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A Note on the Paper "Localization of Zeros of Polar Polynomials on the Unit Disk"

Renato Alvarez-Nodarse, Kenier Castillo

TL;DR

The paper addresses the problem of localizing zeros of the $k$-polar polynomials $Q_{n;k}(z;\xi)$ associated with a finite measure and monic orthogonal polynomials. It critiques a recent claim by CR24 and furnishes a direct, general solution via a binomial-convolution framework, recasting the problem through an invertible linear map and the identity $Q(\xi+\omega)=P(\xi+\omega)\star_{\mathrm{G}} S(\omega)$ with $S(\omega)=\sum_{j=0}^n \binom{n+k}{j+k} \omega^j$ and $R(z)=(z-\xi)^k$. The core contribution is a rigorous exercise-based construction that yields precise zero localization, together with a clarification of the connections to Grace's theorem and OPUC theory, and a critique of CR24 in light of classical references BE95 and M66. The work also discusses a

Abstract

In this note we show that the only result of [Rocky Mountain J. Math. 54 (2024), no. 4, 995--1004] is nothing more than a misformulated version of an exercise from classical texts, presented with a flawed proof. To place the matter on firmer ground, we provide instead a direct solution of a more general problem.

A Note on the Paper "Localization of Zeros of Polar Polynomials on the Unit Disk"

TL;DR

The paper addresses the problem of localizing zeros of the -polar polynomials associated with a finite measure and monic orthogonal polynomials. It critiques a recent claim by CR24 and furnishes a direct, general solution via a binomial-convolution framework, recasting the problem through an invertible linear map and the identity with and . The core contribution is a rigorous exercise-based construction that yields precise zero localization, together with a clarification of the connections to Grace's theorem and OPUC theory, and a critique of CR24 in light of classical references BE95 and M66. The work also discusses a

Abstract

In this note we show that the only result of [Rocky Mountain J. Math. 54 (2024), no. 4, 995--1004] is nothing more than a misformulated version of an exercise from classical texts, presented with a flawed proof. To place the matter on firmer ground, we provide instead a direct solution of a more general problem.

Paper Structure

This paper contains 1 section, 1 theorem, 23 equations.

Table of Contents

  1. Introduction

Key Result

Theorem 1.2

Let be polynomials of degree $n$, and suppose that $K \subset \mathbb{C}$ is an open or closed disk or half-plane, or the open or closed exterior of a disk, that contains all zeros of $P$. If $Q(0) \neq 0$, then each zero $\zeta$ of is of the form $\zeta = -\alpha\beta$ with $\alpha \in K$ and $Q(\beta)=0$. If $Q(0)=0$, then this continues to hold as long as $K$ is not the open or closed exterio

Theorems & Definitions (5)

  • Example 1.1
  • Theorem 1.2
  • proof : Solution
  • Remark 1.4
  • Example 1.5