Minimal Mahler Measure in Cubic Number Fields

TL;DR

This paper investigates the minimal Mahler measure in cubic number fields, establishing sharp bounds related to discriminant growth and providing an algorithm to compute this measure for all cubics within a discriminant bound.

Contribution

It proves the sharpness of lower bounds for the Mahler measure in cubic fields and introduces an algorithm to compute the measure for all such fields with bounded discriminant.

Findings

Lower bounds are sharp for cubic fields as a function of discriminant.

An explicit algorithm is developed for computing the minimal Mahler measure.

The study provides comprehensive data for cubics with discriminant up to a bound N.

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 $N$.