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Problèmes diophantiens sur l’anneau des entiers algébriques. (Diophantine problems over the ring of algebraic integers). (French) Zbl 0618.10014

Sémin. Théor. Nombres, Univ. Bordeaux I 1985-1986, Exp. No. 22, 9 p. (1986).
Let K be a field of algebraic numbers, \(\bar K\) its algebraic closure, \(Z^ S_{K}\) and \(Z^ S_{\bar K}\) their rings of S-integers, \(P_ i(x_ 1,...,x_ n)\) (1\(\leq i\leq r)\) polynomials with coefficients from \(Z^ S_{K}\). The problem of solvability of the system of equations \(P_ i(x_ 1,...,x_ n)=0\) (1\(\leq i\leq r)\) in \(x_ 1,...,x_ n\) from \(Z^ S_{\bar K}\) is investigated (”arithmetical case”). The geometrical analogue of this problem (”geometrical case”) is discussed.
Reviewer: S.Kotov

MSC:

11D88 \(p\)-adic and power series fields
14G20 Local ground fields in algebraic geometry