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Improved well-posedness for the limit flow of differentiation of roots of polynomials

Charles Bertucci, Valentin Pesce

TL;DR

The paper rigorously analyzes the limit flow of roots of polynomials under differentiation on the circle by studying a nonlinear, nonlocal PDE and its integrated primitive form. By introducing a viscosity-solution framework for the primitive equation and a truncated model, the authors establish existence, uniqueness, and stability with respect to initial data, overcoming singularities via a minimum-positivity hypothesis. They further connect the PDE to a discrete system of trigonometric polynomial roots (particle system) and provide a detailed heuristic with partial convergence results toward the viscosity solution, offering insight into Dyson-like dynamics of root flows. Overall, the work advances the rigorous understanding of differentiation-induced root dynamics and their links to free probability and random-polynomial theory.

Abstract

In this paper, we study the partial differential equation on the circle that was heuristically obtained by Steinerberg [32] on the real line and which represents the evolution of the density of the roots of polynomials under differentiation. After integrating the partial differential equation in question, we observe that it can be treated with the theory of viscosity solutions. This equation at hand is a non linear parabolic integro-differential equation which involves the elliptic operator called the half-Laplacian. Due to the singularity of the equation, we restrict our study to strictly positive initial condition. We obtain a comparison principle for solutions of the primitive equation which yields uniqueness, existence, continuity with respect to initial condition. We also present heuristics to justify that the system of particles indeed approximates the solution of the equation.

Improved well-posedness for the limit flow of differentiation of roots of polynomials

TL;DR

The paper rigorously analyzes the limit flow of roots of polynomials under differentiation on the circle by studying a nonlinear, nonlocal PDE and its integrated primitive form. By introducing a viscosity-solution framework for the primitive equation and a truncated model, the authors establish existence, uniqueness, and stability with respect to initial data, overcoming singularities via a minimum-positivity hypothesis. They further connect the PDE to a discrete system of trigonometric polynomial roots (particle system) and provide a detailed heuristic with partial convergence results toward the viscosity solution, offering insight into Dyson-like dynamics of root flows. Overall, the work advances the rigorous understanding of differentiation-induced root dynamics and their links to free probability and random-polynomial theory.

Abstract

In this paper, we study the partial differential equation on the circle that was heuristically obtained by Steinerberg [32] on the real line and which represents the evolution of the density of the roots of polynomials under differentiation. After integrating the partial differential equation in question, we observe that it can be treated with the theory of viscosity solutions. This equation at hand is a non linear parabolic integro-differential equation which involves the elliptic operator called the half-Laplacian. Due to the singularity of the equation, we restrict our study to strictly positive initial condition. We obtain a comparison principle for solutions of the primitive equation which yields uniqueness, existence, continuity with respect to initial condition. We also present heuristics to justify that the system of particles indeed approximates the solution of the equation.
Paper Structure (37 sections, 29 theorems, 191 equations)

This paper contains 37 sections, 29 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. General introduction
  3. Overview of the results of the paper
  4. Structure of the paper
  5. A brief overview of the problem
  6. Location of the roots of the derivative of a polynomial
  7. Known results about random polynomials
  8. Notation
  9. Heuristic of the PDE
  10. Study of the PDE and primitive equation
  11. Notation
  12. Primitive equation
  13. Viscosity solutions of the truncated PDEs
  14. Comparison principles
  15. Consequences of comparison principles for truncated PDEs
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $\mu\in\mathcal{P}(\mathbb T)$ such that $F_\mu$, the cumulative distribution function of $\mu$, satisfies a condition of strict monotony hypothesis min for a $m>0$. Then there exists a viscosity solution of eqPrimitiveintro with initial condition $F_\mu$ satisfying hypothesis min for all time.

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Gauss-Lucas theorem
  • proof
  • Proposition 2.2
  • proof
  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • ...and 46 more