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On the Thue-Mahler equation. II. (English) Zbl 0820.11017

The author improves on existing upper bounds for the number of solutions of Thue-Mahler equations over number fields. Let \(K\) be an algebraic number field and \(S\) a finite set of places on \(K\) of cardinality \(s\), containing all infinite places. Denote by \({\mathcal O}_ S\) the ring of \(S\)-integers and by \({\mathcal U}_ S\) the group of \(S\)-units. (We recall that if \(K=\mathbb{Q}\) and \(S= \{\infty, p_ 1, \dots, p_ t\}\) then \({\mathcal O}_ S= \mathbb{Z} [(p_ 1\dots p_ t)^{-1} ]\) and \({\mathcal U}_ S= \{\pm p_ 1^{z_ 1} \cdots p_ t^{z_ t}\): \(z_ 1,\dots, z_ t\in \mathbb{Z}\}\).) Further, let \(F(X, Y)= a_ 0 X^ r+ a_ 1 X^{r-1} Y+ \cdots+ a_ r Y^ r\) be a binary form of degree \(r\geq 3\) with coefficients in \({\mathcal O}_ S\) which is irreducible over \(K\). The author considers the equation \[ F(x,y)\in {\mathcal U}_ S \qquad \text{in} \quad (x,y)\in {\mathcal O}_ S^ 2. \tag \(*\) \] Two solutions \((x_ 1, y_ 1)\), \((x_ 2, y_ 2)\) are considered equal if \(x_ 1/x_ 2= y_ 1/ y_ 2\in {\mathcal U}_ S\). In [Invent. Math. 75, 561-584 (1984; Zbl 0521.10015)] the reviewer showed that \((*)\) has at most \(7^{4r^ 3 s}\) solutions, by reducing \((*)\) to an \(S\)-unit equation in two unknowns in some finite extension of \(K\) and using an upper bound for the number of solutions of the latter.
In the paper under review, by a rather different method the author improves this bound to \((12r)^{12s}\) for \(r\geq 6\).

MSC:

11D41 Higher degree equations; Fermat’s equation
11D61 Exponential Diophantine equations
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