Bases of relations in one or several variables: fast algorithms and applications. (Bases de relations en une ou plusieurs variables : algorithmes rapides et applications)

In this thesis, we study algorithms for a problem of finding relations in one or severalvariables. It generalizes that of computing a solution to a system of linear modularequations over a polynomial ring, including in particular the computation of Hermite-Padé approximants and bivariate interpolants. Rather than a single solution, we aim atcomputing generators of the solution set which have good properties.Precisely, the input of our problem consists of a finite-dimensional module given bythe action of the variables on its elements, and of some elements of this module; the goalis to compute a Gröbner basis of the module of syzygies between these elements. In termsof linear algebra, the input describes a matrix with a type of Krylov structure, and thegoal is to compute a compact representation of a basis of the nullspace of this matrix.We propose several algorithms in accordance with the structure of the multiplicationmatrices which specify the action of the variables. In the case of a Jordan matrix, weaccelerate the computation of multivariate interpolants under degree constraints; ourresult for a Frobenius matrix leads to a faster algorithm for computing normal forms ofunivariate polynomial matrices. In the case of several dense matrices, we accelerate thechange of monomial order for Gröbner bases of multivariate zero-dimensional ideals.