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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A quadratic approximation to the Sendov radius near the unit circle


Author: Michael J. Miller
Journal: Trans. Amer. Math. Soc. 357 (2005), 851-873
MSC (2000): Primary 30C15
Published electronically: October 19, 2004
MathSciNet review: 2110424
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Abstract: Define $S(n,\beta)$ to be the set of complex polynomials of degree $n\ge2$ with all roots in the unit disk and at least one root at $\beta$. For a polynomial $P$, define $\vert P\vert _\beta$ to be the distance between $\beta$ and the closest root of the derivative $P'$. Finally, define $r_n(\beta)=\sup \{ \vert P\vert _\beta : P \in S(n,\beta) \}$. In this notation, a conjecture of Bl. Sendov claims that $r_n(\beta)\le1$.

In this paper we investigate Sendov's conjecture near the unit circle, by computing constants $C_1$ and $C_2$ (depending only on $n$) such that $r_n(\beta)\sim1+C_1(1-\vert\beta\vert)+C_2(1-\vert\beta\vert)^2$ for $\vert\beta\vert$near $1$. We also consider some consequences of this approximation, including a hint of where one might look for a counterexample to Sendov's conjecture.


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Additional Information

Michael J. Miller
Affiliation: Department of Mathematics, Le Moyne College, Syracuse, New York 13214
Email: millermj@mail.lemoyne.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03766-3
PII: S 0002-9947(04)03766-3
Keywords: Sendov, Ilieff, Ilyeff
Received by editor(s): October 15, 2001
Published electronically: October 19, 2004
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.