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Re: [Axiom-developer] Re: Groebner basis

From: Waldek Hebisch
Subject: Re: [Axiom-developer] Re: Groebner basis
Date: Tue, 5 Feb 2008 16:29:52 +0100 (CET)

> Fabio S. wrote:
> > 
> > I am thinking you are right: I tried the same computations on another 
> > machine: same distribution, same axiom version: it ran for more than 8 
> > hours without problems, but, unluckly, not giving the answer... :-(
> > 
> You may try to do computation modulo a prime number.  For me
> groebner([x ::(Polynomial PF(1663)) for x in eqns])
> finished in about 15 minutes.  For some (unlucky) primes modular
> Groebner basis is quite different than integer Groebner basis, but
> for most primes they share many properties.  In fact, one method
> of computing integer Groebner bases (currently unimplemented in Axiom)
> starts with modular Groebner basis and "lifts" it to an integer one.
> Also, it looks that you are using 'Fraction Polynomial Integer'
> as your domain.  Did you try to use 'DistributedMultivariatePolynomial'
> (which is likely to be much more efficient)?

I now tried the following:

vars := members(reduce(append, [variables(x) for x in eqns]) :: (Set Symbol))
groebner([x :: (HomogeneousDistributedMultivariatePolynomial(vars, Integer)) 
for x in eqns])

It finished in about 30 minutes, using less than 100 Mb of memory.
The difference is that Groebner basis is computed relative to
total degree reverse lexicografic ordering, while your original
code requested lexicographic ordering.

                              Waldek Hebisch

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