No shadowing bounds on Galois orbits in the complex plane

Abstract: For varying pairs of non-isogenous abelian varieties of a given dimension over a given finite field, what is the least possible arclengths sum under a matching of their Frobenius roots? For varying pairs of Salem numbers in $[1,2]$, what is their least possible distance in terms of the sum of their degrees?

We address, and partly answer, these kinds of questions in the seminar, with a particular focus on the two representatives at hand. The method, which is based on potential theory in the complex plane, also establishes the Lehmer conjecture for the integer monic polynomials $P(X)$ that have all their roots limited to the complex disk $|z| < 10^{1/\deg(P)}$: the extremal case where the Galois orbit of algebraic integers is maximally equalized around the unit circle. We also raise a few apparently new questions that our results motivate.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).

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