Model theory of valued D-fields (1997) [10 citations — 7 self]
|
Abstract:
Abstract. The notion of a D-ring, generalizing that of a dierential or a dierence ring, is introduced. Quantier elimination and a version of the AxKochen-Ershov principle is proven for a theory of valued D-elds of residual characteristic zero. The model theory of dierential and dierence elds has been extensively studied (see for example [7, 3]) and valued elds have proven to be amenable to model theoretic analysis (see for example [1, 2]). In this paper we subject a theory of valued elds possessing either a derivation or an automorphism interacting strongly with the valuation to such an analysis. Our theory diers from C. Michaux's theory of henselian dierential elds [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-eld and show that a dierential ring may be regarded as a specialization of a dierence ring. This formal connection supports the view that dierential and dierence algebra are instances of the same theory. We introduce our axioms in Section 5 and prove quantier elimination in Section 7. This provides an example of a non-trivial dierence ring admitting
Citations
| 67 | Model theory, Encyclopedia of Mathematics and – Hodges - 1993 |
| 45 | Valuation theory – Endler - 1972 |
| 32 | Model theory of difference fields – Chatzidakis, Hrushovski - 1999 |
| 26 | Diophantine problems over local fields – Ax, Kochen - 1965 |
| 22 | The Theory of Valuations – Schilling - 1950 |
| 13 | On ordered division rings – Neumann - 1949 |
| 11 | On local uniformization in arbitrary characteristic, The Fields Institute Preprint Series – Kuhlmann - 1997 |
| 8 | The model theory of dierence – Chatzidakis, Hrushovski - 1999 |
| 6 | Model theory of differential fields – Marker - 2000 |
| 5 | Diophantine problems over local – Ax, Kochen - 1965 |
| 5 | Model Theory of Valued D – Scanlon - 1997 |
| 4 | Model Theory, North-Holland, 3rd ed – Chang, Keisler - 1990 |
| 3 | Model theory of dierential – Marker - 1996 |
| 3 | Anneaux locaux henseliens, LNM 169 – Raynaud - 1970 |
| 2 | Quantifier eliminable ordered abelian groups, Algebra and order (Luminy-Marseille – Weispfenning - 1984 |
