SFU Number Theory and Algebraic Geometry Seminar: Gregg Knapp

Distances between the roots of a fixed polynomial appear organically in many places in number theory. For any $f(x) \in \mathbb{Z}[x]$, let $\operatorname{sep}(f)$ denote the minimum distance between distinct roots of $f(x)$. Mahler initiated the study of separation by giving lower bounds on $\operatorname{sep}(f)$ in terms of the degree and Mahler measure of $f(x)$, and these bounds have been improved and generalized in recent years. However, there has been relatively little study concerning upper bounds on $\operatorname{sep}(f)$. In this talk, I will describe recent work with Chi Hoi Yip in which we provide sharp upper bounds on $\operatorname{sep}(f)$ using techniques from the geometry of numbers.