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

  • Claude Levesque (a1) and Michel Waldschmidt (a2)
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 $d$ .

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References
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Mathematika
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