Polynomials on the Cauchy circle

Abstract

Let \(P\) be a complex polynomial of degree \(n\) with \(P(0)=1\) and Cauchy radius 1 about the origin. We discuss the order of magnitude of the minimal number \(N=N(\epsilon _n,n)\) such that\( \min_{1\leq k\leq N}|P({\rm e}^{2\pi {\rm i}k/N})|\leq 1-\epsilon _n. \) Previous estimates of \(N=O(n^{3/2})\) are improved to \(N= O(n\,\log\,n)\). Some other related properties of these polynomials are also exhibited.