The multiple Markov theorem on Angelesco sets

TL;DR

This paper extends Markov's theorem to multiple orthogonal polynomials on Angelesco sets, providing a simple proof that broadens understanding of zero variation in these polynomials.

Contribution

It generalizes Markov's theorem to Angelesco sets for multiple orthogonal polynomials, offering a simple, unrestricted proof without specific assumptions on weights or interval structures.

Findings

Extended Markov's theorem to Angelesco sets

Provided conditions for $ ext{Z}$-matrix to be an $ ext{M}$-matrix

Simplified proof without restrictions on weights or intervals

Abstract

By addressing a long-standing open problem, listed in a highly regarded collection of open questions in the field and described as a "worthwhile research project", this note extends Markov's theorem (Markoff, Math. Ann., 27:177-182, 1886) on the variation of zeros of orthogonal polynomials on the real line to the setting of multiple orthogonal polynomials on Angelesco sets. The analysis reveals that the only distinction from the classical 1886 result lies in establishing sufficient conditions for a given $\mathcal{Z}$-matrix--which, in the Markov case, is the identity matrix--to be an $\mathcal{M}$-matrix. In contrast to most existing studies, which often present highly technical proofs for specific results, this note seeks to provide a simple proof of a general result without imposing restrictions on the weight functions (such as their potential "classical" nature), the number of intervals, or the structure of the partition.