On distribution of zeros of random polynomials in complex plane

Abstract

Let Gn(z)=0+1z+...+n zn be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of Gn(z) are uniformly distributed in [0,2π] asymptotically as n∞. We also prove that the condition (1+|0|)<∞ is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference.