Roots of iterated derivatives of Random polynomials and transport equations.

Andre Galligo

Ploting roots sets of iterated derivatives, of large degree random polynomials, show intringing patterns, suggesting a discretised flow towards the origin and the real axis. In the case when all roots are real, considering their density limite S. Steinerberger recently derived a non-local PDE (relying on Hilbert transform) as a model of the motion. This PDE and an associated kinetic version have interesting mathematical properties. It also turns out that a ``natural'' complex extension of that PDE is the complex Burgers' equation, used in fluid dynamics.