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Arithmetic over the ring of all algebraic integers. (English) Zbl 0581.14014
The 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.

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
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