The Complex Zeros of Random Polynomials

Mark Kac gave an explicit formula for the expectation of the number, νn(Ω), of zeros of a random polynomial, Pn(z) = ∑n−1j=0 ηjzj , in any measurable subset Ω of the reals. Here, η0, . . . , ηn−1 are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function gn for which Eνn(Ω) = ∫Ω gn(x)dx. Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset Ω of the complex plane IC. Namely, we show that Eνn(Ω) = ∫Ω hn(x, y)dxdy + ∫Ω∩IR gn(x)dx, where hn is an explicit intensity function. We also study the asymptotics of hn showing that for large n its mass lies close to, and is uniformly distributed around, the unit circle.