
s-ordering like ds, Ds, ls, ws, Ws or an appropriate matrix
ordering (look at the manual to get information about the possible
@@ -18629,7 +19275,13 @@
The same computation which computes the Milnor, resp.@: the Tjurina,
number, but with ordering @code{dp} instead of @code{ds} (i.e., in
address@hidden
+$K[x_1,\ldots,x_n]$
address@hidden tex
instead of
address@hidden
+$\hbox{Loc}_{(x)}K[x_1,\ldots,x_n])$
address@hidden tex
gives:
@itemize @bullet
@item
@@ -18667,11 +19319,23 @@
@item
The result of the computation here (together with the previous one in
@ref{Milnor and Tjurina}) shows that (for @code{t}=0)
address@hidden
+$\hbox{dim}_K(\hbox{Loc}_{(x,y,z)}K[x,y,z]/\hbox{jacob}(f))$
address@hidden tex
= 250 (previously computed) while
address@hidden
+$\hbox{dim}_K(K[x,y,z]/\hbox{jacob}(f))$
address@hidden tex
= 536. Hence @code{f} has 286 critical points,
counted with multiplicity, outside the origin.
Moreover, since
address@hidden
+$\hbox{dim}_K(\hbox{Loc}_{(x,y,z)}K[x,y,z]/(\hbox{jacob}(f)+(f)))$
address@hidden tex
= 195 =
address@hidden
+$\hbox{dim}_K(K[x,y,z]/(\hbox{jacob}(f)+(f)))$,
address@hidden tex
the affine surface @code{f}=0 is smooth outside the origin.
@end itemize
@@ -18700,27 +19364,72 @@
@cindex Saturation
Since in the example above, the ideal
address@hidden
+$j+(f)$
address@hidden tex
has the same @code{vdim}
in the polynomial ring and in the localization at 0 (each 195),
address@hidden
+$f=0$
address@hidden tex
is smooth outside 0.
Hence
address@hidden
+$j+(f)$
address@hidden tex
contains some power of the maximal ideal
address@hidden
+$m$
address@hidden tex
. We shall
check this in a different manner:
For any two ideals
address@hidden
+$i, j$
address@hidden tex
in the basering
address@hidden
+$R$
address@hidden tex
let
address@hidden
+$$
+\hbox{sat}(i,j)=\{x\in R\;|\; \exists\;n\hbox{ s.t. }
+x\cdot(j^n)\subseteq i\}
+= \bigcup_{n=1}^\infty i:j^n$$
address@hidden tex
@*denote the saturation of
address@hidden
+$i$
address@hidden tex
with respect to
address@hidden
+$j$
address@hidden tex
. This defines,
geometrically, the closure of the complement of V(
address@hidden
+$j$
address@hidden tex
) in V(
address@hidden
+$i$
address@hidden tex
)
(V(
address@hidden
+$i$
address@hidden tex
) denotes the variety defined by
address@hidden
+$i$
address@hidden tex
).
In our case,
address@hidden
+$sat(j+(f),m)$
address@hidden tex
must be the whole ring, hence
generated by 1.
@@ -18860,13 +19569,25 @@
and compute over the ground field Q(t).
We compute the dimension at the generic point,
i.e.,
address@hidden
+$dim_{Q(t)}Q(t)[x,y]/j$.
address@hidden tex
(This gives the
same result as for the deformed ideal above. Hence, the above small
deformation was "generic".)
For almost all
address@hidden
+$a \in Q$
address@hidden tex
this is the same as
address@hidden
+$dim_Q Q[x,y]/j_0$,
address@hidden tex
where
address@hidden
+$j_0=j|_{t=a}$.
address@hidden tex
@smallexample
@c computed example Parameters examples.doc:579
@@ -18900,8 +19621,17 @@
@cindex T2
address@hidden
+$T^1$
address@hidden tex
, resp.@:
address@hidden
+$T^2$
address@hidden tex
, of an ideal
address@hidden
+$j$
address@hidden tex
usually denote the modules of
infinitesimal deformations, resp.@: of obstructions.
In @sc{Singular} there are procedures @code{T_1} and @code{T_2} in
@@ -19071,7 +19801,16 @@
singularity.
@item
The procedure @code{deform} in @code{sing.lib} returns a matrix whose columns
address@hidden
+$h_1,\ldots,h_r$
address@hidden tex
represent all 1st order deformations. More precisely, if
address@hidden
+$I \subset R$ is the ideal generated by $f_1,...,f_s$, then any infinitesimal
+deformation of $R/I$ over $K[\varepsilon]/(\varepsilon^2)$ is given
+by $f+\varepsilon g$,
+where $f=(f_1,...,f_s)$, $g$ a $K$-linear combination of the $h_i$.
address@hidden tex
@item
The procedure @code{versal} in @code{deform.lib} computes a formal
@@ -19191,12 +19930,24 @@
@cindex Finite fields
We define a variety in
address@hidden
+$n$
address@hidden tex
-space of codimension 2 defined by
polynomials of degree
address@hidden
+$d$
address@hidden tex
with generic coefficients over the prime
field
address@hidden
+$Z/p$
address@hidden tex
and look for zeros on the torus. First over the prime
field and then in the finite extension field with
address@hidden
+$p^k$
address@hidden tex
elements.
In general there will be many more solutions in the second case.
(Since the @sc{Singular} language is interpreted, the evaluation of many
@@ -19373,9 +20124,24 @@
Elimination is the algebraic counterpart of the geometric concept of
projection. If
address@hidden
+$f=(f_1,\ldots,f_n):k^r\rightarrow k^n$
address@hidden tex
is a polynomial map,
the Zariski-closure of the image is the zero-set of the ideal
address@hidden
+$$
+\displaylines{
+j=J \cap k[x_1,\ldots,x_n], \;\quad\hbox{\rm where}\cr
+J=(x_1-f_1(t_1,\ldots,t_r),\ldots,x_n-f_n(t_1,\ldots,t_r))\subseteq
+k[t_1,\ldots,t_r,x_1,\ldots,x_n]
+}
+$$
address@hidden tex
i.e, of the ideal j obtained from J by eliminating the variables
address@hidden
+$t_1,\ldots,t_r$.
address@hidden tex
This can be done by computing a standard basis of J with respect to a product
ordering where the block of t-variables precedes the block of
x-variables and then selecting those polynomials which do not contain
@@ -19387,13 +20153,23 @@
@strong{WARNING:} In the case of a local or a mixed ordering, elimination
needs special
care. f may be considered as a map of germs
address@hidden
+$f:(k^r,0)\rightarrow(k^n,0)$,
address@hidden tex
but even
if this map germ is finite, we are in general not able to compute the image
germ because for this we would need an implementation of the Weierstrass
preparation theorem. What we can compute, and what @code{eliminate} actually
does,
is the following: let V(J) be the zero-set of J in
address@hidden
+$k^r\times(k^n,0)$,
address@hidden tex
then the
closure of the image of V(J) under the projection
address@hidden
+$$\hbox{pr}:k^r\times(k^n,0)\rightarrow(k^n,0)$$
+can be computed.
address@hidden tex
Note that this germ contains also those components
of V(J) which meet the fiber of pr outside the origin.
This is achieved by an ordering with the block of t-variables having a
@@ -19416,6 +20192,9 @@
@enumerate
@item
First we compute the equations of the simple space curve
address@hidden
+$\hbox{T}[7]^\prime$
address@hidden tex
consisting of two tangential cusps given in parametric form.
@item
We compute weights for the equations such that the
@@ -19423,6 +20202,9 @@
@item
Then we compute the tangent developable of the rational
normal curve in
address@hidden
+$P^4$.
address@hidden tex
@end enumerate
@smallexample
@@ -19572,11 +20354,20 @@
Now let's look at an example which uses resolutions: The Hilbert-Burch
theorem says that the ideal i of a reduced curve in
address@hidden
+$K^3$
address@hidden tex
has a free resolution of length 2 and that i is given by the 2x2 minors
of the 2nd matrix in the resolution.
We test this for two transversal cusps in
address@hidden
+$K^3$.
address@hidden tex
Afterwards we compute the resolution of the ideal j of the tangent developable
of the rational normal curve in
address@hidden
+$P^4$
address@hidden tex
from above.
Finally we demonstrate the use of the type @code{resolution} in connection with
the @code{lres} command.
@@ -19697,24 +20488,45 @@
@cindex Ext
We start by showing how to calculate the
address@hidden
+$n$
address@hidden tex
-th Ext group of an
ideal. The ingredients to do this are by the definition of Ext the
following: calculate a (minimal) resolution at least up to length
address@hidden
+$n$
address@hidden tex
, apply the Hom-functor, and calculate the
address@hidden
+$n$
address@hidden tex
-th homology
group, that is form the quotient
address@hidden
+$\hbox{\rm ker} / \hbox{\rm Im}$
address@hidden tex
in the resolution sequence.
