| Tue, Sep 01, 2020
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Number Theory Seminar
2:00 PM
Online (Zoom) | | Mahler's measure and some of its behavior under iteration
Mingming Zhang, OSU
E-mail Paul Fili for the Zoom link.
| | | Abstract: For an algebraic number $\alpha$ we denote by $M(\alpha)$ the Mahler measure of $\alpha$. As $M(\alpha)$ is again an algebraic number (indeed, an algebraic integer), $M(\cdot)$ is a self-map on $\overline{\mathbb{Q}}$, and therefore defines a dynamical system. The \emph{orbit size} of $\alpha$, denoted $\# \mathcal{O}_M(\alpha)$, is the cardinality of the forward orbit of $\alpha$ under $M$. In this talk, we will start by introducing the definition of Mahler's measure, then briefly discuss results on the orbit sizes of algebraic numbers with degree at least 3 and non-unit norm, as well as algebraic units of degree 4. Our main goal is to discuss that if $\alpha$ is an algebraic unit of degree $d\geq 5$ such that the Galois group of the Galois closure of $\mathbb{Q}(\alpha)$ contains $A_d$, then the orbit size must be 1, 2 or $\infty$. Furthermore, for every degree at least 12 and divisible by 4, there exist algebraic units of arbitrarily large orbit size. This is joint work with Paul Fili and Lukas Pottmeyer. |
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