[GAP Forum] polynomial system solving

[Dima Pasechnik]

Dear GAP-Forum,

On Thu, Mar 31, 2005 at 11:48:08AM -0700, Alexander Hulpke wrote:
> > I would really appreciate it if anyone could help me find the less
> > comlicated method to solve polynomial systems using GAP.Thank you.
> 
> Not knowing what your mathematical background or your application is makes
> it a bit hard to give a suitable response:
> 
> If solving a system of polynomial equations is your only application, I
> would not recommend to use GAP, but a dedicated system.
> 
> If you need to do it as part of a longer calculation in GAP, you might be
> able to use the built-in Gr"oebner bases, however:
one can presumably use the "vector enumeration" capabilities too,
instead of the Groebner basis. This will give the regular representation
of the ring modulo the ideal. Then one can find a basis of
common eigenvectors; in this basis the matrices representing variables
of the ring will be diagonal, and the solutions will be readily available
as the eigenvalues for each eigenvector.
Certainly, one will need the ideal to be 0-dimensional, 
but otherwise it makes little sense to talk about solving anyway, at least
for infinite fields.

I don't know how much less efficient this is than the Groebner basis
approach. Has anyone ever done any serious experiments (or theory?)
in this respect?
In the non-commutative case vector enumeration behaves as bad as Todd-Coxeter, 
but in the commutative case, I don't know.
I might be even willing to look into this...

Best,
Dmitrii

---
Dr. D.V. Pasechnik
E&OR/FEB
Tilburg University
P.O. Box 90153
5000 LE Tilburg
The Netherlands
http://center.uvt.nl/staff/pasechnik/