The Hom functor is given simply by transposing (hence dualizing) the
module or the corresponding matrix with the command @code{transpose}.
The image of the
address@hidden
+$(n-1)$
address@hidden tex
-st map is generated by the columns of the
corresponding matrix. To calculate the kernel apply the command
@code{syz} at the
address@hidden
+$(n-1)$
address@hidden tex
-st transposed entry of the resolution.
Finally, the quotient is obtained by the command @code{modulo}, which
gives for two modules A = ker, B = Im the module of relations of
address@hidden
+$A/(A \cap B)$
address@hidden tex
in the usual way. As we have a chain complex this is obviously the same
as ker/Im.
@@ -19753,17 +20565,44 @@
example.
If
address@hidden
+$M$
address@hidden tex
is a module, then
address@hidden
+$\hbox{Ext}^1(M,M)$, resp.\ $\hbox{Ext}^2(M,M)$,
address@hidden tex
are the modules of infinitesimal deformations, resp.@: of obstructions, of
address@hidden
+$M$
address@hidden tex
(like T1 and T2 for a singularity). Similar to the treatment
for singularities, the semiuniversal deformation of
address@hidden
+$M$
address@hidden tex
can be
computed (if
address@hidden
+$\hbox{Ext}^1$
address@hidden tex
is finite dimensional) with the help of
address@hidden
+$\hbox{Ext}^1$, $\hbox{Ext}^2$
address@hidden tex
and the cup product. There is an extra procedure for
address@hidden
+$\hbox{Ext}^k(R/J,R)$
address@hidden tex
if
address@hidden
+$J$
address@hidden tex
is an ideal in
address@hidden
+$R$
address@hidden tex
since this is faster than the
general Ext.
@@ -19771,15 +20610,42 @@
@itemize @bullet
@item
the infinitesimal deformations
address@hidden
+($=\hbox{Ext}^1(K,K)$)
address@hidden tex
and obstructions
address@hidden
+($=\hbox{Ext}^2(K,K)$)
address@hidden tex
of the residue field
address@hidden
+$K=R/m$
address@hidden tex
of an ordinary cusp,
address@hidden
+$R=Loc_m K[x,y]/(x^2-y^3)$, $m=(x,y)$.
address@hidden tex
To compute
address@hidden
+$\hbox{Ext}^1(m,m)$
address@hidden tex
we have to apply @code{Ext(1,syz(m),syz(m))} with
@code{syz(m)} the first syzygy module of
address@hidden
+$m$
address@hidden tex
, which is isomorphic to
address@hidden
address@hidden
+$\hbox{Ext}^2(K,K)$.
address@hidden tex
address@hidden
address@hidden
+$\hbox{Ext}^k(R/i,R)$
address@hidden tex
for some ideal
address@hidden
+$i$
address@hidden tex
and with an extra option.
@end itemize
@@ -19875,18 +20741,45 @@
@cindex Polar curves
The polar curve of a hypersurface given by a polynomial
address@hidden
+$f\in k[x_1,\ldots,x_n,t]$
address@hidden tex
with respect to
address@hidden
+$t$
address@hidden tex
(we may consider
address@hidden
+$f=0$
address@hidden tex
as a family of
hypersurfaces parametrized by
address@hidden
+$t$
address@hidden tex
) is defined as the Zariski
closure of
address@hidden
+$V(\partial f/\partial x_1,\ldots,\partial f/\partial x_n) \setminus V(f)$
address@hidden tex
if this happens to be a curve. Some authors consider
address@hidden
+$V(\partial f/\partial x_1,\ldots,\partial f/\partial x_n)$
address@hidden tex
itself as polar curve.
We may consider projective hypersurfaces
address@hidden
+(in $P^n$),
address@hidden tex
affine hypersurfaces
address@hidden
+(in $k^n$)
address@hidden tex
or germs of hypersurfaces
address@hidden
+(in $(k^n,0)$),
address@hidden tex
getting in this way
projective, affine or local polar curves.
@@ -19999,12 +20892,24 @@
@cindex Depth
We compute the depth of the module of Kaehler differentials
address@hidden
+D$_k$(R)
address@hidden tex
of the variety defined by the
address@hidden
+$(m+1)$
address@hidden tex
-minors of a generic symmetric
address@hidden
+$(n \times n)$-matrix.
address@hidden tex
We do this by computing the resolution over the polynomial
ring. Then, by the Auslander-Buchsbaum formula, the depth is equal to
the number of variables minus the length of a minimal resolution. This
example was suggested by U.@: Vetter in order to check whether his bound
address@hidden
+$\hbox{depth}(\hbox{D}_k(R))\geq m(m+1)/2 + m-1$
address@hidden tex
could be improved.
@smallexample
@@ -20173,6 +21078,9 @@
We work in characteristic 0 and use the Lie algebra generated by one
vector field of the form
address@hidden
+$\sum x_i \partial /\partial x_{i+1}$.
address@hidden tex
@smallexample
@c computed example G_a_-Invariants examples.doc:1783
LIB "ainvar.lib";
@@ -20338,6 +21246,9 @@
We compute the Hamburger-Noether expansion of a plane curve
singularity given by a polynomial
address@hidden
+$f$
address@hidden tex
in two variables. This is a
matrix which allows to compute the parametrization (up to a given order)
and all numerical invariants like the
@@ -20355,7 +21266,13 @@
@end itemize
Besides this, the library contains procedures to compute the Newton
polygon of
address@hidden
+$f$
address@hidden tex
, the squarefree part of
address@hidden
+$f$
address@hidden tex
and a procedure to
convert one set of invariants to another.
@@ -20580,9 +21497,15 @@
@section Normalization
@cindex Normalization
The normalization will be computed for a reduced ring
address@hidden
+$R/I$
address@hidden tex
. The
result is a list of rings; ideals are always called @code{norid} in the
rings of this list. The normalization of
address@hidden
+$R/I$
address@hidden tex
is the product of
the factor rings of the rings in the list divided out by the ideals
@code{norid}.
@@ -20786,12 +21709,41 @@
@section Kernel of module homomorphisms
@cindex Kernel of module homomorphisms
Let
address@hidden
+$A$
address@hidden tex
,
address@hidden
+$B$
address@hidden tex
be two matrices of size
address@hidden
+$m\times r$ and $m\times s$
address@hidden tex
over the ring
address@hidden
+$R$
address@hidden tex
and consider the corresponding maps
address@hidden
+$$
+R^r \buildrel{A}\over{\longrightarrow}
+R^m \buildrel{B}\over{\longleftarrow} R^s\;.
+$$
address@hidden tex
We want to compute the kernel of the map
address@hidden
+$R^r \buildrel{A}\over{\longrightarrow}
+R^m\longrightarrow
+R^m/\hbox{Im}(B) \;.$
address@hidden tex
This can be done using the @code{modulo} command:
address@hidden
+$$
+\hbox{\tt modulo}(A,B)=\hbox{ker}(R^r
+\buildrel{A}\over{\longrightarrow}R^m/\hbox{Im}(B)) \; .
+$$
address@hidden tex
@smallexample
@c computed example Kernel_of_module_homomorphisms examples.doc:2196
@@ -20809,22 +21761,56 @@
@section Algebraic dependence
@cindex Algebraic dependence
Let
address@hidden
+$g$, $f_1$, \dots, $f_r\in K[x_1,\ldots,x_n]$.
address@hidden tex
We want to check whether
@enumerate
@item
address@hidden
+$f_1$, \dots, $f_r$
address@hidden tex
are algebraically dependent.
Let
address@hidden
+$I=\langle Y_1-f_1,\ldots,Y_r-f_r \rangle \subseteq
+K[x_1,\ldots,x_n,Y_1,\ldots,Y_r]$.
address@hidden tex
Then
address@hidden
+$I \cap K[Y_1,\ldots,Y_r]$
address@hidden tex
are the algebraic relations between
address@hidden
+$f_1$, \dots, $f_r$.
address@hidden tex
@item
-
address@hidden
+$g \in K [f_1,\ldots,f_r]$.
address@hidden tex
+
address@hidden
+$g \in K[f_1,\ldots,f_r]$
address@hidden tex
if and only if the normal form of
address@hidden
+$g$
address@hidden tex
with respect to
address@hidden
+$I$
address@hidden tex
and a
block ordering with respect to
address@hidden
+$X=(x_1,\ldots,x_n)$ and $Y=(Y_1,\ldots,Y_r)$ with $X>Y$
address@hidden tex
is in
address@hidden
+$K[Y]$
address@hidden tex
.
