A quadratic approximation to the Sendov radius near the unit circle
Author:
Michael J. Miller
Journal:
Trans. Amer. Math. Soc. 357 (2005), 851873
MSC (2000):
Primary 30C15
Published electronically:
October 19, 2004
MathSciNet review:
2110424
Fulltext PDF Free Access
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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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 , On a conjecture about the critical points of a polynomial, Delay Equations, Approximation and Application, Birkhäuser, Basel, 1985, 8393. MR 0899090 (88e:30013)
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 J. Borcea, The Sendov conjecture for polynomials with at most seven distinct zeros, Analysis 16 (1996), 137159. MR 1397576 (97g:30006)
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 J. E. Brown and G. Xiang, Proof of the Sendov conjecture for polynomials of degree at most eight, J. Math. Anal. Appl. 232 (1999), 272292. MR 1683144 (2001b:30007)
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 , Some maximal polynomials must be nonreal, J. Math. Anal. Appl. 214 (1997), 283291. MR 1645480 (99h:41010)
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 Q. I. Rahman, On the zeros of a polynomial and its derivative, Pacific J. Math. 41 (1972), 525528. MR 0308374 (46:7488)
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 Z. Rubinstein, On a problem of Ilyeff, Pacific J. Math. 26 (1968), 159161. MR 0237753 (38:6034)
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 J. V. Uspensky, Theory of Equations, McGrawHill, New York, 1948.
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 V. Vâjâitu and A. Zaharescu, Ilyeff's conjecture on a corona, Bull. London Math. Soc. 25 (1993), 4954. MR 1190363 (94h:30004)
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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/S0002994704037663
PII:
S 00029947(04)037663
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.
