Exponential growth of rank jumps for $A$-hypergeometric systems

Abstract

The dimension of the space of holomorphic solutions at nonsingular points (also called the holonomic rank) of an $A$-hypergeometric system $M_A(\beta )$ is known to be bounded above by $2^{2d}\mathrm{vol}(A)$, where $d$ is the rank of the matrix $A$ and $\mathrm{vol}(A)$ is its normalized volume. This bound was thought to be much too large because it is exponential in $d$. Indeed, all the examples we have found in the literature satisfy $\mathrm{rank}(M_A(\beta ))<2\mathrm{vol}(A)$. We construct here, in a very elementary way, some families of matrices $A_{(d)}\in {\mathbb{Z}}^{d\times n}$ and parameter vectors ${\beta }_{(d)}\in {\mathbb{C}}^d$, $d\ge 2$, such that $\mathrm{rank}(M_{A_{(d)}}({\beta }_{(d)}))\ge a^d\mathrm{vol}(A_{(d)})$ for some $a>1$.