UBC Probability Seminar: Nick Cook

We consider the restriction to the unit circle of random degree-n polynomials with iid normalized coefficients (Kac polynomials). Recent work of Yakir and Zeitouni shows that for Gaussian coefficients, the minimum modulus (suitably rescaled) follows a limiting exponential distribution. We show this is a universal phenomenon, extending their result to arbitrary sub-Gaussian coefficients, such as Rademacher signs. For discrete distributions we must now deal with possible arithmetic structure in the polynomial evaluated at different points of the circle. Our proof divides the circle into major arcs that are well approximated by rationals, which we handle by crude arguments, and complementary minor arcs, for which we obtain strong comparisons with the Gaussian model via sharp decay estimates on characteristic functions. Based on joint work with Hoi Nguyen.