On Rumely's local-global principle
type: Item
instance of: scholarly article
title: On Rumely's local-global principle
zbMATH Open document ID: 0857.11033
author: Barry W. Green
author: Florian Pop
author: Peter Roquette
published in: Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)
publication date: 1997-03-11T00:00:00Z
review text: In 1970 \textit{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 \textit{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.
reviewed by: Alexander Prestel
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 780024
zbMATH Keywords: Hilbert's 10th problem
zbMATH Keywords: algebraic integers
zbMATH Keywords: local-global principle
zbMATH Keywords: solvability of diophantine equations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 780024