The SagbiHomotopy.jl package for solving polynomial systems

TL;DR

The SagbiHomotopy.jl package employs SAGBI homotopies in Julia to efficiently solve polynomial systems, especially those with horizontally parameterized structures, reducing computational effort compared to traditional methods.

Contribution

This paper introduces a novel Julia package that uses SAGBI homotopies for solving polynomial systems, offering an optimal start system and improved efficiency for specific problem classes.

Findings

Reduces the number of solution paths for square horizontally parameterized systems.

Demonstrates effectiveness on chemistry and physics problems.

Outperforms polyhedral homotopies in relevant cases.

Abstract

We present the Julia package SagbiHomotopy.jl for solving systems of polynomial equations using numerical homotopy continuation. The package introduces an optimal choice of a start system based on SAGBI homotopies. For square horizontally parameterized systems, where each equation is a linear combination of a given set of polynomials, SAGBI homotopies significantly reduce the number of solution paths to track compared to polyhedral homotopies currently used by default in most software for numerical homotopy continuation. We illustrate our framework with a variety of examples, including problems arising in chemistry and physics.