Solving systems of polynomial equations with stmmetries using the SAGBI-Gröbner basis

Dans cette thèse, nous proposons une méthode efficace pour résoudre des systèmes polynômiaux dont les équations sont invariantes par l'action d'un groupe fini G. L'idée est calculer simultanément une base de Gröbner SAGBI(une génération des bases de Gröbner à des idéaux de sous algèbres de l'anneau des polynômes) et une base de Gröbner dans l'anneau des invariants symétriques. Plus précisément, nous proposons dans cette thèse deux algorithmes: nous explicitions d'abord un algorithme à la F5 pour calculer efficacement une base de Gröbner SAGBI. Le deuxième algorithme est une version légèrement modifiée de l'algorithme FGLM qui permet de convertir une base de Gröbner SAGBI tronquée d'un idéal de dimension zéro en une base de Gröbner tronquée dans l'anneau des invariants symétriques. Enfin, nous montrons comment ces algorithmes peuvent être combinés pour trouver les racines complexes d'un tel système algébrique.PARIS-BIUSJ-Physique recherche (751052113) / SudocSudocFranceF

A new algorithm for computing regular representations for radicals of parametric differential ideals

The regular representation of the radical of a differential ideal has various applications such as solving the membership problem, computing Taylor expansion of solutions, finding the Lie symmetries, and solving dynamical systems. Presently, there is no algorithm giving all regular representations for all possible values of the parameters for a polynomial differential ideal with parametric coefficients. In this article, we propose a new algorithm that computes all different regular representations with respect to all possible states of the parameters. Also, we present an efficient criterion to reduce some ineffectual computations. Implementing the algorithm in Maple and several examples reported in this article demonstrate the high efficiency of the algorithm

Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases

In this paper, we propose an efficient method to solve polynomial systems whose equations are left invariant by the action of a finite group G. The idea is to simultaneously compute a truncated SAGBI-Gröbner bases (a generalisation of Gröbner bases to ideals of subalgebras of polynomial ring) and a Gröbner basis in the invariant ring K[σ1,..., σn] where σi is the i-th elementary symmetric polynomial. To this end, we provide two algorithms: first, from the F5 algorithm we can derive an efficient and easy to implement algorithm for computing truncated SAGBI–Gröbner bases of the ideals in invariant rings. A first implementation of this algorithm in C enable us to estimate the practical efficiency: for instance, it takes only 92s to compute a SAGBI basis of Cyclic 9 modulo a small prime. The second algorithm is inspired by the FGLM algorithm: from a truncated SAGBI–Gröbner basis of a zero-dimensional ideal we can compute efficiently a Gröbner basis in some invariant rings K[h1,..., hn]. Finally, we will show how this two algorithms can be combined to find the complex roots of such invariant polynomial systems

Efficient Calculation of all Steady States in large-scale overlapping generations models

In this paper, we address the problem of analyzing and computing all steady states of an overlapping generation (OLG) model with production and many generations. The characterization of steady states coincides with a geometrical representation of the algebraic variety of a polynomial ideal, and, in principle, one can apply computational algebraic geometry methods to solve the problem. However, it is infeasible for standard methods to solve problems with a large number of variables and parameters. Instead, we use the specific structure of the economic problem to develop a new algorithm that does not employ the usual steps for the computation of Grobner basis such as the computation of successive S-polynomial and expensive division

F4-invariant Algorithm for Computing SAGBI-Gröbner Bases

International audienceThis paper introduces a new algorithm for computing SAGBI-Gr\"obner bases for ideals of invariant rings of permutation groups. This algorithm is based on \F_4 algorithm. A first implementation of this algorithm has been made in SAGE and MAPLE computer algebra systems and have been successfully tried on a number of examples