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**On equations in two S-units over function fields of characteristic 0.**
*(English)*
Zbl 0632.10015

In an earlier paper [Invent. Math. 75, 561-584 (1984; Zbl 0521.10015)], the author derived an explicit upper bound for the number of solutions of linear equations in two S-units and, as an application of this, for the number of solutions of Thue-Mahler equations with variables in a fixed algebraic number field. These upper bounds have the remarkable property that they are independent of the coefficients of the forms involved in the equations. In the present paper, the author proves analogous results over function fields.

Let K be an F-field, i.e. a field of characteristic 0 with a set of non- Archimedean (multiplicative) valuations \(M_ K\) which satisfies a product formula. Let k be the constant field of K (i.e. the field consisting of \(\alpha\in K\) with \(| \alpha |_ v\leq 1\) for all \(v\in M_ K)\), and let S be a finite subset of \(M_ K\) with cardinality s. The main result of the paper asserts that for each pair \(\lambda\), \(\mu\) in \(K\setminus \{0\}\), the S-unit equation \(\lambda x+\mu y=1\) in S-units x, y has at most \(2\cdot 7^{2s}\) solutions with \(\lambda\) x/\(\mu\) \(y\not\in k.\)

Applications are given to Thue equations and Thue-Mahler equations over F-fields. Further applications can be found in a joint paper of the author and the reviewer [J. Reine Angew. Math. 358, 6-19 (1985; Zbl 0552.10010)].

Let K be an F-field, i.e. a field of characteristic 0 with a set of non- Archimedean (multiplicative) valuations \(M_ K\) which satisfies a product formula. Let k be the constant field of K (i.e. the field consisting of \(\alpha\in K\) with \(| \alpha |_ v\leq 1\) for all \(v\in M_ K)\), and let S be a finite subset of \(M_ K\) with cardinality s. The main result of the paper asserts that for each pair \(\lambda\), \(\mu\) in \(K\setminus \{0\}\), the S-unit equation \(\lambda x+\mu y=1\) in S-units x, y has at most \(2\cdot 7^{2s}\) solutions with \(\lambda\) x/\(\mu\) \(y\not\in k.\)

Applications are given to Thue equations and Thue-Mahler equations over F-fields. Further applications can be found in a joint paper of the author and the reviewer [J. Reine Angew. Math. 358, 6-19 (1985; Zbl 0552.10010)].

Reviewer: K.Györy

### MSC:

11D41 | Higher degree equations; Fermat’s equation |

11R58 | Arithmetic theory of algebraic function fields |

11D57 | Multiplicative and norm form equations |

11R27 | Units and factorization |