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Hilbertʼs tenth problem for rational function fields over p-adic fields

Claudia Degroote (UGent) and Jeroen Demeyer (UGent)
(2012) JOURNAL OF ALGEBRA. 361. p.172-187
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Abstract
Let K be a p-adic field (a finite extension of some Q_p) and let K(t) be the field of rational functions over K. We define a kind of quadratic reciprocity symbol for polynomials over K and apply it to prove isotropy for a certain class of quadratic forms over K(t). Using this result, we give an existential definition for the predicate “v_t(x) >= 0” in K(t). This implies undecidability of diophantine equations over K(t).
Keywords
Undecidability, Hilbert's Tenth Problem, Quadratic form, Valuation, Diophantine set

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

Chicago
Degroote, Claudia, and Jeroen Demeyer. 2012. “Hilbertʼs Tenth Problem for Rational Function Fields over P-adic Fields.” Journal of Algebra 361: 172–187.
APA
Degroote, C., & Demeyer, J. (2012). Hilbertʼs tenth problem for rational function fields over p-adic fields. JOURNAL OF ALGEBRA, 361, 172–187.
Vancouver
1.
Degroote C, Demeyer J. Hilbertʼs tenth problem for rational function fields over p-adic fields. JOURNAL OF ALGEBRA. 2012;361:172–87.
MLA
Degroote, Claudia, and Jeroen Demeyer. “Hilbertʼs Tenth Problem for Rational Function Fields over P-adic Fields.” JOURNAL OF ALGEBRA 361 (2012): 172–187. Print.
@article{4128132,
  abstract     = {Let K be a p-adic field (a finite extension of some Q_p) and let K(t) be the field of rational functions over K. We define a kind of quadratic reciprocity symbol for polynomials over K and apply it to prove isotropy for a certain class of quadratic forms over K(t). Using this result, we give an existential definition for the predicate “v_t(x) >= 0” in K(t). This implies undecidability of diophantine equations over K(t).},
  author       = {Degroote, Claudia and Demeyer, Jeroen},
  issn         = {0021-8693},
  journal      = {JOURNAL OF ALGEBRA},
  keywords     = {Undecidability,Hilbert's Tenth Problem,Quadratic form,Valuation,Diophantine set},
  language     = {eng},
  pages        = {172--187},
  title        = {Hilbertʼs tenth problem for rational function fields over p-adic fields},
  url          = {http://dx.doi.org/10.1016/j.jalgebra.2012.03.039},
  volume       = {361},
  year         = {2012},
}

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