Arbitrary rank jumps for A-hypergeometric systems through Laurent polynomials

We investigate the solution space of hypergeometric systemsof differential equations in the sense of Gel’fand, Graev,Kapranov and Zelevinski. For any integer d 2, we constructa matrix A_(d) ^{d x 2d} and a parameter vector ß_(d)such that the holonomic rank of the A-hypergeometric systemH_A(d)(ß_(d)) exceeds the simplicial volume vol(A_(d))by at least d – 1. The largest previously known gap betweenrank and volume was 2. Our construction gives evidence to the general observation thatrank jumps seem to go hand in hand with the existence of multipleLaurent (or Puiseux) polynomial solutions.