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

TL;DR

This paper analyzes a nonlinear integro-differential PDE modeling the evolution of polynomial roots' density, establishing well-posedness and connecting it to particle system approximations.

Contribution

It provides a rigorous well-posedness framework for a complex PDE related to polynomial root dynamics, using viscosity solutions and comparison principles.

Findings

Established comparison principle for the PDE

Proved existence and uniqueness of solutions

Linked particle system approximations to the PDE

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.