In this paper, we consider the defect (also called ramification deficiency) of finite extensions of valued fields. For a valued field (K, v), we will denote its value group by vK and its residue field by K or by Kv. An extension of valued fields is written as (L|K, v), meaning that v is a valuation on L and K is equipped with the restriction of this valuation. Every finite extension L of a valu...

We show that every henselian valued field L of residue characteristic 0 admits a proper subfield K which is dense in L. We present conditions under which this can be taken such that L|K is transcendental and K is henselian. These results are of interest for the investigation of integer parts of ordered fields. We present examples of real closed fields which are larger than the quotient fields o...

We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also co...

We show that every valued differential field has an immediate strict extension that is spherically complete. We also discuss the issue of uniqueness up to isomorphism of such an extension.

In this paper we develope the fundamentals of the generalized symplectic geometry on the bundle of linear frames LM of an n-dimensional manifold M that follows upon taking the R-valued soldering 1-form θ on LM as a generalized symplectic potential. The development is centered around generalizations of the basic structure equation df = −Xf ω of standard symplectic geometry to LM when the symplec...

We give a presentation of the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan [12]. We limit our attention to a simple major subclass of V -minimal theories of the form ACVF S , that is, the theory of algebraically cl...

We prove the triviality of the Grothendieck ring of a Z-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point. At the Edinburgh meeting on the model theory ...

We present a theorem of Kollár on the density property of valued PAC fields and a theorem of Abraham Robinson on the model completeness of the theory of algebraically closed non-trivial valued fields. Then we prove that the theory T of non-trivial valued fields in an appropriate first order language has a model completion T̃ . The models of T̃ are non-trivial valued fields (K, v) that are ω-imper...

These are lecture notes from a graduate course on p-adic and motivic integration (given at BGU). Themain topics are: Quantifier elimination in the p-adics, rationality of p-adic zeta functions and their motivic analogues, basic model theory of algebraically closed valued fields, motivic integration following Hrushovski and Kazhdan, application to the Milnor fibration. Background: basic model th...

The paper shows elimination of imaginaries for real closed valued fields to the geometric sorts which were introduced in the [6]. We also show that this result is in some sense optimal.