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 ${\mathbb{Z}}_p$ inside ${\mathbb{Q}}_p$ and of ${\mathbb{F}}_p[[t]]$ inside ${\mathbb{F}}_p((t))$, 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 ${\mathbb{Z}}_p$ inside ${\mathbb{Q}}_p$ uniformly for all p. For any fixed finite extension of ${\mathbb{Q}}_p$, we give an existential formula and a universal formula in the ring language which define the valuation ring.