The valence of harmonic polynomials viewed through the probabilistic lens

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.

11 pages. Revised version including a more careful explanation in the concluding remark 4.1

Cited by in corpus (1)

• A note on the expectation of zeros of random harmonic polynomials: The Kac model