L'idéal de Bernstein d'un arrangement libre d'hyperplans linéaires

Abstract

Let $V$ a vector space of dimension $n$. A family $\{H_1,\dots ,H_p\}$ of vectorial hyperplans $V$ defines an arrangement $\mathcal{A}$ of $V$. For $i\in \{1,\dots ,p\}$, let $l_i$ be a linear form on $V$ with $H_i$ as kernel. We denote by $A_V(\mathbf{C})$, the Weyl algebra of algebraic differential operators on $V$. Following J. Bernstein, the ideal constituted by polynomials $b\in \mathbf{C}[s_1,\dots ,s_p]$ such that : $$b(s_1,\dots ,s_p)l_1^{s_1}\dots l_p^{s_p}\in A_n(\mathbf{C})[s_1,\dots ,s_p]l_1^{s_1+1}\dots l_p^{s_p+1},$$ is not reduced to zero. This ideal does not depend on the choice of linear forms $l_i$. The goal of this article is to determine this ideal when $\mathcal{A}$ is a free arrangement constituted by linear hyperplans within the meaning of K. Saito.

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Cited by in corpus (2)

• Bernstein-Sato ideals and hyperplane arrangements
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