Algebras generated by reciprocals of linear forms

Details Algebras generated by reciprocals of linear forms Algebras generated by reciprocals of linear forms

2001-05-11

arxiv.org/abs/math/0105095v3

Journal of Algebra, vol. 250, 549-558 (2002)

Mathematics

Combinatorics

a typo corrected; a footnote added

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 $C(\Delta)$ 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}.

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