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

Motivated by the questions posed by W. V. Li and A. Wei and the conjecture of E. Lundberg and A. Thomack, we study the expected number of zeros of random harmonic polynomials $H_{n,m}(z)=p_n(z)+\overline{q_m(z)}$ with independently and identically distributed Gaussian coefficients. In this paper we verify the conjecture of E. Lundberg and A. Thomack that the expectation is $O(n)$ when $\mathrm{°}p=\alpha \mathrm{°}q$, where $0\le \alpha <1$. This result extends the previous estimates when $m$ is a fixed constant or $m=n$ to more general case.