Exponential growth of rank jumps for A-hypergeometric systems

Rank Jumps in Codimension 2A-hypergeometric Systems

The holonomic rank of the A-hypergeometric system HA(β) is shown to depend on the parameter vector β when the underlying toric ideal IA is a non Cohen Macaulay codimension 2 toric ideal. The set of exceptional parameters is usually infinite.

Combinatorics of Rank Jumps in Simplicial Hypergeometric Systems

Let A be an integer d × n matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d− 1 not containing the origin. It is known that the semigroup ring C[NA] is Cohen–Macaulay if and only if the rank of the GKZ hypergeometric system HA(β) equals the normalized volume of conv(A) for all complex parameters β ∈ Cd (Saito, 2002). Our refinement here shows that HA(β) h...

Rank Jumps in Codimension

The Rank of a Hypergeometric System

The holonomic rank of the hypergeometric system MA(β) equals the simplicial volume of A ⊆ Z for generic parameters β ∈ C; in general, this is only a lower bound. To the toric ring SA we attach the ranking arrangement RA(SA), and use this algebraic invariant along with the exceptional arrangement EA(SA) = {β ∈ C d | rank MA(β) > vol(A)} to construct a combinatorial formula for the rank jump jA(S...

Exponential bounds for the hypergeometric distribution.

We establish exponential bounds for the hypergeometric distribution which include a finite sampling correction factor, but are otherwise analogous to bounds for the binomial distribution due to León and Perron (Statist. Probab. Lett.62 (2003) 345-354) and Talagrand (Ann. Probab.22 (1994) 28-76). We also extend a convex ordering of Kemperman's (Nederl. Akad. Wetensch. Proc. Ser. A76 = Indag. Mat...