Hyperelliptic uniformization of algebraic curves of the third order

TL;DR

This paper develops a hyperelliptic uniformization method for algebraic functions of the third order, aiding in the analysis of Hermite-Padé approximants and their pole distribution in complex analysis.

Contribution

It introduces a hyperelliptic uniformization approach for third-order algebraic functions, enabling parametrization and numerical analysis of the elliptic case.

Findings

Uniformization provides a new parametrization method.

Numerical procedure for elliptic curve determination.

Description of pole distribution limits for approximants.

Abstract

An algebraic function of the third order plays an important role in the problem of asymptotics of Hermite-Pad\'e approximants for two analytic functions with branch points. This algebraic function appears as the Cauchy transform of the limiting measure of the asymptotic distribution of the poles of the approximants. In many cases this algebraic function can be determined by using the given position of the branch points of the functions which are approximated and by the condition that its Abelian integral has purely imaginary periods. In the present paper we obtain a hyperelliptic uniformization of this algebraic function. In the case when each approximated function has only two branch points, the genus of this function can be equal to 0, 1 (elliptic case) or 2 (ultra-elliptic case). We use this uniformization to parametrize the elliptic case. This parametrization allows us to obtain a numerical procedure for finding this elliptic curve and as a result we can describe the limiting measure of the distribution of the poles of the approximants.