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On the number of zeros of certain rational harmonic functions

Author(s): Dmitry Khavinson; Genevra Neumann
Journal: Proc. Amer. Math. Soc. 134 (2006), 1077-1085.
MSC (2000): Primary 26C15; Secondary 30D05, 83C99
Posted: July 25, 2005
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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
Email: dmitry@uark.edu

Genevra Neumann
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email: neumann@math.ksu.edu

DOI: 10.1090/S0002-9939-05-08058-5
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
Posted: July 25, 2005
Additional Notes: The first author was supported by a grant from the National Science Foundation.
Communicated by: Juha M. Heinonen
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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