Id\'eal de Bernstein d'un arrangement central g\'en\'erique d'hyperplans

Abstract

Let $ V $ a vector space of dimension $n$. A $V$ family $ \{H_1, \ldots, H_p \} $ of vectorial hyperplanes being distinct two by two defines an arrangement $ {\cal A}_p = {\cal A} ( H_1, \ldots ,H_p ) $ of $ V $. For $ i \in \{ 1, \ldots, p \} $, let $ l_i $ be a linear form on $V$ with $H_i$ as kernel. This arrangement is generic if the intersection of every sub-family of $n$ hyperplanes of the arranfement is reduced to zero. Let $A_V ({\bf C}) $, be the Weyl algebra of algebraic differential operators with coefficients in the symetric algebra denoted $S$ of the dual of $V$. Following J. Bernstein, the ideal constituted by polynomials $ b \in {\bf C} [s_1, \ldots, s_p] $ such that : $$ \; \; b (s_1, \ldots, s_p) \, l_1^{s_1} \ldots l_p^{s_p} \in A_V ({\bf C}) [s_1, \ldots, s_p] \, l_1^ {s_1 + 1} \ldots l_p^{s_p + 1} $$ is not reduced to zero. This ideal does not depend on the choice of linear forms $l_i $ which define the hypersurfaces $H_i$. The goal of this article is to precise this ideal.