Solving polynomial systems via homotopy continuation and monodromy

TL;DR

This paper introduces a monodromy-based framework for solving polynomial systems efficiently, demonstrating that under certain assumptions, the number of paths tracked is linear in the number of solutions, with competitive software implementation.

Contribution

It presents a novel monodromy-based approach using decorated graphs for solving polynomial systems, with theoretical analysis and practical software implementation.

Findings

Expected number of homotopy paths is linear in solutions

Software implementation is competitive with state-of-the-art methods

Framework effectively describes monodromy-based solvers

Abstract

We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the theoretical assumption that monodromy actions are generated uniformly, we show that the expected number of homotopy paths tracked by an algorithm following this framework is linear in the number of solutions. We demonstrate that our software implementation is competitive with the existing state-of-the-art methods implemented in other software packages.