Families of Thue equations associated with a rank 1 subgroup of totally real units of a number field. (Familles d’équations de Thue associées à un sous-groupe de rang 1 d’unités totalement réelles d’un corps de nombres.) (French. English summary) Zbl 1394.11030
Cojocaru, A. C. (ed.) et al., SCHOLAR – a scientific celebration highlighting open lines of arithmetic research. Conference in honour of M. Ram Murty’s mathematical legacy on his 60th birthday, Centre de Recherches Mathématiques, Université de Montréal, Canada, October 15–17, 2013. Providence, RI: American Mathematical Society (AMS); Montreal: Centre de Recherches Mathématiques (CRM) (ISBN 978-1-4704-1457-3/pbk; 978-1-4704-2843-3/ebook). Contemporary Mathematics 655. Centre de Recherches Mathématiques Proceedings, 117-134 (2015).
Summary: Let \(F\) be an irreducible binary form attached to a number field \(K\) of degree \(\geq 3\). Let \(\varepsilon\not\in \{-1, 1\}\) be a totally real unit of \(K\). By twisting \(F\) with the powers \(\varepsilon^a\) of \(\varepsilon\), (\(a\in\mathbb {Z}\)), we obtain an infinite family \(F_a\) of binary forms. Let \(m\in\mathbb {Z}\). We give an effective bound for \(\max\{|a|, \log|x|, \log|y|\}\) when \(a,x,y\) are rational integers satisfying \(F_a(x,y)=m\) with \(xy\not=0\).
For the entire collection see [Zbl 1334.11003].
For the entire collection see [Zbl 1334.11003].
MSC:
| 11D61 | Exponential Diophantine equations |
| 11D25 | Cubic and quartic Diophantine equations |
| 11D41 | Higher degree equations; Fermat’s equation |
| 11D59 | Thue-Mahler equations |
