Bounding the Number of Roots of Multi-Homogeneous Systems.

Determining the number of solutions of a multi-homogeneous polynomial system is a fundamental problem in algebraic geometry. The multi-homogeneous Bezout (m-Bezout) number bounds from above the number of non-singular solutions of a multi-homogeneous system, but its computation is a #P-hard problem. Recent work related the m-Bezout number of certain multi-homogeneous systems derived from rigidity theory with graph orientations, cf Bartzos et al. [2]. A first generalization applied graph orientations for bounding the root count of a multi-homogeneous system that can be modeled by simple undirected graphs, as shown by three of the authors [3]. Here, we prove that every multi-homogeneous system can be modeled by hypergraphs and the computation of its m-Bezout bound is related to constrained hypergraph orientations. Thus, we convert the algebraic problem of bounding the number of roots of a polynomial system to a purely combinatorial problem of analyzing the structure of a hypergraph. We also provide a formulation of the orientation problem as a constraint satisfaction problem (CSP), hence leading to an algorithm that computes the multi-homogeneous bound by finding constrained hypergraph orientations.