Flow of the zeros of polynomials under iterated differentiation

Martínez-Finkelshtein, Andrei; Rakhmanov, Evguenii A.

doi:10.2140/apde.2026.19.1417

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Flow of the zeros of polynomials under iterated differentiation

Andrei Martínez-Finkelshtein and Evguenii A. Rakhmanov

Vol. 19 (2026), No. 7, 1417–1445

DOI: 10.2140/apde.2026.19.1417

Abstract

For a monic polynomial $Q_{n}$ of degree $n$ , let $Q_{n, k}$ be its $k$ -th derivative normalized to be monic. Under the only assumption that the sequence ${Q_{n}}$ has a weak-* limiting zero distribution (an empirical distribution of zeros) represented by a probability measure $μ_{0}$ with compact support in the complex plane, we show that, as $n, k \to \infty$ such that $k ∕ n \to t \in (0, 1)$ , the Cauchy transform of the normalized zero-counting measure of the polynomials $Q_{n, k}$ converges in a neighborhood of infinity to an analytic function, uniquely determined by $μ_{0}$ and $t$ , that can be written as the Cauchy transform of a measure $μ_{t}$ , not necessarily uniquely determined unless $μ_{0}$ is supported on the real line.

The family of these Cauchy transforms and, when well defined, the corresponding measures $μ_{t}$ , $t \in (0, 1)$ , 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 $μ_{0}$ , or a nonlocal diffusion equation governing the density of $μ_{t}$ on $ℝ$ .

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.

Keywords

polynomials, zeros, empirical distribution of zeros, weak convergence, inviscid Burgers equation, free probability

Mathematical Subject Classification

Primary: 30C15

Secondary: 30C10, 37F10, 46L54, 76B99

Milestones

Received: 1 December 2024

Revised: 16 July 2025

Accepted: 19 September 2025

Published: 18 September 2026

Authors

Andrei Martínez-Finkelshtein
Department of Mathematics Baylor University Waco, TX United States	Department of Mathematics University of Almería Almería Spain

Evguenii A. Rakhmanov
Department of Mathematics University of South Florida Tampa, FL United States

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Vol. 19, No. 7, 2026