Classifying bivariate rational hypergeometric functions

A. DICKENSTEIN

Oberwolfach, Alemania

Workshop; Workshop on algebraic aspects of hypergeometric functions; 2008

MFO Oberwolfach, Alemania

We characterize those codimension-two configurations A whose associated Ahypergeometric system admits, for some integral homogeneity, a rational solution which is stable, i.e. not annihilated by any partial derivative. We prove that as conjectured in: E. Cattani, A. Dickenstein, and B. Sturmfels, Rational Hypergeometric Functions, Compositio Math., 2001, such an A must be a Cayley essential configuration. The proof uses in an essential manner the classification of algebraic univariate hypergeometric functions due to F. Beukers and G. Heckman. Monodromy for the hypergeometric function nFn-1, Invent. Math., 1989. Our classification is applied to the study of the rationality of classical bivariate hypergeometric Laurent series. We moreover show that for any Cayley configuration A of codimension two, a sufficiently high derivative of any rational function solution of the associated Ahypergeometric system is a toric residue. This gives a geometric interpretation of monodromy invariant A-hypergeometric functions for codimension two configurations. Joint work with Eduardo Cattani and Fernando Rodríguez Villegas.