Sharp bounds for the valence of certain harmonic polynomials

Geyer, Lukas

doi:10.1090/S0002-9939-07-08946-0

Sharp bounds for the valence of certain harmonic polynomials
HTML articles powered by AMS MathViewer

by Lukas Geyer

Proc. Amer. Math. Soc. 136 (2008), 549-555

DOI: https://doi.org/10.1090/S0002-9939-07-08946-0

Published electronically: November 2, 2007

PDF | 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

Ben Bielefeld, Yuval Fisher, and John Hubbard, The classification of critically preperiodic polynomials as dynamical systems, J. Amer. Math. Soc. 5 (1992), no. 4, 721–762. MR 1149891, DOI 10.1090/S0894-0347-1992-1149891-3
Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263–297. MR 1251582, DOI 10.1007/BF02392534
Dmitry Khavinson and Genevra Neumann, On the number of zeros of certain rational harmonic functions, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1077–1085. MR 2196041, DOI 10.1090/S0002-9939-05-08058-5
Dmitry Khavinson and Grzegorz Świa̧tek, On the number of zeros of certain harmonic polynomials, Proc. Amer. Math. Soc. 131 (2003), no. 2, 409–414. MR 1933331, DOI 10.1090/S0002-9939-02-06476-6
Levy, S., Critically finite rational maps, Ph.D. Thesis, Princeton University, 1985.
Kevin M. Pilgrim, Canonical Thurston obstructions, Adv. Math. 158 (2001), no. 2, 154–168. MR 1822682, DOI 10.1006/aima.2000.1971
Poirier, A., Hubbard forests, IMS Preprint 92-12, (1992).
Poirier, A., On post critically finite polynomials. Part two: Hubbard trees, IMS Preprint 93-07, (1993).
Rhie, S. H., $n$-point Gravitational Lenses with $5(n-1)$ Images, arXiv:astro-ph/0305166v1.