Algebras generated by reciprocals of linear forms

Abstract

Let $Δ$ 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 $C(Δ)$ of rational functions generated by $\{1/α\mid α\in Δ\}$. Then the ring $\partial(V)$ of differential operators with constant coefficients naturally acts on $C(Δ)$. We study the graded $\partial(V)$-module structure of $C(Δ)$. We especially find standard systems of minimal generators and a combinatorial formula for the Poincaré series of $C(Δ)$. Our proofs are based on a theorem by Brion-Vergne [brv1] and results by Orlik-Terao [ort2}.