Levesque, Claude; Waldschmidt, Michel Families of cubic Thue equations with effective bounds for the solutions. (English) Zbl 1358.11050 Borwein, Jonathan M. (ed.) et al., Number theory and related fields. In memory of Alf van der Poorten. Based on the proceedings of the international number theory conference, Newcastle, Australia, March 12–16, 2012. New York, NY: Springer (ISBN 978-1-4614-6641-3/hbk; 978-1-4614-6642-0/ebook). Springer Proceedings in Mathematics & Statistics 43, 229-243 (2013). Let \(K\) be a cubic number field, which is not totally real, \(\alpha \in K \setminus \mathbb Q\) an algebraic integer and \(\varepsilon \in K \setminus \mathbb Q\) a unit. The authors show that for given \(k \in \mathbb N\) there are only finitely many triples \((n,x,y) \in \mathbb Z ^3\) satisfying \[ |N(x - \varepsilon^n \alpha y)| \leq k, \] where \(N\) denotes the norm from \(K\) to \(\mathbb Q\), and that there are effectively computable bounds for the solutions, which depend only on \(\alpha\). Equivalently, the above question yields an infinite family of cubic Thue inequalities, indexed by \(n \in \mathbb Z\).In the first part of the proof the authors use elementary estimates and results from T. N. Shorey and R. Tijdeman [Exponential Diophantine equations. Cambridge etc.: Cambridge University Press (1986; Zbl 0606.10011)] to obtain \(x - \varepsilon^n \alpha y = \varepsilon^\ell \xi_1\), where the house of \(\xi_1\) is bounded by an effective constant times \(\root 3 \of k\). In the second half of the paper it is shown that there is some effective constant \(\kappa_{21}\) such that \[ |\ell| \leq \kappa_{21} \log k. \] This is achieved by investigating linear forms in logarithms.For the entire collection see [Zbl 1266.11001]. Reviewer: Günter Lettl (Graz) Cited in 3 Documents MSC: 11D25 Cubic and quartic Diophantine equations 11D59 Thue-Mahler equations Keywords:cubic Thue equations; cubic number field; effective bounds for solutions Citations:Zbl 0606.10011 PDFBibTeX XMLCite \textit{C. Levesque} and \textit{M. Waldschmidt}, Springer Proc. Math. Stat. 43, 229--243 (2013; Zbl 1358.11050) Full Text: DOI arXiv