Some elements of Lie-differential algebra and a uniform companion for large Lie-differential fields

In this paper, we develop the beginning of Lie-differential algebra, in the sense of Kolchin (see [E.R. Kolchin, Differential algebra and algebraic groups, in: Pure and Applied Mathematics, vol. 54, Academic Press, 1973]) by using tools introduced by Hubert in [E. Hubert, Differential algebra for derivations with nontrivial commutation rules, J. Pure Appl. Algebra 200 (2005) 163–190].In particular it allows us to adapt the results of Tressl (see [M. Tressl, A uniform companion for large differential fields of characteristic zero, Trans. Amer. Math. Soc. 357 (10) (2005) 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 model complete theory of pure fields. As an application, we introduce the Lie counterpart of classical theories of differential fields in several commuting derivations.