## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 22 (2017), paper no. 74, 18 pp.

### Limiting empirical distribution of zeros and critical points of random polynomials agree in general

#### Abstract

In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences $\{a_n\}_{n\geq 1}$ and $\{b_n\}_{n\geq 1}$ of complex numbers whose limiting empirical measures are the same. By choosing $\xi _n = a_n$ or $b_n$ with equal probability, define the sequence of polynomials by $P_n(z)=(z-\xi _1)\dots (z-\xi _n)$. We show that the limiting measure of zeros and critical points agree for this sequence of random polynomials under some assumption. We also prove a similar result for triangular array of numbers. A similar result for zeros of generalized derivative (can be thought as random rational function) is also proved. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.

#### Article information

**Source**

Electron. J. Probab., Volume 22 (2017), paper no. 74, 18 pp.

**Dates**

Received: 2 September 2016

Accepted: 24 July 2017

First available in Project Euclid: 13 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1505268105

**Digital Object Identifier**

doi:10.1214/17-EJP85

**Mathematical Reviews number (MathSciNet)**

MR3698743

**Zentralblatt MATH identifier**

1376.30003

**Subjects**

Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}

Secondary: 60G57: Random measures 60B10: Convergence of probability measures

**Keywords**

random polynomials random rational functions zeros critical points Gauss-Lucas theorem potential theory

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Reddy, Tulasi Ram. Limiting empirical distribution of zeros and critical points of random polynomials agree in general. Electron. J. Probab. 22 (2017), paper no. 74, 18 pp. doi:10.1214/17-EJP85. https://projecteuclid.org/euclid.ejp/1505268105