The Poincar\'e series of the algebra of rational functions which are regular outside hyperplanes

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 $R(\Delta)$ of rational functions on V which are regular outside $\bigcup_{\alpha\in\Delta} \ker\alpha$. Then the ring $R(\Delta)$ is naturally doubly filtered by the degrees of denominators and of numerators. In this paper we give an explicit combinatorial formula for the Poincar\'e series in two variables of the associated bigraded vector space $\bar{R}(\Delta)$. This generalizes the main theorem of Terao, H.: Algebras generated by reciprocals of linear forms, to appear in J.Algebra (arXiv:math.CO/0105095).