Computing Minimal Multi-Homogeneous B\'ezout Numbers is Hard
Klaus Meer and Gregorio Malajovich
Theory of Computing Systems 2005.

Abstract

The multi-homogeneous B\'ezout 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 B\'ezout number. In this paper, it is proved that the problem of computing, or even estimating the optimal multi-homogeneous B\'ezout 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-homo\-ge\-neous B\'ezout number up to a fixed factor cannot exist even in a randomized setting, unless \$BPP \supseteq NP\$.

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EPrint Type: Article Project Keyword UNSPECIFIED Theory & Algorithms 966 Klaus Meer 28 November 2005