@end enumerate
@@ -21165,35 +22151,83 @@
A vector in @sc{Singular} is always an element of a free module over the
basering. It is given as a list of polynomials in one of the following
formats
address@hidden
+$[f_1,...,f_n]$ or $f_1*gen(1)+...+f_n*gen(n)$, where $gen(i)$
address@hidden tex
denotes the i-th canonical generator of a free module (with 1 at place i and
0 everywhere else).
Both forms are equivalent. A vector is internally represented in
the second form with the
address@hidden
+$gen(i)$
address@hidden tex
being "special" ring variables, ordered accordingly to the monomial ordering.
Therefore, the form
address@hidden
+$[f_1,...,f_n]$
address@hidden tex
is given as output only if the monomial ordering gives priority to the
component, i.e@:., is of the form @code{(c,...)} (see @ref{Module
orderings}). However, in any case the procedure @code{show} from the
library @code{inout.lib} displays the bracket format.
A vector
address@hidden
+$v=[f_1,...,f_n]$
address@hidden tex
should always be considered as a column vector in a free module
of rank equal to
address@hidden
+nrows($v$)
address@hidden tex
where
address@hidden
+nrows($v$)
address@hidden tex
is equal to the maximal index
address@hidden
+$r$
address@hidden tex
such that
address@hidden
+$f_r \not= 0$.
address@hidden tex
This is due to the fact, that internally
address@hidden
+$v$
address@hidden tex
is a polynomial in a sparse representation, i.e.,
address@hidden
+$f_i*gen(i)$
address@hidden tex
is not stored if
address@hidden
+$f_i=0$
address@hidden tex
(for reasons of efficiency), hence the last 0-entries of
address@hidden
+$v$
address@hidden tex
are lost.
Only more complex structures are able to keep the rank.
A module
address@hidden
+$M$
address@hidden tex
in @sc{Singular} is given by a list of vectors
address@hidden
+$v_1,...,v_k$
address@hidden tex
which generate the module as a submodule of the free module of rank
equal to
address@hidden
+nrows($M$)
address@hidden tex
which is the maximum of
address@hidden
+nrows($v_i$).
address@hidden tex
If one wants to create a module with a larger rank than given by its
generators, one has to use the command @code{attrib(M,"rank",r)} (see
@@ -21208,33 +22242,84 @@
By the above remarks it might appear that @sc{Singular} is only able to handle
submodules of a free module. However, this is not true. @sc{Singular}
can compute with any finitely generated module over the basering
address@hidden
+$R$.
address@hidden tex
Such a module, say
address@hidden
+$N$,
address@hidden tex
is not represented by its generators but by its
(generators and) relations. This means that
address@hidden
+$N = R^n/M$ where $n$
address@hidden tex
is the number of generators of
address@hidden
+$N$ and $M \subseteq R^n$
address@hidden tex
is the module of relations.
In other words, defining a module
address@hidden
+$M$
address@hidden tex
as a submodule of a free module
address@hidden
+$R^n$
address@hidden tex
can also be considered as the definition of
address@hidden
+$N = R^n/M$.
address@hidden tex
Note that most functions, when applied to a module
address@hidden
+$M$,
address@hidden tex
really deal with
address@hidden
+$M$.
address@hidden tex
However, there are some functions which deal with
address@hidden
+$N = R^n/M$ instead of $M$.
address@hidden tex
For example, @code{std(M)} computes a standard basis of
address@hidden
+$M$
address@hidden tex
(and thus gives another representation of
address@hidden
+$N$ as $N = R^n/$std($M$)).
address@hidden tex
However, @code{dim(M)}, resp.@: @code{vdim(M)}, returns
address@hidden
+dim$(R^n/M)$, resp.@: dim$_k(R^n/M)$
address@hidden tex
(if M is given by a standard basis).
The function @code{syz(M)} returns the first syzygy module of
address@hidden
+$M$,
address@hidden tex
i.e@:., the module
of relations of the given generators of
address@hidden
+$M$
address@hidden tex
which is equal to the second syzygy module of
address@hidden
+$N$.
address@hidden tex
Refer to the description of each function in
@ref{Functions} to get information which module the function deals with.
The numbering in @code{res} and other commands for computing resolutions
refers to a resolution of
address@hidden
+$N = R^n/M$
address@hidden tex
(see @ref{res}; @ref{Syzygies and resolutions}).
It is possible to compute in any field which is a valid ground field in
@@ -21278,13 +22363,28 @@
flexibility might also be confusing for the novice user. Therefore, we
recommend to those not familiar with monomial orderings to generally use
the ordering @code{dp} for computations in the polynomial ring
address@hidden
+$K[x_1,\ldots,x_n]$,
address@hidden tex
resp.@: @code{ds} for computations in the localization
address@hidden
+$\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$.
address@hidden tex
For inhomogeneous input ideals, standard (resp.@: groebner) bases
computations are generally faster
with the orderings
address@hidden
+$\hbox{Wp}(w_1, \ldots, w_n)$
address@hidden tex
(resp.@:
address@hidden
+$\hbox{Ws}(w_1, \ldots, w_n)$)
address@hidden tex
if the input is quasihomogeneous w.r.t. the weights
address@hidden
+$w_1$, $\ldots$, $w_n$ of $x_1$, $\ldots$, $x_n$.
address@hidden tex
If the output needs to be "triangular" (resp.@: "block-triangular"), the
lexicographical ordering @code{lp} (resp.@: lexicographical
@@ -21299,12 +22399,39 @@
@cindex term orderings
@cindex monomial orderings
address@hidden
+A monomial ordering (term ordering) on $K[x_1, \ldots, x_n]$ is
+a total ordering $<$ on the
+set of monomials (power products) $\{x^\alpha \mid \alpha \in \bf{N}^n\}$
+which is compatible with the
+natural semigroup structure, i.e., $x^\alpha < x^\beta$ implies $x^\gamma
+x^\alpha < x^\gamma x^\beta$ for any $\gamma \in \bf{N}^n$.
+We do not require
+$<$ to be a well ordering.
address@hidden tex
See the literature cited in @ref{References}.
It is known that any monomial ordering can be represented by a matrix
address@hidden
+$M$ in $GL(n,R)$,
address@hidden tex
but, of course, only integer coefficients are of relevance in
practice.
address@hidden
+Global orderings are well orderings (i.e., \hbox{$1 < x_i$} for each variable
+$x_i$), local orderings satisfy $1 > x_i$ for each variable. If some
variables are ordered globally and others locally we
+call it a mixed ordering. Local or mixed orderings are not well orderings.
+
+Let $K$ be the ground field, \hbox{$x = (x_1, \ldots, x_n)$} the
+variables and $<$ a monomial ordering, then Loc $K[x]$ denotes the
+localization of $K[x]$ with respect to the multiplicatively closed set $$\{1 +
+g \mid g = 0 \hbox{ or } g \in K[x]\backslash \{0\} \hbox{ and }L(g) <
+1\}.$$ Here, $L(g)$
+denotes the leading monomial of $g$, i.e., the biggest monomial of $g$ with
+respect to $<$. The result of any computation which uses standard basis
+computations has to be interpreted in Loc $K[x]$.
address@hidden tex
Note that the definition of a ring includes the definition of its
monomial ordering (see
@@ -21318,6 +22445,9 @@
@cindex Global orderings
@cindex orderings, global
address@hidden
+For all these orderings: Loc $K[x]$ = $K[x]$
address@hidden tex
@table @asis
@item lp:
@@ -21325,35 +22455,81 @@
@cindex lp, global ordering
@cindex lexicographical ordering
@*
address@hidden
+$x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
+\alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i <
+\beta_i$.
address@hidden tex
@item rp:
reverse lexicographical ordering:
@cindex rp, global ordering
@cindex reverse lexicographical ordering
@*
address@hidden
+$x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
+\alpha_n = \beta_n,
+ \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
address@hidden tex
@item dp:
degree reverse lexicographical ordering:
@cindex degree reverse lexicographical ordering
@cindex dp, global ordering
@*
address@hidden
+let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
address@hidden tex
address@hidden
+ $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) < \deg(x^\beta)$ or
address@hidden tex
address@hidden
+ \phantom{$x^\alpha < x^\beta \Leftrightarrow $}$ \deg(x^\alpha) =
+ \deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n = \beta_n,
+ \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
address@hidden tex
@item Dp:
degree lexicographical ordering:
@cindex degree lexicographical ordering
@cindex Dp, global ordering
@*
address@hidden
+let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
address@hidden tex
address@hidden
+ $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) < \deg(x^\beta)$ or
address@hidden tex
address@hidden
+ \phantom{ $x^\alpha < x^\beta \Leftrightarrow $} $\deg(x^\alpha) =
+ \deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 = \beta_1,
+ \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$
address@hidden tex
@item wp:
weighted reverse lexicographical ordering:
@cindex weighted reverse lexicographical ordering
@cindex wp, global ordering
@*
address@hidden
+let $w_1, \ldots, w_n$ be positive integers. Then ${\tt wp}(w_1, \ldots,
+w_n)$
address@hidden tex
is defined as @code{dp}
but with
address@hidden
+$\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
address@hidden tex
@item Wp:
weighted lexicographical ordering:
@cindex weighted lexicographical ordering
@cindex WP, global ordering
@*
address@hidden
+let $w_1, \ldots, w_n$ be positive integers. Then ${\tt Wp}(w_1, \ldots,
+w_n)$
address@hidden tex
is defined as @code{Dp}
but with
address@hidden
+$\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
address@hidden tex
@end table
@c --------------------------------------------------------------------------
@node Local orderings, Module orderings, Global orderings, Monomial orderings
@@ -21363,8 +22539,17 @@
For ls, ds, Ds and, if the weights are positive integers, also for ws and
Ws, we have
address@hidden
+Loc $K[x]$ = $K[x]_{(x)}$,
address@hidden tex
the localization of
address@hidden
+$K[x]$
address@hidden tex
at the maximal ideal
address@hidden
+\ $(x_1, ..., x_n)$.
