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AN EXISTENTIAL ∅-DEFINITION OF $F_q [[t]]$ IN $F_q \left( t \right)$

Published online by Cambridge University Press:  12 December 2014

WILL ANSCOMBE  and
JOCHEN KOENIGSMANN
WILL ANSCOMBE
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF LEEDS LEEDS, LS2 9JT, UKE-mail: w.g.anscombe@leeds.ac.uk
JOCHEN KOENIGSMANN
Affiliation:
MATHEMATICAL INSTITUTE RADCLIFFE OBSERVATORY SITE WOODSTOCK ROAD OXFORD, OX2 6GG, UKE-mail: koenigsmann@maths.ox.ac.uk
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Abstract

We show that the valuation ring $F_q [[t]]$ in the local field $F_q \left( t \right)$ is existentially definable in the language of rings with no parameters. The method is to use the definition of the henselian topology following the work of Prestel-Ziegler to give an ∃-$F_q $-definable bounded neighbouhood of 0. Then we “tweak” this set by subtracting, taking roots, and applying Hensel’s Lemma in order to find an ∃-$F_q $-definable subset of $F_q [[t]]$ which contains $tF_q [[t]]$. Finally, we use the fact that $F_q $ is defined by the formula $x^q - x = 0$ to extend the definition to the whole of $F_q [[t]]$ and to rid the definition of parameters.

Several extensions of the theorem are obtained, notably an ∃-∅-definition of the valuation ring of a nontrivial valuation with divisible value group.

Keywords

Valuationshenselian valued fieldsdefinable valuationsmodel theory
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 4 , December 2014 , pp. 1336 - 1343
Copyright
Copyright © The Association for Symbolic Logic 2014 

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References

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