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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
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 | |||||||||
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| 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: |
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| 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: |
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