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- W3093094620 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 $ntoinfty$ 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." @default.
- W3093094620 created "2020-10-22" @default.
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- W3093094620 date "2018-07-30" @default.
- W3093094620 modified "2023-09-27" @default.
- W3093094620 title "Zero-free neighborhood around the unit circle for Kac polynomials" @default.
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