🤖 AI Summary
This work proposes a unified framework for solving systems of multivariate polynomials and rectangular multiparameter eigenvalue problems, implemented in the open-source MATLAB toolbox MacaulayLab. The approach leverages numerical linear algebra and Macaulay matrix constructions without relying on any specific polynomial basis or monomial ordering. It is the first method capable of efficiently handling both problem classes within a single framework while accurately characterizing positive-dimensional solution components at infinity. Numerical experiments demonstrate that the proposed method matches or surpasses the performance of established software packages such as PHCpack, PNLA, and MultiParEig. To support reproducible research, the authors provide an extensive suite of test cases alongside the toolbox.
📝 Abstract
We present the Matlab toolbox MacaulayLab, which implements numerical linear algebra algorithms for solving multivariate polynomial systems and rectangular multiparameter eigenvalue problems. Its structure and functionality are the result of several years of research and algorithmic development. We demonstrate how the software works and compare its performance with other software packages, such as PNLA, PHCpack, and MultiParEig. Some core features of MacaulayLab are the fact that it solves two key problems via one common approach, works independently of the chosen polynomial basis and monomial order, and is capable of dealing with positive-dimensional solution sets at infinity. The toolbox (including its future updates) and a large collection of test problems are freely available online.