The valence of harmonic polynomials viewed through the probabilistic lens

Lundberg, Erik

doi:10.1090/proc/16152

The valence of harmonic polynomials viewed through the probabilistic lens
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by Erik Lundberg;

Proc. Amer. Math. Soc. 151 (2023), 2963-2973

DOI: https://doi.org/10.1090/proc/16152

Published electronically: April 13, 2023

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Abstract:

We prove the existence of complex polynomials $p(z)$ of degree $n$ and $q(z)$ of degree $m<n$ such that the harmonic polynomial $p(z) + \overline {q(z)}$ has at least $\lceil n \sqrt {m} \rceil$ many zeros. This provides an array of new counterexamples to Wilmshurst’s conjecture that the maximum valence of harmonic polynomials $p(z)+\overline {q(z)}$ taken over polynomials $p$ of degree $n$ and $q$ of degree $m$ is $m(m-1)+3n-2$. More broadly, these examples show that there does not exist a linear (in $n$) bound on the valence with a uniform (in $m$) growth rate. The proof of this result uses a probabilistic technique based on estimating the average number of zeros of a certain family of random harmonic polynomials.

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