Algebras generated by reciprocals of linear forms

Abstract

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 Poincaré series of $C(\Delta )$. Our proofs are based on a theorem by Brion-Vergne [brv1] and results by Orlik-Terao [ort2}.

a typo corrected; a footnote added