  Let $\Delta$ be a finite set of nonzero linear forms in several variables
with coefficients in a field $\mathbf K$ of characteristic zero. Consider the
$\mathbf K$-algebra $C(\Delta)$ of rational functions generated by $\{1/\alpha
\mid \alpha \in \Delta \}$. Then the ring $\partial(V)$ of differential
operators with constant coefficients naturally acts on $C(\Delta)$. We study
the graded $\partial(V)$-module structure of $C(\Delta)$. We especially find
standard systems of minimal generators and a combinatorial formula for the
Poincar\'e series of $C(\Delta)$. Our proofs are based on a theorem by
Brion-Vergne [brv1] and results by Orlik-Terao [ort2}.
