## 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

### Citations:

Zbl 0646.10009; Zbl 0521.10015
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