S-unit equations over number fields
Let K be an algebraic number field of degree d and S a finite set of places of K of cardinality s, containing all infinite places. The S-unit equation \((1)\quad a_ 1x_ 1+...+a_ nx_ n=1\) in S-units \(x_ 1,...,x_ n\) has only finitely many solutions without vanishing subsums, i.e. with \(a_{i_ 1}x_{i_ 1}+...+a_{i_ t}x_{i_ t}\neq 0\) for each non-empty subset \(\{i_ 1,...,i_ t\}\) of \(\{\) 1,...,n\(\}\). \textit{K. Györy} and the reviewer [Compos. Math. 66, 329-354 (1988; Zbl 0644.10015)] proved that there is a finite number C such that for all \(a_ 1,...,a_ n\in K^*\) (1) has at most C solutions, without being able to compute explicitly such a C. In [J. Reine Angew. Math. 406, 109- 120 (1990; Zbl 0693.10016)] the author shows that for \(K={\mathbb{Q}}\) one may take \(C=(8s)^{2^{26n+4}s^ 6}.\) In the paper under review, the author shows that for arbitrary K, \(C=(8sd!)^{2^{36nd!}s^ 6}\) can be taken. The idea of his proof is as follows. By using his own quantitative subspace theorem over number fields, the author derives an upper bound for the minimal number of proper linear subspaces of \(K^ n\) containing the set of all solutions of (1). In fact, the author obtains such an upper bound independent of \(a_ 1,...,a_ n\) with the help of some ``gap principle'' developed by himself (independently, Györy and the reviewer (loc. cit.) obtained a similar gap principle for S-unit equations over function fields). Then the proof is finished by induction on n.
scientific article; zbMATH DE number 4168770
scientific article