Upper bounds on polynomial root separation

Abstract: 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.

number theory

Audience: researchers in the discipline

( paper )

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca

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