In a similar vein to these now quite old questions on advice for calculating a Gröbner basis:

Fast computation of a Groebner basis. What is possible?

What is the state of art in Groebner bases

I am interested in advice/tips/tricks for computing a Gröbner basis, but in the case where we likely already know the answer is {1}. In this case the computation is used to verify there are no solutions to a large set of multivariate polynomial constraints.

Macaulay2 seems to systematically build up larger and larger degrees of polynomials, while Mathematica (which is usually slower and is opaque in its computations) can sometimes run dramatically faster than Macaulay2 for this use case (possibly Mathematica does a more "depth first" search which can lead to contradictions quicker?). I've seen cases where Mathematica takes ~ 4 min, while Macaulay takes > 20 hours.

In particular, the RAM requirements for these runs is often the limited factor, when all I really want to know is the "ideal membership problem" for 1. So I wonder if:

Is there a more appropriate calculation, computational package, or algorithm for calculating the Gröbner basis with the "hint" that there is very likely some relationship that collapses everything to {1}?