The complex zeros of random polynomials

Mark Kac gave an explicit formula for the expectation of the number, ${\nu _n}(\Omega )$, of zeros of a random polynomial,

$${P_n}(z) = \sum\limits_{j = 0}^{n - 1} {{\eta _j}{z^j}} ,$$

in any measurable subset $\Omega$ of the reals. Here, ${\eta _0}, \ldots ,{\eta _{n - 1}}$ are independent standard normal random variables. In fact, for each $n > 1$, he obtained an explicit intensity function ${g_n}$ for which

$${\mathbf{E}}{\nu _n}(\Omega ) = \int_\Omega {{g_n}(x)\,dx.}$$

Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset $\Omega$ of the complex plane $\mathbb{C}$. Namely, we show that

$${\mathbf{E}}{\nu _n}(\Omega ) = \int_\Omega {{h_n}(x,y)\,dxdy + \int_{\Omega \cap \mathbb{R}} {{g_n}(x)\,dx,} }$$

where ${h_n}$ is an explicit intensity function. We also study the asymptotics of ${h_n}$ showing that for large $n$ its mass lies close to, and is uniformly distributed around, the unit circle.