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Sharp bounds for the valence of certain harmonic polynomials


Author: Lukas Geyer
Journal: Proc. Amer. Math. Soc. 136 (2008), 549-555
MSC (2000): Primary 26C10, 30C10, 37F10
DOI: https://doi.org/10.1090/S0002-9939-07-08946-0
Published electronically: November 2, 2007
MathSciNet review: 2358495
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Abstract: In Khavinson and Swiatek (2002) it was proved that harmonic polynomials $ z-\overline{p(z)}$, where $ p$ is a holomorphic polynomial of degree $ n > 1$, have at most $ 3n-2$ complex zeros. We show that this bound is sharp for all $ n$ by proving a conjecture of Sarason and Crofoot about the existence of certain extremal polynomials $ p$. We also count the number of equivalence classes of these polynomials.


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Additional Information

Lukas Geyer
Affiliation: Department of Mathematics, Montana State University, P.O. Box 172400, Bozeman, Montana 59717–2400
Email: geyer@math.montana.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08946-0
Received by editor(s): October 26, 2005
Received by editor(s) in revised form: September 27, 2006
Published electronically: November 2, 2007
Additional Notes: The author was partially supported by a Feodor Lynen Fellowship of the Alexander von Humboldt Foundation.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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