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The theory of generic difference fields

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2003
Yıldırım, İrem
A difference field M , is a field with a distinguished endomorphism, is called a generic difference field if it is existentially closed among the models of the theory of difference fields. In the language Ld, by a theorem of Hrushovski, it is characterized by the following: M is an algebraically closed field, s is an automorphism of M, and if W and V are varieties defined over M such that W is a subset of VU s (V ) and the projection maps W to V and W to s(V ) are generically onto, then there is a tuple a in M such that (a, s ( a)) in W. This thesis is a survey on the theory of generic difference fields, called ACFA, which has been studied by Angus Macintyre, Van den Dries, Carol Wood, Ehud Hrushovski and Zoe Chatzidakis. ACFA is the model completion of the theory of algebraically closed difference fields. It is very close to having full quantifier elimination, but it doesn't. We can eliminate quantifiers down to formulas with one quantifier and hence obtain the completions of ACFA. This entails the decidability of the theory ACFA as well as its extensions obtained by specifying the characteristic. The fixed field of s is a pseudo-finite field.