Families of Thue-Mahler equations having only trivial solutions. (Familles d’équations de Thue-Mahler n’ayant que des solutions triviales.) (French. English summary) Zbl 1304.11016
Summary: Let \(K\) be a number field, let \(S\) be a finite set of places of \(K\) containing the archimedean places and let \(\mu\), \(\alpha_1,\alpha_2,\alpha_3\) be non-zero elements in \(K\). Denote by \(\mathcal O_S\) the ring of \(S\)-integers in \(K\) and by \(\mathcal O_S^\times\) the group of \(S\)-units. Then the set of equivalence classes (namely, up to multiplication by \(S\)-units) of the solutions \(x,y,z,\varepsilon_1, \varepsilon_2,\varepsilon_3,\varepsilon\in\mathcal O_S^3\times(\mathcal O_S^\times)^4\) of the Diophantine equation
\[ (X-\alpha_1E_1Y)(-X\alpha_2E_2Y)(X-\alpha_3E_3Y)Z=\mu E, \]
satisfying \(\text{Card}\{\alpha_1\varepsilon_1,\alpha_2\varepsilon_2,\alpha_3\varepsilon_3\}= 3\), is finite. With the help of this last result, we exhibit new families of Thue-Mahler equations having only trivial solutions. Furthermore, we produce an effective upper bound for the number of these solutions. The proofs of this paper rest heavily on Schmidt’s subspace theorem.
\[ (X-\alpha_1E_1Y)(-X\alpha_2E_2Y)(X-\alpha_3E_3Y)Z=\mu E, \]
satisfying \(\text{Card}\{\alpha_1\varepsilon_1,\alpha_2\varepsilon_2,\alpha_3\varepsilon_3\}= 3\), is finite. With the help of this last result, we exhibit new families of Thue-Mahler equations having only trivial solutions. Furthermore, we produce an effective upper bound for the number of these solutions. The proofs of this paper rest heavily on Schmidt’s subspace theorem.
MSC:
11D59 | Thue-Mahler equations |
11D45 | Counting solutions of Diophantine equations |
11D61 | Exponential Diophantine equations |
11D25 | Cubic and quartic Diophantine equations |
11J87 | Schmidt Subspace Theorem and applications |