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


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.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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