large groebner and primdec computations

[Burcin Erocal]

Hi,

someone from this group might be able to answer this question:

http://ask.sagemath.org/question/151/large-groebner-basis-calculations

Here is the summary:

---------------------------------

large groebner basis calculations

I know there are a number of different packages in Sage capable of
computing Groebner basis but I don't have much experience with these
tools. I am wondering which option and specifically which routine might
be most suitable for very large (50,000 generators) but relatively
sparse (polynomials of degree less than 4 and with 4 or fewer terms
over 64 variables). I am interested in basis and prime decomposition in
relation to classifying low dimensional Frobenius algebras. I am
currently using groeber and the primdec library from Singular in Sage
but I am up against significant performance limits.

Also, does any have an recommended methods of pre-processing sparse
sets of generators prior to calling groebner? And are there any
packages capable of exploiting multi-processor architecture?

Bill Page
---------------------------------

Please use the ask.sagemath.org website for responses.

Cheers,
Burcin

[Michael Brickenstein]

Hi!

Sorry, I did not notice the message before.

There exists no 'best variant' for large systems.

The ugly answer is, that you have to try
several variants and the most important
options, at least redTail...
You can also try several orderings.
Usually, the best ordering is dp, but
sometimes you can arrange the variables in a smart block ordering structure,
which makes things much easier.

Maybe, you can also try slimgb, which has had good results in the past
for large system, but systems of these number of variables are reallly
hard.
Having 50000 equations however is awesome and improves the possibility to
compute the GB.

The core algorithms std and slimgb both implement their own preprocessing.

Regardings primary decomposition of such big systems.
This is more difficult than Groebner bases. So you can essentially
forget about it, unless there are some special conditions.
In fact, when you're ideal is such over determined
and you might only have a few solutions with multiplicity 1,
the prime decomposition is equivalent to solving.

I think you did not mention the characteristic.

Cheers,
Michael

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-------------------------------------------
Dr. rer. nat. Michael Brickenstein
Mathematisches Forschungsinstitut Oberwolfach gGmbH
Schwarzwaldstr. 9 - 11
77709 Oberwolfach
Tel.: 07834/979-31
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