Yet another eigenvalue algorithm for solving polynomial systems

Abstract

In recent 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 illustrate the effectiveness of our method with numerous examples and provide an implementation in the proof-of-concept Julia package 𝙴𝚒𝚐𝚎𝚗𝚟𝚊𝚕𝚞𝚎𝚂𝚘𝚕𝚟𝚎𝚛.𝚓𝚕.