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On the decidability of the existential theory of \(\mathbb{F}_p[[t]]\). (English) Zbl 1046.12006

Kuhlmann, Franz-Viktor (ed.) et al., Valuation theory and its applications. Volume II. Proceedings of the international conference and workshop, University of Saskatchewan, Saskatoon, Canada, July 28–August 11, 1999. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3206-9/hbk). Fields Inst. Commun. 33, 43-60 (2003).
Let \({\mathbb F}_p\) be a finite field with \(p\) elements and let \(t\) be a variable. Let \(L\) be the language of rings augmented by a symbol for \(t\). The question addressed by the paper is, “Is the existential theory of \(L\) over the ring \({\mathbb F}_p[[t]]\) (of power series in \(t\) with coefficients in \({\mathbb F}_p\)) decidable – that is, is there an algorithm which, given an existential sentence of \(L\) (formed by existential quantifiers followed by a Boolean combination of polynomial equations in several variables and with coefficients in \({\mathbb F}_p\)), decides whether or not that is true in \({\mathbb F}_p[[t]]\)?”
The answer of the paper is positive (there is an algorithm) provided that there is resolution of singularities in characteristic \(p\) (which is an open problem).
It is known that the full theory of a power series ring over a decidable field of zero characteristic is decidable (Ax-Kochen, Ershov, P. Cohen, Weispfenning), while the theory of \({\widetilde {\mathbb F}}_p[[t]]\) is undecidable in the language which extends \(L\) by a symbol for the elements of \({\widetilde {\mathbb F}}_p\) (Cherlin). The positive-existential theory of \({\mathbb F}_p[[t]]\) is decidable (Denef-Lipshitz). Hence the first important unanswered relevant question is the decidability of the ring-theory of \({\mathbb F}_p[[t]]\), and the decidability of the existential theory will be a first crucial step. The result underlines the importance of resolution of singularities, which, from a logician’s point of view, may be considered as fundamental.
For the entire collection see [Zbl 1021.00011].

MSC:

12L05 Decidability and field theory
03B25 Decidability of theories and sets of sentences
03C60 Model-theoretic algebra
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