The Poincaré series of the algebra of rational functions which are regular outside hyperplanes

Details The Poincaré series of the algebra of rational functions which are regular outside hyperplanes The Poincaré series of the algebra of rational functions which are regular outside hyperplanes

2002-02-28

arxiv.org/abs/math/0202296v2

Journal of Algebra 266 (2003), 169-179

Mathematics

Combinatorics

11 pages

Scientific paper

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).

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