
Statistics Papers
Document Type
Journal Article
Date of this Version
11-1995
Publication Source
Transactions of the American Mathematical Society
Volume
347
Issue
11
Start Page
4365
Last Page
4384
Abstract
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.
Recommended Citation
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