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

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

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