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An undecidability result for power series rings of positive characteristics. II. (English) Zbl 0664.03008
[For part I see ibid. 99, 364-366 (1987; Zbl 0639.03009).]
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.

##### MSC:
 03B25 Decidability of theories and sets of sentences 12L05 Decidability and field theory 13F25 Formal power series rings 11U05 Decidability (number-theoretic aspects) 11D88 $$p$$-adic and power series fields
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##### References:
 [1] James Ax and Simon Kochen, Diophantine problems over local fields. III. Decidable fields, Ann. of Math. (2) 83 (1966), 437 – 456. · Zbl 0223.02050 · doi:10.2307/1970476 · doi.org [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. · Zbl 0452.12013 [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. · Zbl 0574.03017 · doi:10.1007/BFb0099383 · doi.org [4] -, Undecidability of rational function fields in nonzero characteristic, Logic Colloq., no. 82, North-Holland, Amsterdam. · Zbl 0551.03027 [5] Paul J. Cohen, Decision procedures for real and \?-adic fields, Comm. Pure Appl. Math. 22 (1969), 131 – 151. · Zbl 0167.01502 · doi:10.1002/cpa.3160220202 · doi.org [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. · Zbl 0457.12011 [7] J. Denef and L. Lipshitz, An effective form of Greenberg’s Theorem (preprint). · Zbl 0518.12015 [8] Ju. L. Eršov, On the elementary theory of maximal normed fields, Dokl. Akad. Nauk SSSR 165 (1965), 21 – 23 (Russian). [9] Angus Macintyre, On definable subsets of \?-adic fields, J. Symbolic Logic 41 (1976), no. 3, 605 – 610. · Zbl 0362.02046 · doi:10.2307/2272038 · doi.org [10] Ju. V. Matijasevič, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279 – 282 (Russian). [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. · Zbl 0639.03009 [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. · Zbl 0584.03022 · doi:10.1007/BFb0099397 · doi.org
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