address@hidden tex
@table @asis
@item ls:
@@ -21372,36 +22557,81 @@
@cindex negative lexicographical ordering
@cindex ls, local ordering
@*
address@hidden
+$x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
+\alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i >
+\beta_i$.
address@hidden tex
@item ds:
negative degree reverse lexicographical ordering:
@cindex negative degree reverse lexicographical ordering
@cindex ds, local ordering
@*
address@hidden
+let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
address@hidden tex
address@hidden
+ $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) > \deg(x^\beta)$ or
address@hidden tex
address@hidden
+ \phantom{ $x^\alpha < x^\beta \Leftrightarrow$}$ \deg(x^\alpha) =
+ \deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n = \beta_n,
+ \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
address@hidden tex
@item Ds:
negative degree lexicographical ordering:
@cindex negative degree lexicographical ordering
@cindex Ds, local ordering
@*
address@hidden
+let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then
address@hidden tex
address@hidden
+ $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) > \deg(x^\beta)$ or
address@hidden tex
address@hidden
+ \phantom{ $ x^\alpha < x^\beta \Leftrightarrow$}$ \deg(x^\alpha) =
+ \deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 = \beta_1,
+ \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$
address@hidden tex
@item ws:
(general) weighted reverse lexicographical ordering:
@cindex general weighted reverse lexicographical ordering
@cindex local weighted reverse lexicographical ordering
@cindex ws, local ordering
@*
address@hidden
+${\tt ws}(w_1, \ldots, w_n),\; w_1$
address@hidden tex
a nonzero integer,
address@hidden
+$w_2,\ldots,w_n$
address@hidden tex
any integer (including 0),
is defined as @code{ds}
but with
address@hidden
+$\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
address@hidden tex
@item Ws:
(general) weighted lexicographical ordering:
@cindex general weighted lexicographical ordering
@cindex local weighted lexicographical ordering
@cindex Ws, local ordering
@*
address@hidden
+${\tt Ws}(w_1, \ldots, w_n),\; w_1$
address@hidden tex
a nonzero integer,
address@hidden
+$w_2,\ldots,w_n$
address@hidden tex
any integer (including 0),
is defined as @code{Ds}
but with
address@hidden
+$\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
address@hidden tex
@end table
@c --------------------------------------------------------------------------
@@ -21410,22 +22640,42 @@
@cindex Module orderings
@sc{Singular} offers also orderings on the set of ``monomials''
address@hidden
+$\{ x^a e_i \mid a \in N^n, 1 \leq i \leq r \}$ in Loc $K[x]^r$ = Loc
+$K[x]e_1
++ \ldots +$Loc $K[x]e_r$, where $e_1, \ldots, e_r$ denote the canonical
+generators of Loc $K[x]^r$, the r-fold direct sum of Loc $K[x]$.
+(The function {\tt gen(i)} yields $e_i$).
address@hidden tex
We have two possibilities: either to give priority to the component of a
vector in
address@hidden
+ Loc $K[x]^r$
address@hidden tex
or (which is the default in @sc{Singular}) to give priority
to the coefficients.
The orderings @code{(<,c)} and @code{(<,C)} give priority to the
coefficients; whereas
@code{(c,<)} and @code{(C,<)} give priority to the components.
@*Let < be any of the monomial orderings of
address@hidden
+Loc $K[x]$
address@hidden tex
as above.
@table @asis
@item (<,C):
@cindex C, module ordering
@cindex module ordering C
address@hidden
+$<_m = (<,C)$ denotes the module ordering (giving priority to the
coefficients):
address@hidden tex
@*
address@hidden
+\quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow x^\alpha <
+x^\beta$ or ($x^\alpha = x^\beta $ and $ i < j$).
address@hidden tex
@strong{Example:}
@smallexample
@@ -21440,6 +22690,13 @@
@end smallexample
@item (C,<):
address@hidden
+$<_m = (C, <)$ denotes the module ordering (giving priority to the component):
address@hidden tex
address@hidden
+\quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow i < j$ or ($
+i = j $ and $ x^\alpha < x^\beta $).
address@hidden tex
@strong{Example:}
@smallexample
@@ -21455,6 +22712,13 @@
@item (<,c):
@cindex c, module ordering
@cindex module ordering c
address@hidden
+$<_m = (<,c)$ denotes the module ordering (giving priority to the
coefficients):
address@hidden tex
address@hidden
+\quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow x^\alpha <
+x^\beta$ or ($x^\alpha = x^\beta $ and $ i > j$).
address@hidden tex
@strong{Example:}
@smallexample
@@ -21468,6 +22732,13 @@
@end smallexample
@item (c,<):
address@hidden
+$<_m = (c, <)$ denotes the module ordering (giving priority to the component):
address@hidden tex
address@hidden
+\quad \quad $x^\alpha e_i <_m x^\beta e_j \Leftrightarrow i > j$ or ($
+i = j $ and $ x^\alpha < x^\beta $).
address@hidden tex
@strong{Example:}
@smallexample
@@ -21481,7 +22752,14 @@
@end smallexample
@end table
address@hidden
+The output of a vector $v$ in $K[x]^r$ with components $v_1,
+\ldots, v_r$ has the format $v_1 * gen(1) + \ldots + v_r * gen(r)$
address@hidden tex
(up to permutation) unless the ordering starts with @code{c}.
address@hidden
+In this case a vector is written as $[v_1, \ldots, v_r]$.
address@hidden tex
In all cases @sc{Singular} can read input in both formats.
@c --------------------------------------------------------------------------
@@ -21492,25 +22770,152 @@
@cindex M, ordering
Let
address@hidden
+$M$
address@hidden tex
be an invertible
address@hidden
+$(n \times n)$-matrix
address@hidden tex
with integer coefficients and
address@hidden
+$M_1, \ldots, M_n$ the rows of $M$.
address@hidden tex
The M-ordering < is defined as follows:
@*
address@hidden
+\quad \quad $x^a < x^b \Leftrightarrow \exists\ 1 \leq i \leq n :
+M_1 a = \; M_1 b, \ldots, M_{i-1} a = \; M_{i-1} b$ and $M_i a < \; M_i b$.
address@hidden tex
Thus,
address@hidden
+$x^a < x^b$
+if and only if $M a$ is smaller than $M b$
address@hidden tex
with respect to the lexicographical ordering.
