An existential ∅-definition of Fq[[t]]Fq[[t]] IN Fq(t) - Anscombe, W; Koenigsmann, J (Cambridge University Press, 2014)
We show that the valuation ring Fq [[t]] in the local field Fq ((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 ∃-Fq -definable bounded neighbouhood of 0. Then we “tweak” this set by subtracting, taking roots, and applying Hensel’s Lemma in order to find an ∃-Fq -definable subset of Fq [[t]] which contains tFq [[t]]. Finally, we use the fact that Fq is defined by the formula xq − x = 0 to extend the definition to the whole of Fq [[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.