The expected number of complex zeros of complex random polynomials (Q1746528)

\textit{M. Kac} [Bull. Am. Math. Soc. 49, 314--320 (1943; Zbl 0060.28602); ibid. 49, 938 (1943; Zbl 0060.28603)] gave an explicit formula for the expectation \(\mathbb{E}[\nu_n(\Omega)]\) of the number of zeros \(\nu_n(\Omega )\) of a random polynomial \( P_n(z) =\sum\nolimits_{i=1}^{n-1} \eta_j z^j\) in any measurable subset \(\Omega\) of the reals, where the coefficients \(\eta_j \) are independent standard normal random variables. Much later \textit{L. A. Shepp} and \textit{R. J. Vanderbei} [Trans. Am. Math. Soc. 347, No. 11, 4365--4384 (1995; Zbl 0841.30006)] found the expected number \(\mathbb{E}[\nu_n(\Lambda)]\) of complex zeros of complex random polynomials in a measurable subset \(\Lambda\) of the complex plane. Using their technique, the authors obtain an explicit evaluation of the average intensity function in a closed form. The main result of this paper is the following: Let \(\{\alpha_j\}_{j=0}^{n-1}\) and \(\{\beta_j\}_{j=0}^{n-1}\) be sequences of independent and identically distributed real standard normal random variables. Suppose that the coefficients \(\eta_j\) of the random polynomial \(P_n\) are defined by \(\eta_j =\alpha_j +i \beta_j\) for \(j =0, \dots , n-1\). Then for any measurable region \(\Lambda \) of the complex plane and any integer \(n>1\), \[ \mathbb{E}[\nu_n(\Lambda)]= \int_{\Lambda}h_n(x, y) dx dy, \] where the intensity function \(h_n\) is given by \[ h_ n =\frac{1}{\pi}\left(\frac{1}{(1-|z|^2)^{2}}- \frac{n^2|z|^{2n-2}}{(1-|z|^{2n})^{2}}\right). \] The limiting expressions for the intensity function and the expected number of zeros in open circular disks in the complex plane are also derived.