On metric heights
type: Item
instance of: scholarly article
title: On metric heights
zbMATH Open document ID: 1050.11089
DOI: 10.1023/A:1025936026150
published in: Periodica Mathematica Hungarica
publication date: 2004-09-07T00:00:00Z
review text: The naive height \(H(\alpha)\) of an algebraic number \(\alpha\) is defined to be the maximum of the absolute values of the coefficients of its minimal polynomial. Define the metric height of \(\alpha\) by \(\widehat{H}(\alpha) = \inf \{H(\alpha_1) \cdots H(\alpha_s)\}\), where the infimum is taken over all \(s \geq 1\) and all algebraic numbers \(\alpha_j\) such that \(\alpha_1 \cdots \alpha_s = \alpha\). The authors study the basic properties of the metric height and then show that \(\widehat{H}(m \sqrt{n}\,) = mn\) for natural numbers \(m, n\) with \(n > 1\) squarefree; they also manage to compute the metric height of other \(d\)-th roots of integers with few prime factors and observe that general formulas even for cubic surds seem to be difficult to obtain.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11D68
Mathematics Subject Classification ID: 11Y16
zbMATH DE Number: 2099013
zbMATH Keywords: algebraic numbers
zbMATH Keywords: heights
author: A. Dubickas
author: Chris Smyth
MaRDI profile type: Publication
cites work: Mahler Measures Close to an Integer
cites work: On the Remak height, the Mahler measure and conjugate sets of algebraic numbers lying on two circles
cites work: On the metric Mahler measure
cites work: Q4488162
full work available at URL: a:1025936026150
OpenAlex ID: W1514941655
description: scientific article; zbMATH DE number 2099013