Polynome de Bernstein-Sato generique local

Abstract

Given a family of analytic functions near 0 \in C^n parametrized by a smooth space, we study the Bernstein polynomial of the fiber on an irreducible variety V of the space of parameters and we show that it is generically constant. We prove that this polynomial b(s) satisfies a functional equation on V from which we derive a contructible stratification of the space of parameters by the Bernstein polynomial of the fiber. When the hypersurface admits generically a unique singularity at 0 \in C^n, we prove that b(s) is the generic Bernstein polynomial in the sense of Brian\c{c}on-Geandier-Maisonobe. The tools for the proofs are a formal generalization of an algorithm by Oaku and the generic standard bases studied by the author.