Yet another eigenvalue algorithm for solving polynomial systems

TL;DR

This paper introduces a new eigenvalue-based algorithm for solving polynomial systems that is reliable, efficient, and outperforms some existing methods, especially in overdetermined cases, with practical implementation in Julia.

Contribution

The paper presents a novel eigenvalue algorithm that improves solving polynomial systems, particularly overdetermined ones, with better speed and reliability than previous approaches.

Findings

Outperforms homotopy continuation in certain cases

Reliable solution of systems with isolated solutions

Implementation available in Julia package EigenvalueSolver.jl

Abstract

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. We design an algorithm which solves systems with isolated solutions reliably and efficiently. In overdetermined cases, it reduces the task to an eigenvalue problem in a simpler and considerably faster way than in previous methods, and it can outperform the homotopy continuation approach. We provide many examples and an implementation in the proof-of-concept Julia package EigenvalueSolver.jl.