Flow of the zeros of polynomials under iterated differentiation
| Autor: | Martinez-Finkelshtein, Andrei, Rakhmanov, Evgenii A. |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 29 pages |
| Databáze: | arXiv |
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