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 Bézout (m-Bézout) 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-Bézout number of certain multi-homogeneous systems derived from rigidity theory with graph orientations, cf Bartzos et al. (2020). 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 (Bartzos et al., 2021). Here, we prove that every multi-homogeneous system can be modeled by hypergraphs and the computation of its m-Bézout 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.