Contributions à la résolution des systèmes algébriques : réduction, localisation, traitement des singularités ; implantations

Jérémy Berthomieu 1
1 Équipe Max
LIX - Laboratoire d'informatique de l'école polytechnique [Palaiseau]
Abstract : This PhD thesis deals with some particular aspects of the algebraic systems resolution. Firstly, we introduce a way of minimizing the number of additive variables appearing in an algebraic system. For this, we make use of two invariants of variety introduced by Hironaka: the ridge and the directrix. Then, we propose fast arithmetic routines, the so-called relaxed routines, for p-adic integers. These routines allow us, then, to solve efficiently an algebraic system with rational coefficients locally, i.e. over the p-adic integers. In a fourth part, we are interested in the factorization of a bivariate polynomial, which is at the root of the decomposition of hypersurfaces into irreducible components. We propose an algorithm reducing the factorization of the input polynomial to that of a polynomial whose dense size is essentially equivalent to the convex-dense size of the input polynomial. In the last part, we consider real algebraic systems solving in average. We design a probabilistic algorithm computing an approximate complex zero of the real algebraic system given as input.
Document type :
Theses
Symbolic Computation [cs.SC]. Ecole Polytechnique X, 2011. French


https://tel.archives-ouvertes.fr/tel-00666435
Contributor : Jérémy Berthomieu <>
Submitted on : Monday, February 20, 2012 - 10:49:40 AM
Last modification on : Wednesday, October 29, 2014 - 1:23:46 PM

File

Identifiers

  • HAL Id : tel-00666435, version 3

Collections

Citation

Jérémy Berthomieu. Contributions à la résolution des systèmes algébriques : réduction, localisation, traitement des singularités ; implantations. Symbolic Computation [cs.SC]. Ecole Polytechnique X, 2011. French. <tel-00666435v3>

Export

Share

Metrics

Consultation de
la notice

149

Téléchargement du document

175