Distribution of the zeros of polynomials near the unit circle

TL;DR

This paper provides improved estimates on the distribution of polynomial zeros near the unit circle, including bounds within small discs, gear-shaped regions, and annular discrepancies, advancing understanding of zero localization.

Contribution

It extends previous results by offering sharper bounds on zero counts near the unit circle using geometric and discrepancy methods.

Findings

Improved bounds on zeros within small circular discs around the unit circle.

Derived upper bounds for zeros in gear-shaped regions.

Established sharp bounds on the annular discrepancy of zeros.

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.