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- Publisher
- Springer Journals
- Copyright
- Copyright © 2017 by The Author(s)
- Subject
- Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
- ISSN
- 1617-9447
- eISSN
- 2195-3724
- DOI
- 10.1007/s40315-017-0215-1
- Publisher site
- See Article on Publisher Site
Abstract
Given $$H\subseteq \mathbb {C}$$ H ⊆ C two natural objects to study are the set of zeros of polynomials with coefficients in H, $$\begin{aligned} \left\{ z\in \mathbb {C}: \exists k>0,\, \exists (a_n)\in H^{k+1}, \sum _{n=0}^{k}a_{n}z^n=0\right\} , \end{aligned}$$ z ∈ C : ∃ k > 0 , ∃ ( a n ) ∈ H k + 1 , ∑ n = 0 k a n z n = 0 , and the set of zeros of a power series with coefficients in H, $$\begin{aligned} \left\{ z\in \mathbb {C}: \exists (a_n)\in H^{\mathbb {N}}, \sum _{n=0}^{\infty } a_nz^n=0\right\} . \end{aligned}$$ z ∈ C : ∃ ( a n ) ∈ H N , ∑ n = 0 ∞ a n z 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 $$r\in (1/2,1),$$ r ∈ ( 1 / 2 , 1 ) , if H is $$2\cos ^{-1}(\frac{5-4|r|^2}{4})$$ 2 cos - 1 ( 5 - 4 | r | 2 4 ) -dense in $$S^1,$$ S 1 , then the set of zeros of polynomials with coefficients in H is dense in $$\{z\in {\mathbb {C}}: |z|\in [r,r^{-1}]\},$$ { z ∈ C : | z | ∈ [ r , r - 1 ] } , and the set of zeros of power series with coefficients in H contains the annulus $$\{z\in \mathbb {C}: |z|\in [r,1)\}$$ { z ∈ C : | z | ∈ [ 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.
Journal
Computational Methods and Function Theory – Springer Journals
Published: Oct 9, 2017