TOPOLOGICAL DIFFERENTIAL FIELDS

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1 TOPOLOGICAL DIFFERENTIAL FIELDS NICOLAS GUZY AND FRANÇOISE POINT Abstract. First, we 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. It generalizes previous results due to M. Singer for ordered differential fields and of C. Michaux for valued differential fields. Using the proof of that result, we give an Ax-Kochen-Ershov theorem in that differential context. Finally, we consider first-order theories of topological fields admitting a model-companion and their expansion to differential fields and we show that under the same criterion as before, the expansion is still model-complete. 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. MSC: 03C10, 12L12, 12J, 12J10, 12J15, 12H05 Keywords: topological fields, derivation, model-companion, Henselization 1. Introduction We are given an inductive theory which is an expansion of a theory of fields and which has a model-completion or a model-companion. We expand the language with a new unary function symbol which satisfies the axioms of a derivation, then we consider the following question: when does the corresponding expansion of the theory retain the property of having a model-completion (respectively a model-companion)? We will generalize a criterion due to M. Singer [27] for ordered fields, which was later adapted to the case of valued fields by C. Michaux [14]. Both criteria have a topological flavour. As a byproduct we extend the positive answer for Hilbert s Seventeenth Problem for p-adically closed fields of p-rank d. Then, in the same spirit, we will prove an Ax-Kochen-Ershov type result for valued differential fields. Recently, M. Tressl had introduced a more algebraic approach to this problem, also dealing with the case of several commuting derivations. There, the crux of the matter is to decide which finite systems of algebraic differential equations are required to have a solution [28]. His approach (specialized to the case of one derivation) differs from ours since he restricts himself to expansions by definition of the ring language. Date: May 4, Fellow Research at the Fonds National de la Recherche Scientifique. 2 Senior Research Associate at the Fonds National de la Recherche Scientifique. 1

2 2 NICOLAS GUZY AND FRANÇOISE POINT 2. Preliminaries A topological L-structure M, τ is a first-order L-structure M with a Hausdorff topology τ such that every n-ary function symbol f of L is interpreted by a continuous function M n to M, and every m-ary relation symbol R of L is interpreted by an open subset of M m (M n and M m are endowed with the product topology). In the following, we will only deal with topological L-structures which are expansions of fields; we will use the terminology topological L-fields, where L L fields := {+,,., 1, 0, 1}. This is a generalization of the well-known notion of topological fields (see [21] p. 28). Note that in a topological L-field M, a fundamental system V of neighbourhoods of zero determines the topology. Indeed, for each m M, m+v is a fundamental system of neighbourhoods of m. Now a set O is open if and only if it is a neighbourhood of each of its points, if and only if for each m O there exists V V such that m+v O (see [29] p.19, Theorem 3.1). So, we will also use the notation M, V to denote the topological structure M, τ, with V a fundamental system of neighbourhoods of zero. We will be working in a monadic second-order language L t introduced by T.A. McKee ([13]); he restricts L 2 -formulas by imposing that the subformulas are of the form Xψ ( Xψ) where X only occurs negatively in ψ (respectively positively) and X belongs to τ (see also [8]). J. Flum and M. Ziegler have shown that L t -sentences have the property that if B 1 and B 2 are two bases for the same topology then M, B 1 = σ iff M, B 2 = σ, where σ is an L t -sentence (see [8] p. 6). So, in particular L t -sentences cannot distinguish between M, V and M, τ. Definition 2.1. (1) The topological L-field L, ν is called a topological L-extension of the topological L-field K, τ if K is an L-substructure of L and τ is the restriction of ν to K i.e. for all V τ, there exists W ν such that W K = V. We will denote by ν K the trace of ν on K, namely {W ν : V τ W K = V }. (We mention the language L in this definition since we will consider the case of L 1 L 2, K a topological L 2 -field, L a topological L 1 -field which is a topological L 1 -extension of K.) (2) Let K, τ be a topological L-field. A Cauchy sequence in K is a sequence (a µ ) µ<λ of elements of K, indexed by all ordinals less than some ordinal λ, which satisfies: V V λ 0 µ 1, µ 2 ν 0 a µ2 a µ1 V. We say that (a µ ) µ<λ converges to a K if for every V V there exists λ 0 such that for any µ λ 0, a a µ V. (3) A topological L-extension K, τ of K, τ is called a completion of K, τ if the following are satisfied: (a) every Cauchy sequence in K converges to some element of K; (b) K is dense in K. Notation 2.1. (1) For a subset E of a ring, we put E := E {0}.

