Take a (more or less, random) polynomial 𝑃 of degree 𝑁, and differentiate it several times keeping track of the roots of those sequential derivatives, 𝑃,𝑃′,𝑃′′,…. Remarkably, the roots seem to form tracks, as if sliding towards the origin like water drops.
A conceptual explanation (a physicist’s proof) is quite easy: in the standard electrostatic interpretation, the zeros of the derivative of a polynomial are the stationary point of the Coulomb force field corresponding to the ensemble of point charges sitting at the roots of that polynomial, and near a root, the remainder of the charges feel like a homogenized (locally nearly linear) potential.
For a mathematical elaboration of that idea, see Hanin’s paper here.
The goal of this project is to investigate the collective behaviors in more details. If one can pair the roots of a polynomial and its derivative, with high probability, one can form directed forests out of the collective roots of the derivatives. What are the typical shapes of those forests? What are their local statistics?
Familiarity with basic complex analysis or electrostatics, confidence in coding in Python, or Julia, or Mathematica.