Computing Minimal Multi-homogeneous Bézout Numbers Is Hard

Abstract

Computing good estimates for the number of solutions of a polynomial system is of great importance in many areas such as computational geometry, algebraic geometry, mechanical engineering, to mention a few. One prominent and frequently used example of such a bound is the multi-homogeneous Bezout number. It provides a bound for the number of solutions of a system of multi-homogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multi-homogeneous system, one can ask for the optimal multi-homogenization that would minimize the Bezout number. In this paper, it is proved that the problem of computing, or even estimating the optimal multi-homogeneous Bezout number is NP-hard. In terms of approximation theory for combinatorial optimization, the problem of computing the best multi-homogeneous structure does not belong to APX, unless P = NP. As a consequence, polynomial time algorithms for estimating the minimal multi-homogeneous Bezout number up to a fixed factor cannot exist even in a randomized setting, unless BPP⊒NP.