3 TOPOLOGICAL DIFFERENTIAL FIELDS 3 (2) Let K be a valued field with v its valuation map. We denote the valuation ring, the maximal ideal, the residue field and respectively the value group by O K, M K, k K and respectively v(k ); the residue map is denoted by. : O K k K. Sometimes we also denote the valuation ring of K by O v (especially when dealing with several valuations on a given field). (3) Let K, < be an ordered field and A be a subset of K. Then A >0 denotes the set of elements of A which are strictly positive. Examples 2.1. (1) Any field K, +,,., 1, 0, 1,. with an absolute value. taking values in R + (see [21, p. 19]) is a topological L fields -field with as a fundamental system of neighbourhoods of zero the set {x K : x < r; r R >0 }. Let us recall the characterization due to J. Shafarevic and I. Kaplansky, of the topological fields L whose topology is given by an absolute value (see [21, Theorem (4), p. 44]). Let T be the set of nilpotent elements of L, namely those whose powers converge to 0 and let N be the set of elements of L neither nilpotent nor whose inverse is nilpotent. A subset B of L is bounded if for every neighbourhood U of 0 there exists a neighbourhood W of 0 such that B.W U. Then, there exists an absolute value on L which determines the topology iff T is open and T N is bounded. Note that the condition T N bounded implies that every element of K can be written as x.y 1 with x, y belonging to a neighbourhood of 0, y 0. (2) Let K, v be a valued field. We can equip this field with a topology τ v by choosing as fundamental system of neighbourhoods of zero the set of a.m K, with a O K (see [8] p.123). We will denote by K, τ v a topological field equipped with a topology coming from a valuation v. Such field has a completion (see [18, A 4.11]) and in the case K, v is an archimedean valued field (in other words v takes its values in R (see [21] p. 36)), this completion is Henselian (see [21, p. 98]). In this case, one can choose as fundamental system of neighbourhoods of zero, the set of M n K with n ω. (3) Let K, < be an ordered field; we endow it with the interval topology, so a fundamental system of neighbourhoods of zero is: {] a; a[; a K }. It is an example of a topological L fields {<}-field. We will denote by K, τ < a topological field equipped with a topology coming from an order <. In the case K is archimedean, its completion is real-closed (see Theorem (1.23) in [17]). (4) Let K be a field and let K((t)) be the field of Laurent series over K. We may define a valuation v with value group Z as follows: v(k ) = 0 and v(t) = 1. The field K((t)), v is then a complete topological field (see [21, p. 90]). Concerning examples (2) and (3), note that one may characterize in the language L t, those topological fields with a valuation topology (respectively an order topology) (see [8] p.123, 126).

4 4 NICOLAS GUZY AND FRANÇOISE POINT Now we will introduce a subclass of the class of topological fields which resembles the one whose topology is coming from an absolute value. Definition 2.2. Given a topological L-field K and a neighbourhood V V, we will say that K satisfies Hypothesis (V) with respect to V if V.V V, and every non-zero element a of K can be written as a = x.y 1 where x, y V and y 0. Notation 2.2. Let K, V L, W be two topological L-fields, let a, b be two elements of L, then a and b are infinitely close with respect to K (a K b) iff for any neighbourhood V V, there exists W W such that W L = V and a b W. Lemma 2.3. Let K, τ be a topological L-field. Consider the iterated Laurent series field extension K((t 1 ))((t 2 )) ((t n )), for some natural number n 1. Then, we can endow K((t 1 ))((t 2 )) ((t n )) with a topology in such a way that it becomes a L fields -extension of K and whenever K satisfies Hypothesis (V ) with respect to a neighbourhood V of zero, then it satisfies the same hypothesis with respect to a neighbourhood W with W K = V. Proof: It suffices to show the Lemma for the Laurent series field K((t)). We endow K((t)) with the following topology τ extending the topology τ on K, in such a way that K((t)) becomes a topological L fields -field. Let V be a fundamental system of neighbourhoods of zero in K; we define a fundamental system of neighbourhoods of zero in K((t)) as follows. Let W i := { j i a j.t j : a j K} and W i,v k := { j i a j.t j : a k V, a j K}, i k, i, k Z. Then, denote