Optimal factorizations of rational numbers using factorization trees

Abstract

Let mt(α) denote the t-metric Mahler measure of the algebraic number α. Recent work of the first author established that the infimum in mt(α) is attained by a single point [Formula: see text] for all sufficiently large t. Nevertheless, no efficient method for locating [Formula: see text] is known. In this paper, we define a new tree data structure, called a factorization tree, which enables us to find [Formula: see text] when α ∈ ℚ. We establish several basic properties of factorization trees, and use these properties to locate [Formula: see text] in previously unknown cases.