# On the rank of an $A$-hypergeometric $D$-module versus the normalized volume of $A$

## TL;DR

This paper investigates the relationship between the rank of $A$-hypergeometric $D$-modules and the normalized volume of $A$, establishing bounds and structural properties of parameters where the rank exceeds the volume.

## Contribution

It proves that for simple parameters in the exceptional set, the rank-to-volume ratio is at most $d-1$, and characterizes parameters where this ratio is at least 2.

## Key findings

- The rank equals the normalized volume outside the exceptional set.
- For simple parameters, the ratio is bounded above by $d-1$.
- Parameters with ratio at least 2 form an affine subspace arrangement of codimension at least 3.

## Abstract

The rank of an $A$-hypergeometric $D$-module $M_A(\beta)$, associated with a full rank $(d\times n)$-matrix $A$ and a vector of parameters $\beta\in \mathbb{C}^d$, is known to be the normalized volume of $A$, denoted $\mathrm{vol}(A)$, when $\beta$ lies outside the exceptional arrangement $\mathcal{E}(A)$, an affine subspace arrangement of codimension at least two. If $\beta\in \mathcal{E}(A)$ is simple, we prove that $d-1$ is a tight upper bound for the ratio $\mathrm{rank}(M_A(\beta))/\mathrm{vol}(A)$ for any $d\geq 3$. We also prove that the set of parameters $\beta$ such that this ratio is at least $2$ is an affine subspace arrangement of codimension at least $3$.