# Homological methods for hypergeometric families

@article{Matusevich2004HomologicalMF, title={Homological methods for hypergeometric families}, author={Laura Felicia Matusevich and Ezra Miller and Uli Walther}, journal={Journal of the American Mathematical Society}, year={2004}, volume={18}, pages={919-941} }

We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems H_A(\beta) arising from a d x n integer matrix A and a parameter \beta \in \CC^d. To do so we introduce an Euler-Koszul functor for hypergeometric families over \CC^d, whose homology generalizes the notion of a hypergeometric system, and we prove…

## 90 Citations

The rank of a hypergeometric system

- MathematicsCompositio Mathematica
- 2010

Abstract The holonomic rank of the A-hypergeometric system MA(β) is the degree of the toric ideal IA for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we…

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- Mathematics
- 2018

If $\beta\in\CC^d$ is integral but not a strongly resonant parameter for the homogeneous matrix $A\in\ZZ^{d\times n}$ with $\ZZ A=\ZZ^d$, then the associated GKZ-system carries a naturally defined…

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- Mathematics
- 2008

We study the irregularity sheaves attached to the $A$-hypergeometric $D$-module $M_A(\beta)$ introduced by Gel'fand et al., where $A\in\mathbb{Z}^{d\times n}$ is pointed of full rank and…

SLOPES OF HYPERGEOMETRIC SYSTEMS ALONG COORDINATE VARIETIES

- Mathematics
- 2006

Let A be an integer d × n matrix defining a positive semigroup NA and � ∈ C d a complex parameter vector. This article gives a complete combinatorial description of the critical indices (or slopes)…

Binomial D-modules Ima Preprint Series # 2137 Institute for Mathematics and Its Applications Binomial D-modules

- Mathematics
- 2006

We study quotients of the Weyl algebra by left ideals whose g enerators consist of an arbitraryZ-graded binomial ideal I in C[∂1, . . . , ∂n] along with Euler operators defined by the grading and a…

Combinatorics of rank jumps in simplicial hypergeometric systems

- Mathematics
- 2004

Let A be an integer d x 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…

Exponential growth of rank jumps for A-hypergeometric systems

- Mathematics
- 2012

The dimension of the space of holomorphic solutions at nonsingular points (also called the holonomic rank) of a $A$--hypergeometric system $M_A (\beta)$ is known to be bounded above by $…

Duality and monodromy reducibility of A-hypergeometric systems

- Mathematics
- 2005

We study hypergeometric systems HA(β) in the sense of Gelfand, Kapranov and Zelevinsky under two aspects: the structure of their holonomically dual system, and reducibility of their rank module. We…

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