Minimal Mahler Measure in Cubic Number Fields

Abstract

The minimal integral Mahler measure of a number field $K$, $M({\mathcal{O}}_K)$, is the minimal Mahler measure of a non-torsion primitive element of ${\mathcal{O}}_K$. Upper and lower bounds, which depend on the discriminant, are known. We show that for cubics, the lower bounds are sharp with respect to its growth as a function of discriminant. We construct an algorithm to compute $M({\mathcal{O}}_K)$ for all cubics with absolute value of the discriminant bounded by $N$.

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