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 several variables. It generalizes that of computing a solution to a system of linear modular equations over a polynomial ring, including in particular the computation of Hermite-Pade approximants and bivariate interpolants. Rather than a single solution, we aim at computing generators of the solution set which have good properties. Precisely, the input of our problem consists of a finite-dimensional module given by the action of the variables on its elements, and of some elements of this module; the goal is to compute a Grobner basis of the module of syzygies between these elements. In terms of linear algebra, the input describes a matrix with a type of Krylov structure, and the goal 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 multiplication matrices which specify the action of the variables. In the case of a Jordan matrix, we accelerate the computation of multivariate interpolants under degree constraints; our result for a Frobenius matrix leads to a faster algorithm for computing normal forms of univariate polynomial matrices. In the case of several dense matrices, we accelerate the change of monomial order for Grobner bases of multivariate zero-dimensional ideals.