Fast Rational Univariate Representation via Gaussian Elimination

2026-07-07

AI Summary

This work proposes an efficient method for computing Rational Univariate Representations (RUR) of zero-dimensional polynomial systems by leveraging dense linear algebra and Gaussian elimination. Building upon classical FGLM-type algorithms, the approach replaces conventional steps with Gaussian elimination, thereby significantly enhancing computational efficiency for large-scale systems while rigorously preserving theoretical correctness. Experimental results demonstrate that the proposed method correctly parameterizes zero-dimensional ideals with thousands of solutions in just a few seconds. The implementation is publicly available as the open-source Julia package RationalUnivariateRepresentation.jl.

Abstract

In this note, we present RationalUnivariateRepresentation.jl (https://newrur.gitlabpages.inria.fr/RationalUnivariateRepresentation.jl/), a Julia package for computing rational univariate representations of zero-dimensional polynomial systems. The package uses dense linear algebra and Gaussian elimination for the FGLM-like stage. The purpose of this contribution is to advocate for this choice and explain the implementation details that turn the algorithm into practical software. In particular, we show that our implementation can compute guaranteedly correct parametrizations of ideals with thousands of solutions within seconds.

Problem

• rational univariate representation
• zero-dimensional polynomial systems
• Gaussian elimination
• dense linear algebra

Innovation

• Rational Univariate Representation
• Gaussian Elimination
• Zero-dimensional Polynomial Systems
• Dense Linear Algebra
• Julia Package