## 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, *ν _{n}*(Ω), of zeros of a random polynomial,

P_{n}(z) = ∑^{n−1}_{j=0} η_{j}z^{j} ,

in any measurable subset Ω of the reals. Here, η_{0}, . . . , 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

**Date Posted:** 27 November 2017