15 November 2021 Random polynomials: Central limit theorems for the real roots
Oanh Nguyen, Van Vu
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Duke Math. J. 170(17): 3745-3813 (15 November 2021). DOI: 10.1215/00127094-2020-0089


The number of real roots has been a central subject in the theory of random polynomials and random functions since the fundamental papers of Littlewood, Offord, and Kac in the 1940s. The main task here is to determine the limiting distribution of this random variable. In 1974, Maslova famously proved a central limit theorem (CLT) for the number of real roots of Kac polynomials. It has remained the only limiting theorem available for the number of real roots for more than four decades. In this paper, using a new approach, we derive a general CLT for the number of real roots of a large class of random polynomials with coefficients growing polynomially. Our result both generalizes and strengthens Maslova’s theorem.


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Oanh Nguyen. Van Vu. "Random polynomials: Central limit theorems for the real roots." Duke Math. J. 170 (17) 3745 - 3813, 15 November 2021. https://doi.org/10.1215/00127094-2020-0089


Received: 14 April 2019; Revised: 16 November 2020; Published: 15 November 2021
First available in Project Euclid: 18 November 2021

MathSciNet: MR4340724
zbMATH: 1482.60041
Digital Object Identifier: 10.1215/00127094-2020-0089

Primary: 60F05
Secondary: 97K60

Keywords: central limit theorem , random polynomial , Universality

Rights: Copyright © 2021 Duke University Press


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Vol.170 • No. 17 • 15 November 2021
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