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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 $H_i$ askernel. We denote by $A_V ({\bf C}) $, the Weyl algebra of algebraicdifferential operators on $V$. Following J. Bernstein, the ideal constituted bypolynomials $ 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. Thisideal does not depend on the choice of linear forms $ l_i $. The goal of thisarticle is to determine this ideal when $ {\cal A} $ is a free arrangementconstituted by linear hyperplans within the meaning of K. Saito.

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