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Counting Solutions to Binomial Complete Intersections

Eduardo Cattani·Alicia Dickenstein
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Abstract

We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.

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Eduardo Cattani, Alicia Dickenstein. 2006-04-27. Counting Solutions to Binomial Complete Intersections. https://arxiv.org/abs/math/0510520

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