On the Complexity of Computing Multi-Homogeneous Bézout Numbers

We study the question how difficult it is to compute the multi-homogeneous B\'ezout number for a polynomial system of given number $n$ of variables and given support $A$ of monomials with non-zero coefficients. We show that this number is NP-hard to compute. It cannot even be efficiently approximated within an arbitrary, fixed factor unless P = NP. This is joint work with Gregorio Malajovich.