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The first part corresponds to the symbolic-numeric study

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Academic year: 2022

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PhD Thesis

Title :

“Des m´ethodes symboliques-num´eriques et exactes pour la factorisation absolue des polynˆomes en deux variables”

Jury :

M. Manuel BRONSTEIN INRIA - Sophia Antipolis Examinateur M. Andr´e GALLIGO Universit´e de Nice Sophia Antopolis Directeur M. Marc GIUSTI CNRS - ´Ecole Polytechnique Rapporteur M. Gr´egoire LECERF CNRS - Universit´e de Versailles Examinateur M. Maurice MIGNOTTE Universit´e de Strasbourg Rapporteur

M. Bruno SALVY INRIA - Rocquencourt Rapporteur

This thesis is about absolute factorization algorithms. The first chapter is a survey on this topic, and then we describe our contributions. They are divided in two parts.

The first part corresponds to the symbolic-numeric study. We give a method which allows us to get an exact absolute factorization from an approximate one. Next, this method is used to get an absolute polynomial factorization algorithm. This algorithm uses ideas developped by A. Galligo, D. Rupprecht, and M. van Hoeij. Thanks to the LLL algorithm, it allows us to get the absolute factorization for polynomials with big degree (bigger than 100). It was impossible before.

In the second part, we give two algorithms. The first one adapt a “lifting and recombination”

scheme to the Gao’s algorithm. We get then a new algorithm with a better complexity than the Gao’s one. The second one is a Las Vegas algorithm. It is an absolute irreducibility test for polynomials with integer coefficients. This algorithm use modular computations, and the shape of the Newton’s polytope.

Keywords : polynomials with several variables, absolute factorization, symbolic-numeric algo- rithm, symbolic algorithm, LLL, Newton’s polytope.

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