Arbitrary rank jumps forA-hypergeometric systems through Laurent polynomials

Abstract

We investigate the solution space of hypergeometric systems of differential equations in the sense of Gel?fand, Graev, Kapranov and Zelevinski. For any integer d 2, we construct a matrix A(d) d x 2d and a parameter vector s(d) such that the holonomic rank of the A-hypergeometric system HA(d)(s(d)) exceeds the simplicial volume vol(A(d)) by at least d ? 1. The largest previously known gap between rank and volume was 2. 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