by W := { j n W i,v k j, V j V, i k, i, k Z, n ω }. Note that W i W i+1, t i W i \ W i+1, and W i,v k K = {0}, i, k ω, i k. One has to check it gives us a field topology (see [8] p.120). Moreover, whenever V is a neighbourhood of zero in K such that K satisfies Hypothesis (V ), then K((t)) satisfies Hypothesis (V ) with respect to W 0,V 0. Notation 2.3. Let W be a neighbourhood of 0 in K. We will denote by W [X 0,, X n ], n ω, the set of polynomials in K[X 0,, X n ] whose coefficients are in W. Definition 2.4. An inductive class C of topological L-fields satisfies Hypothesis (I) if every element K of C has a neighbourhood V of 0 such that: (1) K satisfies Hypothesis (V) with respect to V, and (2) given an iterated Laurent series L fields -extension L = K((t 1 )) ((t n )) and a polynomial f(x) W [X], where W is a neighbourhood of zero in L and W K = V, such that for some element a W, we have f(a) K 0 and f 2 (a) K c with c K, then there exists a topological L fields -extension L of L belonging to C which is also a topological L-extension of K, such that for some element b of L, f(b) = 0 and a K b. Now we will revisit our previous examples of classes of topological L-fields and show they do satisfy the above hypotheses. Examples 2.2. (1) If K, τ < is a topological L fields {<}-field, equipped with the order topology, then we will use the fact that it has a real closure which

5 TOPOLOGICAL DIFFERENTIAL FIELDS 5 will have the desired property too. Indeed, we use the Taylor expansion of f and the intermediate value property. (Write f(a + h) = f(a) + h.f (a)+ higher order terms and note that for small h in K we get a change of signs). In this case, we may take V = K and again the class C < of topological fields equipped with the order topology satisfies Hypothesis (I). We extend the order on the Laurent series field K((t)) by choosing t positive and infinitely small with respect to K >0 for the order topology (namely 0 < t < K >0 ). (2) If K, τ v is a topological L fields -field equipped with the valuation topology, then we may take V = O K and we use the fact that it always has an Henselization (see [21, p. 131]) which is an immediate extension and satisfies Hensel s Lemma ( or one of its equivalent form, for instance Newton s Lemma (see [21, p. 98,100]) ). Denote by C v this class of topological fields equipped with the valuation topology. Then C v satisfies Hypothesis (I) using Taylor expansion and the fact that we can extend the valuation on K to K((t)) such that v(t) > v(c) for any c K. (3) We will show in Section 8 that large fields fit in our framework. Recall that a field K is large if every smooth integral curve defined over K that has an K- rational point has infinitely many K-rational points (see [16, p. 2]). Examples of large fields are P AC, P RC and P pc fields (see [16, Section (3)]). 3. Differential lifting In this section, we will consider expansions of topological L-fields to L {D}-fields, where D is a new unary function symbol which will satisfy the axioms of a derivation. We will write down a scheme (DL) of axioms in the language of L {D} t expressing the fact that if a differential polynomial has a zero, while considered as an ordinary algebraic polynomial, then it has a differential solution which in addition is close to the algebraic solution. Then, we will show that any element of a class of topological L-fields satisfying Hypothesis (I) can be embedded in another element of that class satisfying this scheme (DL). We first recall some differential algebra terminology (see [9, p. 75]). In the following, R, D will denote a non-zero differential domain and C R will denote the set of elements with zero derivative (i.e. constant elements). Although in this section, we will only deal with the case when R is a field, we will directly place ourselves in a more general setting that will be used in Section 9. Definition 3.1. Let R{X} be the ring of differential polynomials over R in one differential indeterminate X over R. Let f(x) R{X} and suppose f(x) = f (X,..., X (n) ) for some polynomial f in n + 1 variables with coefficients in R. We say that X (N) appears in f R{X} if X (N) appears in f. The leader u f of f in R{X} \ R is the variable X (N), with the largest possible natural N, which appears in f. So N is the order of the differential polynomial f and deg uf f is the degree of the differential polynomial f.

