On the non-Archimedean metric Mahler measure

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 $M_{\mathrm{\infty }}$, and prove that $M_{\mathrm{\infty }}(\alpha )=1$ if and only if $\alpha$ is a root of unity. We further show that $M_{\mathrm{\infty }}$ defines a projective height on $\bar{\mathbb Q}^\times/ \bar{\mathbb Q}^\times_$\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}$$asavectorspaceover$\mathbb Q$. Finally, we demonstrate how to compute $M_{\mathrm{\infty }}(\alpha )$ when $\alpha$ is a surd.