The rank of a hypergeometric system

TL;DR

This paper develops a combinatorial formula for the rank jump of A-hypergeometric systems, refining the understanding of their parameter space stratification and extending results to generalized systems.

Contribution

It introduces the ranking arrangement and exceptional arrangement to compute rank jumps and refines the stratification of the parameter space for A-hypergeometric systems.

Findings

Derived a combinatorial formula for rank jumps.

Refined the stratification of the exceptional arrangement.

Extended results to generalized A-hypergeometric systems.

Abstract

The holonomic rank of the A-hypergeometric system M_A(\beta) is the degree of the toric ideal I_A for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of nongeneric parameters to construct a combinatorial formula for the rank jump of M_A(\beta). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of M_A(\beta) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the parameter space. These results hold for generalized A-hypergeometric systems as well, where the semigroup ring of A is replaced by a nontrivial weakly toric module M contained in \CC[\ZZ A]. We also provide a direct proof of the result of M. Saito and W. Traves regarding the isomorphism classes of M_A(\beta).