Exceptional parameters for generic A-hypergeometric systems

Abstract

The holonomic rank of an A-hypergeometric system $H_A(β)$ is conjectured to be independent of the parameter vector $β$ if and only if the toric ideal $I_A$ is Cohen Macaulay. We prove this conjecture in the case that $I_A$ is generic by explicitly constructing more than $\vol(A)$ many linearly independent hypergeometric functions for parameters $β$ coming from embedded primes of certain initial ideals of $I_A$.