# Zero-free neighborhoods around the unit circle for Kac polynomials

Gerardo Barrera, Paulo Manrique-Mirón

Research output: Contribution to journalArticlepeer-review

## 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 non-degenerate real random variables. It is well-known that the roots of a Kac polynomial concentrate around the unit circle as $n\to \infty$ if and only if $\mathbb{E}[\log(1 +|\xi_0|)] <\infty$. Under the finiteness of the second moment, we show that there exists an annulus of width $O(n^{-2}(log n)^{-3})$ around the unit circle which is free of roots with probability $1 - O((log n)^{-1/2})$. The proof relies on small ball probability inequalities and the least common denominator.

Original language English Periodica Mathematica Hungarica 1-18 18 0031-5303 https://doi.org/10.1007/s10998-021-00409-7 Published - 7 Aug 2021 A1 Journal article-refereed

## Fields of Science

• 111 Mathematics
• Salem–Zygmund type inequalities
• Zeros of random polynomials
• 112 Statistics and probability
• Small ball probability
• Locally sub-Gaussian random variables
• Locally sub-Gaussian random variables
• Salem-Zygmund type inequalities
• Small ball probability
• Zeros of random polynomials
• Complex zeros