A Collection of Metric Mahler Measures

Let M(α) denote the Mahler measure of the algebraic number α. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative group of algebraic numbers. Later, Fili and the author used similar techniques to study a non-Archimedean version.We show how to generalize the above constructions in order to associate, to each point in (0,∞], a metric version M_x of the Mahler measure, each having a triangle inequality of a different strength. We are able to compute M_x (α) for sufficiently small x, identifying, in the process, a function ¯M with certain minimality properties. Further, we show that the map x↦M_x (α) defines a continuous function on the positive real numbers.