Exponential growth of rank jumps for A-hypergeometric systems

TL;DR

This paper demonstrates that the holonomic rank of A-hypergeometric systems can grow exponentially with the dimension, challenging previous assumptions of a more modest bound related to the volume of A.

Contribution

The authors construct explicit examples showing exponential growth of the rank in A-hypergeometric systems, revealing that the upper bound can be significantly exceeded.

Findings

Rank can grow exponentially with dimension d

Constructed families of matrices exhibit rank exceeding 2^d times volume

Challenges previous beliefs about the bound on holonomic rank

Abstract

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