The complex zeros of random polynomials

Mark Kac gave an explicit formula for the expectation of the number, vn(Ω), of zeros of a random polynomial, in any measurable subset Ci of the reals. Here, η_o,., η_n-1are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function g for which 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 ℂ. Namely, we show that where h_nis an explicit intensity function. We also study the asymptotics of h_nshowing that for large n its mass lies close to, and is uniformly distributed around, the unit circle.