Arbitrary rank jumps for $A$-hypergeometric systems through Laurent polynomials

Abstract

We investigate the solution space of hypergeometric systems of differential equations in the sense of Gelfand, Graev, Kapranov and Zelevinsky. For any integer $d\ge 2$ we construct a matrix $A_d\in {\mathbb{N}}^{d\times 2d}$ and a parameter vector ${\beta }_d$ such that the holonomic rank of the $A$-hypergeometric system $H_{A_d}({\beta }_d)$ exceeds the simplicial volume $\vol(A_d)$ by at least $d-1$. The largest previously known gap between rank and volume was two. Our argument is elementary in that it uses only linear algebra, and our construction gives evidence to the general observation that rank-jumps seem to go hand in hand with the existence of multiple Laurent (or Puiseux) polynomial solutions.