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2017 Limiting empirical distribution of zeros and critical points of random polynomials agree in general
Tulasi Ram Reddy
Electron. J. Probab. 22(none): 1-18 (2017). DOI: 10.1214/17-EJP85

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.

Citation

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Tulasi Ram Reddy. "Limiting empirical distribution of zeros and critical points of random polynomials agree in general." Electron. J. Probab. 22 1 - 18, 2017. https://doi.org/10.1214/17-EJP85

Information

Received: 2 September 2016; Accepted: 24 July 2017; Published: 2017
First available in Project Euclid: 13 September 2017

zbMATH: 1376.30003
MathSciNet: MR3698743
Digital Object Identifier: 10.1214/17-EJP85

Subjects:
Primary: 30C15
Secondary: 60B10, 60G57

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