Powers of roots in linear spaces
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)]).
scientific article; zbMATH DE number 5554698
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