6 6 NICOLAS GUZY AND FRANÇOISE POINT Definition 3.2. Let f be in R{X} \ R, f = f d (X (N) ) d + + f 1 X (N) + f 0 where f 0,, f d are elements of R[X, X, X (N 1) ], f d 0 and X (N) = u f. The initial i f of f is defined as f d. The separant s f (X) of f is defined as: s f (X) = f X (N). Lemma 3.3. ([9], Lemma 2, p. 167, [28] Corollary 2.10). Let P be a differential prime ideal of R{X} with P R = {0}. Let f be a non-zero differential polynomial of minimal order and then of minimal degree belonging to P. Then P = {g R{X} n N such that (i f s f ) n g f } where f is the differential ideal of R{X} generated by f. So we can define the notion of generic polynomial. Definition 3.4. Let R 1 be a differential domain containing R and let a be in R 1 \ R. The set of elements of R{X} vanishing on a is a prime differential ideal of R{X}, denoted by I(a, R), whose intersection with R is {0}. If I(a, R) = {0}, then we say that a is differentially transcendental over R, if I(a, R) 0, then we say that a is differentially algebraic over R. In this last case, let f be a non-zero element of this ideal which is of minimal order and then of minimal degree. We will call f a generic polynomial of a over R. When R is a field, we get I(a, R) = I(f) := {g R{X} : s f k.g < f > for some k ω }, and such element f is unique up to multiplication by a non-zero element of R ( see [12, Lemma (1.4)] ). Let L be the expansion of the language L by a unary function symbol D. To ease notations, we shall denote the i-th iterate D i (x) of the derivation of an element x, by x (i). Definition 3.5. (1) A differential topological L-field K, τ is an L -struture K satisfying the following axioms ( ): a b D(a + b) = D(a) + D(b), a b D(a.b) = a.d(b) + D(a).b, such that the restriction K, τ is a topological L-field. There is no requirement of any interaction between the derivation D and the topology τ. (2) A differential topological L-extension is an L -first-order extension which is in addition a topological L-extension (see Definition 2.1). (3) We call (DL) (differential lifting) the following scheme of axioms in the language L t. Let L be a differential topological L-field and assume that L satisfies Hypothesis (V ) where V is a neighbourhood of 0. Then, L satisfies (DL) if for each differential polynomial f(x) = f (X, X,..., X (n) ) belonging to V {X}, for every W V,

7 TOPOLOGICAL DIFFERENTIAL FIELDS 7 ( α 0,..., α n V )(f (α 0,..., α n ) = 0 s f(α 0,..., α n ) 0) ( ( z) ( n f(z) = 0 s f (z) 0 (z (i) α i W ) )). i=0 Note that we could have added in the definition of the scheme (DL) the requirement that f is irreducible. Anyway, if f(c) = 0, s f (c) 0 ( ) for an element c V and f = g.h, then one of the factors of f satisfies ( ). Moreover, multiplying f by a non-zero element of V, we may assume that each of the factors have coefficients in V. Then g or h will have order n and satisfy the hypothesis ( ). If we apply the scheme (DL) to one of these polynomials then f will satisfy the conclusions of the scheme (DL). Let us show an easy topological lemma. Lemma 3.6. Let K, V be a topological L-field satisfying Hypothesis (V ) with respect to V, let L, V be a topological L-extension of K, V and let f(x 0,, X n ) be a polynomial in (n + 1) indeterminates over K such that f(ᾱ) = β where ᾱ = (α 0,, α n ) and α 0,, α n, β K. Proof: (1) Let t 0,, t n belong to L, be such that t i K 0, 0 i n, then f(ᾱ+ t) K β. Moreover, if f(ᾱ) 0 then f(ᾱ + t) 0. (2) Assume now that f(x 0,, X n ) belong to V [X 0,, X n ] and that α 0,, α n V. Write f(x 0,, X n ) = m i=0 g i(x 0,, X n 1 ).Xn. i Then there exists an element d V such that d.g i (X 0,, X n 1 ) V [X 0,, X n 1 ] and d g i (ᾱ) V. (1) Let U be an open neighbourhood of 0 in K and U a neighbourhood of 0 in L such