Computing Minimal Multi-Homogeneous Bezout Numbers Is Hard

The multi-homogeneous Bezout number is 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 actually 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. Moreover, 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.

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