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DIOPHANTINE SETS OF POLYNOMIALS OVER ALGEBRAIC EXTENSIONS OF THE RATIONALS

Published online by Cambridge University Press:  18 August 2014

CLAUDIA DEGROOTE  and
JEROEN DEMEYER
CLAUDIA DEGROOTE
Affiliation:
GHENT UNIVERSITY, DEPARTMENT OF MATHEMATICS KRIJGSLAAN 281 9000 GENT, BELGIUME-mail: [email protected]: [email protected]
JEROEN DEMEYER
Affiliation:
GHENT UNIVERSITY, DEPARTMENT OF MATHEMATICS KRIJGSLAAN 281 9000 GENT, BELGIUME-mail: [email protected]: [email protected]
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Abstract

Let L be a recursive algebraic extension of ℚ. Assume that, given α ∈ L, we can compute the roots in L of its minimal polynomial over ℚ and we can determine which roots are Aut(L)-conjugate to α. We prove that there exists a pair of polynomials that characterizes the Aut(L)-conjugates of α, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in ℝ, or in a finite extension of ℚp (with p an odd prime). Then we show that subsets of L[X]k that are recursively enumerable for every recursive presentation of L[X], are diophantine over L[X].

Keywords

Hilbert’s tenth problemrecursively enumerable setsdiophantine sets
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 3 , September 2014 , pp. 733 - 747
Copyright
Copyright © The Association for Symbolic Logic 2014 

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