Polynomial factorization and curve decomposition algorithms

Cristina Bertone 1, 2, 3
2 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : Affine algebraic curves are a tool applied in different fields, for instance CAGD. They are defined using polynomials, but they often have several different irreducible components. In this thesis we develop efficient algorithms to decompose a curve defined by rational polynomials. In the first part we present an absolute factorization algorithm for bivariate polynomials (this problem is equivalent to the decomposition of a curve in the plane). We start from the existing algorithm TKTD and we improve the definition of the algebraic extension needed for the factorization, using modular techniques and the LLL algorithm to identify an algebraic number form its p-adic approximation. In the second part we pass to the problem of decomposing a curve in the three-dimensional space: the corresponding technique of the factorization for the case of the plan is the primary decomposition of an ideal for the three-dimensional case. At first, we show some bounds on the degrees of the surfaces separating the different components, using some classical results of algebraic geometry, as the "Lifting problem" or the Castelnuovo-Mumford regularity. After this, we apply consider a classical algorithm of decomposition, which is not efficient for computations, and we apply on it the modular techniques. We obtain a modular algorithm giving the Hilbert function for the reduced components of the curve. The two main algorithms were tested on several examples and compared with the executions times of other softwares.
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Cristina Bertone. Polynomial factorization and curve decomposition algorithms. Mathematics [math]. Université Nice Sophia Antipolis; Università degli studi di Torino, 2010. English. ⟨tel-00560802⟩

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