We show that the theory of the non-standard Frobenius automorphism, acting on an algebraically closed valued field of equal characteristic 0, is NTP2. More generally, in the contractive as well as in the isometric case, we prove that a σ-Henselian valued difference field of equicharacteristic 0 is NTP2, provided both the residue difference field and the value group (as an ordered difference gro...

A quantifier elimination procedure for a class M over a language L is an algorithm that given an L-formula φ computes a quantifier-free L-formula φ′ such that M |= φ′ ←→ φ. An L-formula is called linear if it contains no products or reciprocals of quantified variables. Quantifier elimination methods for valued fields have been extensively investigated in the past. The existence of a quantifier ...

We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure, o-minimality is shown. For Henselian valued fields, both the model theory and the analytic theory are developed. We give a list of examples that comprises, to ...

We give a proposal for future development of the model theory of valued fields. We also summarize recent results on p-adic numbers. Let K be a valued field with a valuation map v : K → G ∪ {∞} to an ordered group 1 G; this is a map satisfying (i) v(x) = ∞ if and only if x = 0; (ii) v(xy) = v(x) + v(y) for all x, y ∈ K; (iii) v(x + y) ≥ min{v(x), v(y)} for all x, y ∈ K. We write R for the valuat...

These notes focus mainly on the model theory of algebraically closed valued fields (loosely referred to as ACVF). This subject begins with work by A. Robinson in the 1950s (see the proof of model completeness of algebraically closed valued fields in [41]). Thus, it predates the major work of Ax-Kochen and Ershov around 1963; and, unlike the latter (and much subsequent work on quantifier elimina...

These lecture notes are a (slightly) extended version of my lecture course ‘Valued Fields’ as given during the first week of the Münster Month in Model Theory. They are heavily based on the book Valued Fields by Engler and Prestel, as well as (unpublished) lectures given by Jochen Koenigsmann at the University of Oxford in Hilary 2010. Many of the proofs presented are taken from (or at least in...

We study model theoretic properties of valued fields (equipped with a real-valued multiplicative valuation), viewed as metric structures in continuous first order logic. For technical reasons we prefer to consider not the valued field (K, |·|) directly, but rather the associated projective spaces KP, as bounded metric structures. We show that the class of (projective spaces over) metric valued ...