Exponential growth of rank jumps for $A$-hypergeometric systems

Abstract

AbstractThe dimension of the space of holomorphic solutions at nonsingularpoints (also called the holonomic rank) of a A–hypergeometric systemM A (β) is known to be bounded above by 2 2d vol(A) [SST00], where d is therank of the matrix A and vol(A) is its normalized volume. This bound wasthought to be very vast because it is exponential on d. Indeed, all the ex-amples we have found in the literature verify that rank(M A (β)) 1. 1 Introduction Let A = (a ij ) = (a 1 a 2 ···a n ) be a full rank matrix with columns a j ∈ Z d andd ≤ n. Following Gel’fand, Graev, Kapranov and Zelevinsky (see [GGZ87] and[GZK89]) we can define the A–hypergeometric system with parameter β ∈ C d asthe left ideal H A (β) of the Weyl algebra D = C[x 1 ,...,x n ]h∂ 1 ,...,∂ n i generatedby the following set of differential operators: