Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Diophantine problem for polynomial rings and fields of rational functions


Author: J. Denef
Journal: Trans. Amer. Math. Soc. 242 (1978), 391-399
MSC: Primary 10N05; Secondary 10B30
DOI: https://doi.org/10.1090/S0002-9947-1978-0491583-7
MathSciNet review: 0491583
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the diophantine problem for a ring of polynomials over an integral domain of characteristic zero or for a field of rational functions over a formally real field is unsolvable.


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

  • [1] J. Ax, On the undecidability of power series fields, Proc. Amer. Math. Soc. 16 (1965), 846. MR 0177890 (31:2148)
  • [2] J. Becker and L. Lipshitz, Remarks on the elementary theories of formal and convergent power series, Fund. Math. (to appear). MR 580584 (82i:14004a)
  • [3] J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193-291. MR 0199150 (33:7299)
  • [4] M. Davis, Hilbert's tenth problem is unsolvable, Amer. Math. Monthly 80 (1973), 233-269. MR 0317916 (47:6465)
  • [5] M. Davis and H. Putnam, Diophantine sets over polynomial rings, Illinois J. Math. 7 (1963), 251-256. MR 0147387 (26:4903)
  • [6] M. Davis, Yu. Matijasevič and J. Robinson, Diophantine equations: Positive aspects of a negative solution, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R.I., 1976, pp. 323-378. MR 0432534 (55:5522)
  • [7] J. Denef, Hilbert's tenth problem for quadratic rings, Proc. Amer. Math. Soc. 48 (1975), 214-220. MR 0360513 (50:12961)
  • [8] -, Diophantine sets over $ {\textbf{Z}}[T]$ , Proc. Amer. Math. Soc. 69 (1978), 148-150. MR 0462934 (57:2899)
  • [9] Yu. Eršov, Undecidability of certain fields, Dokl. Akad. Nauk SSSR 161 (1965), 349-352. MR 0175785 (31:61)
  • [10] -, New examples of undecidable theories, Algebra i. Logika 5 (1966), 37-47. MR 0207560 (34:7375)
  • [11] W. Fulton, Algebraic curves, Benjamin, New York, 1969. MR 0313252 (47:1807)
  • [12] S. Lang, Elliptic functions, Addison-Wesley, London, 1973. MR 0409362 (53:13117)
  • [13] L. Lipshitz, Undecidable problems for addition and divisibility in algebraic number rings. II, Proc. Amer. Math. Soc. 64 (1977), 122-128. MR 0536659 (58:27375b)
  • [14] A. I. Malcev, On the undecidability of the elementary theories of certain fields, Sibirsk Mat. Ž. 1 (1960), 71-77; ibid 2 (1961), 639; English transl., Amer. Math. Soc. Transl. (2) 48 (1965). MR 0125797 (23:A3094)
  • [15] Yu. Penzin, Undecidability of fields of rational functions over fields of characteristic 2, Algebra i. Logika 12 (1973), 205-210; 244. MR 0389875 (52:10705)
  • [16] Y. Pourchet, Sur la représentation en somme de carrés des polynômes à une indéterminé sur un corps de nombres algébriques, Acta Arith. 19 (1971), 89-104. MR 0289442 (44:6632)
  • [17] J. Robinson, Definability and decision problems in arithmetic, J. Symbolic Logic 14 (1949), 98-114. MR 0031446 (11:151f)
  • [18] -, The undecidability of algebraic rings and fields, Proc. Amer. Math. Soc. 10 (1959), 950-957. MR 0112842 (22:3691)
  • [19] -, On the decision problem for algebraic rings, Studies in Math. Anal. and Related Topics, Stanford, 1962, pp. 297-304. MR 0146083 (26:3609)
  • [20] -, The decision problem for fields, Sympos. on the Theory of Models, North-Holland, Amsterdam, 1965, pp. 299-311. MR 0200163 (34:62)
  • [21] R. M. Robinson, Undecidable rings, Trans. Amer. Math. Soc. 70 (1951), 137-159. MR 0041081 (12:791b)
  • [22] -, The undecidability of pure transcendental extensions of real fields, Z. Math. Logik Grundlagen Math. 10 (1964), 275-282. MR 0172803 (30:3021)
  • [23] A. Tarski, The elementary undecidability of pure transcendental extensions of real closed fields, Notices Amer. Math. Soc. 10 (1963), A-355.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 10N05, 10B30

Retrieve articles in all journals with MSC: 10N05, 10B30


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0491583-7
Keywords: Hilbert's tenth problem, unsolvable problems, diophantine equations
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society