Modeling complex root motion of real random polynomials under differentiation

TL;DR

This paper develops nonlocal, nonlinear PDE models to describe the anisotropic dynamics of complex roots of high-degree random polynomials under differentiation, extending previous 1D models to 2D for complex roots.

Contribution

It introduces the first PDE model for the complex roots of random polynomials in 2D, generalizing previous 1D models and assuming a regular distribution with local homogeneity.

Findings

Derived coupled PDEs for complex root dynamics in 2D

Validated models with computational examples using Maple

Extended understanding of root behavior under differentiation

Abstract

In this paper, we consider nonlocal, nonlinear partial differential equations to model anisotropic dynamics of complex root sets of random polynomials under differentiation. These equations aim to generalise the recent PDE obtained by Stefan Steinerberger (2019) in the real case, and the PDE obtained by Sean O'Rourke and Stefan Steinerberger (2020) in the radial case, which amounts to work in 1D. These PDEs approximate dynamics of the complex roots for random polynomials of sufficiently high degree n. The unit of the time t corresponds to n differentiations, and the increment $\Delta$t corresponds to 1 n. The general situation in 2D, in particular for complex roots of real polynomials, was not yet addressed. The purpose of this paper is to present a first attempt in that direction. We assume that the roots are distributed according to a regular distribution with a local homogeneity property (defined in the text), and that this property is maintained under differentiation. This allows us to derive a system of two coupled equations to model the motion. Our system could be interesting for other applications. The paper is illustrated with examples computed with the Maple system.