An asymptotic refinement of the Gauss-Lucas Theorem for random polynomials with i.i.d. roots

Sean O'Rourke, Noah Williams
{"title":"An asymptotic refinement of the Gauss-Lucas Theorem for random polynomials with i.i.d. roots","authors":"Sean O'Rourke, Noah Williams","doi":"arxiv-2409.09538","DOIUrl":null,"url":null,"abstract":"If $p:\\mathbb{C} \\to \\mathbb{C}$ is a non-constant polynomial, the\nGauss--Lucas theorem asserts that its critical points are contained in the\nconvex hull of its roots. We consider the case when $p$ is a random polynomial\nof degree $n$ with roots chosen independently from a radially symmetric,\ncompactly supported probably measure $\\mu$ in the complex plane. We show that\nthe largest (in magnitude) critical points are closely paired with the largest\nroots of $p$. This allows us to compute the asymptotic fluctuations of the\nlargest critical points as the degree $n$ tends to infinity. We show that the\nlimiting distribution of the fluctuations is described by either a Gaussian\ndistribution or a heavy-tailed stable distribution, depending on the behavior\nof $\\mu$ near the edge of its support. As a corollary, we obtain an asymptotic\nrefinement to the Gauss--Lucas theorem for random polynomials.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"40 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09538","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

If $p:\mathbb{C} \to \mathbb{C}$ is a non-constant polynomial, the Gauss--Lucas theorem asserts that its critical points are contained in the convex hull of its roots. We consider the case when $p$ is a random polynomial of degree $n$ with roots chosen independently from a radially symmetric, compactly supported probably measure $\mu$ in the complex plane. We show that the largest (in magnitude) critical points are closely paired with the largest roots of $p$. This allows us to compute the asymptotic fluctuations of the largest critical points as the degree $n$ tends to infinity. We show that the limiting distribution of the fluctuations is described by either a Gaussian distribution or a heavy-tailed stable distribution, depending on the behavior of $\mu$ near the edge of its support. As a corollary, we obtain an asymptotic refinement to the Gauss--Lucas theorem for random polynomials.
具有 i.i.d. 根的随机多项式的高斯-卢卡斯定理的渐近改进
如果 $p:\mathbb{C}\到 \mathbb{C}$ 是一个非常数多项式,高斯-卢卡斯定理断言其临界点包含在其根的凸壳中。我们考虑的情况是,当 $p$ 是阶数为 $n$ 的随机多项式时,其根是从复平面中一个径向对称、紧凑支撑的可能度量 $\mu$ 中独立选择的。我们证明,最大(幅度)临界点与 $p$ 的最大根密切相关。这使我们能够计算最大临界点在阶数 $n$ 趋于无穷大时的渐近波动。我们证明,波动的极限分布可以用高斯分布或重尾稳定分布来描述,这取决于 $\mu$ 在其支持边缘附近的行为。作为推论,我们得到了随机多项式的高斯-卢卡斯定理的渐近修正。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信