Arithmetic over the ring of all algebraic integers.

*(English)*Zbl 0581.14014The main result of this paper is the proof of a conjecture of David Cantor, giving a Hasse principle for the ring of all algebraic integers: an absolutely irreducible affine variety defined over a number field K has points rational over the ring of all algebraic integers if and only if it has such points locally, for every finite place v of K. The proof is a generalization of a method due to D. C. Cantor and P. Roquette [J. Number Theory 18, 1-26 (1984; Zbl 0538.12014)], and is an application of capacity theory on algebraic curves as developed by the author. By combining the Hasse principle with constructive algebraic geometry and a decision procedure for completely valued algebraically closed fields, one obtains a positive solution to Hilbert’s tenth problem over the ring of algebraic integers: a decision procedure for diophantine equations and inequalities over this ring.

##### MSC:

14G05 | Rational points |

12L05 | Decidability and field theory |

14H25 | Arithmetic ground fields for curves |

11D75 | Diophantine inequalities |

11D99 | Diophantine equations |

03B25 | Decidability of theories and sets of sentences |