The valence of harmonic polynomials viewed through the probabilistic lens

Abstract

We prove the existence of complex polynomials p ( z ) p(z) of degree n n and q ( z ) q(z) of degree m > n m>n such that the harmonic polynomial p ( z ) + q ( z ) ¯ p(z) + \overline {q(z)} has at least ⌈ n m ⌉ \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 ) + q ( z ) ¯ p(z)+\overline {q(z)} taken over polynomials p p of degree n n and q q of degree m m is m ( m − 1 ) + 3 n − 2 m(m-1)+3n-2 . More broadly, these examples show that there does not exist a linear (in n n ) bound on the valence with a uniform (in m 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.