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

This is joint work with Raf Cluckers, Eva Leenknegt, and Angus Macintyre.

We give a first-order definition, in the ring language, of the ring of p-adic integers inside the field p-adic numbers which works uniformly for all p and for valuation rings of all finite field extensions and of all local fields of positive characteristic p, and in many other Henselian valued fields as well. The formula canbe taken existential-universal in the ring language. 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 any fixed general p-adic field we give an existential formula in the ring language which defines the valuation ring.

We also state some connections to some open problems.