On distribution of zeros of random polynomials in complex plane

Details On distribution of zeros of random polynomials in complex plane On distribution of zeros of random polynomials in complex plane

2011-02-17

arxiv.org/abs/1102.3517v1

Mathematics

Probability

Scientific paper

Let $G_n(z)=\xi_0+\xi_1z+...+\xi_n z^n$ be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of $G_n(z)$ are uniformly distributed in $[0,2\pi]$ asymptotically as $n\to\infty$. We also prove that the condition $\E\ln(1+|\xi_0|)<\infty$ is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference.

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