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FAMILIES OF THUE EQUATIONS ASSOCIATED WITH A RANK ONE SUBGROUP OF THE UNIT GROUP OF A NUMBER FIELD

Published online by Cambridge University Press:  29 November 2017

Claude Levesque
Affiliation:
Département de mathématiques et de statistique, Université Laval, Québec (Québec), Canada G1V 0A6, Canada email claude.levesque@mat.ulaval.ca
Michel Waldschmidt
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, UMR 7586 IMJ-PRG, F-75005 Paris, France email michel.waldschmidt@imj-prg.fr
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Abstract

Let $K$ be an algebraic number field of degree $d\geqslant 3$, $\unicode[STIX]{x1D70E}_{1},\unicode[STIX]{x1D70E}_{2},\ldots ,\unicode[STIX]{x1D70E}_{d}$ the embeddings of $K$ into $\mathbb{C}$, $\unicode[STIX]{x1D6FC}$ a non-zero element in $K$, $a_{0}\in \mathbb{Z}$, $a_{0}>0$ and

$$\begin{eqnarray}F_{0}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC})Y).\end{eqnarray}$$
Let $\unicode[STIX]{x1D710}$ be a unit in $K$. For $a\in \mathbb{Z}$, we twist the binary form $F_{0}(X,Y)\in \mathbb{Z}[X,Y]$ by the powers $\unicode[STIX]{x1D710}^{a}$ ($a\in \mathbb{Z}$) of $\unicode[STIX]{x1D710}$ by setting
$$\begin{eqnarray}F_{a}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})Y).\end{eqnarray}$$
Given $m>0$, our main result is an effective upper bound for the size of solutions $(x,y,a)\in \mathbb{Z}^{3}$ of the Diophantine inequalities
$$\begin{eqnarray}0<|F_{a}(x,y)|\leqslant m\end{eqnarray}$$
for which $xy\not =0$ and $\mathbb{Q}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})=K$. Our estimate is explicit in terms of its dependence on $m$, the regulator of $K$ and the heights of $F_{0}$ and of $\unicode[STIX]{x1D710}$; it also involves an effectively computable constant depending only on $d$.

Type
Research Article
Copyright
Copyright © University College London 2017 

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References

Bugeaud, Y. and Győry, K., Bounds for the solutions of Thue–Mahler equations and norm form equations. Acta Arith. 74 1996, 273292.CrossRefGoogle Scholar
Evertse, J.-H. and Győry, K., Unit Equations in Diophantine Number Theory (Cambridge Studies in Advanced Mathematics 146 ), Cambridge University Press (Cambridge, 2015).CrossRefGoogle Scholar
Gourdon, X. and Salvy, B., Effective asymptotics of linear recurrences with rational coefficients. Discrete Math. 153 1996, 145163.CrossRefGoogle Scholar
Levesque, C. and Waldschmidt, M., Familles d’équations de Thue–Mahler n’ayant que des solutions triviales. Acta Arith. 155 2012, 117138.CrossRefGoogle Scholar
Levesque, C. and Waldschmidt, M., Families of cubic Thue equations with effective bounds for the solutions. Springer Proc. Math. Statist. 43 2013, 229243.Google Scholar
Levesque, C. and Waldschmidt, M., Solving effectively some families of Thue Diophantine equations. Moscow J. Comb. Number Theory 3(3–4) 2013, 118144.Google Scholar
Levesque, C. and Waldschmidt, M., Familles d’équations de Thue associées à un sous-groupe de rang 1 d’unités totalement réelles d’un corps de nombres. In SCHOLAR—a Scientific Celebration Highlighting Open Lines of Arithmetic Research (volume dedicated to Ram Murty) (CRM collection Contemporary Mathematics 655 ), American Mathematical Society (Providence, RI, 2015), 117134.CrossRefGoogle Scholar
Levesque, C. and Waldschmidt, M., A family of Thue equations involving powers of units of the simplest cubic fields. J. Théor. Nombres Bordeaux 27(2) 2015, 537563.CrossRefGoogle Scholar
Levesque, C. and Waldschmidt, M., Solving Simultaneously Thue Diophantine Equations: Almost Totally Imaginary Case (Lecture Notes Series 23 ), Ramanujan Mathematical Society (2016), 137156.Google Scholar
Shorey, T. N. and Tijdeman, R., Exponential Diophantine Equations (Cambridge Tracts in Mathematics 87 ), Cambridge University Press (Cambridge, 1986).CrossRefGoogle Scholar
Siegel, C. L., Abschätzung von Einheiten. In Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1969), 7186. Reproduced in Gesammelte Abhandlungen IV, Springer Collected Works in Mathematics, §88 (1979), 66–81.Google Scholar
Waldschmidt, M., Diophantine Approximation on Linear Algebraic Groups (Grundlehren der Mathematischen Wissenschaften 326 ), Springer (Berlin, 2000).CrossRefGoogle Scholar