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

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Baker, Simon and Yu, Han (2018) Root sets of polynomials and power series with finite choices of coefficients. Computational Methods and Function Theory, 18 (1). pp. 89-97. doi:10.1007/s40315-017-0215-1 ISSN 1617-9447.

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Official URL: https://doi.org/10.1007/s40315-017-0215-1

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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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Polynomials
Journal or Publication Title: Computational Methods and Function Theory
Publisher: Heldermann Verlag
ISSN: 1617-9447
Official Date: March 2018
Dates:
DateEvent
March 2018Published
9 October 2017Available
6 June 2017Accepted
Volume: 18
Number: 1
Page Range: pp. 89-97
DOI: 10.1007/s40315-017-0215-1
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access (Creative Commons)
Date of first compliant deposit: 2 October 2017
Date of first compliant Open Access: 28 March 2018
Grant number: EP/M001903/1.
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
EP/M001903/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
UNSPECIFIEDUniversity of St Andrewshttp://dx.doi.org/10.13039/501100000740
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