S-unit equations over number fields
type: Item
instance of: scholarly article
title: S-unit equations over number fields
zbMATH Open document ID: 0711.11017
DOI: 10.1007/BF01233421
published in: Inventiones Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143828
review text: 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.
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 4168770
zbMATH Keywords: gap principle
zbMATH Keywords: S-unit equations
author: Hans Peter Schlickewei
reviewed by: Jan-Hendrik Evertse
MaRDI profile type: Publication
cites work: Q4106278
cites work: Q3781167
cites work: On equations in S-units and the Thue-Mahler equation
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cites work: Q3786527
cites work: Zur Approximation algebraischer Zahlen. I: Über den grössten Primteiler binärer Formen
cites work: The \(\mathfrak p\)-adic Thue-Siegel-Roth-Schmidt theorem
cites work: The number of subspaces occurring in the p-adic subspace theorem in diophantine approximation.
cites work: An explicit upper bound for the number of solutions of the S-unit equation.
cites work: Q4009397
cites work: Q4732523
OpenAlex ID: W2004723509
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description: scientific article; zbMATH DE number 4168770