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Reading Rational Univariate Representations on lexicographic Groebner bases

Alexander Demin, Fabrice Rouillier, Joao Ruiz

TL;DR

This work presents a general Las Vegas framework for computing Rational Univariate Representations of zero-dimensional polynomial systems by certifying a separating linear form and reading RURs directly from lexicographic Gröbner bases of bivariate elimination ideals. It develops a bivariate-read approach that recovers coordinate parametrizations from a lex order basis, and then assembles them into a full RUR for V(I), with complexity bounds that improve upon prior Las Vegas methods in the general case. The method remains effective even when the original ideal is not in shape position, and experiments with Maple and Julia demonstrate competitive performance against state-of-the-art Monte Carlo implementations while providing separation-certification. The paper also analyzes the influence of the separating element on computation time, sparsity, and real root isolation, highlighting practical strategies for selecting separating forms to optimize performance and output size.

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.

Reading Rational Univariate Representations on lexicographic Groebner bases

TL;DR

This work presents a general Las Vegas framework for computing Rational Univariate Representations of zero-dimensional polynomial systems by certifying a separating linear form and reading RURs directly from lexicographic Gröbner bases of bivariate elimination ideals. It develops a bivariate-read approach that recovers coordinate parametrizations from a lex order basis, and then assembles them into a full RUR for V(I), with complexity bounds that improve upon prior Las Vegas methods in the general case. The method remains effective even when the original ideal is not in shape position, and experiments with Maple and Julia demonstrate competitive performance against state-of-the-art Monte Carlo implementations while providing separation-certification. The paper also analyzes the influence of the separating element on computation time, sparsity, and real root isolation, highlighting practical strategies for selecting separating forms to optimize performance and output size.

Abstract

In this contribution, we consider a zero-dimensional polynomial system in variables defined over a field . 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.
Paper Structure (19 sections, 8 theorems, 18 equations, 3 tables, 6 algorithms)

This paper contains 19 sections, 8 theorems, 18 equations, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. State of the art
  3. Our contributions
  4. Lexicographic Gröbner bases and RUR candidates
  5. Reading parameterizations candidates on lexicographic Gröbner basis
  6. The bivariate case
  7. Proof:
  8. The computation of the required bivariate lexicographic Gröbner bases
  9. The computation of RUR-Candidates
  10. Complexity analysis
  11. Proof:
  12. RUR-Candidates from a unique Lexicographic Gröbner basis
  13. Separating elements and RUR
  14. Proof:
  15. Influences of separating elements
  16. ...and 4 more sections

Key Result

Corollary 1

Let $\tilde{f}(T),h_{i,1}(T)X_i+h_{i,0}(T),i=1\ldots n$ be a parameterization of $V(I)$ associated to some separating linear form $t$ with $\tilde{f}$ monic. Set $\overline{f}(T)$ the squarefree part of $\tilde{f}(T)$ and $f_{X_i}=\frac{-1}{\deg(\overline{f})}h_{j,0}(T)h_{j,1}(T)^{-1}\overline{f}'\m

Theorems & Definitions (19)

  • Definition 1: Rational Univariate Representation
  • Definition 2
  • Definition 3
  • Remark 1
  • Corollary 1
  • Remark 2
  • Proposition 1: Last corollary in Gia89
  • Proposition 2
  • Remark 3
  • Corollary 2
  • ...and 9 more