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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On the number of zeros of certain rational harmonic functions

Authors: Dmitry Khavinson and Genevra Neumann
Journal: Proc. Amer. Math. Soc. 134 (2006), 1077-1085
MSC (2000): Primary 26C15; Secondary 30D05, 83C99
Published electronically: July 25, 2005
MathSciNet review: 2196041
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Abstract: Extending a result of Khavinson and Swiatek (2003) we show that the rational harmonic function $\overline{r(z)} - z$, where $r(z)$ is a rational function of degree $n > 1$, has no more than $5n - 5$ complex zeros. Applications to gravitational lensing are discussed. In particular, this result settles a conjecture by Rhie concerning the maximum number of lensed images due to an $n$-point gravitational lens.

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Additional Information

Dmitry Khavinson
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701

Genevra Neumann
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

PII: S 0002-9939(05)08058-5
Keywords: Rational harmonic mappings, fixed points, argument principle, gravitational lenses
Received by editor(s): January 22, 2004
Received by editor(s) in revised form: October 28, 2004
Published electronically: July 25, 2005
Additional Notes: The first author was supported by a grant from the National Science Foundation.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.