Convergence of two-dimensional hypergeometric series for algebraic functions

Abstract

Description of convergence domains for multiple power series is a quite difficult problem. In 1889 J.Horn showed that the case of hypergeomteric series is more favourable. He found a parameterization formula for surfaces of conjugative radii of such series. But until recently almost nothing was known about the description of convergence domains in terms of functional inequalities ρj(|a1|, . . . , |am|) < 0 relatively moduli |ai| of series variables. In this paper we give a such description for hypergeometric series representing solutions to tetranomial algebraic equations. In our study we use the remarkable observation by M. Kapranov (16) consisting in the fact that the Horn’s formulae give a parameterization of discriminant locus for a corresponding A-discriminant. We prove that usually the considered convergence domains are determined by a signle or two inequalities ρ(|at |, |as|) ≶ 0, where ρ is a reduced discriminant.