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
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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.

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