On the average number of zeros of random harmonic polynomials with i.i.d. coefficients: precise asymptotics

Addressing a problem posed by W. Li and A. Wei (2009), we investigate the average number of (complex) zeros of a random harmonic polynomial $p(z) + \overline{q(z)}$ sampled from the Kac ensemble, i.e., where the coefficients are independent identically distributed centered complex Gaussian random variables. We establish a precise asymptotic, showing that when $\text{deg} p = \text{deg} q = n$ tends to infinity the average number of zeros is asymptotic to $\frac{1}{2} n \log n$. We further consider the average number of zeros restricted to various regions in the complex plane leading to interesting comparisons with the classically studied case of analytic Kac polynomials. We also consider deterministic extremal problems for harmonic polynomials with coefficient constraints; using an indirect probabilistic method we show the existence of harmonic polynomials with unimodular coefficients having at least $\frac{2}{\pi} n \log n + O(n)$ zeros. We conclude with a list of open problems.