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fp be polynomials in n variables with coefficients in a field K. We associate with these polynomials a number of functional equations and related ideals B, B j and B Σ of K[s 1 ,. .. , sp] called Bernstein–Sato ideals. Using standard basis techniques, our aim is to present an algorithm for computing generators of B j and B Σ .
In 1987, C. Sabbah proved the existence of Bernstein-Sato polynomials associated with several analytic functions. The purpose of this article is to give a more elementary and constructive proof of the result of C. Sabbah based on the notion of the analytic Gröbner fan of a D-module.
Keywords: Bernstein–Sato ideal D-module Gröbner base Primary decomposition a b s t r a c t Let k be a field of characteristic 0. Given a polynomial mapping f = n to k p , the local Bernstein–Sato ideal of f at a point a ∈ k n is defined as an ideal of the ring of polynomi-als in s = (s 1 ,. .. , s p). We propose an algorithm for computing local… (More)
This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in the case of one function. We also make an extensive study of an example for which we give an expression of a generic… (More)
Given p polynomials with coefficients in a commutative unitary integral ring C containing Q, we define the notion of a generic Bernstein-Sato polynomial on an irreducible affine scheme V ⊂ Spec(C). We prove the existence of such a non zero rational polynomial which covers and generalizes previous existing results by H. Biosca. When C is the ring of an… (More)
This is the first part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. The main result of this part is a constructibility result for the analytic Gröbner fan of a parametric ideal in the ring of analytic differential operators. In this part, the main tool is the notion of generic reduced… (More)
The contribution of this paper lies in two aspects. The first one deals with a natural notion of generic Gröbner (or standard) bases on an irreducible affine scheme for an ideal depending on parameters. This takes place in rings of differential operators and concerns the algebraic and the formal case. Thus we obtain a generalization of some known results in… (More)
Parametric Gröbner bases have been studied for more than 15 years and are now a further developed subject. Here we propose a general study of parametric standard bases, that is with local orders. We mainly focus on the commutative case but we also treat the case of differential operators rings. We will be concerned by two aspects: a theoretical aspect with… (More)
This note summarizes results concerning local Gröbner fans. Any Gröbner fan of an ideal in a power series ring, or in an analytic or formal differential operator ring, is found to be a polyhedral fan. We also compare the notions of global and local Gröbner fans, and discuss applications of our results. Résumé ´ Eventail de Gröbner local. Cette note résume… (More)
Cette note résume des résultats portant sur l'´ eventail de Gröbner local. Nous montrons que l'´ eventail de Gröbner associéà un idéal d'un anneau de séries ou de l'anneau des opérateurs différentiels formels ou analytiques est unéventail polyédral. Nous comparonségalement les notions d'´ eventail de Gröbner global et local et discutons des applications de… (More)