that U K = U. Let us associate to the polynomial f(x 0,, X n ) K[X 0,, X n ], the function f L : L n+1 L : (x 0,, x n ) f(x 0,, x n ) (this is well-defined since f has coefficients in K); define g L := f L β and f K (respectively g K ) to be the restriction of f L (respectively g L ) to K. Then g 1 K (U) is an open neighbourhood of Kn+1 containing ᾱ and g 1 L (U ) is an open neighbourhood of L n+1 containing also ᾱ. Since g L is defined over K, we have g 1 K (U) = g 1 L (U ) K n+1. Since t i K 0, 0 i n, we obtain (ᾱ + t) g 1 L (U ) and so, f(ᾱ + t) β U. We conclude that f(ᾱ + t) K β. For the second part, assuming now that β K. Let W be an open neighbourhood of β in K which does not contain 0 and let W be an open neighbourhood in L such that W L = W (note 0 / W ). Hence f 1 L (W ) is an open subset of L containing ᾱ (because f(ᾱ) = β 0) and also ᾱ + t because f 1 L (W ) K = f 1 K (W ). So f(ᾱ+ t) belongs to W ; hence f(ᾱ+ t) 0. m (2) Write f(x 0,, X n ) as g i (X 0,, X n 1 ) Xn. i For each i {0,, m}, i=0 we have that g i (α 0,, α n 1 ) K. So, there exist elements d i V such that d i g i (α 0,, α n 1 ) V. Set d = d i=0 d i V.

8 8 NICOLAS GUZY AND FRANÇOISE POINT Then, d f (α 0,, α n 1, X n ) = m (d g i (α 0,, α n 1 )) Xn i V [X n ] where i=0 d g i (X 0,, X n 1 ) V [X 0,, X n 1 ]. Now, we will prove a crucial lemma. Lemma 3.7. Let K, V be a differential topological L-field satisfying Hypothesis (I) (with respect to a class C and a neighbourhood V of 0). Let f(x) = f (X,..., X (n) ) be a differential polynomial with coefficients in V. Suppose that f (X 0,..., X n ) = 0 has a solution α 0,..., α n V which satisfies s f (α 0,..., α n ) 0. Then there exists a differential topological L-extension L, V of K in C so that f (X,..., X (n) ) = 0 has a solution (z,..., z (n) ) which satisfies n ( ) s f (z) 0 z (i) K α i. i=0 Proof: We will proceed as in [27, p. 85] and [14, p. 34], M. Singer uses the intermediate value property for real closed fields and C. Michaux uses Hensel s Lemma for Henselian valued fields. Let us consider the topological L fields -extension K 1 := K((t 0 ))... ((t n 1 )), where the t i s are algebraically independent on K and the t i s are infinitely close to zero with respect to K. Let U K 1 be such that U K = V and K 1 satisfies Hypothesis (V ) with respect to U (see Lemma 2.3). Let c i = α i +t i for i {0,..., n 1}. Let us note that we have that c i K α i. Assuming f (α 0,..., α n ) = 0, s f (α 0,..., α n ) 0 (so s f 2 (α 0,..., α n ) = β K ) and t 0,, t n 1 K 0, we get that If we show that s f (c 0,, c n 1, α n ) K s f (α 0,, α n 1, α n ) = β K and f (c 0,, c n 1, α n ) K 0 by Lemma (3.6)(1). d f (c 0,..., c n 1, α n ) K 0, s d f 2 (c 0,..., c n 1, α n ) K β and d f (c 0,, c n 1, X n ) U[X n ] for some d U and β K 1, then we can apply Hypothesis (I). It suffices to apply Lemma (3.6)(2) (with respect to K 1 ) together with the fact that the t i s are infinitely close to 0 with respect to K. Since K C and by Hypothesis (I), there exists an L fields -extension L C of K 1, which is in addition an L-extension of K, and an element c n L which satisfies f (c 0,, c n 1, c n ) = 0 and s f (c 0,, c n ) 0 such that c n α n K 0. We first extend the derivation to the algebraic closure of K 1 in L (call this subfield L 1 ) and then, we can choose a transcendence basis of L over L 1, extend D on this basis and then again to the algebraic closure. Since any derivation D of K 1 uniquely extends to L 1 then, in order to make L 1 a differential L-extension of K, we only need to extend the derivation of K to K 1. This can be done by setting D(c 0 ) = c 1,..., D(c n 1 ) = c n, the derivative D(c n ) is then uniquely determined by the equation f (c 0,..., c n ) = 0 since the separant s f is non-zero.

