An existential 0-definition of Fq[[t]] in Fq((t))
Abstract
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 existential-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 existential-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 existential 0-definition of the valuation ring of a non-trivial valuation with divisible value group.