The following matrices represent (for 3 variables) the global and
local orderings defined above (note that the matrix is not uniquely determined
by the ordering):
address@hidden
+
+$\quad$ lp:
+$\left(\matrix{
+ 1 & 0 & 0 \cr
+ 0 & 1 & 0 \cr
+ 0 & 0 & 1 \cr
+ }\right)$
+\quad dp:
+$\left(\matrix{
+ 1 & 1 & 1 \cr
+ 0 & 0 &-1 \cr
+ 0 &-1 & 0 \cr
+ }\right)$
+\quad Dp:
+$\left(\matrix{
+ 1 & 1 & 1 \cr
+ 1 & 0 & 0 \cr
+ 0 & 1 & 0 \cr
+ }\right)$
+
+$\quad$ wp(1,2,3):
+$\left(\matrix{
+ 1 & 2 & 3 \cr
+ 0 & 0 &-1 \cr
+ 0 &-1 & 0 \cr
+ }\right)$
+\quad Wp(1,2,3):
+$\left(\matrix{
+ 1 & 2 & 3 \cr
+ 1 & 0 & 0 \cr
+ 0 & 1 & 0 \cr
+ }\right)$
+
+$\quad$ ls:
+$\left(\matrix{
+-1 & 0 & 0 \cr
+ 0 &-1 & 0 \cr
+ 0 & 0 &-1 \cr
+ }\right)$
+\quad ds:
+$\left(\matrix{
+-1 &-1 &-1 \cr
+ 0 & 0 &-1 \cr
+ 0 &-1 & 0 \cr
+ }\right)$
+\quad Ds:
+$\left(\matrix{
+-1 &-1 &-1 \cr
+ 1 & 0 & 0 \cr
+ 0 & 1 & 0 \cr
+ }\right)$
+
+$\quad$ ws(1,2,3):
+$\left(\matrix{
+-1 &-2 &-3 \cr
+ 0 & 0 &-1 \cr
+ 0 &-1 & 0 \cr
+ }\right)$
+\quad Ws(1,2,3):
+$\left(\matrix{
+-1 &-2 &-3 \cr
+ 1 & 0 & 0 \cr
+ 0 & 1 & 0 \cr
+ }\right)$
address@hidden tex
Product orderings (see next section) represented by a matrix:
address@hidden
+$\quad$ (dp(3), wp(1,2,3)):
+$\left(\matrix{
+1& 1& 1& 0& 0& 0 \cr
+0& 0& -1& 0& 0& 0 \cr
+0& -1& 0& 0& 0& 0 \cr
+0& 0& 0& 1& 2& 3 \cr
+0& 0& 0& 0& 0& -1 \cr
+0& 0& 0& 0& -1& 0 \cr
+ }\right)$
+
+$\quad$ (Dp(3), ds(3)):
+$\left(\matrix{
+1& 1& 1& 0& 0& 0 \cr
+1& 0& 0& 0& 0& 0 \cr
+0& 1& 0& 0& 0& 0 \cr
+0& 0& 0& -1& -1& -1 \cr
+0& 0& 0& 0& 0& -1 \cr
+0& 0& 0& 0& -1& 0 \cr
+ }\right)$
address@hidden tex
Orderings with extra weight vector (see below) represented by a matrix:
address@hidden
+$\quad$ (dp(3), a(1,2,3),dp(3)):
+$\left(\matrix{
+1& 1& 1& 0& 0& 0 \cr
+0& 0& -1& 0& 0& 0 \cr
+0& -1& 0& 0& 0& 0 \cr
+0& 0& 0& 1& 2& 3 \cr
+0& 0& 0& 1& 1& 1 \cr
+0& 0& 0& 0& 0& -1 \cr
+0& 0& 0& 0& -1& 0 \cr
+ }\right)$
+
+$\quad$ (a(1,2,3,4,5),Dp(3), ds(3)):
+$\left(\matrix{
+1& 2& 3& 4& 5& 0 \cr
+1& 1& 1& 0& 0& 0 \cr
+1& 0& 0& 0& 0& 0 \cr
+0& 1& 0& 0& 0& 0 \cr
+0& 0& 0& -1& -1& -1 \cr
+0& 0& 0& 0& 0 & -1 \cr
+0& 0& 0& 0& -1& 0 \cr
+ }\right)$
address@hidden tex
@address@hidden:
@smallexample
@@ -21540,7 +22945,13 @@
@end smallexample
If the ring has
address@hidden
+$n$
address@hidden tex
variables and the matrix contains less than
address@hidden
+$n \times n$
address@hidden tex
entries an error message is given, if there are more entries,
the last ones are ignored.
@@ -21561,6 +22972,9 @@
@cindex orderings, product
Let
address@hidden
+$x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_m)$
address@hidden tex
be two ordered sets of variables,
Inductively one defines the product ordering of more than two monomial
@@ -21582,9 +22996,24 @@
@cindex a, ordering
@cindex orderings, a
address@hidden
+${\tt a}(w_1, \ldots, w_n),\; $
address@hidden tex
address@hidden
+$w_1,\ldots,w_n$
address@hidden tex
any integers (including 0), defines
address@hidden
+$\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n$
address@hidden tex
and
@*
address@hidden
+ $$\deg(x^\alpha) < \deg(x^\beta) \Rightarrow x^\alpha < x^\beta,$$
address@hidden tex
address@hidden
+ $$\deg(x^\alpha) > \deg(x^\beta) \Rightarrow x^\alpha > x^\beta. $$
address@hidden tex
@*An extra weight vector does not define a monomial ordering by itself:
it can only be used in combination with other orderings
to insert an extra line of weights into the ordering
@@ -21633,14 +23062,44 @@
@cindex Standard bases
@subheading Definition
address@hidden
+Let $R = \hbox{Loc}_< K[\underline{x}]$ and let $I$ be a submodule of $R^r$.
+Note that for r=1 this means that $I$ is an ideal in $R$.
+Denote by $L(I)$ the submodule of $R^r$ generated by the leading terms
+of elements of $I$, i.e. by $\left\{L(f) \mid f \in I\right\}$.
+Then $f_1, \ldots, f_s \in I$ is called a {\bf standard basis} of $I$
+if $L(f_1), \ldots, L(f_s)$ generate $L(I)$.
address@hidden tex
@subheading Properties
@table @asis
@item normal form:
@cindex Normal form
address@hidden
+A function $\hbox{NF} : R^r \times \{G \mid G\ \hbox{ a standard
+basis}\} \to R^r, (p,G) \mapsto \hbox{NF}(p|G)$, is called a {\bf normal
+form} if for any $p \in R^r$ and any standard basis $G$ the following
+holds: if $\hbox{NF}(p|G) \not= 0$ then $L(g)$ does not divide
+$L(\hbox{NF}(p|G))$ for all $g \in G$.
+
+\noindent
+$\hbox{NF}(p|G)$ is called a {\bf normal form of} $p$ {\bf with
+respect to} $G$ (note that such a function is not unique).
address@hidden tex
@item ideal membership:
@cindex Ideal membership
address@hidden
+For a standard basis $G$ of $I$ the following holds:
+$f \in I$ if and only if $\hbox{NF}(f,G) = 0$.
address@hidden tex
@item Hilbert function:
address@hidden
+Let \hbox{$I \subseteq K[\underline{x}]^r$} be a homogeneous module, then the
Hilbert function
+$H_I$ of $I$ (see below)
+and the Hilbert function $H_{L(I)}$ of the leading module $L(I)$
+coincide, i.e.,
+$H_I=H_{L(I)}$.
address@hidden tex
@end table
@c ---------------------------------------------------------------------------
@@ -21648,8 +23107,32 @@
@section Hilbert function
@cindex Hilbert function
@cindex Hilbert series
address@hidden
+Let M $=\bigoplus_i M_i$ be a graded module over $K[x_1,..,x_n]$ with
+respect to weights $(w_1,..w_n)$.
+The {\bf Hilbert function} of $M$, $H_M$, is defined (on the integers) by
+$$H_M(k) :=dim_K M_k.$$
+The {\bf Hilbert-Poincare series} of $M$ is the power series
+$$\hbox{HP}_M(t) :=\sum_{i=-\infty}^\infty
+H_M(i)t^i=\sum_{i=-\infty}^\infty dim_K M_i \cdot t^i.$$
+It turns out that $\hbox{HP}_M(t)$ can be written in two useful ways
+for weights $(1,..,1)$:
+$$\hbox{HP}_M(t)={Q(t)\over (1-t)^n}={P(t)\over (1-t)^{dim(M)}}$$
+where $Q(t)$ and $P(t)$ are polynomials in ${\bf Z}[t]$.
+$Q(t)$ is called the {\bf first Hilbert series},
+and $P(t)$ the {\bf second Hilbert series}.
+If \hbox{$P(t)=\sum_{k=0}^N a_k t^k$}, and \hbox{$d = dim(M)$},
+then \hbox{$H_M(s)=\sum_{k=0}^N a_k$ ${d+s-k-1}\choose{d-1}$}
+(the {\bf Hilbert polynomial}) for $s \ge N$.
address@hidden tex
@*
@*
address@hidden
+Generalizing these to quasihomogeneous modules we get
+$$\hbox{HP}_M(t)={Q(t)\over {\Pi_{i=1}^n(1-t^{w_i})}}$$
+where $Q(t)$ is a polynomial in ${\bf Z}[t]$.
+$Q(t)$ is called the {\bf first (weighted) Hilbert series} of M.
address@hidden tex
@c ---------------------------------------------------------------------------
@node Syzygies and resolutions, Characteristic sets, Hilbert function,
Mathematical background
@@ -21657,11 +23140,22 @@
@cindex Syzygies and resolutions
@subheading Syzygies
address@hidden
+Let $R$ be a quotient of $\hbox{Loc}_< K[\underline{x}]$ and let
\hbox{$I=(g_1, ..., g_s)$} be a submodule of $R^r$.
