L'id\'eal de Bernstein d'un arrangement libre d'hyperplans lin\'eaires

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