{"title":"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","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":"{\"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}","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}
An asymptotic refinement of the Gauss-Lucas Theorem for random polynomials with i.i.d. roots
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.