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Preprint Number 847

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847. Will Anscombe
One-dimensional F-definable sets in F((t))
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Submission date: 19 March 2015.

Abstract:

We study definable sets in power series fields with perfect residue fields. We show that certain `one-dimensional' definable sets are in fact existentially definable. This allows us to apply results from previous work about existentially definable sets to one-dimensional definable sets.
More precisely, let F be a perfect field and let a be a tuple from F((t)) of transcendence degree 1 over F. Using the description of F-automorphisms of F((t)) given by Schilling, we show that the orbit of a under F-automorphisms is existentially definable in the ring language with parameters from F(t).
We deduce the following corollary. Let X be an F-definable subset of F((t)) which is not contained in F, then the subfield generated by X is equal to F((t^{p^n})), for some n<ω.

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Full text arXiv 1503.05803: pdf, ps.


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