+Then the {\bf module of syzygies} (or {\bf 1st syzygy module}, {\bf module of
relations}) of $I$, syz($I$), is defined to be the kernel of the map \hbox{$R^s
\rightarrow R^r,\; \sum_{i=1}^s w_ie_i \mapsto \sum_{i=1}^s w_ig_i$.}
address@hidden tex
The @strong{k-th syzygy module} is defined inductively to be the module
of syzygies of the
address@hidden
+$(k-1)$-st
address@hidden tex
syzygy module.
address@hidden
+Note, that the syzygy modules of $I$ depend on a choice of generators $g_1,
..., g_s$.
+But one can show that they depend on $I$ uniquely up to direct summands.
address@hidden tex
@table @code
@item @strong{Example:}
@@ -21679,10 +23173,26 @@
@end table
@subheading Free resolutions
address@hidden
+Let $I=(g_1,...,g_s)\subseteq R^r$ and $M= R^r/I$.
+A {\bf free resolution of $M$} is a long exact sequence
+$$...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F_1
+\buildrel{A_1}\over{\longrightarrow} F_0\longrightarrow M\longrightarrow
+0,$$
address@hidden tex
@*where the columns of the matrix
address@hidden
+$A_1$
address@hidden tex
generate
address@hidden
+$I$
address@hidden tex
. Note, that resolutions need not to be finite (i.e., of
finite length). The Hilbert Syzygy Theorem states, that for
address@hidden
+$R=\hbox{Loc}_< K[\underline{x}]$
address@hidden tex
there exists a ("minimal") resolution of length not exceeding the number of
variables.
@@ -21715,11 +23225,37 @@
@subheading Betti numbers and regularity
@cindex Betti number
@cindex regularity
address@hidden
+Let $R$ be a graded ring (e.g., $R = \hbox{Loc}_< K[\underline{x}]$) and
+let $I \subset R^r$ be a graded submodule. Let
+$$
+ R^r = \bigoplus_a R\cdot e_{a,0} \buildrel A_1 \over \longleftarrow
+ \bigoplus_a R\cdot e_{a,1} \longleftarrow \ldots \longleftarrow
+ \bigoplus_a R\cdot e_{a,n} \longleftarrow 0
+$$
+be a minimal free resolution of $R^n/I$ considered with homogeneous maps
+of degree 0. Then the {\bf graded Betti number} $b_{i,j}$ of $R^r/I$ is
+the minimal number of generators $e_{a,j}$ in degree $i+j$ of the $j$-th
+syzygy module of $R^r/I$ (i.e., the $(j-1)$-st syzygy module of
+$I$). Note, that by definition the $0$-th syzygy module of $R^r/I$ is $R^r$
+and the 1st syzygy module of $R^r/I$ is $I$.
address@hidden tex
The @strong{regularity} of
address@hidden
+$I$
address@hidden tex
is the smallest integer
address@hidden
+$s$
address@hidden tex
such that
address@hidden
+$$
+ \hbox{deg}(e_{a,j}) \le s+j-1 \quad \hbox{for all $j$.}
+$$
address@hidden tex
@table @code
@item @strong{Example:}
@@ -21752,6 +23288,43 @@
@section Characteristic sets
@cindex Characteristic sets
address@hidden
+Let $<$ be the lexicographical ordering on $R=K[x_1,...,x_n]$ with $x_1
+< ... < x_n$.
+For $f \in R$ let lvar($f$) (the leading variable of $f$) be the largest
+variable in $f$,
+i.e., if $f=a_s(x_1,...,x_{k-1})x_k^s+...+a_0(x_1,...,x_{k-1})$ for some
+$k \leq n$ then lvar$(f)=x_k$.
+
+Moreover, let
+\hbox{ini}$(f):=a_s(x_1,...,x_{k-1})$. The pseudo remainder
+$r=\hbox{prem}(g,f)$ of $g$ with respect to $f$ is
+defined by the equality $\hbox{ini}(f)^a\cdot g = qf+r$ with
+$\hbox{deg}_{lvar(f)}(r)<\hbox{deg}_{lvar(f)}(f)$ and $a$
+minimal.
+
+A set $T=\{f_1,...,f_r\} \subset R$ is called triangular if
+$\hbox{lvar}(f_1)<...<\hbox{lvar}(f_r)$. Moreover, let $ U \subset T $,
+then $(T,U)$ is called a triangular system, if $T$ is a triangular set
+such that $\hbox{ini}(T)$ does not vanish on $V(T) \setminus V(U)
+(=:V(T\setminus U))$.
+
+$T$ is called irreducible if for every $i$ there are no
+$d_i$,$f_i'$,$f_i''$ such that
+$$ \hbox{lvar}(d_i)<\hbox{lvar}(f_i) =
+\hbox{lvar}(f_i')=\hbox{lvar}(f_i''),$$
+$$ 0 \not\in \hbox{prem}(\{ d_i, \hbox{ini}(f_i'),
+\hbox{ini}(f_i'')\},\{ f_1,...,f_{i-1}\}),$$
+$$\hbox{prem}(d_if_i-f_i'f_i'',\{f_1,...,f_{i-1}\})=0.$$
+Furthermore, $(T,U)$ is called irreducible if $T$ is irreducible.
+
+The main result on triangular sets is the following:
+let $G=\{g_1,...,g_s\} \subset R$ then there are irreducible triangular sets
$T_1,...,T_l$
+such that $V(G)=\bigcup_{i=1}^{l}(V(T_i\setminus I_i))$
+where $I_i=\{\hbox{ini}(f) \mid f \in T_i \}$. Such a set
+$\{T_1,...,T_l\}$ is called an {\bf irreducible characteristic series} of
+the ideal $(G)$.
address@hidden tex
@table @code
@item @strong{Example:}
@@ -21776,14 +23349,61 @@
@c tex and info versions of it. It end just before the introducing text
@c to the first example.
address@hidden
+Let $f\colon(C^{n+1},0)\rightarrow(C,0)$ be a complex isolated hypersurface
singularity given by a polynomial with algebraic coefficients which we also
denote by $f$.
+Let $O=C[x_0,\ldots,x_n]_{(x_0,\ldots,x_n)}$ be the local ring at the origin
and $J_f$ the Jacobian ideal of $f$.
+
+A {\bf Milnor representative} of $f$ defines a differentiable fibre bundle
over the punctured disc with fibres of homotopy type of $\mu$ $n$-spheres.
+The $n$-th cohomology bundle is a flat vector bundle of dimension $n$ and
carries a natural flat connection with covariant derivative $\partial_t$.
+The {\bf monodromy operator} is the action of a positively oriented generator
of the fundamental group of the puctured disc on the Milnor fibre.
+Sections in the cohomology bundle of {\bf moderate growth} at $0$ form a
regular $D=C\{t\}[\partial_t]$-module $G$, the {\bf Gauss-Manin connection}.
+
+By integrating along flat multivalued families of cycles, one can consider
fibrewise global holomorphic differential forms as elements of $G$.
+This factors through an inclusion of the {\bf Brieskorn lattice}
$H'':=\Omega^{n+1}_{C^{n+1},0}/df\wedge d\Omega^{n-1}_{C^{n+1},0}$ in $G$.
+
+The $D$-module structure defines the {\bf V-filtration} $V$ on $G$ by
$V^\alpha:=\sum_{\beta\ge\alpha}C\{t\}ker(t\partial_t-\beta)^{n+1}$.
+The Brieskorn lattice defines the {\bf Hodge filtration} $F$ on $G$ by
$F_k=\partial_t^kH''$ which comes from the {\bf mixed Hodge structure} on the
Milnor fibre.
+Note that $F_{-1}=H'$.
+
+The induced V-filtration on the Brieskorn lattice determines the {\bf
singularity spectrum} $Sp$ by $Sp(\alpha):=\dim_CGr_V^\alpha Gr^F_0G$.
+The spectrum consists of $\mu$ rational numbers $\alpha_1,\dots,\alpha_\mu$
such that $e^{2\pi i\alpha_1},\dots,e^{2\pi i\alpha_\mu}$ are the eigenvalues
of the monodromy.
+These {\bf spectral numbers} lie in the open interval $(-1,n)$, symmetric
about the midpoint $(n-1)/2$.
+
+The spectrum is constant under $\mu$-constant deformations and has the
following semicontinuity property:
+The number of spectral numbers in an interval $(a,a+1]$ of all singularities
of a small deformation of $f$ is greater or equal to that of f in this interval.
+For semiquasihomogeneous singularities, this also holds for intervals of the
form $(a,a+1)$.
+
+Two given isolated singularities $f$ and $g$ determine two spectra and from
these spectra we get an integer.
+This integer is the maximal positive integer $k$ such that the semicontinuity
holds for the spectrum of $f$ and $k$ times the spectrum of $g$.
+These numbers give bounds for the maximal number of isolated singularities of
a specific type on a hypersurface $X\subset{P}^n$ of degree $d$:
+such a hypersurface has a smooth hyperplane section, and the complement is a
small deformation of a cone over this hyperplane section.
