An existential 0-definition of F_q[[t]] in F_q((t))

TL;DR

The paper proves that the valuation ring F_q[[t]] in the local field F_q((t)) can be defined existentially within the language of rings without parameters, using topological and algebraic methods.

Contribution

It introduces a novel existential definability of F_q[[t]] in F_q((t)) without parameters, extending to valuation rings with divisible value groups.

Findings

F_q[[t]] is existentially definable in F_q((t))

The method uses henselian topology and Hensel's Lemma

Extension to valuation rings with divisible value groups

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.