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 | References | Similar Articles | Additional Information
Abstract: In Khavinson and Swiatek (2002) it was proved that harmonic polynomials , where
is a holomorphic polynomial of degree
, have at most
complex zeros. We show that this bound is sharp for all
by proving a conjecture of Sarason and Crofoot about the existence of certain extremal polynomials
. 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.