Idéal de Bernstein d'un arrangement central générique d'hyperplans

Abstract

Let $V$ a vector space of dimension $n$. A $V$ family $\{H_1,\dots ,H_p\}$ of vectorial hyperplanes being distinct two by two defines an arrangement ${\mathcal{A}}_p=\mathcal{A}(H_1,\dots ,H_p)$ of $V$. For $i\in \{1,\dots ,p\}$, 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 $A_V(\mathbf{C})$, 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 $b\in \mathbf{C}[s_1,\dots ,s_p]$ such that : $$b(s_1,\dots ,s_p)l_1^{s_1}\dots l_p^{s_p}\in A_V(\mathbf{C})[s_1,\dots ,s_p]l_1^{s_1+1}\dots l_p^{s_p+1}$$ 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.

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