Polynomials on the Cauchy circle (Q1347037)

Let \({\mathcal P}_ n^ c\) consist of the polynomials \[ P(z) = 1 + \sum_{k=1}^ n a_ kz^ k, \quad \sum_{k=1}^ n | a_ k | = 1. \] For \(\varepsilon_ n \geq 0\) the order of magnitude of the number \[ N = N(\varepsilon_ n,n) : = \min \Bigl\{ \nu \in \mathbb{N} : \forall P \in {\mathcal P}_ n^ c \quad \min_{1 \leq k \leq n} \bigl | P(e^{2 \pi ik/ \nu}) \bigr | \leq 1 - \varepsilon_ n \Bigr\} \] is discussed. For \(\varepsilon_ n = O(1/n)\) previous estimates of \(N = O(n^{3/2})\) are improved to \(N = O(n \log n)\). For \(n\leq 199\) the best upper bounds for \(N({1 \over 16n}, n)\) are presented. Moreover, an upper bound for the modulus of the zero of least modulus of \(P \in {\mathcal P}_ n^ c\) is established.