Root sets of polynomials and power series with finite choices of coefficients

Han Yu

Published 2016-12-12Version 1

In this paper we consider the root set of power series with finite choices of coefficients: $\{z\in\mathbb{C}| \exists a_n\in H, \sum_{n=0}^{\infty} a_nz^n=0\}$ where $H\subset\mathbb{C}$ is a finite subset. For $H=\{\pm 1\}$, the root set of Littlewood series is under consideration for a long while. Here we study the case when $H=\{e^{i\frac{2\pi}{p}k},k=0,1,2,...,p-1\}$ is the set of all roots of unity with prime order. We show that the correponding root set contains annulus $\{z| l_p<|z|\leq 1\}$ where $l_p\to \frac{1}{2}$ for $p\to\infty$. Intuitively speaking, power series with more coefficients has significantly larger root set. A corollary of this result is that for large $p$, the root set of polynomials with coefficients in $H$ is dense in $\{z| l_p<|z|\leq \frac{1}{l_q}\}$ where $l_p\to \frac{1}{2}$ for $p\to\infty$.