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The rank of a hypergeometric system

Part of: Homological methods Semigroups Differential algebra Special varieties

Published online by Cambridge University Press:  17 August 2010

Christine Berkesch
Christine Berkesch*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA (email: cberkesc@math.purdue.edu)
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Abstract

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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 attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of non-generic parameters to construct a combinatorial formula for the rank jump of MA(β). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of MA(β) 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 non-trivial weakly toric module M⊆ℂ[ℤA] . We also provide a direct proof of the main result in [M. Saito, Isomorphism classes of A-hypergeometric systems, Compositio Math. 128 (2001), 323–338] regarding the isomorphism classes of MA (β) .

Keywords

hypergeometricD-moduletoricholonomicrankEuler–Koszulcombinatorial

MSC classification

Secondary: 33C70: Other hypergeometric functions and integrals in several variables 14M25: Toric varieties, Newton polyhedra 16E30: Homological functors on modules (Tor, Ext, etc.) 20M25: Semigroup rings, multiplicative semigroups of rings 13N10: Rings of differential operators and their modules
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 1 , January 2011 , pp. 284 - 318
Copyright
Copyright © Foundation Compositio Mathematica 2010

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