Certifying Galois/monodromy Actions via Homotopy Graphs
This provides a reliable computational tool for mathematicians studying algebraic structures in polynomial systems, though it builds incrementally on existing homotopy graph frameworks.
The researchers developed a certified numerical algorithm for computing Galois/monodromy groups of parametrized polynomial systems, using certified homotopy path tracking to guarantee correctness. They implemented and tested the algorithm on examples from pure and applied mathematics, successfully certifying properties of several notable groups.
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.