Heat flow and repeated differentiation of polynomials with i.i.d. roots
Abstract
How do the zeros of a polynomial evolve under the holomorphic heat flow or repeated differentiation? In this work, we develop a unified probabilistic proof to three conjectures on the evolution of polynomial zeros under holomorphic heat flow and repeated differentiation. We study these two evolutions for random polynomials with i.i.d.~roots z1,…,zn∼μ0, where μ0 on ℂ satisfies suitable 2+δ logarithmic moment conditions. For the heat flow of small time t>0 and Lipschitz continuous Stieltjes transform of μ0, we identify the limit distribution μt as explicit push-forward of μ0 under a transport map. For rotationally invariant μ0, we characterize the limit after ⌊ tn⌋ differentiations through its radial quantiles. For instance, the heat flow evolves the circular law into the elliptic and semicircle law, while differentiation moves surviving mass towards the origin. We also show that o(n) derivatives preserve the initial distribution μ0. In fact, all convergences hold almost surely. The common proof strategy crucially relies on recursion identities from leaving out a root, and concentration inequalities, which lead to self-consistent equations for Stieltjes transforms. This brings a method familiar from random matrix theory to polynomial evolutions. Moreover, it allows for generalizations beyond rotational symmetric distributions and quantitative stability estimates for o(n) differentiations and vanishing heat-flow times.
Related
- Repeated differentiation of random polynomials with i.i.d. rotationally invariant roots
- Real-rooted flow polynomials have only integral roots
- A combinatorial criterion for detecting roots of Bernstein-Sato polynomials
- Convergence for Small-Order Derivatives of Random Polynomials with Independent Roots
- On the Roots of Connected Domination Polynomials
BibTeX
@misc{jalowy2026heat,
title = {{Heat flow and repeated differentiation of polynomials with i.i.d. roots}},
author = {Jonas Jalowy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.24867},
eprint = {2609.24867},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24867},
}
RIS
TY - UNPB
TI - Heat flow and repeated differentiation of polynomials with i.i.d. roots
AU - Jonas Jalowy
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.24867
UR - https://arxiv.org/abs/2609.24867
AB - How do the zeros of a polynomial evolve under the holomorphic heat flow or repeated differentiation? In this work, we develop a unified probabilistic proof to three conjectures on the evolution of polynomial zeros under holomorphic heat flow and repeated differentiation. We study these two evolutions for random polynomials with i.i.d.~roots z_1,…,z_n∼μ_0, where μ_0 on ℂ satisfies suitable 2+δ logarithmic moment conditions. For the heat flow of small time t>0 and Lipschitz continuous Stieltjes transform of μ_0, we identify the limit distribution μ_t as explicit push-forward of μ_0 under a transport map. For rotationally invariant μ_0, we characterize the limit after ⌊ tn⌋ differentiations through its radial quantiles. For instance, the heat flow evolves the circular law into the elliptic and semicircle law, while differentiation moves surviving mass towards the origin. We also show that o(n) derivatives preserve the initial distribution μ_0. In fact, all convergences hold almost surely. The common proof strategy crucially relies on recursion identities from leaving out a root, and concentration inequalities, which lead to self-consistent equations for Stieltjes transforms. This brings a method familiar from random matrix theory to polynomial evolutions. Moreover, it allows for generalizations beyond rotational symmetric distributions and quantitative stability estimates for o(n) differentiations and vanishing heat-flow times.
ER -