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 | References | Similar Articles | Additional Information

Abstract: Define to be the set of complex polynomials of degree with all roots in the unit disk and at least one root at . For a polynomial , define to be the distance between and the closest root of the derivative . Finally, define . In this notation, a conjecture of Bl. Sendov claims that .

In this paper we investigate Sendov's conjecture near the unit circle, by computing constants and (depending only on ) such that for near . 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:
https://doi.org/10.1090/S0002-9947-04-03766-3

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.