Modular Las Vegas Algorithms for Polynomial Absolute Factorization

Let f(X,Y) \in \ZZ[X,Y] be an irreducible polynomial over \QQ. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of $f$, or more precisely, of $f$ modulo some prime integer $p$. The same idea of choosing a $p$ satisfying some prescribed properties together with $LLL$ is used to provide a new strategy for absolute factorization of $f(X,Y)$. We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to factorize some polynomials of degree up to 400

A lifting and recombination algorithm for rational factorization of sparse polynomials

We propose a new lifting and recombination scheme for rational bivariate polynomial factorization that takes advantage of the Newton polytope geometry. We obtain a deterministic algorithm that can be seen as a sparse version of an algorithm of Lecerf, with now a polynomial complexity in the volume of the Newton polytope. We adopt a geometrical point of view, the main tool being derived from some algebraic osculation criterions in toric varieties.Comment: 22 page

Efficient Computation of the Characteristic Polynomial

This article deals with the computation of the characteristic polynomial of dense matrices over small finite fields and over the integers. We first present two algorithms for the finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third algorithm could improve both complexity and computational time. We use these results as a basis for the computation of the characteristic polynomial of integer matrices. We first use early termination and Chinese remaindering for dense matrices. Then a probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices

Resolving zero-divisors using Hensel lifting

Algorithms which compute modulo triangular sets must respect the presence of zero-divisors. We present Hensel lifting as a tool for dealing with them. We give an application: a modular algorithm for computing GCDs of univariate polynomials with coefficients modulo a radical triangular set over the rationals. Our modular algorithm naturally generalizes previous work from algebraic number theory. We have implemented our algorithm using Maple's RECDEN package. We compare our implementation with the procedure RegularGcd in the RegularChains package.Comment: Shorter version to appear in Proceedings of SYNASC 201

Factoring bivariate polynomials using adjoints

One relates factorization of bivariate polynomials to singularities of projective plane curves. One proves that adjoint polynomials permit to solve the recombinations of the modular factors induced by the absolute and rational factorizations, and so without using Hensel's lifting. One establishes in such a way the relations between the algorithm of Duval-Ragot (locally constant functions) and of Ch\`eze-Lecerf (lifting and recombinations), and one shows that a fast computation of adjoint polynomials leads to a fast factorization. The proof is based on cohomological sequences and residue theory.Comment: 22 pages, 2 figures. Extended version of arXiv.1201.578