Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

We give a de nition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp((t)), which works uniformly for all p and all nite eld extensions of these elds, and in many other Henselian valued elds as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modi cation of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform de nition by an existential formula and neither by a universal formula for the valuation rings of all the nite extensions of a given Henselian valued eld. We also show that there is no existential formula of the ring language de ning Zp inside Qp uniformly for all p. For any xed nite extension of Qp, we give an existential formula and a universal formula in the ring language which de ne the valuation ring.