4 GUILLAUME DUVAL differential extensions. This leads us to a new proof of a result due to Drach and Kolchin, namely Theorem 26. The key point to prove this result is the auxiliary Proposition 25(ii), for which we give a valuative proof in section 4.5. 1.3.3. Continuity of derivation: geometry and examples. There are several points of view relating valuations to differential equations: in [19] and [28], Rosen- licht and Singer used valuations as algebraic tools to compute particular solutions of differential equations, (see [3] and references included for a more precise account on the subject). In [22] and [23], Rosenlicht discovered that valuations are adapted to measure asymptotic growth of functions belonging to Hardy fields. Here, we will be mainly concerned with this point of view. Indeed, we shall try to show in section 4 that this notion of continuity of a derivation w.r.t. a valuation is a necessary condition for this valuation to be able to describe the analytic behaviour of functions. Seidenberg’s main idea in [24] was to attach valuations to solutions of non- singular vector fields, that is to prove a “Cauchy type theorem” for valuations whose rings remain stable under the Lie derivative associated to the vector field. To introduce his work he said, Roughly, derivations are related to contact, and so are valuations, so one may ask for a study connecting derivations and valuations.” Following this strategy, for singular planar vector fields, Fortuny discovered in [7], that the valuations attached to the solutions are precisely l’Hopital’s ones. Corollary 38 will give a flavour of Seidenberg and Fortuny’s works. 1.3.4. Continuity and field extensions. In light of what has been said previously, we study here the permanence of the continuity of a derivation by field extensions. Our main results in section 5 are: Theorem 3. Let (F/K, ∂) be an algebraic extension of differential field of characteristic zero, such that K contains all the roots of unity. Let ν be a non- trivial valuation of F . If the restriction of to K is continuous w.r.t. ν, then is continuous with the same bound as its restriction to K. Theorem 4. Let (F/K, ∂) be a Liouvillian Picard-Vessiot field extension with algebraically closed constant field C. Let ν be any Gal∂(F/K)-invariant valuation of F/C such that the restriction of to K is continuous with respect to ν. Then is continuous on F with respect to ν on F . Theorem 3 asserts that the continuity of a derivation, w.r.t. a valuation is preserved by algebraic extension of the ground field. This property is rather natural. It says that a function which is algebraic over a given ground field shares the same asymptotic and infinitesimal properties as the functions belonging to this given ground field. However, its proof is quite complicated and essentially uses the idea that the derivation behaves better w.r.t. invariant valuations than the other ones. Theorem 4 is an illustration of the previous idea. Indeed, by Proposition 39, when tr . deg(F/K) 1 there are some valuations on F for which is not contin- uous, despite that the restriction of to K is continuous with respect to ν. By Theorem 4, such valuations are not invariant.
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