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On the number of zeros of certain harmonic polynomials
Author(s):
Dmitry
Khavinson;
Grzegorz
Swiatek
Journal:
Proc. Amer. Math. Soc.
131
(2003),
409-414.
MSC (2000):
Primary 26C10
Posted:
September 17, 2002
MathSciNet review:
1933331
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Abstract:
Using techinques of complex dynamics we prove the conjecture of Sheil-Small and Wilmshurst that the harmonic polynomial  ,  , has at most  complex zeros.
References:
-
- 1.
- D. Bshouty, W. Hengartner & T. Suez: The exact bound of the number of zeros of harmonic polynomials, J. d'Analyse Math. 67 (1995), 207-218 MR 97f:30025
- 2.
- L. Carleson & T. Gamelin: Complex Dynamics, Springer-Verlag, New York-Berlin-Heidelberg (1993) MR 94h:30033
- 3.
- M. Cristea: A generalization of the argument principle, Compl. Var. Theory Appl. 42 (2000), 335-345 MR 2001d:30085
- 4.
- P. Duren, W. Hengartner & R.S. Langesen: The argument principle for harmonic functions, Amer. Math. Monthly 103 (1996), 411-415 MR 97f:30002
- 5.
- D. Sarason, written communication, Feb. 1999,
Oct. 2000 - 6.
- T. Sheil-Small in Tagesbericht, Mathematisches Forsch. Inst. Oberwolfach, Funktionentheorie, 16-22.2.1992, 19
- 7.
- A.S. Wilmshurst, The valence of harmonic polynomials, Proc. AMS 126 (1998), 2077-2081 MR 98h:30029
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Additional Information:
Dmitry
Khavinson
Affiliation:
Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701
Email:
dmitry@comp.uark.edu
Grzegorz
Swiatek
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email:
swiatek@math.psu.edu
DOI:
10.1090/S0002-9939-02-06476-6
PII:
S 0002-9939(02)06476-6
Received by editor(s):
May 1, 2001
Posted:
September 17, 2002
Additional Notes:
The first author was partially supported by an NSF grant DMS-0139008
The second author was partially supported by an NSF grant DMS-0072312
Communicated by:
Juha M. Heinonen
Copyright of article:
Copyright
2002,
American Mathematical Society
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