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

DOI:
https://doi.org/10.1090/S0002-9939-05-08058-5

Published electronically:
July 25, 2005

MathSciNet review:
2196041

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Extending a result of Khavinson and Swiatek (2003) we show that the rational harmonic function , where is a rational function of degree , has no more than 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 -point gravitational lens.

**[APOD 01]**NASA Astronomy Picture of the Day,*Giant cluster bends, breaks images - 2001 June 10*, http://antwrp.gsfc.nasa.gov/apod/ap010610.html.**[APOD 03]**NASA Astronomy Picture of the Day,*Dark matter map - 2003 August 14*, http://antwrp.gsfc.nasa.gov/apod/ap030814.html.**[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 http://cfa-www.harvard.edu/ narayan/papers/JeruLect.ps.**[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,*-point gravitational lenses with 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, http://www.livingreviews.org/Articles/Volume1/1998-12wamb/.**[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.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
26C15,
30D05,
83C99

Retrieve articles in all journals with MSC (2000): 26C15, 30D05, 83C99

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:
https://doi.org/10.1090/S0002-9939-05-08058-5

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.