Zero distribution for Angelesco Hermite--Pad\'e polynomials

TL;DR

This paper studies the asymptotic zero distribution of Hermite--Padé polynomials for vector functions with separated branch points, establishing a connection with vector equilibrium problems and Riemann surfaces.

Contribution

It proves a limit zero distribution theorem for Angelesco Hermite--Padé polynomials and introduces an alternative characterization via Riemann surfaces, extending previous methods.

Findings

Limit measures are characterized by vector equilibrium problems.

The zero distribution is proven to converge under Angelesco conditions.

A conjecture is proposed for more general cases without Angelesco separation.

Abstract

We consider the problem of zero distribution of the first kind Hermite--Pad\'e polynomials associated with a vector function $\vec f = (f_1, \dots, f_s)$ whose components $f_k$ are functions with a finite number of branch points in plane. We assume that branch sets of component functions are well enough separated (which constitute the Angelesco case). Under this condition we prove a theorem on limit zero distribution for such polynomials. The limit measures are defined in terms of a known vector equilibrium problem. Proof of the theorem is based on the methods developed by H.~Stahl, A.~A.~Gonchar and the author. These methods obtained some further generalization in the paper in application to systems of polynomials defined by systems of complex orthogonality relations. Together with the characterization of the limit zero distributions of Hermite--Pad\'e polynomials by a vector equilibrium problem we consider an alternative characterization using a Riemann surface $\mathcal R(\vec f)$ associated with $\vec f$. In this terms we present a more general (without Angelesco condition) conjecture on the zero distribution of Hermite--Pad\'e polynomials. Bibliography: 72 items.