On a new approach to the problem of the zero distribution of Hermite-Pad\'e polynomials for a Nikishin system

TL;DR

This paper introduces a novel scalar equilibrium problem approach on a Riemann surface to analyze the zero distribution of Hermite-Padé polynomials for Nikishin systems, offering an alternative to traditional vector methods.

Contribution

It presents a new scalar equilibrium problem framework on a Riemann surface for studying zero distributions, diverging from conventional vector approaches.

Findings

Scalar equilibrium problem effectively describes zero distribution

Approach simplifies analysis compared to vector methods

Results applicable to Hermite-Padé polynomials in Nikishin systems

Abstract

A new approach to the problem of the zero distribution of Hermite-Pad\'e polynomials of type I for a pair of functions $f_1,f_2$ forming a Nikishin system is discussed. Unlike the traditional vector approach, we give an answer in terms of a scalar equilibrium problem with harmonic external field, which is posed on a two-sheeted Riemann surface.