Minimal Mahler Measure in Cubic Number Fields

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 resp...

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Bibliographic Details
Published in:arXiv.org 2022-03
Main Authors: Eldredge, Lydia, Petersen, Kathleen
Format: Article
Language:English
Subjects:
Algorithms
Lower bounds
Number theory
Online Access:Get full text
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Summary: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 \(N\).
ISSN:2331-8422