The flow of polynomial roots under differentiation
The question of how roots of polynomials move under differentiation is
classical. Contributions to this subject have been made by Gauss, Lucas,
Marcel Riesz, Polya and many others.
In 2018, Stefan Steinerberger derived formally a PDE that should
describe the dynamics of roots under differentiation in certain
situations. The PDE in question is of hydrodynamic
type and bears a striking resemblance to the models used in mathematical
biology to describe collective behavior and flocking of various species
- such as fish, birds or ants.
The equation is critical, but due to strongly nonlinear form of its
coefficients, proving global regularity for its solutions is harder than
for equations such as Burgers, SQG or Euler alignment model.
I will discuss joint work with Changhui Tan in which we establish global
regularity of Steinerberger's equation and make a rigorous connection
between its solutions and evolution of roots under
differentiation for a class of trigonometric polynomials.