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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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