# 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}
}
• Published 18 June 2004
• Mathematics
• Journal of the American Mathematical Society
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…
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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 $Duality and monodromy reducibility of A-hypergeometric systems We study hypergeometric systems HA(β) in the sense of Gelfand, Kapranov and Zelevinsky under two aspects: the structure of their holonomically dual system, and reducibility of their rank module. We ## References SHOWING 1-10 OF 57 REFERENCES Combinatorics of rank jumps in simplicial hypergeometric systems • Mathematics • 2004 Let A be an integer d x n matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d - 1 not containing the origin. It is known that the semigroup ring C[NA] is Rational Solutions of the A-hypergeometric System Associated with a Monomial Curve Introduction. Inthis paper we make a detailed analysis of the -hypergeometric system (or GKZ system) associated with a monomial curve and integral, hence resonant, exponents. We describe all rational Exceptional parameters for generic A-hypergeometric systems The holonomic rank of an A-hypergeometric system$H_A(\beta)$is conjectured to be independent of the parameter vector$\beta$if and only if the toric ideal$I_A\$ is Cohen Macaulay. We prove this
Cohen-Macaulay rings
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• 1993
In this chapter we introduce the class of Cohen–Macaulay rings and two subclasses, the regular rings and the complete intersections. The definition of Cohen–Macaulay ring is sufficiently general to
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Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors