2024/08/25 by Andrei Martı́nez-Finkelshtein, Martinez-Finkelshtein, Andrei, Евгений Андреевич Рахманов +1 · 2 citations
Mathematics · #Iterative Methods for Nonlinear Equations
paper · pdf · doi:10.48550/arxiv.2408.13851
For a monic polynomial Qn of degree n, let Qn, k be its k-th derivative normalized to be monic. Under the only assumption that the sequence \Qn\ has a weak* limiting zero distribution (an empirical distribution of zeros) represented by a probability measure μ0 with compact support in the complex plane, we show that as n, k → ∞ such that k / n → t ∈(0,1), the Cauchy transform of the normalized zero-counting measure of the polynomials Qn, k converges in a neighborhood of infinity to an analytic function, uniquely determined by μ0 and t, that can be written as the Cauchy transform of a measure μt, not necessarily uniquely determined unless μ0 is supported on the real line. The family of these Cauchy transforms and, when well defined, the corresponding measures μt , t ∈(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 μ0, or a nonlocal diffusion equation governing the density of μt on \mathbb R. We provide an elementary and unified approach that not only recovers, but also explains various phenomena observed in prior works - from Burgers-type PDEs to free probability limits.