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

TL;DR

This paper provides uniform definitions of valuation rings in Henselian valued fields with finite or pseudo-finite residue fields using ring language formulas, including existential-universal forms, and discusses limitations of such definitions.

Contribution

It introduces uniform ring language definitions for valuation rings in various Henselian fields, including explicit formulas and negative results on definability limitations.

Findings

Uniform definitions for Z_p in Q_p and F_p[[t]] in F_p((t))

Existential-universal formulas in ring language for valuation rings

Negative results on uniform existential and universal definitions

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.