On the non-Archimedean metric Mahler measure

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_∞, and prove that M_∞(α)=1 if and only if α is a root of unity. We further show that M_∞ defines a projective height on as a vector space over Q. Finally, we demonstrate how to compute M_∞(α) when α is a surd.