The complex zeros of random polynomials

Abstract:

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.