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A difference field is a field with a distinguished automorphism σ. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on(More)
The questions this manuscript addresses arose in the course of an investigation of the imaginary sorts in ultraproducts of p-adic fields. These were shown to be understandable given the imaginary sorts of certain finite-dimensional vector spaces over the residue field. The residue field is pseudo-finite, and the imaginary elements there were previously(More)
We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a(More)
We study forking, Lascar strong types, Keisler measures and defin-able groups, under an assumption of N IP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking(More)
This paper contains a series of easy constructions and observations relating to the Lascar group and to simple theories. 1 In x1 we review basic model theoretic ideas, relating mostly to model completion and saturated models. We do so in order to introduce a framework very slightly more general than the usual rst-order one that will be useful to us, and(More)