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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Diophantine sets of polynomials over number fields


Author: Jeroen Demeyer
Journal: Proc. Amer. Math. Soc. 138 (2010), 2715-2728
MSC (2010): Primary 11U09; Secondary 03D25, 11D99, 11R09, 12E10
Published electronically: April 5, 2010
MathSciNet review: 2644887
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Abstract: Let $ \mathcal{R}$ be a number field or a recursive subring of a number field and consider the polynomial ring $ \mathcal{R}[T]$. We show that the set of polynomials with integer coefficients is diophantine over $ \mathcal{R}[T]$. Applying a result by Denef, this implies that every recursively enumerable subset of $ \mathcal{R}[T]^k$ is diophantine over $ \mathcal{R}[T]$.


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Additional Information

Jeroen Demeyer
Affiliation: Department of Mathematics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
Email: jdemeyer@cage.ugent.be

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10329-3
PII: S 0002-9939(10)10329-3
Keywords: Diophantine set, recursively enumerable set, Hilbert's tenth problem.
Received by editor(s): June 1, 2009
Received by editor(s) in revised form: December 10, 2009
Published electronically: April 5, 2010
Additional Notes: The author is a Postdoctoral Fellow of the Research Foundation—Flanders (FWO)
Communicated by: Julia Knight
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.