Optimal factorizations of rational numbers using factorization trees

Abstract

Let $m_t(\alpha )$ denote the $t$-metric Mahler measure of the algebraic number $\alpha$. Recent work of the first author established that the infimum in $m_t(\alpha )$ is attained by a single point $\bar{\alpha }=({\alpha }_1,\dots ,{\alpha }_N)\in {\overline{\mathbb{Q}}}^N$ for all sufficiently large $t$. Nevertheless, no efficient method for locating $\bar{\alpha }$ is known. In this article, we define a new tree data structure, called a factorization tree, which enables us to find $\bar{\alpha }$ when $\alpha \in \mathbb{Q}$. We establish several basic properties of factorization trees, and use these properties to locate $\bar{\alpha }$ in previously unknown cases.

References in corpus (4)

• An effective lower bound for the height of algebraic numbers
• The parametrized family of metric Mahler measures
• The $t$-metric Mahler measures of surds and rational numbers
• Metric Heights on an Abelian Group

Cited by in corpus (1)

• Continued fraction expansions in connection with the metric Mahler measure