A-Hypergeometric ranks for toric threefolds

Abstract

We study the holonomic rank for A-hypergeometric systems. Especially, we are interested in describing the holonomic rank explicitly. We define homological objects that generalize A-hypergeometric systems: the A-hypergeometric D-complexes. These complexes have proved to be a powerful tool to investigate holonomic ranks, and may also be useful to study other D-invariants. We provide explicit rank formulas, including a complete computation in the case that the rank of the matrix A is three.