Type-definable NIP fields are Artin–Schreier closed

Groups Definable in Separably Closed Fields

We consider the groups which are infinitely definable in separably closed fields of finite degree of imperfection. We prove in particular that no new definable groups arise in this way: we show that any group definable in such a field L is definably isomorphic to the group of L-rational points of an algebraic group defined over L.

Definable sets in algebraically closed valued fields: elimination of imaginaries

It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in K of certain definable R-submodules of K (for all n ≥ 1). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definabl...

Definable and Invariant Types in Enrichments of NIP Theories

Type decomposition in NIP theories

A first order theory is NIP if all definable families of subsets have finite VCdimension. We provide a justification for the intuition that NIP structures should be a combination of stable and order-like components. More precisely, we prove that any type in an NIP theory can be decomposed into a stable part (a generically stable partial type) and an order-like quotient.

Definable Sets in Valued Fields