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. © 2006 Elsevier Inc. All rights reserved.
Cattani, E., & Dickenstein, A. (2007). Counting solutions to binomial complete intersections. Journal of Complexity, 23(1), 82–107. https://doi.org/10.1016/j.jco.2006.04.004