Date of this Version
Transactions of the American Mathematical Society
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.
Shepp, L. A., & Vanderbei, R. J. (1995). The Complex Zeros of Random Polynomials. Transactions of the American Mathematical Society, 347 (11), 4365-4384. Retrieved from https://repository.upenn.edu/statistics_papers/187
Date Posted: 27 November 2017