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

Simon Baker; Han Yu

doi:10.1007/s40315-017-0215-1

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

Simon Baker, Han Yu

Pure Mathematics

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Abstract

Given H⊆C two natural objects to study are the set of zeros of polynomials with coefficients in H,
{z∈C:∃k>0,∃(an)∈Hk+1,∑n=0kanzn=0},
and the set of zeros of a power series with coefficients in H,
{z∈C:∃(an)∈HN,∑n=0∞anzn=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∈(1/2,1), if H is 2cos−1(5−4|r|24)-dense in S1, then the set of zeros of polynomials with coefficients in H is dense in {z∈C:|z|∈[r,r−1]}, and the set of zeros of power series with coefficients in H contains the annulus {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.

Original language	English
Pages (from-to)	89-97
Number of pages	9
Journal	Computational Methods and Function Theory
Volume	18
Issue number	1
Early online date	9 Oct 2017
DOIs	https://doi.org/10.1007/s40315-017-0215-1
Publication status	Published - Mar 2018

Keywords

Root set
Littlewood polynomials
Unimodular polynomials

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10.1007/s40315-017-0215-1Licence: CC BY

Yu_CMFT_PowerSeries_CC
© The Author(s) 2017. Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Final published version, 283 KBLicence: CC BY

Cite this

@article{b272abf6cd6d4ee8ba50d4910716d1cc,

title = "Root sets of polynomials and power series with finite choice of coefficients",

abstract = "Given H⊆C two natural objects to study are the set of zeros of polynomials with coefficients in H,{z∈C:∃k>0,∃(an)∈Hk+1,∑n=0kanzn=0},and the set of zeros of a power series with coefficients in H,{z∈C:∃(an)∈HN,∑n=0∞anzn=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∈(1/2,1), if H is 2cos−1(5−4|r|24)-dense in S1, then the set of zeros of polynomials with coefficients in H is dense in {z∈C:|z|∈[r,r−1]}, and the set of zeros of power series with coefficients in H contains the annulus {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.",

keywords = "Root set, Littlewood polynomials, Unimodular polynomials",

author = "Simon Baker and Han Yu",

note = "The first author is supported by the EPSRC Grant EP/M001903/1. The second author is supported by a PhD scholarship provided by the School of Mathematics in the University of St Andrews. ",

year = "2018",

month = mar,

doi = "10.1007/s40315-017-0215-1",

language = "English",

volume = "18",

pages = "89--97",

journal = "Computational Methods and Function Theory",

issn = "1617-9447",

publisher = "Springer",

number = "1",

}

TY - JOUR

T1 - Root sets of polynomials and power series with finite choice of coefficients

AU - Baker, Simon

AU - Yu, Han

N1 - The first author is supported by the EPSRC Grant EP/M001903/1. The second author is supported by a PhD scholarship provided by the School of Mathematics in the University of St Andrews.

PY - 2018/3

Y1 - 2018/3

N2 - Given H⊆C two natural objects to study are the set of zeros of polynomials with coefficients in H,{z∈C:∃k>0,∃(an)∈Hk+1,∑n=0kanzn=0},and the set of zeros of a power series with coefficients in H,{z∈C:∃(an)∈HN,∑n=0∞anzn=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∈(1/2,1), if H is 2cos−1(5−4|r|24)-dense in S1, then the set of zeros of polynomials with coefficients in H is dense in {z∈C:|z|∈[r,r−1]}, and the set of zeros of power series with coefficients in H contains the annulus {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.

AB - Given H⊆C two natural objects to study are the set of zeros of polynomials with coefficients in H,{z∈C:∃k>0,∃(an)∈Hk+1,∑n=0kanzn=0},and the set of zeros of a power series with coefficients in H,{z∈C:∃(an)∈HN,∑n=0∞anzn=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∈(1/2,1), if H is 2cos−1(5−4|r|24)-dense in S1, then the set of zeros of polynomials with coefficients in H is dense in {z∈C:|z|∈[r,r−1]}, and the set of zeros of power series with coefficients in H contains the annulus {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.

KW - Root set

KW - Littlewood polynomials

KW - Unimodular polynomials

U2 - 10.1007/s40315-017-0215-1

DO - 10.1007/s40315-017-0215-1

M3 - Article

SN - 1617-9447

VL - 18

SP - 89

EP - 97

JO - Computational Methods and Function Theory

JF - Computational Methods and Function Theory

IS - 1

ER -

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

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