On the zeros of random harmonic polynomials: the naive model

Abstract

A complex harmonic polynomial is the sum of a complex polynomial and a conjugated complex polynomial, of degrees $n$ and $m$ respectively. Li and Wei (2009) presented a formula for the expected number of zeros of a random harmonic polynomial in $\mathbb{C}$. In this paper we prove that if $m$ is a fixed number, this expectation is asymptotically $n$ as $n\rightarrow\infty$, and if $m=n$ we find a lower and upper bound of order $n\log n$ for this expectation for sufficiently large $n$.