On the Distribution of Complex Roots of Random Polynomials with Heavy-tailed Coefficients

Friedrich Götze, Dmitry Zaporozhets

Published 2011-04-28Version 1

Consider a random polynomial $G_n(z)=\xi_nz^n+...+\xi_1z+\xi_0$ with i.i.d. complex-valued coefficients. Suppose that the distribution of $\log(1+\log(1+|\xi_0|))$ has a slowly varying tail. Then the distribution of the complex roots of $G_n$ concentrates in probability, as $n\to\infty$, to two centered circles and is uniform in the argument as $n\to\infty$. The radii of the circles are $|\xi_0/\xi_\tau|^{1/\tau}$ and $|\xi_\tau/\xi_n|^{1/(n-\tau)}$, where $\xi_\tau$ denotes the coefficient with the maximum modulus.