We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety . Our starting point is a homogeneous ideal in the Cox ring of , 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 . We study these properties and provide bounds on the size of the matrices in our approach when is a complete intersection.

, , , , ,
Mathematical Computation
Networks and Optimization

Bender, M.R, & Telen, S.J.L. (2022). Toric eigenvalue methods for solving sparse polynomial systems. Mathematical Computation, 91, 2397–2429. doi:10.1090/mcom/3744