On the rank of an A-hypergeometric D-module versus the normalized volume of A

Abstract

The rank of an A-hypergeometric D-module MA(β), associated with a full rank (d×n)-matrix A and a vector of parameters β∈Cd, is known to be the normalized volume of A, denoted vol(A), when β lies outside the exceptional arrangement E(A), an affine subspace arrangement of codimension at least two. If β∈E(A) is simple, we prove that d−1 is a tight upper bound for the ratio rank(MA(β))/vol(A) for any d≥3. We also prove that the set of parameters β such that this ratio is at least 2 is an affine subspace arrangement of codimension at least 3.