Polynomial Homotopies for Dense, Sparse and Determinantal Systems

TL;DR

This paper introduces numerical homotopy continuation methods tailored for dense, sparse, and determinantal polynomial systems, ensuring optimal paths and accurate root counting for generic instances.

Contribution

It presents a unified approach for solving different classes of polynomial systems using homotopy continuation, including software and application insights.

Findings

Paths lead to solutions for generic systems

Homotopy is optimal for the classes considered

Root counting aligns with generic system resolution

Abstract

Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution of a generic system that is used to start up the deformations. Software and applications are discussed.