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The zeros of random polynomials cluster uniformly near the unit circle

Published online by Cambridge University Press:  01 May 2008

C. P. Hughes
Affiliation:
Department of Mathematics, University of York, York, YO10 5DD, UK (email: ch540@york.ac.uk)
A. Nikeghbali
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland (email: ashkan.nikeghbali@math.unizh.ch)
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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.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2008