Andrei Martinez-Finkelshtein, Evgenii A. Rakhmanov
{"title":"迭代微分下多项式零点的流动","authors":"Andrei Martinez-Finkelshtein, Evgenii A. Rakhmanov","doi":"arxiv-2408.13851","DOIUrl":null,"url":null,"abstract":"Given a sequence of polynomials $Q_n$ of degree $n$, we consider the\ntriangular table of derivatives $Q_{n, k}(x)=d^k Q_n(x) /d x^k$. Under the only\nassumption that the sequence $\\{Q_n\\}$ has a weak* limiting zero distribution\n(an empirical distribution of zeros) represented by a unit measure $\\mu_0$ with\ncompact support in the complex plane, we show that as $n, k \\rightarrow \\infty$\nsuch that $k / n \\rightarrow t \\in(0,1)$, the Cauchy transform of the\nzero-counting measure of the polynomials $Q_{n, k}$ converges in a neighborhood\nof infinity to the Cauchy transform of a measure $\\mu_t$. The family of measures $\\mu_t $, $t \\in(0,1)$, whose dependence on the\nparameter $t$ can be interpreted as a flow of the zeros under iterated\ndifferentiation, has several interesting connections with the inviscid Burgers\nequation, the fractional free convolution of $\\mu_0$, or a nonlocal diffusion\nequation governing the density of $\\mu_t$ on $\\mathbb R$. The main goal of this paper is to provide a streamlined and elementary proof\nof all these facts.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"22 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Flow of the zeros of polynomials under iterated differentiation\",\"authors\":\"Andrei Martinez-Finkelshtein, Evgenii A. Rakhmanov\",\"doi\":\"arxiv-2408.13851\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given a sequence of polynomials $Q_n$ of degree $n$, we consider the\\ntriangular table of derivatives $Q_{n, k}(x)=d^k Q_n(x) /d x^k$. Under the only\\nassumption that the sequence $\\\\{Q_n\\\\}$ has a weak* limiting zero distribution\\n(an empirical distribution of zeros) represented by a unit measure $\\\\mu_0$ with\\ncompact support in the complex plane, we show that as $n, k \\\\rightarrow \\\\infty$\\nsuch that $k / n \\\\rightarrow t \\\\in(0,1)$, the Cauchy transform of the\\nzero-counting measure of the polynomials $Q_{n, k}$ converges in a neighborhood\\nof infinity to the Cauchy transform of a measure $\\\\mu_t$. The family of measures $\\\\mu_t $, $t \\\\in(0,1)$, whose dependence on the\\nparameter $t$ can be interpreted as a flow of the zeros under iterated\\ndifferentiation, has several interesting connections with the inviscid Burgers\\nequation, the fractional free convolution of $\\\\mu_0$, or a nonlocal diffusion\\nequation governing the density of $\\\\mu_t$ on $\\\\mathbb R$. The main goal of this paper is to provide a streamlined and elementary proof\\nof all these facts.\",\"PeriodicalId\":501145,\"journal\":{\"name\":\"arXiv - MATH - Classical Analysis and ODEs\",\"volume\":\"22 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Classical Analysis and ODEs\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.13851\",\"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 - Classical Analysis and ODEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.13851","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Flow of the zeros of polynomials under iterated differentiation
Given a sequence of polynomials $Q_n$ of degree $n$, we consider the
triangular table of derivatives $Q_{n, k}(x)=d^k Q_n(x) /d x^k$. Under the only
assumption that the sequence $\{Q_n\}$ has a weak* limiting zero distribution
(an empirical distribution of zeros) represented by a unit measure $\mu_0$ with
compact support in the complex plane, we show that as $n, k \rightarrow \infty$
such that $k / n \rightarrow t \in(0,1)$, the Cauchy transform of the
zero-counting measure of the polynomials $Q_{n, k}$ converges in a neighborhood
of infinity to the Cauchy transform of a measure $\mu_t$. The family of measures $\mu_t $, $t \in(0,1)$, whose dependence on the
parameter $t$ can be interpreted as a flow of the zeros under iterated
differentiation, has several interesting connections with the inviscid Burgers
equation, the fractional free convolution of $\mu_0$, or a nonlocal diffusion
equation governing the density of $\mu_t$ on $\mathbb R$. The main goal of this paper is to provide a streamlined and elementary proof
of all these facts.