Toric Eigenvalue Methods for Solving Sparse Polynomial Systems

TL;DR

This paper introduces a new eigenvalue-based numerical method for solving sparse polynomial systems on toric varieties, leveraging multigraded regularity to handle solutions with multiplicities.

Contribution

It develops a novel eigenvalue theorem in the toric setting and provides bounds on matrix sizes for complete intersections, improving robustness and applicability.

Findings

The method effectively computes solutions with arbitrary multiplicities.

Provides bounds on matrix sizes for complete intersection systems.

Enhances numerical stability in 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 $X$. Our starting point is a homogeneous ideal $I$ in the Cox ring of $X$, 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 $I$. We study these properties and provide bounds on the size of the matrices appearing in our approach when $I$ is a complete intersection.