A quadratic approximation to the Sendov radius near the unit circle
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- by Michael J. Miller PDF
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Abstract:
Define $S(n,\beta )$ to be the set of complex polynomials of degree $n\ge 2$ with all roots in the unit disk and at least one root at $\beta$. For a polynomial $P$, define $|P|_\beta$ to be the distance between $\beta$ and the closest root of the derivative $P’$. Finally, define $r_n(\beta )=\sup \{ |P|_\beta : P \in S(n,\beta ) \}$. In this notation, a conjecture of Bl. Sendov claims that $r_n(\beta )\le 1$. 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 $r_n(\beta )\sim 1+C_1(1-|\beta |)+C_2(1-|\beta |)^2$ for $|\beta |$ 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.References
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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
- Received by editor(s): October 15, 2001
- Published electronically: October 19, 2004
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 357 (2005), 851-873
- MSC (2000): Primary 30C15
- DOI: https://doi.org/10.1090/S0002-9947-04-03766-3
- MathSciNet review: 2110424