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. Interestingly, Dimitri Shlyakhtenko and Terry Tao have later formally obtained the same PDE as the evolution equation for free fractional convolution of a measure, an object appearing in free probability. 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.