Dmitry Zaporozhets

On Distribution of Zeros of Random Polynomials in Complex Plane

Let \(G_{n}(z) = \xi _{0} + \xi _{1}z + \cdots + \xi _{n}{z}^{n}\) 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, 2π] asymptotically as \(n\,\rightarrow \,\infty \). We also prove that the condition \(\mathbf{E}\,\ln (1 + \vert \xi _{0}\vert )\,<\,\infty \) is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference.

Asymptotic distribution of complex zeros of random analytic functions

Let

Roots of random polynomials whose coefficients have logarithmic tails

\xi_0,\xi_1,\ldots

Convex hulls of random walks, hyperplane arrangements, and Weyl chambers

be independent identically distributed complex- valued random variables such that

Correlation functions of real zeros of random polynomials

\mathbb{E}\log(1+|\xi _0|)<\infty

ON THE DISTRIBUTION OF COMPLEX ROOTS OF RANDOM POLYNOMIALS WITH HEAVY-TAILED COEFFICIENTS

. We consider random analytic functions of the form \[\mathbf{G}_n(z)=\sum_{k=0}^{\infty}\xi_kf_{k,n}z^k,\] where

Mean width of regular polytopes and expected maxima of correlated Gaussian variables

f_{k,n}

UNIVERSALITY FOR ZEROS OF RANDOM ANALYTIC FUNCTIONS

are deterministic complex coefficients. Let

CONVEX HULLS OF MULTIDIMENSIONAL RANDOM WALKS

\mu_n

On the distribution of the number of real zeros of a random polynomial

be the random measure counting the complex zeros of