You may be surprised to learn that differentiation of polynomials is still an active and even growing area of research. We will look at some recent work of Steinerberger answering the title question for polynomials with real roots, and the surprising connection to eigenvalues of matrices. Time permitting we may discuss the challenges of the complex root case. While there are interesting connections to probability, combinatorics, and operator algebras we will try to keep the vast majority of the talk at the level of calculus, linear algebra, and undergraduate analysis.
How do polynomial roots move under repeated differentiation?