On the rank of an Ahypergeometric Dmodule versus the normalized volume of A
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 $d1$ 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 $3$.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1907.08669
 Bibcode:
 2019arXiv190708669B
 Keywords:

 Mathematics  Algebraic Geometry;
 13N10;
 32C38;
 33C70;
 14M25
 EPrint:
 8 pages