Combinatorics of rank jumps in simplicial hypergeometric systems Academic Article

Let A A be an integer d n d imes n matrix, and assume that the convex hull conv ( A ) operatorname {conv}(A) of its columns is a simplex of dimension d 1 d-1 not containing the origin. It is known that the semigroup ring C [ N A ] mathbb {C}[mathbb {N} A] is CohenMacaulay if and only if the rank of the GKZ hypergeometric system H A ( ) H_A(\beta ) equals the normalized volume of conv ( A ) operatorname {conv}(A) for all complex parameters C d \beta in mathbb {C}^d (Saito, 2002). Our refinement here shows that H A ( ) H_A(\beta ) has rank strictly larger than the volume of conv ( A ) operatorname {conv}(A) if and only if \beta lies in the Zariski closure (in