I am trying to solve a system of polynomial equations (with rational number coefficients). For this I am computing its Gröbner basis using the F4 algorithm implemented in Maple. (`Groebner[Basis]`).
As far as I understand it, the algorithms solves the problem modulo one or more prime number and later reconstructs the full (rational number) solution. Usually it only takes a handfull of primes. But in my case it by now solved the problem successfully modulo about ~300 different primes, each time doing exactly the same computation (according to the log files running with `infolevel[GroebnerBasis]:=5`).
Could somebody enlighten me what this means? In particular, can I interpret this as indication that the system of equations does have or does not have a solution?
- My system of equations is rather large. Each individual solve modulo a prime takes half an hour on a powerful workstation pc. Other computeralgebra systems without Faugère's algorithms cant solve it at all.
- (part of) the output can be found here: https://gist.github.com/krox/484252f075eb19edd0ac865099a564ba . The curious thing to me is that each solve behaves exactly the same. So why is maple repeating it with different primes over and over again?