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Diophantine sets of polynomials over algebraic extensions of the rationals

Claudia Degroote (UGent) and Jeroen Demeyer (UGent)
(2014) JOURNAL OF SYMBOLIC LOGIC. 79(3). p.733-747
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Abstract
Let L be a recursive algebraic extension of Q. Assume that, given alpha is an element of 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 alpha. We prove that there exists a pair of polynomials that characterizes the Aut(L)-conjugates of alpha, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in R, or in a finite extension of Q(p) (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
FUNCTION-FIELDS, HILBERTS 10TH PROBLEM, diophantine sets, Hilbert's tenth problem, recursively enumerable sets

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Citation

Please use this url to cite or link to this publication:

Chicago
Degroote, Claudia, and Jeroen Demeyer. 2014. “Diophantine Sets of Polynomials over Algebraic Extensions of the Rationals.” Journal of Symbolic Logic 79 (3): 733–747.
APA
Degroote, C., & Demeyer, J. (2014). Diophantine sets of polynomials over algebraic extensions of the rationals. JOURNAL OF SYMBOLIC LOGIC, 79(3), 733–747.
Vancouver
1.
Degroote C, Demeyer J. Diophantine sets of polynomials over algebraic extensions of the rationals. JOURNAL OF SYMBOLIC LOGIC. 2014;79(3):733–47.
MLA
Degroote, Claudia, and Jeroen Demeyer. “Diophantine Sets of Polynomials over Algebraic Extensions of the Rationals.” JOURNAL OF SYMBOLIC LOGIC 79.3 (2014): 733–747. Print.
@article{4128161,
  abstract     = {Let L be a recursive algebraic extension of Q. Assume that, given alpha is an element of 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 alpha. We prove that there exists a pair of polynomials that characterizes the Aut(L)-conjugates of alpha, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in R, or in a finite extension of Q(p) (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].},
  author       = {Degroote, Claudia and Demeyer, Jeroen},
  issn         = {0022-4812},
  journal      = {JOURNAL OF SYMBOLIC LOGIC},
  language     = {eng},
  number       = {3},
  pages        = {733--747},
  title        = {Diophantine sets of polynomials over algebraic extensions of the rationals},
  url          = {http://dx.doi.org/10.1017/jsl.2013.9},
  volume       = {79},
  year         = {2014},
}

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