Toric eigenvalue methods for solving sparse polynomial systems

Bender, Matías; Telen, Simon

doi:10.1090/mcom/3744

Toric eigenvalue methods for solving sparse polynomial systems
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by Matías R. Bender and Simon Telen;

Math. Comp. 91 (2022), 2397-2429

DOI: https://doi.org/10.1090/mcom/3744

Published electronically: June 30, 2022

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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 in our approach when $I$ is a complete intersection.

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