Numerical computation of Galois groups

TL;DR

This paper introduces numerical methods to compute Galois groups of geometric problems, enabling analysis beyond the full symmetric group case, with algorithms for generators and structural insights, demonstrated via a Macaulay2 package.

Contribution

It presents novel numerical algorithms for computing Galois groups and analyzing their structure, extending capabilities beyond existing methods limited to full symmetric groups.

Findings

Algorithms successfully compute Galois group generators.

Methods provide structural information about Galois groups.

Implementation demonstrated with a Macaulay2 package.

Abstract

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. Computing monodromy permutations using numerical algebraic geometry gives information about the group, but can only determine it when it is the full symmetric group. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators while the other gives information on its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations.