[GAP Forum] Buchberger algorithm, real and complex data type

[Alexander Hulpke]

Dear Forum, Shengyu Shen asked:

> ===========================================================
> Note  that  GAP  at  the  moment  only  includes  a  naïve implementation of
>   Buchberger's  algorithm  (which  is  mainly intended as a teaching tool). It
>   might not be sufficient for serious problems.
> ===========================================================
> 
> So it this still true for current release of GAP?

Yes. We felt that we did not have the expertise to implement a Groebner basis algorithm that would be plausibly comparable with that of dedicated systems. The warning in the manual is there to alert users to not take Groebner bassi performance as an indication of the general system performance. (This concerns runtime, not correctness.)

The existing functionality in GAP should be sufficient for small (or illustration) examples but it is easy to construct examples why (say) Singular will  solve in seconds and which take hours or more in GAP.

(S)he also asked:

> I find that there are no real and complex type in GAP. 
> 
> So how can I handle those polynomials with real or complex solutions?

Yes, GAP has no generic real/complex data type. If you need factorizations of polynomials, you would have to use a suitable algebraic extension. As first try I would see whether the cyclotomic numbers would work, as their arithmetic is more efficient than that of generic algebraic extensions.

Regards,

   Alexander Hulpke