We discuss definability of henselian valuation rings in the Macintyre
language LMacβ, 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 β-β -definable in
LMacβ, and henselian valuation rings with value group
Z are uniformly ββ-β -definable in the ring
language, but not uniformly β-β -definable in
LMacβ. We apply these results to local fields Qpβ
and Fpβ((t)), as well as to higher dimensional local fields