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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Sharp bounds for the valence of certain harmonic polynomials
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by Lukas Geyer PDF
Proc. Amer. Math. Soc. 136 (2008), 549-555 Request permission

Abstract:

In Khavinson and Świa̧tek (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.
References
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Additional Information
  • Lukas Geyer
  • Affiliation: Department of Mathematics, Montana State University, P.O. Box 172400, Bozeman, Montana 59717–2400
  • MR Author ID: 638391
  • Email: geyer@math.montana.edu
  • 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
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2358495