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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)
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 n of certain definable R-submodules of K n (for all n ≥ 1). The proof involves the development of a theory of independence for unary types, which play the role of 1-types,(More)