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Diophantine sets of polynomials over number fields

Jeroen Demeyer (UGent)
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Abstract
Let R be a number field or a recursive subring of a number field and consider the polynomial ring R[T]. We show that the set of polynomials with integer coefficients is diophantine over R[7]. Applying a result by Denef, this implies that every recursively enumerable subset of R[T](k) is diophantine over R[T].
Keywords
Hilbert's Tenth Problem, RECURSIVELY-ENUMERABLE SETS, Recursively enumerable set, Diophantine set, HILBERTS 10TH PROBLEM, P-ADIC FIELDS, FINITE-FIELD, RINGS

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Citation

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

Chicago
Demeyer, Jeroen. 2010. “Diophantine Sets of Polynomials over Number Fields.” Proceedings of the American Mathematical Society 138 (8): 2715–2728.
APA
Demeyer, Jeroen. (2010). Diophantine sets of polynomials over number fields. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 138(8), 2715–2728.
Vancouver
1.
Demeyer J. Diophantine sets of polynomials over number fields. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. 2010;138(8):2715–28.
MLA
Demeyer, Jeroen. “Diophantine Sets of Polynomials over Number Fields.” PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 138.8 (2010): 2715–2728. Print.
@article{1073291,
  abstract     = {Let R be a number field or a recursive subring of a number field and consider the polynomial ring R[T]. We show that the set of polynomials with integer coefficients is diophantine over R[7]. Applying a result by Denef, this implies that every recursively enumerable subset of R[T](k) is diophantine over R[T].},
  author       = {Demeyer, Jeroen},
  issn         = {0002-9939},
  journal      = {PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY},
  keyword      = {Hilbert's Tenth Problem,RECURSIVELY-ENUMERABLE SETS,Recursively enumerable set,Diophantine set,HILBERTS 10TH PROBLEM,P-ADIC FIELDS,FINITE-FIELD,RINGS},
  language     = {eng},
  number       = {8},
  pages        = {2715--2728},
  title        = {Diophantine sets of polynomials over number fields},
  url          = {http://dx.doi.org/10.1090/S0002-9939-10-10329-3},
  volume       = {138},
  year         = {2010},
}

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