# Computing Groebner basis for a complicated systems of polynomials

I am trying to solve complicated systems of polynomial equations. The first step is to determine maximal sets of independent variables for the solution manifold (ideal) or the number of isolated solutions using Gröbner bases. In some cases, Mathematica, Maple and SymPy do not seem to be able to determine the Gröbner basis (in reasonable time).

Do you recommend some tools for this task?

Let me give a test case for which Mathematica, Maple and SymPy did not finish in reasonable time (~<10h) on my computer. I used grevlex monomial order. Do you know a tool that can compute the Gröbner basis for this case? There are 21 polynomials and 8 variables u, q1, r1, q2, r2, q3, r3, q4 (using Gram-Schmidt, one can reduce to 8 polynomials): https://pastebin.com/mpqZUQqC

• Have you seen the post Benchmarks for Gröbner bases and polynomial system solution at scicomp.SE? Feb 5, 2019 at 19:07
• Thanks for pointing out this post! Yes, I had seen it. There are several tools that can compute Gröbner bases (in principle). As I am reaching systems of equations where Mathematica, Maple, and SymPy seem to fail, I am hoping that someone could paste the stated system of equations into his favorite tool and let me know whether it manages to give the basis. Feb 5, 2019 at 19:17
• You might try Singular, but given the size of your system I wouldn't be too optimistic. Feb 5, 2019 at 19:20
• In addition to Singular, you should try Macaulay2. Feb 6, 2019 at 10:01
• For calculations modulo a prime see also mathic : github.com/broune/mathic, github.com/broune/memtailor, github.com/broune/mathicgb . Feb 6, 2019 at 17:38

infolevel[Groebner] := 4:  # see what is going on