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On Rumely’s local-global principle. (English) Zbl 0857.11033
In 1970 Yu. Matiyasevich [Sov. Math., Dokl. 11, 354-358 (1970); translation from Dokl. Akad. Nauk SSSR 191, 279-282 (1970; Zbl 0212.33401)] completed the proof of a negative answer to Hilbert’s 10th Problem: there is no algorithm for deciding the solvability of diophantine equations over the rational integers. It was then conjectured by Julia Robinson that the solvability in algebraic integers (of arbitrary degree over the rationals) should be decidable. This was finally proved by R. Rumely [J. Reine Angew. Math. 368, 127-133 (1986; Zbl 0581.14014)] by introducing a ‘capacity theory on algebraic curves’.
In the present paper the authors give a direct algebraic proof for Rumely’s result: Let \(R\) be the ring of all algebraic integers in the algebraic closure \(K\) of the rationals and let \(V\) be an affine algebraic \(R\)-variety, irreducible over \(K\). If for every prime ideal \(p\) of \(R\) the variety \(V\) admits a point with coordinates in the localization \(R_p\), then \(V\) admits a point with coordinates in \(R\). This local-global principle together with the fact that the theory of the pair \((K,R_p)\) for each \(p\) is decidable gives an algorithm for deciding the solvability of diophantine equations over \(R\).
The authors also strengthen the above local-global principle by including archimedean primes and certain rationality conditions.

MSC:
11G35 Varieties over global fields
11R58 Arithmetic theory of algebraic function fields
14G25 Global ground fields in algebraic geometry
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