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On diophantine equations over the ring of all algebraic integers. (English) Zbl 0538.12014
Let K be an algebraic number field of finite degree and \({\mathfrak O}\) its ring of algebraic integers. Moreover, let \(\tilde K\) be the algebraic closure of K and \(\tilde {\mathfrak O}\) the integral closure of \({\mathfrak O}\) in \(\tilde K\). Given m rational functions \(f_ i(X_ 1,...,X_ n)\in K(X_ 1,...,X_ n) (1\leq i\leq m),\) the diophantine problem - called a Skolem problem - discussed is as follows: under what conditions does there exist \(z_ 1,...,z_ n\in \tilde K\) such that all \(f_ i(z_ 1,...,z_ n)\) belong to \(\tilde {\mathfrak O}\). For a non-Archimedean prime v of K the corresponding problem for \(K_ v (=\) completion of K with respect to v) and \({\mathfrak O}_ v (=\) canonical valuation ring of \(K_ v)\) is called the ’local problem’. The authors prove a local-global principle for the solvability of such Skolem problems, i.e. the problem is solvable in (K,\({\mathfrak O})\) if and only if it is solvable in \((K_ v,{\mathfrak O}_ v)\) for all non-Archimedean primes v of K. Actually, only a finite number of local conditions has to be checked: as the authors show, the local problem is always solvable, if the minimum value of the coefficients of \(f_ 1,...,f_ m\) is non-negative. This local-global principle implies that there is a decision procedure for Skolem problems, because the local problems are decidable due to a result of A. Robinson.
Reviewer: A.Prestel

12L05 Decidability and field theory
11D99 Diophantine equations
12L12 Model theory of fields
11R99 Algebraic number theory: global fields
11S99 Algebraic number theory: local and \(p\)-adic fields
12J25 Non-Archimedean valued fields
Full Text: DOI
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