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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
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Abstract: Using techinques of complex dynamics we prove the conjecture of Sheil-Small and Wilmshurst that the harmonic polynomial $z-\overline{p(z)}$, $\deg p = n > 1$, has at most $3n-2$ complex zeros.

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L. Carleson & T. Gamelin: Complex Dynamics, Springer-Verlag, New York-Berlin-Heidelberg (1993) MR 94h:30033

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M. Cristea: A generalization of the argument principle, Compl. Var. Theory Appl. 42 (2000), 335-345 MR 2001d:30085

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P. Duren, W. Hengartner & R.S. Langesen: The argument principle for harmonic functions, Amer. Math. Monthly 103 (1996), 411-415 MR 97f:30002

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D. Sarason, written communication, Feb. 1999,

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T. Sheil-Small in Tagesbericht, Mathematisches Forsch. Inst. Oberwolfach, Funktionentheorie, 16-22.2.1992, 19

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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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