  We investigate the solution space of hypergeometric systems of differential
equations in the sense of Gelfand, Graev, Kapranov and Zelevinsky. For any
integer $d \geq 2$ we construct a matrix $A_d \in \N^{d \times 2d}$ 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.
