  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.
