[Axiom-developer] Re: severe (!) bug in normalize

[Martin Rubey]

Dear Waldek,

thanks for your explanations.

Waldek Hebisch <address@hidden> writes:

> Martin Rubey wrote:
> > Waldek Hebisch <address@hidden> writes:

> > I guess that the main problem with EXPR and friends is, that it is not
> > clear what the variables are. Do you know the assumptions needed for
> > RischNormalize?

> I do not know what problems with variables you see here

I meant that it is not clear what the coefficient field for rischNormalize is:
rischNormalize works over F, which is a "FunctionSpace R". 

To fix things, suppose that F is EXPR INT and the expression we consider
contains only one variable. I guess the field of constants is the field of
"numeric" elements from EXPR INT, i.e., expressions that do not contain a
variable.

Of course, this is still not a computable coefficient field. However, in this
case it is at least clear mathematically what sqrt(2)*sqrt(3)-sqrt(6) is
supposed to be.

Now, if we switch to a field of constants containing some variables, what about
sqrt(x)*sqrt(y)-sqrt(x*y)?

> But rischNormalize uses operations implemented by other domains to perform
> actual computuations, so there are extra assumptions.  One is that if an
> expression does not depend on a kernel (which is checked using derivatives)
> simplifier should eliminate this kernel from the expression.  For example
> representing 'y' as 'x - x + y' is legal for computable fields, 

There are some related axioms defined in attreg.spad (namely
canonicalUnitNormal, unitsKnown and canonical). In any case, I think that there
should be a way to check whether equality is "mathematical" or only
heuristic.

> but rischNormalize would not tolerate such representation.  Of course, since
> expressions are represented as rational functions of kernels such simple
> problem can not happen in Axiom. But (sqrt(2)*sqrt(3)-sqrt(6))*exp(x) already
> hints towards possible problems.


By the way, are there any interesting computable coefficient fields for
rischNormalize?

> (I am aware of tread about Polynomial Expression Integer, but I think it is a
> separate problem).  

Yes. In fact, I meanwhile think that it is not a problem, apart from the fact
that axiom lacks domains UnivariateExpression and MultivariateExpression
analogous to the polynomial domains.

Martin