Computing Minimal Multi-Homogeneous B\'ezout Numbers is Hard ## AbstractThe 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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