A polynomial time test to detect numbers with many exceptional points

For each algebraic number [Formula: see text] and each positive real number [Formula: see text], the [Formula: see text]-metric Mahler measure [Formula: see text] creates an extremal problem whose solution varies depending on the value of [Formula: see text]. The second author studied the points [Formula: see text] at which the solution changes, called exceptional points for[Formula: see text] . Although each algebraic number has only finitely many exceptional points, it is conjectured that, for every [Formula: see text], there exists a number having at least [Formula: see text] exceptional points. In this paper, we describe a polynomial time algorithm for establishing the existence of numbers with at least [Formula: see text] exceptional points. Our work constitutes an improvement over the best known existing algorithm which requires exponential time. We apply our main result to show that there exist numbers with at least [Formula: see text] exceptional points, another improvement over previous work which was only able to reach [Formula: see text] exceptional points.

Optimal factorizations of rational numbers using factorization trees

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.

The Infimum in the Metric Mahler Measure

Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number α by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure.