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Rational Hypergeometric Functions

Published online by Cambridge University Press:  04 December 2007

Eduardo Cattani ,
Alicia Dickenstein  and
Bernd Sturmfels
Eduardo Cattani
Affiliation:
University of Massachusetts, Amherst, U.S.A. E-mail: cattani@math.umass.edu
Alicia Dickenstein
Affiliation:
FCEyN Universidad de Buenos Aires, Argentina. E-mail: alidick@dm.uba.ar
Bernd Sturmfels
Affiliation:
University of California Berkeley, U.S.A. E-mail: bernd@math.berkeley.edu
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Abstract

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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 projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator of some rational hypergeometric function.

Keywords

hypergeometric functionstoric varietiesdiscriminantsresultantsresidues
Type
Research Article
Information
Compositio Mathematica , Volume 128 , Issue 2 , September 2001 , pp. 217 - 240
Copyright
© 2001 Kluwer Academic Publishers