a b s t r a c tWe consider first-order theories of topological fields admitting a model-completion and their expansion to differential fields (requiring no interaction between the derivation and the other primitives of the language). We give a criterion under which the expansion still admits a model-completion which we axiomatize. It generalizes previous results due to M. Singer for ordered differential fields and of C. Michaux for valued differential fields. As a corollary, we show a transfer result for the NIP property. We also give a geometrical axiomatization of that model-completion. Then, for certain differential valued fields, we extend the positive answer of Hilbert's seventeenth problem and we prove an Ax-Kochen-Ershov theorem. Similarly, we consider first-order theories of topological fields admitting a model-companion and their expansion to differential fields, and under a similar criterion as before, we show that the expansion still admits a model-companion. This last result can be compared with those of M. Tressl: on one hand we are only dealing with a single derivation whereas he is dealing with several, on the other hand we are not restricting ourselves to definable expansions of the ring language, taking advantage of our topological context. We apply our results to fields endowed with several valuations (respectively several orders).

In [12]. T. Scanlon proved a quantifier elimination result for valued D-fields in a three-sorted language by using angular component functions. Here we prove an analogous theorem in a different language which was introduced by F. Delon in her thesis. This language allows us to lift the quantifier elimination result to a one-sorted language by a process described in the Appendix. As a byproduct, we state and prove a “positivstellensatz” theorem for the differential analogue of the theory of real-series closed fields in the valued D-field setting.

In (J. Symbolic Logic 56(2) (1991) 539), Bélair developed a theory analogous to the theory of real closed rings in the p-adic context, namely the theory of p-adically closed integral rings. Firstly we use the property proved in Lemma 2.4 in (J. Symbolic Logic 60(2) (1995) 484) to express this theory in a language including a p-adic divisibility relation and we show that this theory admits definable Skolem functions in this language (in the sense of (J. Symbolic Logic 49 (1984) 625)). Secondly, we are interested in dealing with some questions similar to that of (Z. Math. Logik Grundlag. 29(5) (1983) 417); e.g., results about integral-definite polynomials over a p-adically closed integral ring A and a kind of "Nullstellensatz" using the notion of M A -radical.

We construct a fibered dimension function in some topological differential fields.

No abstract