On Irregular Binomial $D$-modules

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 $β\in \CC^d$ 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.

17 pages, Theorems 4.8 and 5.1, Examples 4.5 and 4.6, some remarks and some references added

References in corpus (3)

• Irregularity of hypergeometric systems via slopes along coordinate subspaces
• Equivariant D-modules
• Gevrey solutions of irregular hypergeometric systems in two variables