Root Sets of Polynomials and Power Series with Finite Choices of Coefficients

Abstract

Given $$H\subseteq \mathbb {C}$$ two natural objects to study are the set of zeros of polynomials with coefficients in H, $$\begin{aligned} \left\{ z\in \mathbb {C}: \exists k>0,\, \exists (a_n)\in H^{k+1}, \sum _{n=0}^{k}a_{n}z^n=0\right\} , \end{aligned}$$ and the set of zeros of a power series with coefficients in H, $$\begin{aligned} \left\{ z\in \mathbb {C}: \exists (a_n)\in H^{\mathbb {N}}, \sum _{n=0}^{\infty } a_nz^n=0\right\} . \end{aligned}$$ 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 progressively more dense.

Locations

• Computational Methods and Function Theory - View - PDF