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

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. (C) 2013 Elsevier B.V. All rights reserved.