Zero-free neighborhoods around the unit circle for Kac polynomials

Gerardo Barrera, Paulo Manrique-Mirón

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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.

TidskriftPeriodica Mathematica Hungarica
Sidor (från-till)1-18
Antal sidor18
StatusPublicerad - 7 aug 2021
MoE-publikationstypA1 Tidskriftsartikel-refererad


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