Minimal Mahler measure in cubic number fields

Abstract

The minimal integral Mahler measure of a number field [Formula: see text], [Formula: see text], is the minimal Mahler measure of an integral generator of [Formula: see text]. Upper and lower bounds, which depend on the discriminant and degree of [Formula: see text], are known. We show that for three natural families of cubics, the lower bounds are sharp with respect to its growth as a function of discriminant. We construct an algorithm to compute [Formula: see text] for all cubics with absolute value of the discriminant bounded by [Formula: see text] and show the resulting data for [Formula: see text].