Existentially closed ordered difference fields and rings

We describe classes of existentially closed ordered difference fields and rings. We show an Ax-Kochen type result for a class of valued ordered difference fields. 1. Existentially closed real-closed difference fields. In the first part of this paper we will consider on one hand difference totally ordered fields, namely totally ordered fields with a distinguished automorphism σ and on the other hand preordered difference fields. By a well-known theorem of A. Tarski, the theory RCF of real-closed fields is the model-companion of the theory of the ordered fields and a direct consequence of results of H. Kikyo and S. Shelah, is that the theory of real-closed ordered difference fields, RCFσ does not have a model-companion (see [18]). Note that in a difference field (K, σ), one has automatically a pair of fields, namely (K,Fix(σ)), where Fix(σ) denotes the subfield of elements of K fixed by σ and if K is real-closed, then so is Fix(σ). W. Baur showed that the theory of all pairs of real-closed fields (K,L) with a predicate for a subfield is undecidable ([1]). However, he also showed that the theory of the pairs (K,L) such that, adding to the language of ordered rings a new function symbol for a valuation v, v is convex, the residue field of L is dense in the residue field of K and each finite-dimensional L-vector space of K has a basis a1, · · · , an satisfying for all bi ∈ L that v( ∑ i bi.ai) = mini{v(bi.ai)}, becomes decidable ([1]). First, we describe a class of existentially closed totally ordered difference fields (even though it is not an elementary class). We also consider the case of a proper preordering, using former results of A. Prestel and L. van den Dries. Then, we consider valued ordered fields and we assume on one hand that σ is strictly increasing on the set of elements of strictly positive valuation and on the other hand that in the pair (K,Fix(σ)), the residue field of K and the residue field of Fix(σ) coincide (and so we are trivially in the Baur setting). We proceed as for the case of valued difference fields with an ω-increasing automorphism treated by E. Hrushovski ([7]) and we show an Ax-Kochen-Ersov type result.