A tropical method for solving parametrized polynomial systems

TL;DR

This paper introduces a tropical geometric framework for efficiently solving parametrized polynomial systems by constructing optimal homotopies, demonstrated through applications in chemistry, physics, and graph theory.

Contribution

It presents a novel tropical approach for generating optimal homotopies tailored to two types of parametrized systems, improving solution efficiency.

Findings

Efficient computation of tropical data for parametrized systems.

Construction of homotopies matching the generic number of solutions.

Case studies demonstrating practical applications in various scientific fields.

Abstract

We give a framework for constructing generically optimal homotopies for parametrized polynomial systems from tropical data. Here, generically optimal means that the number of paths tracked is equal to the generic number of solutions. We focus on two types of parametrized systems -- vertically parametrized and horizontally parametrized systems -- and discuss techniques for computing the tropical data efficiently. We end the paper with several case studies, where we analyze systems arising from chemical reaction networks, coupled oscillators, and rigid graphs.