Polynomial Systems, Homotopy Continuation, and Applications

Abstract

Systems of multivariate polynomial equations are ubiquitous throughout mathematics. They also appear prominently in scientific applications such as kinematics [20,22], computer vision [11,15], power flow engineering [18], and statistics [12]. Numerical homotopy continuation methods are a fundamental tool for both solving these systems and determining more refined information about their structure.In this article, we offer a brief glimpse of polynomial homotopy continuation methods: the general theory, a few applications, and some software packages that implement these methods. Our aim is to spark the reader's interest in this exciting and broad area of research. We invite those looking to learn more to join us at the AMS Short Course: Polynomial systems, homotopy continuation, and applications, to be held January 2-3 at the 2023 Joint Mathematics Meetings in Boston.