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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

MSC:
 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
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