Uniform definability of henselian valuation rings in the Macintyre language

TL;DR

This paper investigates how henselian valuation rings can be uniformly defined in the Macintyre language, revealing conditions under which such definability is possible and applying results to local fields.

Contribution

It establishes uniform definability results for henselian valuation rings in the Macintyre language, including cases with finite, Hilbertian residue fields, and explores definability in various local fields.

Findings

Henselian valuation rings with finite or Hilbertian residue fields are uniformly ∃-definable in the Macintyre language.

Valuation rings with value group ℤ are ∃∀-definable in ring language but not uniformly ∃-definable in Macintyre language.

Results apply to local fields like ℚₚ and ℱₚ((t)), as well as higher dimensional local fields.

Abstract

We discuss definability of henselian valuation rings in the Macintyre language $\mathcal{L}_{\rm Mac}$, the language of rings expanded by n-th power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$, and henselian valuation rings with value group $\mathbb{Z}$ are uniformly $\exists\forall$-$\emptyset$-definable in the ring language, but not uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$. We apply these results to local fields $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$, as well as to higher dimensional local fields.