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A PDE describing Roots of Polynomials under Differentiation (S. Steinerberger, University of Washington, USA)

Jan 13
13-01-2021 16:00 Uhr bis 17:00 Uhr
Online

A PDE describing Roots of Polynomials under Differentiation

Speaker: Prof. Dr. Stefan Steinerberger
Affiliation: University of Washington, USA
Zoom:

Abstract: Suppose you have a polynomial p_n (think of n as being quite large) and suppose you know where the roots are. What can you say about the roots of the derivative p_n‘? Clearly, one could compute them but if n is large, that is not so easy — can you make a softer statement, predicting „roughly“ where they are? This question goes back to Gauss who proved a pretty Theorem about it. We will ask the question of what happens when one keeps differentiating: if the roots of p_n look like, say, a Gaussian, what can you say about the roots of the polynomial after you have differentiated 0.1*n times? This leads to some very fun equations and some fascinating new connections to Probability Theory, Potential Theory and Partial Differential Equations. In particular, there is a nice nonlocal PDE that seems to describe everything. I promise nice pictures!

https://en.www.math.fau.de/applied-analysis/