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December 2010 A quantitative result on polynomials with zeros in the unit disk
Tomohiro Chijiwa
Proc. Japan Acad. Ser. A Math. Sci. 86(10): 165-168 (December 2010). DOI: 10.3792/pjaa.86.165

Abstract

On Sendov's conjecture, M. J. Miller states the following in his paper [10,11]; if a zero $\beta$ of a polynomial which has all the zeros in the closed unit disk is sufficiently close to the unit circle, then the distance from $\beta$ to the closest critical point is less than or equal to 1. It is desirable to quantify this assertion. In this paper, we estimate the radius of the disk with center at 0 containing all the critical points and estimate the range of the zero $\beta$ satisfying the above for the first step. This result, moreover, implies that if Sendov’s conjecture is false, then the polynomial must be close to an extremal one.

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Tomohiro Chijiwa. "A quantitative result on polynomials with zeros in the unit disk." Proc. Japan Acad. Ser. A Math. Sci. 86 (10) 165 - 168, December 2010. https://doi.org/10.3792/pjaa.86.165

Information

Published: December 2010
First available in Project Euclid: 6 December 2010

zbMATH: 1220.30006
MathSciNet: MR2779829
Digital Object Identifier: 10.3792/pjaa.86.165

Subjects:
Primary: 12D10 , 26C10 , 30C15

Keywords: critical point , polynomial , Sendov’s conjecture , zero

Rights: Copyright © 2010 The Japan Academy

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Vol.86 • No. 10 • December 2010
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