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I have a system of multilinear polynomial constraints that I believe has no solution. Unfortunately, the number of variables and number of constraints is too large to afford direct computation of a Groebner basis to prove there are no solutions.

However the system of constraints has some nice symmetry which allows arguing that if solutions did exist, the surface of these solutions must have dimension at least 15. So this produces a large jump/gap in dimensions depending on whether solutions exist.

This made me wonder:
Is there some way to estimate the dimension of the algebraic variety, without calculating the Groebner basis, and with small enough range that I could use this to prove no solutions exist?

In my case, the polynomial constraints are multilinear. Does this open up any additional techniques?

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You can try to use numerical algebraic geometry (aka homotopy continuation). It may be more efficient than Groebner bases and you will be able to take advantage of multilinearity (via the multilinear homotopies). Here are some software packages:

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