Exponential growth of rank jumps for A-hypergeometric systems

Abstract

The dimension of the space of holomorphic solutions at nonsingular points (also called the holonomic rank) of a A–hypergeometric system MA(β) is known to be bounded above by 22d vol(A) [SST00], where d is the rank of the matrix A and vol(A) is its normalized volume. This bound was thought to be very vast because it is exponential on d. Indeed, all the examples we have found in the literature verify that rank(MA(β)) < 2vol(A). We construct here, in a very elementary way, some families of matrices A(d) ∈ Z d×n and parameter vectors β(d) ∈ C d , d ≥ 2, such that rank(MA(d) (β(d) )) ≥ a dvol(A(d) ) for certain a > 1.