Existential fragments of theories of henselian valued fields
Sylvy Anscombe, Arno Fehm
TL;DR
The paper investigates existential fragments of the theories of henselian valued fields in positive and mixed characteristic, developing monotonicity principles that connect residue-field theories to whole-field theories. It introduces and analyzes several fragments ($\exists_n$, $\exists_n\exists_1$, $\exists^n$) and establishes translation tools from valued-field language to ring language, enabling effective axiomatics and decidability results. A key technical advance is the use of resolution of singularities (up to dimension $3$) to obtain unconditional results for $\exists_3$ and $\exists_4$, including decidability of the $\exists_3$-theory of $\mathbb{F}_q((t))$ in the $v$-language with a $t$-parameter. The framework yields uniform, fragment-wise decidability and axiomatization across several residue theories, offering new model-theoretic consequences and avenues for further exploration in positive characteristic local fields. Overall, the work provides a principled method to obtain decidability and axiomatization results for existential theories of henselian valued fields through monotonicity, definability in ring language, and singularity-resolution-based assumptions.
Abstract
We study fragments of the existential theory of henselian valued fields with parameters. This includes the $\exists_n$-fragment in the equicharacteristic or unramified mixed characteristic case, the $\exists_n\exists_1$-fragment in the equicharacteristic case, and the $\exists_n$-fragment in the residue characteristic zero case. For example, we obtain an unconditional axiomatization (and thereby decidability) of the $\exists_3$-theory of $\mathbb{F}_{q}(\!(t)\!)$ in the language of valued fields with a parameter for $t$.