### No shadowing bounds on Galois orbits in the complex plane

Heilbronn Number Theory Seminar

30th September 2020, 4:00 pm – 5:00 pm

Zoom,

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.

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