Equilibrium of vector potentials and uniformization of the algebraic curves of genus 0

摘要

We consider equilibrium problems for the logarithmic vector potential related to the asymptotics of the Hermite–Padé approximants. Solutions of such problems can be expressed by means of algebraic functions. The goal of this paper is to describe a procedure for determining the algebraic equation for this function in the case when the genus of this algebraic function is equal zero. Using the coefficients of the equation we compute the extremal cuts of the Riemann surfaces. These cuts are attractive sets for the poles of the Hermite–Padé approximants. We demonstrate the method by an example of the equilibrium problem related to a special system that is called the Angelesco system.