The Diophantine problem for polynomial rings and fields of rational functions
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- by J. Denef PDF
- Trans. Amer. Math. Soc. 242 (1978), 391-399 Request permission
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
- James Ax, On the undecidability of power series fields, Proc. Amer. Math. Soc. 16 (1965), 846. MR 177890, DOI 10.1090/S0002-9939-1965-0177890-2
- Joseph Becker and Leonard Lipshitz, Remarks on the elementary theories of formal and convergent power series, Fund. Math. 105 (1979/80), no. 3, 229–239. MR 580584, DOI 10.4064/fm-105-3-229-239
- J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193–291. MR 199150, DOI 10.1112/jlms/s1-41.1.193
- Martin Davis, Hilbert’s tenth problem is unsolvable, Amer. Math. Monthly 80 (1973), 233–269. MR 317916, DOI 10.2307/2318447
- Martin Davis and Hilary Putnam, Diophantine sets over polynomial rings, Illinois J. Math. 7 (1963), 251–256. MR 147387
- Martin Davis, Yuri Matijasevič, and Julia Robinson, Hilbert’s tenth problem: Diophantine equations: positive aspects of a negative solution, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R.I., 1976, pp. 323–378. (loose erratum). MR 0432534
- J. Denef, Hilbert’s tenth problem for quadratic rings, Proc. Amer. Math. Soc. 48 (1975), 214–220. MR 360513, DOI 10.1090/S0002-9939-1975-0360513-3
- J. Denef, Diophantine sets over $\textbf {Z}[T]$, Proc. Amer. Math. Soc. 69 (1978), no. 1, 148–150. MR 462934, DOI 10.1090/S0002-9939-1978-0462934-X
- Ju. L. Eršov, The undecidability of certain fields, Dokl. Akad. Nauk SSSR 161 (1965), 27–29 (Russian). MR 0175785
- Ju. L. Eršov, New examples of undecidable theories, Algebra i Logika Sem. 5 (1966), no. 5, 37–47 (Russian). MR 0207560
- William Fulton, Algebraic curves. An introduction to algebraic geometry, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss. MR 0313252
- Serge Lang, Elliptic functions, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Amsterdam, 1973. With an appendix by J. Tate. MR 0409362
- L. Lipshitz, Undecidable existential problems for addition and divisibility in algebraic number rings, Trans. Amer. Math. Soc. 241 (1978), 121–128. MR 536658, DOI 10.1090/S0002-9947-1978-0536658-9
- A. I. Mal′cev, The undecidability of the elementary theories of certain fields, Sibirsk. Mat. Ž. 1 (1960), 71–77 (Russian). MR 0125797
- Ju. G. Penzin, Undecidability of fields of rational functions over fields of characteristic $2$, Algebra i Logika 12 (1973), 205–210, 244 (Russian). MR 0389875
- 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 (French). MR 289442, DOI 10.4064/aa-19-1-89-104
- Julia Robinson, Definability and decision problems in arithmetic, J. Symbolic Logic 14 (1949), 98–114. MR 31446, DOI 10.2307/2266510
- Julia Robinson, The undecidability of algebraic rings and fields, Proc. Amer. Math. Soc. 10 (1959), 950–957. MR 112842, DOI 10.1090/S0002-9939-1959-0112842-7
- Julia Robinson, On the decision problem for algebraic rings, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 297–304. MR 0146083
- Julia Robinson, The decision problem for fields, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 299–311. MR 0200163
- Raphael M. Robinson, Undecidable rings, Trans. Amer. Math. Soc. 70 (1951), 137–159. MR 41081, DOI 10.1090/S0002-9947-1951-0041081-0
- Raphael M. Robinson, The undecidability of pure transcendental extensions of real fields, Z. Math. Logik Grundlagen Math. 10 (1964), 275–282. MR 172803, DOI 10.1002/malq.19640101803 A. Tarski, The elementary undecidability of pure transcendental extensions of real closed fields, Notices Amer. Math. Soc. 10 (1963), A-355.
Additional Information
- © Copyright 1978 American Mathematical Society
- 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