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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.

References [Enhancements On Off] (What's this?)

  • [APOD 01] NASA Astronomy Picture of the Day, Giant cluster bends, breaks images - 2001 June 10,
  • [APOD 03] NASA Astronomy Picture of the Day, Dark matter map - 2003 August 14,
  • [BL 04] D. Bshouty and A. Lyzzaik, On Crofoot-Sarason's conjecture for harmonic polynomials, Comput. Methods Funct. Theory 4 (2004), 35-41. MR 2081663 (2005e:30008)
  • [Bu 81] W. L. Burke, Multiple gravitational imaging by distributed masses, Astrophys. J. 244 (1981), L1.
  • [CG 93] L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, New York, (1993). MR 1230383 (94h:30033)
  • [Da 74] P. J. Davis, The Schwarz function and its applications, The Carus Mathematical Monographs, No. 17, The Mathematical Association of America, Buffalo, NY, 1974. MR 0407252 (53 #11031)
  • [Fo 81] O. Forster, Lectures on Riemann Surfaces, Translated from the German by Bruce Gilligan, Graduate Texts in Mathematics, 81, Springer-Verlag, New York, Berlin (1981). MR 0648106 (83d:30046)
  • [Ge 03] L. Geyer, Sharp bounds for the valence of certain harmonic polynomials, preprint (2003).
  • [KS 03] D. Khavinson and G. Swiatek, On the number of zeros of certain harmonic polynomials, Proc. Amer. Math. Soc. 131 (2003), 409-414. MR 1933331 (2003j:30015)
  • [MPW 97] S. Mao, A. O. Petters, and H. J. Witt, Properties of point mass lenses on a regular polygon and the problem of maximum number of images, ``Proceeding of the Eighth Marcel Grossmann Meeting on General Relativity (Jerusalem, Israel, 1997)'', edited by T. Piran, World Scientific, Singapore (1998), 1494-1496, MR 1892087 (2003d:58058)
  • [NB 96] R. Narayan and M. Bartelmann, Lectures on gravitational lensing, in ``Proceedings of the 1995 Jerusalem Winter School'' (1995); online version $^\sim$ narayan/papers/
  • [Pe 92] A. O. Petters, Morse theory and gravitational microlensing, J. Math. Phys. 33 (1992), 1915-1931. MR 1159012 (93c:58060)
  • [PLW 01] A. O. Petters, H. Levine, and J. Wambsganss, Singularity theory and gravitational lensing, Birkhäuser, Boston (2001). MR 1836154 (2002m:83127)
  • [Rh 01] S. H. Rhie, Can a gravitational quadruple lens produce 17 images?, arXiv:astro-ph/0103463.
  • [Rh 03] S. H. Rhie, $n$-point gravitational lenses with $5 (n - 1)$ images, arXiv:astro-ph/0305166.
  • [SS 02] T. Sheil-Small, Complex Polynomials, Cambridge Studies in Advanced Mathematics 73, Cambridge University Press (2002). MR 1962935 (2004b:30001)
  • [SS 92] T. Sheil-Small in Tagesbericht, Mathematisches Forsch. Inst. Oberwolfach, Funktionentheorie, 16-22.2.1992, 19.
  • [St 97] N. Straumann, Complex formulation of lensing theory and applications, Helvetica Physica Acta 70 (1997), 894-908. MR 1481298 (98k:85001)
  • [ST 00] T. J. Suffridge and J. W. Thompson, Local behavior of harmonic mappings, Complex Variables Theory Appl. 41 (2000), 63-80. MR 1758598 (2001a:30019)
  • [Wa 98] J. Wambsganss, Gravitational lensing in astronomy, Living Rev. Relativity 1 (1998) [Online Article]: cited on January 26, 2004,
  • [Wil 98] A. S. Wilmshurst, The valence of harmonic polynomials, Proc. Amer. Math. Soc. 126 (1998), 2077-2081. MR 1443416 (98h:30029)
  • [Wit 90] H. J. Witt, Investigation of high amplification events in light curves of gravitationally lensed quasars, Astron. Astrophys. 236 (1990), 311-322.

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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

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.

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