Uniform definability of henselian valuation rings in the Macintyre language

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