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arXiv · 2407.15306

Distribution of the zeros of polynomials near the unit circle

Mithun Kumar Das
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Abstract

We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~\cite{BE1} in 2008. Furthermore, by combining this result with Euclidean geometry, we derive an upper bound on the number of zeros of such a polynomial within a region resembling a gear wheel. Additionally, we obtain a sharp upper bound on the annular discrepancy of such zeros near the unit circle. Our approach builds upon a modified version of the method described in \cite{BE1}, combined with the refined version of the best-known upper bound for angular discrepancy of zeros of polynomials.

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Mithun Kumar Das. 2024-07-22. Distribution of the zeros of polynomials near the unit circle. https://arxiv.org/abs/2407.15306

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