Optimal factorizations of rational numbers using factorization trees

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 (alpha) over bar = (alpha(1),..., alpha(N)) is an element of (Q) over bar (N) for all sufficiently large t. Nevertheless, no efficient method for locating (alpha) over bar is known. In this paper, we define a new tree data structure, called a factorization tree, which enables us to find (alpha) over bar when alpha is an element of Q. We establish several basic properties of factorization trees, and use these properties to locate (alpha) over bar in previously unknown cases.