An undecidability result for power series rings of positive characteristic
Author:
Thanases Pheidas
Journal:
Proc. Amer. Math. Soc. 99 (1987), 364366
MSC:
Primary 03D35; Secondary 12L05, 13L05
MathSciNet review:
870802
Fulltext 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 of positive characteristic, with cross section, is undecidable whenever does not contain an such that . For example, the result is valid if (the element field where is a prime).
 [1]
James
Ax and Simon
Kochen, Diophantine problems over local fields. III. Decidable
fields, Ann. of Math. (2) 83 (1966), 437–456.
MR
0201378 (34 #1262)
 [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,
BerlinNew York, 1980, pp. 1–9. MR 606776
(83a:13013)
 [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
(86h:03059), http://dx.doi.org/10.1007/BFb0099383
 [4]
Paul
J. Cohen, Decision procedures for real and 𝑝adic
fields, Comm. Pure Appl. Math. 22 (1969),
131–151. MR 0244025
(39 #5342)
 [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, NorthHolland, AmsterdamNew York, 1979,
pp. 131–145. MR 567668
(81h:03090)
 [7]
Angus
Macintyre, On definable subsets of 𝑝adic fields, J.
Symbolic Logic 41 (1976), no. 3, 605–610. MR 0485335
(58 #5182)
 [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
(86m:03059), http://dx.doi.org/10.1007/BFb0099397
 [1]
 J. Ax and S. Kochen, Diophantine problems over local fields. III, Decidable fields, Ann. of Math. (2) 83 (1966), 437456. MR 0201378 (34:1262)
 [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. Karpacz, Poland, 1979), Lecture Notes in Math., vol. 834, SpringerVerlag, 1980, pp. 19. MR 606776 (83a:13013)
 [3]
 G. L. Cherlin, Definability in power series rings of nonzero characteristic, Models and Sets, Lecture Notes in Math., vol. 1103, SpringerVerlag, 1984, pp. 102112. MR 775690 (86h:03059)
 [4]
 P. J. Cohen, Decision procedures for real and adic fields, Comm. Pure Appl. Math. 22 (1969), 131151. MR 0244025 (39:5342)
 [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 Colloq., no. 78, NorthHolland, 1979. MR 567668 (81h:03090)
 [7]
 A. Macintyre, On definable subsets of adic fields, J. Symbolic Logic 41 (1976), 605610. MR 0485335 (58:5182)
 [8]
 V. Weispfenning, Quantifier elimination and decision procedures for valued fields, Logic Colloq., no. 83, Aachen, Lecture Notes in Math., SpringerVerlag, 1986, pp. 419472. MR 775704 (86m:03059)
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
DOI:
http://dx.doi.org/10.1090/S0002993919870870802X
PII:
S 00029939(1987)0870802X
Article copyright:
© Copyright 1987
American Mathematical Society