+The cone itself being a $\mu$-constant deformation of $x_0^d+\dots+x_n^d=0$,
the singularities are bounded by the spectrum of $x_0^d+\dots+x_n^d$.
+
+Using the library {\tt gaussman.lib} one can compute the {\bf monodromy}, the
V-filtration on $H''/H'$, and the spectrum.
address@hidden tex
Let us consider as an example
address@hidden
+$f=x^5+x^2y^2+y^5$
address@hidden tex
.
First, we compute a matrix
address@hidden
+$M$
address@hidden tex
such that
address@hidden
+$\exp(2\pi iM)$
address@hidden tex
is a monodromy matrix of
address@hidden
+$f$
address@hidden tex
and the Jordan normal form of
address@hidden
+$M$
address@hidden tex
:
@smallexample
@c reused example Gauss-Manin_connection math.doc:505
@@ -21808,6 +23428,9 @@
@end smallexample
Now, we compute the V-filtration on
address@hidden
+$H''/H'$
address@hidden tex
and the spectrum:
@smallexample
@c reused example Gauss-Manin_connection_1 math.doc:517
@@ -21859,17 +23482,36 @@
@c end example Gauss-Manin_connection_1 math.doc:517
@end smallexample
Here @code{l[1]} contains the spectral numbers, @code{l[2]} the corresponding
multiplicities, @code{l[3]} a
address@hidden
+$C$
address@hidden tex
-basis of the V-filtration on
address@hidden
+$H''/H'$
address@hidden tex
in terms of the monomial basis of
address@hidden
+$O/J_f\cong H''/H'$
address@hidden tex
in @code{l[4]}.
address@hidden
+If the principal part of $f$ is $C$-nondegenerate, one can compute the
spectrum using the library {\tt spectrum.lib}.
+In this case, the V-filtration on $H''$ coincides with the Newton-filtration
on $H''$ which allows to compute the spectrum more efficiently.
address@hidden tex
Let us calculate one specific example, the maximal number
of triple points of type
address@hidden
+$\tilde{E}_6$ on a surface $X\subset{P}^3$
address@hidden tex
of degree seven.
This calculation can be done over the rationals.
So choose a local ordering on
address@hidden
+$Q[x,y,z]$
address@hidden tex
. Here we take the
negative degree lexicographical ordering which is denoted
@code{ds} in @sc{Singular}:
@@ -21904,21 +23546,44 @@
@end smallexample
The command @code{spectrumnd(f)} computes the spectrum of
address@hidden
+$f$
address@hidden tex
and
returns a list with six entries:
The Milnor number
address@hidden
+$\mu(f)$, the geometric genus $p_g(f)$
address@hidden tex
and the number of different spectrum numbers.
The other three entries are of type @code{intvec}.
They contain the numerators, denominators and
multiplicities of the spectrum numbers. So
address@hidden
+$x^7+y^7+z^7=0$
address@hidden tex
has Milnor number 216 and geometrical
genus 35. Its spectrum consists of the 16 different rationals
@*
address@hidden
+${3 \over 7}, {4 \over 7}, {5 \over 7}, {6 \over 7}, {1 \over 1},
+{8 \over 7}, {9 \over 7}, {10 \over 7}, {11 \over 7}, {12 \over 7},
+{13 \over 7}, {2 \over 1}, {15 \over 7}, {16 \over 7}, {17 \over 7},
+{18 \over 7}$
address@hidden tex
@*appearing with multiplicities
@*1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1.
address@hidden
+The singularities of type $\tilde{E}_6$ form a
+$\mu$-constant one parameter family given by
+$x^3+y^3+z^3+\lambda xyz=0,\quad \lambda^3\neq-27$.
address@hidden tex
Therefore they have all the same spectrum, which we compute
for
address@hidden
+$x^3+y^3+z^3$.
address@hidden tex
@smallexample
poly g=x^3+y^3+z^3;
@@ -21944,6 +23609,9 @@
@end smallexample
This tells us that there are at most 18 singularities of type
address@hidden
+$\tilde{E}_6$ on a septic in $P^3$. But $x^7+y^7+z^7$
address@hidden tex
is semiquasihomogeneous (sqh), so we can also apply the stronger
form of semicontinuity:
@@ -21953,12 +23621,21 @@
@end smallexample
So in fact a septic has at most 17 triple points of type
address@hidden
+$\tilde{E}_6$.
address@hidden tex
Note that @code{spectrumnd(f)} works only if
address@hidden
+$f$
address@hidden tex
has nondegenerate
principal part. In fact @code{spectrumnd} will detect a degenerate
principal part in many cases and print out an error message.
However if it is known in advance that
address@hidden
+$f$
address@hidden tex
has nondegenerate
principal part, then the spectrum may be computed much faster
using @code{spectrumnd(f,1)}.
@@ -21982,10 +23659,33 @@
@comment DO NOT EDIT DIRECTLY, BUT EDIT ti_ip.doc INSTEAD
@cindex ideal, toric
address@hidden
+Let $A$ denote an $m\times n$ matrix with integral coefficients. For $u
+\in Z\!\!\! Z^n$, we define $u^+,u^-$ to be the uniquely determined
+vectors with nonnegative coefficients and disjoint support (i.e.,
+$u_i^+=0$ or $u_i^-=0$ for each component $i$) such that
+$u=u^+-u^-$. For $u\geq 0$ component-wise, let $x^u$ denote the monomial
+$x_1^{u_1}\cdot\ldots\cdot x_n^{u_n}\in K[x_1,\ldots,x_n]$.
+
+The ideal
+$$ I_A:=<x^{u^+}-x^{u^-} | u\in\ker(A)\cap Z\!\!\! Z^n>\ \subset
+K[x_1,\ldots,x_n] $$
+is called a \bf toric ideal. \rm
+
+The first problem in computing toric ideals is to find a finite
+generating set: Let $v_1,\ldots,v_r$ be a lattice basis of $\ker(A)\cap
+Z\!\!\! Z^n$ (i.e, a basis of the $Z\!\!\! Z$-module). Then
+$$ I_A:=I:(x_1\cdot\ldots\cdot x_n)^\infty $$
+where
+$$ I=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
address@hidden tex
The required lattice basis can be computed using the LLL-algorithm
(@pxref{[Coh93]}). For the computation of the saturation, there are various
possibilities described in the
address@hidden
+section Algorithms.
address@hidden tex
@menu
* Algorithms:: Various algorithms for computing toric ideals.
@@ -22013,6 +23713,23 @@
The algorithm of Conti and Traverso (@pxref{[CoTr91]})
address@hidden
+computes $I_A$ via the
+extended matrix $B=(I_m|A)$,
+where $I_m$ is the $m\times m$ unity matrix. A lattice basis of $B$ is
+given by the set of vectors $(a^j,-e_j)\in Z\!\!\! Z^{m+n}$, where $a^j$
+is the $j$-th row of $A$ and $e_j$ the $j$-th coordinate vector. We
+look at the ideal in $K[y_1,\ldots,y_m,x_1,\ldots,x_n]$ corresponding to
+these vectors, namely
+$$ I_1=<y^{a_j^+}- x_j y^{a_j^-} | j=1,\ldots, n>.$$
+We introduce a further variable $t$ and adjoin the binomial $t\cdot
+y_1\cdot\ldots\cdot y_m -1$ to the generating set of $I_1$, obtaining
+an ideal $I_2$ in the polynomial ring $K[t,
+y_1,\ldots,y_m,x_1,\ldots,x_n]$. $I_2$ is saturated w.r.t. all
+variables because all variables are invertible modulo $I_2$. Now $I_A$
+can be computed from $I_2$ by eliminating the variables
+$t,y_1,\ldots,y_m$.
address@hidden tex
Because of the big number of auxiliary variables needed to compute a
toric ideal, this algorithm is rather slow in practice. However, it has
@@ -22027,6 +23744,16 @@
The algorithm of Pottier (@pxref{[Pot94]}) starts by computing a lattice
address@hidden
+basis $v_1,\ldots,v_r$ for the integer kernel of $A$ using the
+LLL-algorithm. The ideal corresponding to the lattice basis vectors
+$$ I_1=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
+is saturated -- as in the algorithm of Conti and Traverso -- by
+inversion of all variables: One adds an auxiliary variable $t$ and the
+generator $t\cdot x_1\cdot\ldots\cdot x_n -1$ to obtain an ideal $I_2$
+in $K[t,x_1,\ldots,x_n]$ from which one computes $I_A$ by elimination of
+$t$.
address@hidden tex
@node Hosten and Sturmfels, Di Biase and Urbanke, Pottier, Algorithms
@@ -22037,6 +23764,33 @@
The algorithm of Hosten and Sturmfels (@pxref{[HoSt95]}) allows to
address@hidden
+compute $I_A$ without any auxiliary variables, provided that $A$ contains a
vector $w$
+with positive coefficients in its row space. This is a real restriction,
+i.e., the algorithm will not necessarily work in the general case.
