Irregular hypergeometric D-modules

TL;DR

This paper investigates the irregularity of hypergeometric D-modules by constructing Gevrey series solutions, extending existing slope characterizations, and providing bounds and exact values for solution spaces.

Contribution

It extends the combinatorial characterization of slopes of hypergeometric D-modules to all matrices A without assumptions and provides bounds and exact dimensions for Gevrey solution spaces.

Findings

Characterization of slopes along coordinate hyperplanes holds generally.

Lower bounds for dimensions of Gevrey solution spaces are established.

Equality of dimensions is proven for very generic parameters.

Abstract

We study the irregularity of hypergeometric D-modules $\mathcal{M}_A (\beta )$ via the explicit construction of Gevrey series solutions along coordinate subspaces in $X =\mathbb{C}^n$. As a consequence, we prove that along coordinate hyperplanes the combinatorial characterization of the slopes of $\mathcal{M}_A (\beta)$ given by M. Schulze and U. Walther in [21] still holds without any assumption on the matrix A. We also provide a lower bound for the dimensions of the spaces of Gevrey solutions along coordinate subspaces in terms of volumes of polytopes and prove the equality for very generic parameters. Holomorphic solutions outside the singular locus of $\mathcal{M}_A (\beta)$ can be understood as Gevrey solutions of order one along X at generic points and so they are included as a particular case.