1. Geometrical Axiomatization for Model Complete Theories of Differential Topological Fields.Nicolas Guzy & Cédric Rivière - 2006 - Notre Dame Journal of Formal Logic 47 (3):331-341.
    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 van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF₀ given by Pierce and Pillay. Moreover, it provides a geometric alternative to the axiomatizations obtained by (...)
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    Topological Differential Fields.Nicolas Guzy & Françoise Point - 2010 - Annals of Pure and Applied Logic 161 (4):570-598.
    We consider first-order theories of topological fields admitting a model-completion and their expansion to differential fields . 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 (...)
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    Topological Differential Fields and Dimension Functions.Nicolas Guzy & Françoise Point - 2012 - Journal of Symbolic Logic 77 (4):1147-1164.
    We construct a fibered dimension function in some topological differential fields.
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    0-D-Valued Fields.Nicolas Guzy - 2006 - Journal of Symbolic Logic 71 (2):639 - 660.
    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 L₂ 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 (...)
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    Some Elements of Lie-Differential Algebra and a Uniform Companion for Large Lie-Differential Fields.Nicolas Guzy - 2007 - Annals of Pure and Applied Logic 150 (1):66-78.
    In this paper, we develop the beginning of Lie-differential algebra, in the sense of Kolchin by using tools introduced by Hubert in [E. Hubert, Differential algebra for derivations with nontrivial commutation rules, J. Pure Appl. Algebra 200 163–190]. In particular it allows us to adapt the results of Tressl 3933–3951]) by showing the existence of a theory of Lie-differential fields of characteristic zero. This theory will serve as a model companion for every theory of large and Lie-differential fields extending a (...)
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