Uniform definability of henselian valuation rings in the Macintyre language

Abstract

We discuss definability of henselian valuation rings in the Macintyre language ${\mathcal{L}}_{\mathrm{Mac}}$, 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 $\exists$-$\varnothing$-definable in ${\mathcal{L}}_{\mathrm{Mac}}$, and henselian valuation rings with value group $\mathbb{Z}$ are uniformly $\exists \forall$-$\varnothing$-definable in the ring language, but not uniformly $\exists$-$\varnothing$-definable in ${\mathcal{L}}_{\mathrm{Mac}}$. We apply these results to local fields ${\mathbb{Q}}_p$ and ${\mathbb{F}}_p((t))$, as well as to higher dimensional local fields.

Cited by in corpus (2)

• Characterizing diophantine henselian valuation rings and valuation ideals
• A definable henselian valuation with high quantifier complexity