Last updated
2024-04-21T12:03:14.13+01:00
Abstract
We show that the valuation ring F_q[[t]] in the local field F_q((t)) 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 existential-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 existential-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 existential
0-definition of the valuation ring of a non-trivial valuation with divisible
value group.
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 existential-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 existential-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 existential
0-definition of the valuation ring of a non-trivial valuation with divisible
value group.
Symplectic ID
414384
Download URL
http://arxiv.org/abs/1306.6760v1
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Publication type
Journal Article
Publication date
28 Jun 2013