Uniform definability of henselian valuation rings in the Macintyre language:

Abstract

We discuss definability of henselian valuation rings in the Macintyre language L Mac , the language of rings expanded by nth power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly ∃- ∅-definable in L Mac , and henselian valuation rings with value group Z are uniformly ∃ ∀ - ∅-definable in the ring language, but not uniformly ∃- ∅-definable in L Mac . We apply these results to local fields Q p and F p ( ( t ) ) , as well as to higher dimensional local fields.