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

TL;DR

This paper investigates the distribution of zeros of polynomials and power series with coefficients from a dense subset of the unit circle, showing how their zeros fill out specific annuli as the coefficient set becomes more dense.

Contribution

It establishes conditions under which zeros of polynomials and power series with coefficients in a dense subset of the unit circle densely fill annuli in the complex plane.

Findings

Zeros of polynomials are dense in certain annuli.

Zeros of power series include entire annuli.

Density of coefficient set influences zero distribution.

Abstract

Given $H\subseteq \mathbb{C}$ two natural objects to study are the set of zeros of polynomials with coefficients in $H$, $$\{z\in \mathbb{C}: \exists k>0,\, \exists (a_n)\in H^{k+1}, \sum_{n=0}^{k}a_{n}z^n=0\},$$ and the set of zeros of power series with coefficients in $H$, $$\{z\in\mathbb{C}: \exists (a_n)\in H^{\mathbb{N}}, \sum_{n=0}^{\infty} a_nz^n=0\}.$$ In this paper we consider the case where each element of $H$ has modulus $1$. The main result of this paper states that for any $r\in(1/2,1),$ if $H$ is $2\cos^{-1}(\frac{5-4|r|^2}{4})$-dense in $S^1,$ then the set of zeros of polynomials with coefficients in $H$ is dense in $\{z\in \mathbb{C}: |z|\in [r,r^{-1}]\},$ and the set of zeros of power series with coefficients in $H$ contains the annulus $\{z\in \mathbb{C}: |z|\in[r,1)\}$. These two statements demonstrate quantitatively how the set of polynomial zeros/power series zeros fill out the natural annulus containing them as $H$ becomes progessively more dense.