On the non-Archimedean metric Mahler measure

TL;DR

This paper introduces a non-Archimedean version of the metric Mahler measure, proving its properties and providing methods for computation, thereby extending the understanding of heights in algebraic number theory.

Contribution

It defines a non-Archimedean metric Mahler measure, proves its key properties, and offers computational techniques for surds, expanding the framework of algebraic heights.

Findings

$M_ infty( alpha ) = 1$ iff $ alpha $ is a root of unity

$M_ infty$ defines a projective height on $ar{Q}^ imes/ ar{Q}^ imes_ ext{tors}$

Methods for computing $M_ infty( alpha )$ for surds

Abstract

Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric na\"ive height on the multiplicative group of algebraic numbers. We give a non-Archimedean version of the metric Mahler measure, denoted $M_\infty$, and prove that $M_\infty(\alpha) = 1$ if and only if $\alpha$ is a root of unity. We further show that $M_\infty$ defines a projective height on $\bar{\mathbb Q}^\times/ \bar{\mathbb Q}^\times_\mathrm{tors}$ as a vector space over $\mathbb Q$. Finally, we demonstrate how to compute $M_\infty(\alpha)$ when $\alpha$ is a surd.