Monodromy Solver: Sequential and Parallel

TL;DR

This paper introduces a homotopy continuation-based algorithm for solving polynomial systems that is robust to failures and optimized for parallel execution, supported by theoretical analysis and extensive simulations.

Contribution

It presents a novel, failure-tolerant, parallelizable algorithm for polynomial system solving using monodromy and homotopy continuation, with probabilistic analysis and simulation tools.

Findings

Algorithm effectively handles many failures during homotopy continuation.

Parallelization significantly improves computational efficiency.

Simulation results validate theoretical analysis and demonstrate scalability.

Abstract

We describe, study, and experiment with an algorithm for finding all solutions of systems of polynomial equations using homotopy continuation and monodromy. This algorithm follows a framework developed in previous work and can operate in the presence of a large number of failures of the homotopy continuation subroutine. We give special attention to parallelization and probabilistic analysis of a model adapted to parallelization and failures. Apart from theoretical results, we developed a simulator that allows us to run a large number of experiments without recomputing the outcomes of the continuation subroutine.