Computing Multi-Homogeneous Bezout Numbers is Hard

TL;DR

This paper proves that computing or approximating the minimal multi-homogeneous Bezout number for polynomial systems is NP-hard and unlikely to be efficiently approximable, highlighting fundamental computational complexity barriers.

Contribution

It establishes the NP-hardness of computing and approximating the multi-homogeneous Bezout number for arbitrary polynomial systems.

Findings

Computing the optimal multi-homogeneous Bezout number is NP-hard.

Approximating the minimal Bezout number within any fixed factor is unlikely in polynomial time.

The problem does not belong to APX unless P = NP.

Abstract

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 contains NP.