ScholarlyCommons

Title

The Complex Zeros of Random Polynomials

Author(s)

Larry A. Shepp, University of Pennsylvania
Robert J. Vanderbei

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, ν_n(Ω), of zeros of a random polynomial,

P_n(z) = ∑ⁿ⁻¹_j=0 η_jz^j ,

in any measurable subset Ω of the reals. Here, η₀, . . . , η_n−1 are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function g_n for which

Eν_n(Ω) = ∫_Ω g_n(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

Eν_n(Ω) = ∫_Ω h_n(x, y)dxdy + ∫_Ω∩IR g_n(x)dx,

where h_n is an explicit intensity function. We also study the asymptotics of h_n 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