RIEMANN-HILBERT PROBLEM AND ALGEBRAIC CURVES

Abstract

There are many problems in pure and applied mathematics that can be solved in terms of a Riemann-Hilbert (R-H problem). The list includes the remarkable class of nonlinear integrable equations, namely nonlinear equations that can be written as the compatibility conditions of linear equations. This class contains a large variety of equations: ODE’s, PDE’s difference equations, etc. Furthermore, the R-H problem formulation provides a powerful technique for obtaining asymptotic results for solutions of ODE’s and PDE’s of this class [1]. A remarkable application of this asymptotic technique is the derivation of asymptotic for orthogonal polynomials which is related to the universality conjecture in one-matrix models (see [2] and [3] and reference therein). In this case the associated rank two R-H problems are formulated on hyperelliptic Riemann surfaces. The integrable structure of multiorthogonal polynomials and in particular biorthogonal polynomials was pointed out in [4]. Asymptotic results for multiorthogonal polynomials necessarily involves nonhyperelliptic curves and higher rank R-H problems, which now attract much attention, because of their application to multimatrix models and approximation theory. Regarding two-matrix models, some asymptotic results have been obtained for the genus zero case (namely the analog of the one-cut case in one-matrix models) and the corresponding genus zero nonhyperelliptic Riemann surface has been derived in terms of the external potential [5]. Regarding approximation theory for multiple orthogonal polynomials some results have been obtained in [6], [7] and more recently, using Riemann-Hilbert techniques in [8]. The principal aim of our investigation is to give effective and explicit solutions to a class of higher rank R-H problems associated with nonhyperelliptic curves. This article is a review of the paper [9]. The Riemann-Hilbert problem in its classical formulation consists of deriving a linear differential equation of Fuchsian type with a given set D of singular