A polynomial time test to detect numbers with many exceptional points

Abstract

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.