Optimal factorizations of rational numbers using factorization trees

TL;DR

This paper introduces factorization trees, a new data structure that helps find the optimal factorization of rational numbers with respect to the $t$-metric Mahler measure, especially for rational numbers where previous methods were inefficient.

Contribution

The paper defines factorization trees and demonstrates their effectiveness in locating the optimal factorization point for rational numbers, advancing computational methods in this area.

Findings

Factorization trees are a useful tool for analyzing rational numbers.

They enable locating the infimum point in the $t$-metric Mahler measure.

New cases of optimal factorizations are identified using these 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 $\bar\alpha = (\alpha_1,\ldots,\alpha_N)\in \overline{\mathbb Q}^N$ 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.