The GKZ hypergeometric $\mathcal D$-module

TL;DR

This paper introduces a stable version of the GKZ hypergeometric $\\mathcal{D}$-module, proves its holonomicity, and describes its rank and geometric properties related to Newton polytopes.

Contribution

It defines the stable GKZ hypergeometric $\mathcal{D}$-module via cohomological functors and establishes its fundamental properties and relations to existing systems.

Findings

Proves the stable GKZ hypergeometric $\mathcal{D}$-module is holonomic.

Shows it forms an integrable connection of specific rank.

Connects the module's rank to the volume of the Newton polytope.

Abstract

For an $(n\times N)$-matrix $A$ of rank $n$ with integer entries, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, called the $A$-hypergeometric system. We define the stable GKZ hypergeometric $\mathcal D$-module using cohomological functors, which is closely related to the $A$-hypergeometric $\mathcal D$-module and the $\mathcal D$-module underlying the better behaved GKZ system introduced by Borisov and Horja. We prove the stable GKZ hypergeometric $\mathcal D$-module is holonomic and is an integrable connection of rank $n!\mathrm{vol}(\Delta_\infty)$ on the Zariski open subset parametrizing nondegenerate Laurent polynomials, where $\Delta_\infty$ is the Newton polytope at $\infty$.