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

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.

💡 Research Summary

Abstract and Motivation
The paper addresses a nonlinear, nonlocal partial differential equation (PDE) that describes how the density of polynomial roots evolves under repeated differentiation. This PDE, originally derived heuristically by Steinerberg on the real line and later considered on the unit circle, reads
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