Compositio Mathematica

Rational Hypergeometric Functions

Eduardo Cattani a1, Alicia Dickenstein a2 and Bernd Sturmfels a3
a1 University of Massachusetts, Amherst, U.S.A. E-mail:
a2 FCEyN Universidad de Buenos Aires, Argentina. E-mail:
a3 University of California Berkeley, U.S.A. E-mail:

Article author query
cattani e   [Google Scholar] 
dickenstein a   [Google Scholar] 
sturmfels b   [Google Scholar] 


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.

Key Words: hypergeometric functions; toric varieties; discriminants; resultants; residues.