Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An undecidability result for power series rings of positive characteristic

Author: Thanases Pheidas
Journal: Proc. Amer. Math. Soc. 99 (1987), 364-366
MSC: Primary 03D35; Secondary 12L05, 13L05
MathSciNet review: 870802
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the existential theory of a power series ring in one variable over an integral domain $ F$ of positive characteristic, with cross section, is undecidable whenever $ F$ does not contain an $ e$ such that $ {e^p} - e = 1$. For example, the result is valid if $ F = {Z_p}$ (the $ p$-element field where $ p$ is a prime).

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

  • [1] James Ax and Simon Kochen, Diophantine problems over local fields. III. Decidable fields, Ann. of Math. (2) 83 (1966), 437–456. MR 0201378
  • [2] J. Becker, J. Denef, and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979), Lecture Notes in Math., vol. 834, Springer, Berlin-New York, 1980, pp. 1–9. MR 606776
  • [3] G. L. Cherlin, Definability in power series rings of nonzero characteristic, Models and sets (Aachen, 1983) Lecture Notes in Math., vol. 1103, Springer, Berlin, 1984, pp. 102–112. MR 775690, 10.1007/BFb0099383
  • [4] Paul J. Cohen, Decision procedures for real and 𝑝-adic fields, Comm. Pure Appl. Math. 22 (1969), 131–151. MR 0244025
  • [5] J. Denef and L. Lipshitz, A constructive analogue of Greenberg's Theorem in positive characteristic, preprint.
  • [6] J. Denef, The Diophantine problem for polynomial rings of positive characteristic, Logic Colloquium ’78 (Mons, 1978) Stud. Logic Foundations Math., vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 131–145. MR 567668
  • [7] Angus Macintyre, On definable subsets of 𝑝-adic fields, J. Symbolic Logic 41 (1976), no. 3, 605–610. MR 0485335
  • [8] Volker Weispfenning, Quantifier elimination and decision procedures for valued fields, Models and sets (Aachen, 1983) Lecture Notes in Math., vol. 1103, Springer, Berlin, 1984, pp. 419–472. MR 775704, 10.1007/BFb0099397

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 03D35, 12L05, 13L05

Retrieve articles in all journals with MSC: 03D35, 12L05, 13L05

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society