Bounding the number of remarkable values via Jouanolou's theorem

International audienceIn this article we bound the number of remarkable values of a polynomial vector field. The proof is short and based on Jouanolou's theorem about rational first integrals of planar polynomial derivations. Our bound is given in term of the size of a Newton polygon associated to the vector field. We prove that this bound is almost reached

A recombination algorithm for the decomposition of multivariate rational functions

International audienceIn this paper we show how we can compute in a deterministic way the decomposition of a multivariate rational function with a recombination strategy. The key point of our recombination strategy is the used of Darboux polynomials. We study the complexity of this strategy and we show that this method improves the previous ones. In appendix, we explain how the strategy proposed recently by J. Berthomieu and G. Lecerf for the sparse factorization can be used in the decomposition setting. Then we deduce a decomposition algorithm in the sparse bivariate case and we give its complexit

Noether's forms for the study of non-composite rational functions and their spectrum

International audienceIn this paper, the spectrum and the decomposability of a multivariate rational function are studied by means of the effective Noether's irreducibility theorem given by Ruppert. With this approach, some new effective results are obtained. In particular, we show that the reduction modulo p of the spectrum of a given integer multivariate rational function r coincides with the spectrum of the reduction of r modulo p for p a prime integer greater or equal to an explicit bound. This bound is given in terms of the degree, the height and the number of variables of r. With the same strategy, we also study the decomposability of r modulo p. Some similar explicit results are also provided for the case of polynomials with coefficients in a polynomial ring

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

Efficient Algorithms for Computing Rational First Integrals and Darboux Polynomials of Planar Polynomial Vector Fields

International audienceWe present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach builds upon a method proposed by Ferragut and Giacomini, whose main ingredients are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating this power series. We provide explicit bounds on the number of terms needed in the power series. This enables us to transform their method into a certified algorithm computing rational first integrals via systems of linear equations. We then significantly improve upon this first algorithm by building a probabilistic algorithm with arithmetic complexity $\~O(N^{2 \omega})$ and a deterministic algorithm solving the problem in at most $\~O(d^2N^{2 \omega+1})$ arithmetic operations, where~$N$ denotes the given bound for the degree of the rational first integral, and where $d \leq N$ is the degree of the vector field, and $\omega$ the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in $\~O(N^{\omega+2})$ arithmetic operations. By comparison, the best previous algorithm uses at least $d^{\omega+1}\, N^{4\omega +4}$ arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package RationalFirstIntegrals which is available to interested readers with examples showing its efficiency

Computing Nearest Gcd with Certification

International audienceA bisection method, based on exclusion and inclusion tests, is used to address the nearest univariate gcd problem formulated as a bivariate real minimization problem of a rational fraction. The paper presents an algorithm, a first implementation and a complexity analysis relying on Smale's $\alpha$-theory. We report its behavior on an illustrative example

Computing Nearest Gcd with Certification

International audienceA bisection method, based on exclusion and inclusion tests, is used to address the nearest univariate gcd problem formulated as a bivariate real minimization problem of a rational fraction. The paper presents an algorithm, a first implementation and a complexity analysis relying on Smale's $\alpha$-theory. We report its behavior on an illustrative example