Reading Rational Univariate Representations on lexicographic Groebner bases

TL;DR

This paper presents a new method to compute Rational Univariate Representations (RUR) from lexicographic Groebner bases for zero-dimensional polynomial systems, enabling certification and calculation without ideal shape restrictions.

Contribution

It introduces a formula to derive RUR directly from Groebner bases, even for non-shape position ideals, simplifying the process and extending applicability.

Findings

Method is competitive with state-of-the-art implementations.

Allows certification of solutions without ideal shape restrictions.

Uses straightforward Gaussian reductions and Groebner bases.

Abstract

In this contribution, we consider a zero-dimensional polynomial system in $n$ variables defined over a field $\mathbb{K}$. In the context of computing a Rational Univariate Representation (RUR) of its solutions, we address the problem of certifying a separating linear form and, once certified, calculating the RUR that comes from it, without any condition on the ideal else than being zero-dimensional. Our key result is that the RUR can be read (closed formula) from lexicographic Groebner bases of bivariate elimination ideals, even in the case where the original ideal that is not in shape position, so that one can use the same core as the well known FGLM method to propose a simple algorithm. Our first experiments, either with a very short code (300 lines) written in Maple or with a Julia code using straightforward implementations performing only classical Gaussian reductions in addition to Groebner bases for the degree reverse lexicographic ordering, show that this new method is already competitive with sophisticated state of the art implementations which do not certify the parameterizations.