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The zeros of random polynomials cluster uniformly near the unit circle

Part of: Geometric function theory

Published online by Cambridge University Press:  01 May 2008

C. P. Hughes  and
A. Nikeghbali
C. P. Hughes
Affiliation:
Department of Mathematics, University of York, York, YO10 5DD, UK (email: [email protected])
A. Nikeghbali
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland (email: [email protected])
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Abstract

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In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdős and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomial.

Keywords

random polynomialsrootsuniform clustering

MSC classification

Secondary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral)
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 3 , May 2008 , pp. 734 - 746
Copyright
Copyright © Foundation Compositio Mathematica 2008