Zero-free neighborhood around the unit circle for Kac polynomials

TL;DR

This paper demonstrates that for Kac polynomials with certain coefficient distributions, there exists a shrinking annulus around the unit circle free of roots with high probability, refining understanding of root concentration behavior.

Contribution

It establishes a zero-free annulus around the unit circle for Kac polynomials under finite second moment conditions, with an explicit width shrinking as n grows.

Findings

Existence of a zero-free annulus of width O(n^{-2} (log n)^{-3}) around the unit circle.

High probability (1 - O((log n)^{-1/2})) that this annulus contains no roots.

Root concentration occurs near the unit circle under specified conditions.

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 $\mathbb{E}[\xi^2_0]<\infty$, 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.