Certifying Galois/monodromy Actions via Homotopy Graphs

2026-03-17

AI Summary

This work addresses the reliable computation of Galois and monodromy groups for parametrized polynomial systems. To this end, it introduces a novel framework that integrates certified homotopy path tracking with homotopy graphs, enabling—for the first time—the rigorous numerical verification of monodromy group actions. By combining certified numerical algorithms with techniques from numerical algebraic geometry, the proposed method guarantees the mathematical correctness of its computational results. The approach has been successfully validated on a range of examples drawn from both pure and applied mathematics, demonstrating its effectiveness, reliability, and practical utility in analyzing the group-theoretic structures of complex polynomial systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Applications

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Galois group
• monodromy group
• parametrized polynomial systems
• certified computation
• homotopy graphs

Innovation

• certified numerical algorithm
• Galois groups
• monodromy groups
• homotopy path tracking
• homotopy graphs