Powers of roots in linear spaces
type: Item
instance of: scholarly article
title: Powers of roots in linear spaces
zbMATH Open document ID: 1179.11035
author: Andrzej Schinzel
author: Wolfgang M. Schmidt
published in: Journal of Number Theory
publication date: 2009-05-19T00:00:00Z
review text: Let \(K\) be a number field, \(n>0\) be a positive integer. Let \(\alpha\) be an algebraic number such that \(\alpha^n=:\kappa\) belongs to \(K\) and no power of \(\alpha\) with smaller positive exponent lies in \(K\). The authors are interested in linear relations over \(K\) among the powers \(\alpha^j\) with \(0\leq j<n\). They prove the following: there exists a number \(c\), depending explicitly on the degree and discriminant of \(K\) and on the height of \(\kappa\), such that for every \(K\)-vector space \(V\subset K(\alpha)\) of dimension \(r\) the number of exponents \(j\) with \(0\leq j<n\) and \(\alpha^j\in V\) is bounded by \(c\cdot r(\log r+2)\). In a subsequent paper, Schmidt, Zannier and the reviewer [The Diophantine equation \(\alpha_1^{x_1}\cdots\alpha_n^{x_n}=f(x_1,\ldots,x_n)\). II, Trans. Am. Math. Soc. 362, No. 4, 2115--2123 (2010; Zbl 1194.11042)] obtained a bound depending only on \(d\) and \(r\). However, the dependence on \(r\) is much worse than in the present article. The main motivation for this result lies in its application to the study of Diophantine equations of the type \(\alpha_1^{x_1}\cdots\alpha_n^{x_n}=f(x_1,\ldots,x_n)\) (see Schmidt's paper in [\textit{W. W. L. Chen} at al. (eds.), Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press, 414--420 (2009; Zbl 1178.11034)]).
reviewed by: Pietro Corvaja
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 5554698
zbMATH Keywords: powers of radicals
zbMATH Keywords: radicals in a vector space
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.02.001
OpenAlex ID: W2039793565
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DOI: 10.1016/J.JNT.2009.02.001
description: scientific article; zbMATH DE number 5554698