Revista Matemática Iberoamericana

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Volume 29, Issue 4, 2013, pp. 1397–1404
DOI: 10.4171/RMI/761

Published online: 2013-12-15

Exponential growth of rank jumps for A-hypergeometric systems

María-Cruz Fernández-Fernández[1]

(1) Universidad de Sevilla, Spain

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}\operatorname{vol}(A)$, where $d$ is the rank of the matrix $A$ and $\operatorname{vol}(A)$ 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 $\operatorname{rank}(M_A (\beta ))<2 \operatorname{vol}(A)$. We construct here, in a very elementary way, some families of matrices $A_{(d)}\in \mathbb{Z}^{d \times n}$ and parameter vectors $\beta_{(d)} \in \mathbb{C}^d$, $d\geq 2$, such that $\operatorname{rank} ( M_{A_{(d)}} (\beta_{(d)} ))\geq a^d \operatorname{vol}(A_{(d)})$ for some $a>1$.

Keywords: Hypergeometric, $D$-module, holonomic rank

Fernández-Fernández María-Cruz: Exponential growth of rank jumps for A-hypergeometric systems. Rev. Mat. Iberoamericana 29 (2013), 1397-1404. doi: 10.4171/RMI/761