In latest years, several advancements have been made in symbolic-numerical eigenvalue techniques for solving polynomial systems. In this article, we add to this list by reducing the task to an eigenvalue problem in a considerably faster and simpler w
ay than in previous methods. This results in an algorithm which solves systems with isolated solutions in a reliable and efficient way, outperforming homotopy methods in overdetermined cases. We provide an implementation in the proof-of-concept Julia package EigenvalueSolver.jl.
We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety $X$. Our starting point is a homogeneous ideal $I$ in the Cox ring of $X$, which in practice might arise from h
omogenizing 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 $I$. We study these properties and provide bounds on the size of the matrices involved in our approach in the case where $I$ is a complete intersection.
