论文标题
随机多项式的单位圆附近的真正根部
Real roots near the unit circle of random polynomials
论文作者
论文摘要
令$ f_n（z）= \ sum_ {k = 0}^n \ varepsilon_k z^k $是一个随机的多项式，其中$ \ varepsilon_0，\ ldots，\ varepsilon_n $是i.i.d.随机变量带有$ \ mathbb {e} \ varepsilon_1 = 0 $和$ \ mathbb {e} \ varepsilon_1^2 = 1 $。让$ r_1，r_2，\ ldots，r_k $表示$ f_n $的真实根，我们表明$ \ {| r_1 |定义的点过程-1，\ ldots，| r_k | -1 \} $收敛到$ n^{ - 1} $的规模的非频繁限制为$ n \ to \ infty $。此外，我们表明，对于每个$δ> 0 $，$ f_n $在单位圆的$θ_δ（1/n）内具有至少$ 1-δ$的真实根。从1995年开始，这解决了Shepp和Vanderbei的猜想，通过确认其最弱的形式并反驳其最强的形式。
Let $f_n(z) = \sum_{k = 0}^n \varepsilon_k z^k$ be a random polynomial where $\varepsilon_0,\ldots,\varepsilon_n$ are i.i.d. random variables with $\mathbb{E} \varepsilon_1 = 0$ and $\mathbb{E} \varepsilon_1^2 = 1$. Letting $r_1, r_2,\ldots, r_k$ denote the real roots of $f_n$, we show that the point process defined by $\{|r_1| - 1,\ldots, |r_k| - 1 \}$ converges to a non-Poissonian limit on the scale of $n^{-1}$ as $n \to \infty$. Further, we show that for each $δ> 0$, $f_n$ has a real root within $Θ_δ(1/n)$ of the unit circle with probability at least $1 - δ$. This resolves a conjecture of Shepp and Vanderbei from 1995 by confirming its weakest form and refuting its strongest form.
