  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.
