Lab Lunch: 24 November 2020 - Kyriakos Kalorkoti

Title: Expected Number of Roots of Littlewood Polynomials Around $\pm1$

Abstract:

We consider Littlewood polynomials drawn uniformly at random and provide upper bounds for the expected number of roots in disks centred at $\pm1$.  In particular the expected number for polynomials of degree~$n$ in a disk of radius O(1/n) is bounded by a constant (depending on the radius constant). For larger discs we cannot expect such a result as indicated by a theorem of Borwein and Littmann.