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

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 Erdélyi, 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.

Comment on arXiv 2409.00156 that has been published in https://doi.org/10.1216/rmj.2024.54.995