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Sendov conjecture for high degree polynomials


Author: Jérôme Dégot
Journal: Proc. Amer. Math. Soc. 142 (2014), 1337-1349
MSC (2010): Primary 30C10, 30C15; Secondary 12D10
DOI: https://doi.org/10.1090/S0002-9939-2014-11888-0
Published electronically: January 29, 2014
MathSciNet review: 3162254
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Abstract: Sendov's conjecture says that if all zeros of a complex polynomial $ P$ lie in the closed unit disk and $ a$ denotes one of them, then the closed disk of center $ a$ and radius $ 1$ contains a critical point of $ P$ (i.e. a zero of its derivative $ P'$). The main result of this paper is to prove that, for each $ a$, there exists an integer $ N$ such that the disk $ \vert\zeta - a\vert\leq 1$ contains a critical point of $ P$ when the degree of $ P$ is larger than $ N$. We obtain this by studying the geometry of the zeros and critical points of a polynomial which would eventually contradict Sendov's conjecture.


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Additional Information

Jérôme Dégot
Affiliation: Lycée Louis-le-Grand, 123 rue St Jacques, 75 005 Paris, France
Email: jerome.degot@numericable.fr

DOI: https://doi.org/10.1090/S0002-9939-2014-11888-0
Keywords: Sendov conjecture, polynomial, geometry of polynomial, zeroes, inequalities
Received by editor(s): November 16, 2011
Received by editor(s) in revised form: May 18, 2012
Published electronically: January 29, 2014
Communicated by: Richard Rochberg
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.