## The Annals of Probability

### Roots of random polynomials with coefficients of polynomial growth

Yen Do, Oanh Nguyen, and Van Vu

#### Abstract

In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coefficients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.

#### Article information

Source
Ann. Probab., Volume 46, Number 5 (2018), 2407-2494.

Dates
Revised: July 2017
First available in Project Euclid: 24 August 2018

https://projecteuclid.org/euclid.aop/1535097632

Digital Object Identifier
doi:10.1214/17-AOP1219

Mathematical Reviews number (MathSciNet)
MR3846831

Zentralblatt MATH identifier
06964341

#### Citation

Do, Yen; Nguyen, Oanh; Vu, Van. Roots of random polynomials with coefficients of polynomial growth. Ann. Probab. 46 (2018), no. 5, 2407--2494. doi:10.1214/17-AOP1219. https://projecteuclid.org/euclid.aop/1535097632

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