On irregular binomial D-modules

Abstract

We prove that a holonomic binomial D-module M A (I, β) is regular if and only if certain associated primes of I determined by the parameter vector $${\beta\in \mathbb{C}^d}$$ are homogeneous. We further describe the slopes of M A (I, β) along a coordinate subspace in terms of the known slopes of some related hypergeometric D-modules that also depend on β. When the parameter β is generic, we also compute the dimension of the generic stalk of the irregularity of M A (I, β) along a coordinate hyperplane and provide some remarks about the construction of its Gevrey solutions.