迭代微分下多项式零点的流动

Andrei Martinez-Finkelshtein, Evgenii A. Rakhmanov
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引用次数: 0

摘要

给定一个阶数为 $n$ 的多项式序列 $Q_n$,我们考虑导数 $Q_{n, k}(x)=d^k Q_n(x) /d x^k$ 的三角形表。在序列 $\{Q_n\}$ 具有弱*极限零分布(零的经验分布)这一唯一假设下,我们证明当 $n、k \rightarrow \infty$ 使得 $k / n \rightarrow t \in(0,1)$ 时,多项式 $Q_{n, k}$ 的计零度量的考奇变换在无穷邻域收敛于度量 $\mu_t$ 的考奇变换。量$\mu_t$,$t \in(0,1)$的族,其对参数$t$的依赖性可以解释为迭代微分下的零点流,与不粘性布尔格序列、$\mu_0$的分数自由卷积或$\mathbb R$上控制$\mu_t$密度的非局部扩散方程有一些有趣的联系。本文的主要目的是对所有这些事实提供一个简化的基本证明。
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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.
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