On Irregular Binomial $D$-modules

TL;DR

This paper characterizes the regularity of holonomic binomial $D$-modules using associated primes and describes their slopes and irregularity, linking them to hypergeometric $D$-modules and Gevrey solutions.

Contribution

It provides a criterion for regularity of binomial $D$-modules and relates their slopes and irregularity to hypergeometric modules, advancing understanding of their structure.

Findings

Regularity characterized by associated primes homogeneity.

Slopes along coordinate subspaces described via hypergeometric modules.

Dimension of irregularity stalk computed for generic parameters.

Abstract

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