Compositio Mathematica

Research Article

The zeros of random polynomials cluster uniformly near the unit circle

C. P. Hughesa1 and A. Nikeghbalia2

a1 Department of Mathematics, University of York, York, YO10 5DD, UK (email: ch540@york.ac.uk)

a2 Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland (email: ashkan.nikeghbali@math.unizh.ch)

Abstract

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.

(Received July 03 2006)

(Accepted August 24 2007)

(Online publication March 14 2008)

Keywords

  • random polynomials;
  • roots;
  • uniform clustering

2000 Mathematics subject classification

  • 30C15