Fast Rational Univariate Representation via Gaussian Elimination

In this note, we present RationalUnivariateRepresentation$\mathrm{.}$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.