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

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Motivated by the questions posed by W. V. Li and A. Wei [Proc. Amer. Math. Soc. 137 (2009), pp. 195–204] and the conjecture of E. Lundberg and A. Thomack [On the average number of zeros of random harmonic polynomials with iid coefficients: precise asymptotics, Preprint, https://arxiv.org/ abs/2308.10333, 2023], we study the expected number of zeros of random harmonic polynomials H n , m ( z ) = p n ( z ) + q m ( z ) ¯ 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 ) O(n) when deg ⁡ p = α deg ⁡ q \deg p = \alpha \deg q , where 0 ≤ α > 1 0\leq \alpha >1 . This result extends the previous estimates when m m is a fixed constant or m = n m=n to more general case.