THE COMPLEX ZEROS OF RANDOM POLYNOMIALS

Mark Kac gave an explicit formula for the expectation of the number, n u(n)(Omega), of zeros of a random polynomial, GRAPHICS in any measur able subset Omega of the reals. Here, eta(0), ..., eta(n-1) are indepe ndent standard normal random variables. In fact, for each n > 1, he ob tained an explicit intensity function g(n) for which E nu(n)(Omega) = integral(Omega) g(n)(x) dx. Here, we extend this formula to obtain an explicit formula far the expected number of zeros in any measurable su bset Omega of the complex plane C. Namely, we show that E nu(n)(Omega) = integral(Omega) h(n)(x, y) dxdy + integral(Omega boolean AND R) g(n )(x) dx, where h(n) is an explicit intensity function. We also study t he asymptotics of h(n) showing that for large n its mass lies close to , and is uniformly distributed around, the unit circle.