Optimal factorizations of rational numbers using factorization trees
Charles L. Samuels, Tanner J. Strunk
TL;DR
The paper studies the $t$-metric Mahler measure $m_t(\alpha)$ and develops a combinatorial framework based on factorization trees to efficiently locate optimal representations when $\alpha$ is rational. It introduces digraph data structures, primitive and canonical optimal factorization trees $\mathcal{P}_\alpha$ and $\mathcal{O}_\alpha$, and measure class graphs to organize factorizations by their Mahler-measure vectors, proving key uniqueness and embedding properties (e.g., $\mathcal{O}_\alpha$ is optimal and injects into $\mathcal{P}_\alpha$). The results show that all optimal factorizations can be captured at leaves or ancestors within these trees and their measure-class quotients, enabling systematic search for optimal representations. The authors validate the approach with concrete examples, including $\alpha=\frac{30}{7}$, $\frac{851}{858}$, and $\frac{316889}{549010}$, demonstrating efficient identification of unique optimal factorizations and compact measure representations. Overall, factorization trees provide a structured, computable route to understanding $m_t(\alpha)$ for rational numbers and may influence practical algorithms related to Lehmer-type problems.
Abstract
Let $m_t(α)$ denote the $t$-metric Mahler measure of the algebraic number $α$. Recent work of the first author established that the infimum in $m_t(α)$ is attained by a single point $\barα= (α_1,\ldots,α_N)\in \overline{\mathbb Q}^N$ for all sufficiently large $t$. Nevertheless, no efficient method for locating $\bar α$ is known. In this article, we define a new tree data structure, called a factorization tree, which enables us to find $\barα$ when $α\in \mathbb Q$. We establish several basic properties of factorization trees, and use these properties to locate $\barα$ in previously unknown cases.