On the non-Archimedean metric Mahler measure

Abstract

Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric naïve 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 ${\bar{\mathbb{Q}}}^{\times }/{\bar{\mathbb{Q}}}_{\mathrm{tors}}^{\times }$ as a vector space over $\mathbb{Q}$. Finally, we demonstrate how to compute $M_{\infty }(\alpha )$ when $\alpha$ is a surd.

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• The parametrized family of metric Mahler measures
• The $t$-metric Mahler measures of surds and rational numbers
• Metric Heights on an Abelian Group
• The infimum in the metric Mahler measure
• A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus
• Orthogonal decomposition of the space of algebraic numbers and Lehmer's problem
• The finiteness of computing the ultrametric Mahler measure
• A collection of metric Mahler measures
• Norms extremal with respect to the Mahler measure
• Metric Mahler measures over number fields