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arXiv · 2510.01703

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

Renato Alvarez-Nodarse·Kenier Castillo
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Abstract

We show that none of the principal claims of Costas-Santos and Rehouma, Rocky Mountain J. Math. 54 (2024), 995-1004, survives as a new and valid result. The main localisation theorem is false as stated; under the missing hypothesis of orthogonality on the unit circle, it is a classical exercise recorded by Marden and by Borwein and Erd\'elyi, while the published proof omits its decisive finite-degree estimate. The annular claim and Sendov application are unproved, and some reported numerical values contradict the latter under its required hypothesis. We establish instead a measure-independent Grace--Szego theorem with sharp affine and optimal disc bounds, and characterise exactly when the convolution method extends to a general factor. This provides the necessary correction to the published record.

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Renato Alvarez-Nodarse, Kenier Castillo. 2025-10-02. A Note on the Paper "Localization of Zeros of Polar Polynomials on the Unit Disk". https://arxiv.org/abs/2510.01703

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