The Flow of Polynomial Roots Under Differentiation

The question about behavior of gaps between zeros polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger [42] formally derived a nonlocal nonlinear partial differential equation which models dynamics roots differentiation. In this paper, we connect rigorously solutions Steinerberger’s PDE evolution for class trigonometric polynomials. Namely, prove that the distribution derivatives polynomial corresponding remain close all times. global in time control follows from analysis propagation errors equation, turns out be fractional heat with main term similar modulated discretized Laplacian $$(-\Delta )^{1/2}$$ .