Algebras Generated by Reciprocals of Linear Forms

Let Δ be a finite set of nonzero linear forms in several variables with coefficients in a field K of characteristic zero. Consider the K-algebra C(Δ) of rational functions generated by {1/α α Δ}. Then the ring ∂(V) of differential operators with constant coefficients naturally acts on C(Δ). We study the graded ∂(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 [4] and results by Orlik–Terao [9].