A Model Complete Theory of Valued D-Fields

The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the AxKochen-Ershov principle is proven for a theory of valued D-fields of residual characteristic zero. The model theory of differential and difference fields has been extensively studied (see for example [7, 3]) and valued fields have proven to be amenable to model theoretic analysis (see for example [1, 2]). In this paper we subject a theory of valued fields possessing either a derivation or an automorphism interacting strongly with the valuation to such an analysis. Our theory differs from C. Michaux's theory of henselian differential fields [8] on this last point: in his theory, the valuation and derivation have a very weak interaction. In Section 1 we introduce the notion of a D-field and show that a differential ring may be regarded as a specialization of a difference ring. This formal connection supports the view that differential and dif...

• Type: Journal
• Year: 2000
• Where: JSYML
• Authors: Thomas Scanlon
• Keywords: Difference Ring, JSYML 2000, Quantifier Elimination, Theory