Mahler and Koksma classification of points in \(\mathbb R^n\) and \(\mathbb C^n\)
type: Item
instance of: scholarly article
title: Mahler and Koksma classification of points in \(\mathbb R^n\) and \(\mathbb C^n\)
zbMATH Open document ID: 1192.11042
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2007-03-20T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1229442630
review text: In order to classify the transcendental numbers, Mahler and Koksma introduced exponents of approximation, \(\omega_d(\xi)\) and \(\omega_d^*(\xi)\), for each real or complex number \(\xi\). The paper under review deals with higher dimensional variants of these quantities and classifications. Let \({\mathcal P}_d\) denote the set of non-zero integer polynomials in \(n\) variables with total degree at most \(d\). Also, let \(|P|\) denote the maximum modulus among the coefficients of \(P\). For a point \(\xi \in {\mathbb C}^n\), \(\psi_d(\xi)\) is defined to be the supremum of all \(\psi\) for which \[ 0 < | P(\xi) | / | P | < | P |^{-\psi} \] has infinitely many solutions \(P \in {\mathcal P}_d\). Similarly, \(\psi_d^*(\xi)\) is defined to be the supremum of all \(\psi^*\) for which there are infinitely many \(P \in {\mathcal P}_d\) with a zero \(\alpha \in {\mathbb C}^n\) such that \(P(\xi) \neq 0\) and \[ | \xi - \alpha | < | P |^{-\psi^*}. \] Here, the absolute value on the left hand side is the maximum modulus of the coordinates of \(\xi - \alpha\). When \(n = 1\), \(\psi_d(\xi) = \omega_d(\xi) + 1\) and \(\psi_d^*(\xi) = \omega_d^*(\xi) + 1\), where \(\omega_d(\xi)\) and \(\omega_d^*(\xi)\) are the exponents defined by Mahler and Koksma respectively. In the present paper, a classification of points in \({\mathbb C}^n\) and \({\mathbb R}^n\) is given by the new exponents in analogy with the classifications of Mahler and Koksma. We call two points \(\eta \in {\mathbb C}^n\) or \({\mathbb R}^n\) and \(\xi \in {\mathbb C}^l\) or \({\mathbb R}^l\) algebraically related if \(\overline{{\mathbb Q}(\eta)} =\overline{{\mathbb Q}(\xi)}\), \textit{i.e}, if the algebraic closures of \({\mathbb Q}(\eta)\) and \({\mathbb Q}(\xi)\) coincide. It was shown by \textit{K. Yu} [J. Reine Angew. Math. 377, 113--126 (1987; Zbl 0614.10031)] that points which are algebraically related are in the same class in the generalized Mahler sense, \textit{i.e.}, in the classification given by the…
reviewed by: Simon Kristensen
Mathematics Subject Classification ID: 11J82
Mathematics Subject Classification ID: 11J83
zbMATH DE Number: 5135178
zbMATH Keywords: generalized Mahler classification
zbMATH Keywords: generalized Koksma classification
zbMATH Keywords: measures of algebraic independence
MaRDI profile type: Publication
OpenAlex ID: W2020698235
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DOI: 10.7169/FACM/1229442630
author: Wolfgang M. Schmidt
description: scientific article; zbMATH DE number 5135178