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Diophantine sets over $ {\bf Z}[T]$


Author: J. Denef
Journal: Proc. Amer. Math. Soc. 69 (1978), 148-150
MSC: Primary 02F50; Secondary 10B99, 10N99
DOI: https://doi.org/10.1090/S0002-9939-1978-0462934-X
MathSciNet review: 0462934
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Abstract: Let $ {\mathbf{Z}}[T]$ be the ring of polynomials with integer coefficients. We prove that every recursively enumerable subset of $ {\mathbf{Z}}[T]$ is diophantine over $ {\mathbf{Z}}[T]$. This extends a theorem of Davis and Putnam which states that every recursively enumerable subset of Z is diophantine over $ {\mathbf{Z}}[T]$.


References [Enhancements On Off] (What's this?)

  • [1] Martin Davis, Hilbert's tenth problem is unsolvable, Amer. Math. Monthly 80 (1973), 233-269. MR 0317916 (47:6465)
  • [2] Martin Davis and Hilary Putnam, Diophantine sets over polynomial rings, Illinois J. Math. 7 (1963), 251-256. MR 0147387 (26:4903)
  • [3] Y. Pourchet, Sur la représentation en somme de carrés des polynômes à une indéterminée sur un corps de nombres algébriques, Acta Arith. 19 (1971), 89-104. MR 0289442 (44:6632)
  • [4] M. O. Rabin, Computable algebra, general theory and theory of computable fields, Trans. Amer. Math. Soc. 95 (1960), 341-360. MR 0113807 (22:4639)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0462934-X
Keywords: Diophantine sets, Hilbert's tenth problem, unsolvable problems, recursively enumerable sets, diophantine equations
Article copyright: © Copyright 1978 American Mathematical Society

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