Sharp bounds for the valence of certain harmonic polynomials

By Lukas Geyer. 2005

51 citations recorded by the source corpus.

In Khavinson and Światek (2002) it was proved that harmonic polynomials z - 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2ed64d0ec1522f8908e5da14990613cfbc5801d1

How researchers see this paper

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In Applied Mathematics

1. Seung-Yeop Lee
2. Lei Tan
3. G. Cui
4. Nikolai G. Makarov
5. Alexei Poltoratski
6. Mishko Mitkovski

In Geometry and Topology

1. Seung-Yeop Lee
2. Lei Tan
3. G. Cui
4. Nikolai G. Makarov

Citations in the corpus

• Topology of quadrature domains (2013, result): “We also explain in [37] that the sharpness results for UQDs in the cases n = d and n = 1 are closely related to the sharpness results by Rhie [45] and, respectively, by Geyer [21] concerning the maximal number of solutions to the equation z = r(z) where r is a rational function.”
• A characterization of hyperbolic rational maps (2011, background): “Finally L. Geyer ([ 16 ]) settled the sharpness for all n at once, by constructing ‘a la Thurston’ a polynomial p of degree n with real coefficients and with mutually distinct critical points z1 ,z 2 ,...,z n−1 such that p(zj ) = zj .”