Lionel Ducos

Optimizations of the subresultant algorithm

Abstract The subresultant algorithm is the most universal and used tool to compute the resultant or the greatest common divisor of two polynomials with coefficients in an integral ring (see (A.G. Akritas, Elements of Computer Algebra with Applications, Wiley, New York, 1989, H. Cohen, A Course in Computational Algebraic Number Theory, Ch. 3, Springer, Berlin, 1993, S.R. Czapor, K.O. Geddes, G. Labahn, Algorithms for Computer Algebra, Kluwer Academic Publishers, Dordrecht, 1992.)). Nevertheless, there exists several notable ameliorations of this algorithm (see (L. Ducos, Algorithme de Bareiss, Algorithme des sous-resultants, Theoret. Inform. Appl. 30 (4) (1996) 319–347, T. Lickteig, M.-F. Roy, Cauchy index computation. Manuscrit non publie (a paraitre), Novembre 1996.)). I propose in this article two improvements in the parts of the subresultant algorithm where the calculations are most costly. The computing-time decreases in a spectacular way.

Constructive Krull Dimension. I: Integral Extensions.

We give a constructive approach to the well known classical theorem saying that an integral extension doesn’t change the Krull dimension.

Construction de corps de décomposition grace aux facteurs de résolvantes

Traditional methods of research for the Galois group of a polynomial fwith coefficients in a field Kare strongly related to factorization of absolute or relative resolvents. In this paper, thanks to resolvents and the Galois theory for rings, we describe an effective algorithm for building a splitting field Eof fover K, i.e. to find a finite system of generators for the kernel of a surjective morphism .

Computing the V-saturation of finitely-generated submodules of V X m where V is a valuation domain

Recently, Lombardi, Quitte and Yengui have given a Grobner-free algorithm which computes the V-saturation of any finitely generated submodule of V X n , where V is a valuation domain. The goal of this paper is to clarify this algorithm, to give precise complexity bounds, and a complete submodule membership test for the saturation. As application, we give precise degree bounds on syzygies over V X .