Computation of conformal representations of compact Riemann surfaces

TL;DR

This paper develops a method to explicitly compute conformal maps of certain three-sheeted Riemann surfaces, which are important for understanding asymptotics of multiple orthogonal polynomials related to Nikishin systems.

Contribution

It introduces a system of polynomial equations that enables explicit conformal representations of specific Riemann surfaces, advancing the computational tools in complex analysis.

Findings

Explicit polynomial equations for conformal maps of three-sheeted surfaces

Application to asymptotic analysis of multiple orthogonal polynomials

Enhanced computational framework for Riemann surface representations

Abstract

We find a system of two polynomial equations in two unknowns, whose solution allows to give an explicit expression of the conformal representation of a simply connected three sheeted compact Riemann surface onto the extended complex plane. This function appears in the description of the ratio asymptotic of multiple orthogonal polynomials with respect to so called Nikishin systems of two measures.