Zero-free neighborhood around the unit circle for Kac polynomials

Abstract

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

New version. The title of the manuscript was updated

References in corpus (2)

• Asymptotic distribution of complex zeros of random analytic functions
• Salem-Zygmund Inequality for locally sub-Gaussian random variables, random trigonometric polynomials, and random circulant matrices