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

Abstract We give a definition, in the ring language, of Z p inside Q p and of F p [ [ t ] ] inside F p ( ( t ) ) , which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Z p inside Q p uniformly for all p. For any fixed finite extension of Q p , we give an existential formula and a universal formula in the ring language which define the valuation ring.