395 research outputs found
Rank jumps in Codimension 2 A-Hypergeometric Systems
The holonomic rank of the A-hypergeometric system H_A(\beta) is shown to
depend on the parameter vector \beta when the underlying toric ideal I_A is a
non Cohen Macaulay codimension 2 toric ideal. The set of exceptional parameters
is usually infinite.Comment: 24 page
Binomial D-modules
We study quotients of the Weyl algebra by left ideals whose generators
consist of an arbitrary Z^d-graded binomial ideal I along with Euler operators
defined by the grading and a parameter in C^d. We determine the parameters for
which these D-modules (i) are holonomic (equivalently, regular holonomic, when
I is standard-graded); (ii) decompose as direct sums indexed by the primary
components of I; and (iii) have holonomic rank greater than the generic rank.
In each of these three cases, the parameters in question are precisely those
outside of a certain explicitly described affine subspace arrangement in C^d.
In the special case of Horn hypergeometric D-modules, when I is a lattice basis
ideal, we furthermore compute the generic holonomic rank combinatorially and
write down a basis of solutions in terms of associated A-hypergeometric
functions. This study relies fundamentally on the explicit lattice point
description of the primary components of an arbitrary binomial ideal in
characteristic zero, which we derive in our companion article arxiv:0803.3846.Comment: This version is shorter than v2. The material on binomial primary
decomposition has been split off and now appears in its own paper
arxiv:0803.384
Nilsson solutions for irregular A-hypergeometric systems
We study the solutions of irregular A-hypergeometric systems that are
constructed from Gr\"obner degenerations with respect to generic positive
weight vectors. These are formal logarithmic Puiseux series that belong to
explicitly described Nilsson rings, and are therefore called (formal) Nilsson
series. When the weight vector is a perturbation of (1,...,1), these series
converge and provide a basis for the (multivalued) holomorphic hypergeometric
functions in a specific open subset of complex n-space. Our results are more
explicit when the parameters are generic or when the solutions studied are
logarithm-free. We also give an alternative proof of a result of Schulze and
Walther that inhomogeneous A-hypergeometric systems have irregular
singularities.Comment: Terminology changed: see Definition 2.6 in current version.
Corrections made to Theorem 6.6, Corollary 6.7 and Corollary 6.8 in version 1
(now Theorem 6.7, Corollary 6.9 and Corollary 6.10, respectively). Added
Corollary 6.3 and Example 6.8. Some stylistic changes, some typos correcte
- …