Solving bivariate systems using Rational Univariate Representations - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Article Dans Une Revue Journal of Complexity Année : 2016

Solving bivariate systems using Rational Univariate Representations

Résumé

Given two coprime polynomials $P$ and $Q$ in $\Z[x,y]$ of degree bounded by $d$ and bitsize bounded by $\tau$, we address the problem of solving the system $\{P,Q\}$. We are interested in certified numerical approximations or, more precisely, isolating boxes of the solutions. We are also interested in computing, as intermediate symbolic objects, rational parameterizations of the solutions, and in particular Rational Univariate Representations (RURs), which can easily turn many queries on the system into queries on univariate polynomials. Such representations require the computation of a separating form for the system, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system. We present new algorithms for computing linear separating forms, RUR decompositions and isolating boxes of the solutions. We show that these three algorithms have worst-case bit complexity $\widetilde{O}_B(d^6+d^5\tau)$, where $\widetilde{O}$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity. We also present probabilistic Las Vegas variants of our two first algorithms, which have expected bit complexity $\widetilde{O}_B(d^5+d^4\tau)$. A key ingredient of our proofs of complexity is an amortized analysis of the triangular decomposition algorithm via subresultants, which is of independent interest.
Fichier principal
Vignette du fichier
JoC.pdf (615.83 Ko) Télécharger le fichier
Vignette du fichier
2016 Solving bivariate systems.png (50.19 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Format : Figure, Image
Loading...

Dates et versions

hal-01342211 , version 1 (05-07-2016)
hal-01342211 , version 2 (06-07-2016)

Identifiants

Citer

Yacine Bouzidi, Sylvain Lazard, Guillaume Moroz, Marc Pouget, Fabrice Rouillier, et al.. Solving bivariate systems using Rational Univariate Representations. Journal of Complexity, 2016, 37, pp.34--75. ⟨10.1016/j.jco.2016.07.002⟩. ⟨hal-01342211v2⟩
392 Consultations
342 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More