Geometrical Axiomatization for Model Complete Theories of Differential Topological Fields

In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of [vdD1, 2.11], and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF0 given by D. Pierce and A. Pillay in [PP]. Moreover it provides a geometric alternative to the axiomatizations obtained in [Tr] and [GP] where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space. 1 Basic algebraic geometry In what follows K is a field of characteristic zero, Ω is a sufficiently saturated elementary extension of K (in particular Ω is of infinite transcendence degree over K) and Ω̃ denotes the algebraic closure of Ω. ∗The first author is supported by a FNRS grant. The second author is supported by a FRIA grant.