论文标题
随机多项式：最接近单位圆的根
Random polynomials: the closest roots to the unit circle
论文作者
论文摘要
令$ f = \ sum_ {k = 0}^n \ varepsilon_k z^k $为一个随机多项式，其中$ \ varepsilon_0，\ ldots，\ varepsilon_n $是iid standard Gaussian Gaussian随机变量，以及让$ q em em em em em em em em em em em em em em em \ ldots，$ eme $ ememe $ demote。我们表明，点过程由根的大小$ \ {1- |ζ_1|，\ ldots，1- |ζ_n|确定。 \} $倾向于以$ n^{ - 2} $为$ n \ rightarrow \ infty $的泊松点过程。结果的结果之一是它决定了最接近单位圆的根的大小。特别是，我们表明\ [\ min_ {k} ||ζ_K| -1 | n^2 \ rightArrow \ mathrm {exp}（1/6），\]，其中$ \ mathrm {exp}（λ）$表示指数的随机变量，该变量的均值$λ^{ - 1} $。从1995年开始，这解决了Shepp和Vanderbei的猜想，后来由Konyagin和Schlag研究。
Let $f = \sum_{k=0}^n \varepsilon_k z^k$ be a random polynomial, where $\varepsilon_0,\ldots ,\varepsilon_n$ are iid standard Gaussian random variables, and let $ζ_1,\ldots,ζ_n$ denote the roots of $f$. We show that the point process determined by the magnitude of the roots $\{ 1-|ζ_1|,\ldots, 1-|ζ_n| \}$ tends to a Poisson point process at the scale $n^{-2}$ as $n\rightarrow \infty$. One consequence of this result is that it determines the magnitude of the closest root to the unit circle. In particular, we show that \[ \min_{k} ||ζ_k| - 1|n^2 \rightarrow \mathrm{Exp}(1/6),\] in distribution, where $\mathrm{Exp}(λ)$ denotes an exponential random variable of mean $λ^{-1}$. This resolves a conjecture of Shepp and Vanderbei from 1995 that was later studied by Konyagin and Schlag.