+
+A lattice basis $v_1,\ldots,v_r$ is again computed via the
+LLL-algorithm. The saturation step is performed in the following way:
+First note that $w$ induces a positive grading w.r.t. which the ideal
+$$ I=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
+corresponding to our lattice basis is homogeneous. We use the following
+lemma:
+
+Let $I$ be a homogeneous ideal w.r.t. the weighted reverse
+lexicographical ordering with weight vector $w$ and variable order $x_1
+> x_2 > \ldots > x_n$. Let $G$ denote a Groebner basis of $I$ w.r.t. to
+this ordering. Then a Groebner basis of $(I:x_n^\infty)$ is obtained by
+dividing each element of $G$ by the highest possible power of $x_n$.
+
+From this fact, we can successively compute
+$$ I_A= I:(x_1\cdot\ldots\cdot x_n)^\infty
+=(((I:x_1^\infty):x_2^\infty):\ldots :x_n^\infty); $$
+in the $i$-th step we take $x_i$ as the cheapest variable and apply the
+lemma with $x_i$ instead of $x_n$.
+
+This procedure involves $n$ Groebner basis computations. Actually, this
+number can be reduced to at most $n/2$
address@hidden tex
(@pxref{[HoSh98]}), and the single
computations -- except from the first one -- show to be easy and fast in
practice.
@@ -22049,6 +23803,38 @@
Like the algorithm of Hosten and Sturmfels, the algorithm of Di Biase
and Urbanke (@pxref{[DBUr95]}) performs up
address@hidden
+to $n/2$ Groebner basis
+computations. It needs no auxiliary variables, but a supplementary
+precondition; namely, the existence of a vector without zero components
+in the kernel of $A$.
+
+The main idea comes from the following observation:
+
+Let $B$ be an integer matrix, $u_1,\ldots,u_r$ a lattice basis of the
+integer kernel of $B$. Assume that all components of $u_1$ are
+positive. Then
+$$ I_B=<x^{u_i^+}-x^{u_i^-}|i=1,\ldots,r>, $$
+i.e., the ideal on the right is already saturated w.r.t. all variables.
+
+The algorithm starts by finding a lattice basis $v_1,\ldots,v_r$ of the
+kernel of $A$ such that $v_1$ has no zero component. Let
+$\{i_1,\ldots,i_l\}$ be the set of indices $i$ with
+$v_{1,i}<0$. Multiplying the components $i_1,\ldots,i_l$ of
+$v_1,\ldots,v_r$ and the columns $i_1,\ldots,i_l$ of $A$ by $-1$ yields
+a matrix $B$ and a lattice basis $u_1,\ldots,u_r$ of the kernel of $B$
+that fulfill the assumption of the observation above. We are then able
+to compute a generating set of $I_A$ by applying the following
+``variable flip'' successively to $i=i_1,\ldots,i_l$:
+
+Let $>$ be an elimination ordering for $x_i$. Let $A_i$ be the matrix
+obtained by multiplying the $i$-th column of $A$ with $-1$. Let
+$$\{x_i^{r_j} x^{a_j} - x^{b_j} | j\in J \}$$
+be a Groebner basis of $I_{A_i}$ w.r.t. $>$ (where $x_i$ is neither
+involved in $x^{a_j}$ nor in $x^{b_j}$). Then
+$$\{x^{a_j} - x_i^{r_j} x^{b_j} | j\in J \}$$
+is a generating set for $I_A$.
address@hidden tex
@node Bigatti and La Scala and Robbiano, , Di Biase and Urbanke, Algorithms
@@ -22059,6 +23845,12 @@
The algorithm of Bigatti, La Scala and Robbiano (@pxref{[BLR98]}) combines the
ideas of
the algorithms of Pottier and of Hosten and Sturmfels. The
computations are performed on a graded ideal with one auxiliary
address@hidden
+variable $u$ and one supplementary generator $x_1\cdot\ldots\cdot x_n -
+u$ (instead of the generator $t\cdot x_1\cdot\ldots\cdot x_n -1$ in
+the algorithm of Pottier). The algorithm uses a quite unusual technique to
+get rid of the variable $u$ again.
address@hidden tex
There is another algorithm of the authors which tries to parallelize
the computations (but which is not implemented in this library).
@@ -22083,6 +23875,25 @@
@subsection Integer programming
@cindex integer programming
address@hidden
+Let $A$ be an $m\times n$ matrix with integral coefficients, $b\in
+Z\!\!\! Z^m$ and $c\in Z\!\!\! Z^n$. The problem
+$$ \min\{c^T x | x\in Z\!\!\! Z^n, Ax=b, x\geq 0\hbox{
+component-wise}\} $$
+is called an instance of the \bf integer programming problem \rm or
+\bf IP problem. \rm
+
+The IP problem is very hard; namely, it is NP-complete.
+
+For the following discussion let $c\geq 0$ (component-wise). We
+consider $c$ as a weight vector; because of its non-negativity, $c$ can
+be refined into a monomial ordering $>_c$. It turns out that we can
+solve such an IP instance with the help of toric ideals:
+
+First we assume that an initial solution $v$ (i.e., $v\in Z\!\!\!
+Z^n, v\geq 0, Av=b$) is already known. We obtain the optimal solution
+$v_0$ (i.e., with $c^T v_0$ minimal) by the following procedure:
address@hidden tex
@c \begin{itemize}
@c \item (1) Compute the toric ideal $I_A$ using one of the algorithms in the
@c previous section.
@@ -22097,11 +23908,23 @@
@itemize @bullet
@item (1) Compute the toric ideal I(A) using one of the algorithms in the
previous section.
@item (2) Compute the reduced Groebner basis G(c) of I(A) w.r.t.@:
address@hidden
+$>_c$
address@hidden tex
.
@item (3) Reduce
address@hidden
+$x^v$
address@hidden tex
modulo G(c) using the Hironaka division algorithm.
If the result of this reduction is
address@hidden
+$x^(v_0)$
address@hidden tex
, then
address@hidden
+$v_0$
address@hidden tex
is an optimal
solution of the given instance.
@end itemize
@@ -22121,6 +23944,9 @@
methods seem to be faster in general than the methods using toric
ideals. But the latter have one great advantage: If one wants to solve
various instances that differ only by the vector
address@hidden
+$b$
address@hidden tex
, one has to
perform steps (1) and (2) above only once. As the running time of step (3)
is very short, solving all the instances is not much harder than
@@ -22263,6 +24089,9 @@
Symbolic Computation
@item
address@hidden
+Faug\`ere,
address@hidden tex
J. C.; Gianni, P.; Lazard, D.; Mora, T.: Efficient computation
of zero-dimensional
Gr@"obner bases by change of ordering. Journal of Symbolic Computation, 1989
@@ -47813,6 +49642,9 @@
@item @strong{Warnings:}
G should satisfy
address@hidden
+$ 2*genus-2 < deg(G) < size(D) $
address@hidden tex
, which is
not checked by the algorithm.
@*G and D should have disjoint supports (checked by the algorithm).
@@ -47877,10 +49709,16 @@
for more details)address@hidden
The code computes the residues of a vector space basis of
address@hidden
+$\Omega(G-D)$
address@hidden tex
at the rational places given by D.
@item @strong{Warnings:}
G should satisfy
address@hidden
+$ 2*genus-2 < deg(G) < size(D) $
address@hidden tex
, which is
not checked by the algorithm.
@*G and D should have disjoint supports (checked by the algorithm).
@@ -47937,8 +49775,14 @@
E[2] ... E[n+2]: matrices used in the procedure decodeSV
E[n+3]: intvec with
E[n+3][1]: correction capacity
address@hidden
+$epsilon$
address@hidden tex
of the algorithm
E[n+3][2]: designed Goppa distance
address@hidden
+$delta$
address@hidden tex
of the current AG code
@end format
@@ -47954,6 +49798,9 @@
The current AG code is @code{AGcode_Omega(G,D,EC)address@hidden
If you know the exact minimum distance d and you want to use it in
@code{decodeSV} instead of
address@hidden
+$delta$
address@hidden tex
, you can change the value
of E[n+3][2] to d before applying decodeSV.
@*If you have a systematic encoding for the current code and want to
@@ -47964,10 +49811,19 @@
@item @strong{Warnings:}
F must be a divisor with support disjoint from the support of D and
with degree
address@hidden
+$epsilon + genus$
address@hidden tex
, where
address@hidden
+$epsilon:=[(deg(G)-3*genus+1)/2]$
address@hidden tex
address@hidden
G should satisfy
address@hidden
+$ 2*genus-2 < deg(G) < size(D) $
address@hidden tex
, which is
not checked by the algorithm.
@*G and D should also have disjoint supports (checked by the