Some model theory of Bezout difference rings- a survey

This survey article is based on a joint work with Ehud Hrushovski on some Bezout difference rings ([19]). We study in a model-theoretic point of view certain classes of difference rings, namely rings with a distinguished endomorphism and more particularly rings of sequences over a field with a shift. First we will recall some results on difference fields. A difference field is a difference ring which is a field (in this case, the endomorphism is necessarily injective); and one can show that any difference field can be embedded in an inversive difference field, namely a field with an automorphism ([5]). The model theory of difference fields started in the nineties; one motivation was to understand the model-theory of non principal ultraproducts of the algebraic closure Fp of the prime fields endowed with the Frobenius maps. The existence of a model-companion for the theory of difference fields was shown by L. van den Dries and A. Macintyre ([27]). Here, we will recall the geometric axiomatization ACFA of the class of existentially closed difference fields given by Z. Chatzidakis and E. Hrushovski ([7]); they identified the different completions of ACFA, and deduce its decidability. The proofs that the non principal ultraproducts of the Fp endowed with the Frobenius maps, are models of ACFA, are much more difficult (?). An easy consequence of the decidability of ACFA is the decidability of some von Neumann commutative regular difference rings. Indeed, these are Boolean products of fields and whenever the endomorphism σ fixes the prime spectrum of the ring, we may apply transfer results due to S. Burris and H. Werner in these products ([4]). More generally we will consider Bezout difference rings in the point of view