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The number of solutions of the Thue-Mahler equation. (English) Zbl 0861.11021
Let \(K\) be an algebraic number field and \(S\) a set of places on \(K\) of finite cardinality \(s\), containing all infinite places. Denote by \({\mathcal O}_S\) the ring of \(S\)-integers \(\{x \in K: |x |_v\leq 1\) for \(v \notin S\}\) and by \({\mathcal O}_S^*\) the group of \(S\)-units, \(\{x\in K : |x |_v=1\) for \(v \notin S\}\). Here we deal with the Thue-Mahler equation over \(K\), \[ F(x,y) \in {\mathcal O}_S^* \qquad\text{in }x,y\in {\mathcal O}_S,\tag{*} \] where \(F(x,y)\) is a binary form with coefficients in \({\mathcal O}_S\) of degree \(r\geq 3\) which is irreducible over \(K\). The set of solutions of (*) can be divided naturally into \({\mathcal O}_S^*\)-cosets, where an \({\mathcal O}_S^*\)-coset is a set of the type \(\{\varepsilon (x,y): \varepsilon \in {\mathcal O}_S^*\}\) with \((x,y)\) a fixed solution of (*). Improving on results of Mahler, Lewis and Mahler and the author, E. Bombieri [Acta Arith. 67, 69-96 (1994; Zbl 0820.11017)] was the first to obtain an upper bound for the number of \({\mathcal O}_S^*\)-cosets of solutions of (*) depending only polynomially on \(r\), namely \((12r)^{12s}\). In the present paper, we improve this to \((5 \times 10^6r)^s\).
Like Bombieri, we distinguish between “large” and “not large” \({\mathcal O}_S^*\)-cosets of solutions of (*) and treat the large cosets by applying the Thue-Siegel method. Our treatment of the not large cosets is rather different from Bombieri’s. Bombieri heavily uses that the number of \({\mathcal O}_S^*\)-cosets of solutions of (*) does not change when \(F\) is replaced by an equivalent form, where equivalence is defined by means of transformations from \(GL_2({\mathcal O}_S)\), and in his proof he uses some complicated notion of reduction of binary forms.
Instead, we use a method which we developed in [Invent. Math. 122, 559-601 (1995; Zbl 0851.11019)] to treat the small \({\mathcal O}^*_S\)-cosets of solutions of norm form equations and decomposable form equations, and work out the refinements which are possible in the special case of Thue-Mahler equations. It is easily seen that there is no loss of generality to assume that \[ F(X,Y)= (X+c^{(1)}Y) \cdots (X+c^{(r)}Y) \] where \(c^{(1)}, \dots, c^{(r)}\) are the conjugates over \(K\) of some algebraic number \(c\). The substance of our method is that, unlike Bombieri, we do not apply the diophantine approximation techniques to a solution \((x,y)\) of (*) but to the number \(u: =x+cy\) and that we work with the absolute Weil height \(H({\mathbf u})\) of the vector \({\mathbf u} = (u^{(1)}, \dots, u^{(r)})\) consisting of all conjugates of \(u\). In particular, we reduce equation (*) to certain diophantine inequalities in terms of \(u\) and \(H({\mathbf u})\) and prove a gap principle for these inequalities.

MSC:
11D41 Higher degree equations; Fermat’s equation
11D61 Exponential Diophantine equations
11D75 Diophantine inequalities
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