Brian C. Hall
University of Notre Dame
Title
Evolution of the zeros of polynomials under repeated differentiation
Abstract
A basic question in the study of polynomials is how the roots of the derivative of a polynomial are related to the roots of the original polynomial. For high-degree polynomials, various results indicate that the distribution of roots does not change much when taking a single derivative. We will therefore look at how the roots evolve when the number of derivatives is proportional to the degree of the polynomial.
In the case of real roots, the evolution of roots under repeated differentiation can be described using a construction from random matrix theory. For complex roots, the situation is more complicated, but still very interesting. I will present recent results of mine with Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko about repeated differentiation of *random* polynomials. In this case, we obtain a very precise description of how the (complex) roots evolve, with connections to PDE and random matrix theory. The talk will be self-contained and have lots of pictures and animations.