Zero-free neighborhoods around the unit circle for Kac polynomials

Abstract

In this paper, we study how the roots of the Kac polynomials W_n(z) = sum _{k=0}^{n-1} xi _k z^k concentrate around the unit circle when the coefficients of W_n are independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as nrightarrow infty if and only if {mathbb {E}}[log ( 1+ |xi _0|)]<infty . Under the condition {mathbb {E}}[xi ^2_0]<infty , we show that there exists an annulus of width {text {O}}(n^{-2}(log n)^{-3}) around the unit circle which is free of roots with probability 1-{text {O}}({(log n)^{-{1}/{2}}}). The proof relies on small ball probability inequalities and the least common denominator used in [17].