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An existential 0-definition of F_q[[t]] in F_q((t)) | | Will Anscombe
; Jochen Koenigsmann
; | | Date: |
28 Jun 2013 | | 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. | | Source: | arXiv, 1306.6760 | | Services: | Forum | Review | PDF | Favorites | |
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