Polynomials on the Cauchy circle

Summary.

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.