高斯Weyl多项式实零点数目的Berry-Esseen界
Berry-Esseen Bounds for the Number of Real Zeros of Gaussian Weyl Polynomials
- School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文为高斯Weyl多项式实根数目建立Berry-Esseen界，通过平稳逼近与截断移动平均表示，证明标准化根数分布与高斯分布的Kolmogorov距离有界，并导出中心极限定理。
AI中文摘要：
我们建立了高斯Weyl多项式$P_n$实根数目的Berry-Esseen界，其中$n$表示次数并假设其足够大。对于每个高于绝对阈值的固定$B$，令$I_n=[-\sqrt n+B\sqrt{\log n}, \sqrt n-B\sqrt{\log n}]$。一致地对于确定性紧区间$I\subseteq I_n$，其长度$\ell$足够大且依赖于$n$，$I$中标准化实根数目的分布与标准高斯分布的Kolmogorov距离至多为$C\log\ell/\sqrt\ell$。因此，每个满足$\ell\to\infty$的此类区间序列都满足中心极限定理。特别地，取$I=I_n$给出界$C\log n/n^{1/4}$。相同的界也适用于$\mathbb R$上标准化实根数目的分布。关键思想是将多项式零点计数近似为局部依赖随机变量之和。我们首先将多项式与平稳高斯过程耦合，然后截断该过程的移动平均表示以获得有限范围依赖性。这一策略为其他具有高斯系数且存在此类平稳逼近和定量截断的随机多项式提供了获得Berry-Esseen界的途径。
英文摘要：
We establish Berry-Esseen bounds for the number of real roots of Gaussian Weyl polynomials $P_n$, where $n$ denotes the degree and is assumed to be sufficiently large. For each fixed $B$ above an absolute threshold, let $I_n=[-\sqrt n+B\sqrt{\log n}, \sqrt n-B\sqrt{\log n}]$. Uniformly over deterministic compact intervals $I\subseteq I_n$ whose length $\ell$ is sufficiently large and depends on $n$, the distribution of the standardized number of real roots in $I$ has Kolmogorov distance at most $C\log\ell/\sqrt\ell$ from the standard Gaussian distribution. Consequently, every such interval sequence with $\ell\to\infty$ satisfies a central limit theorem. In particular, taking $I=I_n$ gives the bound $C\log n/n^{1/4}$. The same bound holds for the distribution of the standardized number of real roots on $\mathbb R$. The key idea is to approximate the polynomial zero count by a sum of locally dependent random variables. We first couple the polynomial to a stationary Gaussian process and then truncate a moving-average representation of that process to obtain finite-range dependence. This strategy provides a route to Berry-Esseen bounds for other random polynomials with Gaussian coefficients whenever such a stationary approximation and quantitative truncation are available.
