An undecidability result for power series rings of positive characteristic. II

Author:
Thanases Pheidas

Journal:
Proc. Amer. Math. Soc. **100** (1987), 526-530

MSC:
Primary 03D35; Secondary 13F25, 13J05, 13L05

MathSciNet review:
891158

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the existential problem for a power series ring over an integral domain of positive characteristic with a predicate which represents the powers of the indeterminate is undecidable. We prove the same result for any ring which is contained in a power series ring and contains the corresponding ring of polynomials.

**[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]**-,*Undecidability of rational function fields in nonzero characteristic*, Logic Colloq., no. 82, North-Holland, Amsterdam.**[5]**Paul J. Cohen,*Decision procedures for real and 𝑝-adic fields*, Comm. Pure Appl. Math.**22**(1969), 131–151. MR**0244025****[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]**J. Denef and L. Lipshitz,*An effective form of Greenberg's Theorem*(preprint).**[8]**Ju. L. Eršov,*On the elementary theory of maximal normed fields*, Dokl. Akad. Nauk SSSR**165**(1965), 21–23 (Russian). MR**0190140****[9]**Angus Macintyre,*On definable subsets of 𝑝-adic fields*, J. Symbolic Logic**41**(1976), no. 3, 605–610. MR**0485335****[10]**Ju. V. Matijasevič,*The Diophantineness of enumerable sets*, Dokl. Akad. Nauk SSSR**191**(1970), 279–282 (Russian). MR**0258744****[11]**T. Pheidas,*The Diophantine problem for addition and divisibility in polynomial rings*, Thesis, Purdue Univ., 1985.**[12]**Thanases Pheidas,*An undecidability result for power series rings of positive characteristic*, Proc. Amer. Math. Soc.**99**(1987), no. 2, 364–366. MR**870802**, 10.1090/S0002-9939-1987-0870802-X**[13]**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

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
03D35,
13F25,
13J05,
13L05

Retrieve articles in all journals with MSC: 03D35, 13F25, 13J05, 13L05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0891158-2

Article copyright:
© Copyright 1987
American Mathematical Society