2024/07/22 by Das, Mithun Kumar
#11K38 #30C15 #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2407.15306
We estimate the number of zeros of a polynomial in ℂ[z] within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erdélyi, and Littmann~\citeBE1 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 \citeBE1, combined with the refined version of the best-known upper bound for angular discrepancy of zeros of polynomials.