Uniform definability of henselian valuation rings in the Macintyre language

fehm, arno; prestel, alexander;

We discuss definability of henselian valuation rings in the Macintyre language LMac, 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 LMac, and henselian valuation rings with value group Z are uniformly ∃∀-∅-definable in the ring language, but not uniformly ∃-∅-definable in LMac. We apply these results to local fields Qp and Fp((t)), as well as to higher dimensional local fields.