# Metric Mahler measures over number fields

## TL;DR

This paper extends the understanding of metric Mahler measures from rational numbers to imaginary quadratic fields with class number one, providing new theoretical insights and partial results for other number fields.

## Contribution

It generalizes the attainment of the infimum in metric Mahler measures from rational numbers to certain imaginary quadratic fields and explores applicability to other number fields.

## Key findings

- Infimum is attained by rational points in imaginary quadratic fields with class number one.
- Established analogs of previous results for these fields.
- Presented partial results for other 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.