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

2013/06/28 by Anscombe, Will, Jochen Koenigsmann, Koenigsmann, Jochen · 1 citation
Mathematics · #03C60 (Primary) 12L12 (Secondary) #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1306.6760

openalex publication_date 2013/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Construction of a valued field whose valuation ring is definable but neither ∃∀∃ nor ∀∃∀-definable in the language of rings
• Definable henselian valuations

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