Certifying Galois/monodromy Actions via Homotopy Graphs

TL;DR

This paper introduces a certified numerical algorithm that uses homotopy graphs and path tracking to accurately compute and verify Galois/monodromy groups of parametrized polynomial systems, with extensive experimental validation.

Contribution

It presents a novel certified algorithm combining homotopy path tracking and graph frameworks to compute and verify Galois/monodromy groups of polynomial systems.

Findings

Successfully certified properties of Galois/monodromy groups in various mathematical examples

Demonstrated the algorithm's effectiveness through extensive experiments

Validated the correctness of monodromy actions in complex polynomial systems

Abstract

We develop a certified numerical algorithm for computing Galois/monodromy groups of parametrized polynomial systems. Our approach employs certified homotopy path tracking to guarantee the correctness of the monodromy action produced by the algorithm, and builds on previous ``homotopy graph" frameworks. We conduct extensive experiments with an implementation of this algorithm, which we have used to certify properties of several notable Galois/monodromy groups which arise in several examples drawn from pure and applied mathematics.