The rank of a hypergeometric system

Understand

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].