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In this paper, we consider nonlocal, nonlinear partial differential equationsto model anisotropic dynamics of complex root sets of random polynomials underdifferentiation. These equations aim to generalise the recent PDE obtained byStefan Steinerberger (2019) in the real case, and the PDE obtained by SeanO'Rourke and Stefan Steinerberger (2020) in the radial case, which amounts towork in 1D. These PDEs approximate dynamics of the complex roots for randompolynomials of sufficiently high degree n. The unit of the time t correspondsto n differentiations, and the increment $\Delta$t corresponds to 1 n. Thegeneral 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 attemptin that direction. We assume that the roots are distributed according to aregular distribution with a local homogeneity property (defined in the text),and that this property is maintained under differentiation. This allows us toderive a system of two coupled equations to model the motion. Our system couldbe interesting for other applications. The paper is illustrated with examplescomputed with the Maple system.

Modeling complex root motion of real random polynomials under differentiation