Introductory notes on the model theory of valued fields

Abstract

These notes will give some very basic definitions and results from model theory. They contain many examples, and in particular discuss extensively the various languages used to study valued fields. They are intended as giving the necessary background to read the papers by Cluckers, Delon, Halupczok, Kowalski, Loeser and Macintyre in this volume. We also mention a few recent results or directions of research in the model theory of valued fields, but omit completely those themes which will be discussed elsewhere in this volume. So for instance, we do not even mention motivic integration.

Analytic Nullstellensätze and the model theory of valued fields

“…A concise presentation of all the general model theory needed here is in [3,Appendix B]. A textbook, which also pays attention to valued fields is [47]; for more specialized expositions of the model theory of valued fields, see [11,22]. A (one-sorted) language  is a pair ( r ,  f ), where  r ,  f are disjoint (possibly empty, or infinite) collections of relation symbols and function symbols, respectively.…”

“…(This way of treating valued fields as model-theoretic structures is essentially equivalent to that in [47,Chapter 4], which employs valuation divisibilities instead of dominance relations. For other choices, see [11,22]. )…”

Computable valued fields

“…The theory ACVF of algebraically closed valued fields is axiomatized by saying that (K, v) is a valued field which is algebraically closed as a field (and recalling that we assumed that the valuation map is surjective). For a reference on algebraically closed valued fields, see [Cha11]. The theory is complete (after naming the characteristic and the characteristic of the residue field), decidable, and admits quantifier elimination.…”