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RE: [Axiom-developer] about Expression Integer

From: Bill Page
Subject: RE: [Axiom-developer] about Expression Integer
Date: Mon, 20 Feb 2006 15:16:14 -0500

On February 19, 2006 4:45 PM Martin Rubey wrote:
> Bill Page writes:
> >  
> > > In other words, consider the following example:
> > > >   P := DistributedMultivariatePolynomial([x,y],Expression Integer)
> > > >   b := monomial(x,[0,0]::DirectProduct(2,NNI))$P
> > > > 

> ...
> I wrote that it could *lead* to "surprising" results. I.e.,
> I could imagine that some innocent looking computation would
> not be correct in the situation above. For example,
> differentiate(b+x,x)
> gives one, but 
> differentiate(b+x,x)@EXPR INT 
> gives two.

(1) -> P := DistributedMultivariatePolynomial([x,y],Expression Integer)

   (1)  DistributedMultivariatePolynomial([x,y],Expression Integer)
   Type: Domain

(2) -> b := monomial(x,[0,0]::DirectProduct(2,NNI))$P

   (2)  x
   Type: DistributedMultivariatePolynomial([x,y],Expression Integer)

(3) -> differentiate(b+x,x)

   (3)  1
   Type: DistributedMultivariatePolynomial([x,y],Expression Integer)

(4) -> differentiate(b+x,x)@EXPR INT

   (4)  2
   Type: Expression Integer

> Note that I did not even need to ask to coerce b+x to EXPR INT...
> Of course, as we are discussing this now, this does not come as 
> a surprise. But I bet that even for quite an experienced axiom
> user, who did not follow this thread for some reason, it will be
> difficult to spot the mistake. Especially if he doesn't realize
> that she wanted to compute something else...

I would call the result in (3) a bug. It seems to me that perhaps
'differentiate$DMP' is making a unwarranted assumption that the
coefficients of DMP are constant. I cannot imagine a properp
mathematical definition of 'differentiate' that depended on the
distinction of polynomial coefficients versus variables.

> > Part of the reason for the confusion, I think, is because the
> > common OutputForm for polynomials does not typographically 
> > identify polynomial variables versus coefficients. But if, say
> > the polynomial variables were always printed in bold face roman
> > type but the coefficients were printed in italic, it might be
> > more clear what is going on.
> I think that this is a very nice idee. I guess this could be 
> even realized on text terminals, couldn't it?

To do this in LaTeX output is obviously quite straight forward.
To do it on text terminals I think would require support of
'curses' or a similar library. I am not sure that Axiom text
output (really a function of the underlying lisp - GCL, I guess)
supports 'curses'. Do you think suport for these conventions in
text output are really that important?

Bill Page.

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