Toric eigenvalue methods for solving sparse polynomial systems

Abstract

We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety X X . Our starting point is a homogeneous ideal I I in the Cox ring of X X , which in practice might arise from homogenizing a sparse polynomial system. We prove a new eigenvalue theorem in the toric compact setting, which leads to a novel, robust numerical approach for solving this problem. Our method works in particular for systems having isolated solutions with arbitrary multiplicities. It depends on the multigraded regularity properties of I I . We study these properties and provide bounds on the size of the matrices in our approach when I I is a complete intersection.