Metric Mahler measures over number fields

Abstract

For an algebraic number $\alpha$, the metric Mahler measure $m_1(\alpha )$ was first studied by Dubickas and Smyth in 2001 and was later generalized to the $t$-metric Mahler measure $m_t(\alpha )$ by the author in 2010. The definition of $m_t(\alpha )$ involves taking an infimum over a certain collection $N$-tuples of points in $\overline{\mathbb{Q}}$, and from previous work of Jankauskas and the author, the infimum in the definition of $m_t(\alpha )$ is attained by rational points when $\alpha \in \mathbb{Q}$. As a consequence of our main theorem in this article, we obtain an analog of this result when $\mathbb{Q}$ is replaced with any imaginary quadratic number field of class number equal to $1$. Further, we study examples of other number fields to which our methods may be applied, and we establish various partial results in those cases.

12 pages

References in corpus (8)

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