9 TOPOLOGICAL DIFFERENTIAL FIELDS 9 Indeed, let f (X,..., X (n) ) = d l=0 g l(x,, X (n 1) ).X (n)l, then f (X,..., X (n) ) (1) = d l=0 g l(x,..., X (n 1) ) (1).X (n)l + d l>0 g l(x,, X (n 1) ).l.x (n)l 1.X (n+1). Now, if we evaluate this polynomial at (c 0, D(c 0 ),..., D(c n 1 )), since s f (c) is non-zero, we get that c (1) n = ( d l g l (c 0,, c (n 1) 0 ) (1).c (n) l 0 ).sf (c) 1. Let z = c 0, by construction, we have z (i) = c i for all i {0,..., n}. As c i α i = t i K 0, we get (z (i) α i ) K 0 for all i {0,, n 1} and by letting t n = c n α n, we get z (n) α n K 0. Let us note that if g(x) = g (X,..., X (m) ) is a differential polynomial with n m and g (α 0,..., α m ) 0 then we have the following additional property: g(z) K g (α 0,..., α m ) and g(z) 0 using Lemma (3.6) (1). Proposition 3.8. Every differential topological L-field K, V satisfying Hypothesis (I) with respect to C and a neighbourhood V of zero, has a differential topological L-extension K belonging to C, satisfying the scheme (DL). Proof: Let (f δ (X)) δ λ, for some ordinal λ, be an enumeration of differential polynomials of order n(δ) ω in V {X}. Using Lemma (3.7), we are building an increasing sequence K (δ), V (δ) δ λ of differential topological L-extensions which satisfy Hypothesis (V ) with respect to a neighbourhood V (δ) of zero. Let K (0) := K and V (0) := V ; assume that K (δ) has been constructed and consider f δ (X) V (δ). Then the following property holds in K (δ+1) : ( α 0,..., α n(δ) V (δ) )(f δ (α 0,..., α n(δ) ) = 0 s f δ (α 0,..., α n(δ) ) 0) ( ( z K (δ+1) )(f δ (z,..., z (n(δ)) ) = 0 s f δ (z,..., z (n(δ)) ) 0 n(δ) i=0 ( (z (i) α ) ) K(δ) i. Then we let K 1, τ 1 be the differential topological L-field δ λ K (δ), V (δ). We have to check that V 1 := δ V (δ) is a neighbourhood of zero with respect to which K 1 satisfies Hypothesis (V ). We iterate this process replacing K by K 1 and V by V 1. So, we obtain a sequence of differential topological L-extensions ( K n, V n ) n ω and K, V := n ω K n, V n will be the desired differential topological L-extension of K. Again, we have to check that V := n V n is a neighbourhood of zero with respect to which K satisfies Hypothesis (V ). Lemma 3.9. If K, τ is a differential topological L-field which satisfies Hypothesis (V ) with respect to a neighbourhood V of zero, and the corresponding scheme (DL) then (1) for any non-zero natural number n, any element of V is an n th primitive of some element in a neighbourhood of zero, in particular K has a non-zero derivation, (2) the field of the constants C K is dense in V,

10 10 NICOLAS GUZY AND FRANÇOISE POINT (3) let f(x 1,, X n ) V {X 1,, X n } be a differential polynomial in n differential indeterminates vanishing on V, then f vanishes on V. Proof: To show the first assertion, we apply the scheme (DL) to the differential polynomial (X (n) a). Let (0,, 0, a) be the corresponding algebraic zero, where a V, n ω and s X (n) a = 1. Therefore, for n = 1, this shows that K has a non-zero derivation. Then, to show that C K is dense in V, we apply the scheme (DL) to the differential polynomial X (1) with the algebraic zero of the form (a, 0) where a V and s X (1) = 1. Finally, let f(x 1,, X n ) be a differential polynomial with coefficients in K vanishing on K. Let m i be the order of X i in f and set N := n i=1 m i. By the way of contradiction assume that f does not vanish on V, namely let k V be such that f ( k) 0. Then consider the differential polynomial X (N+1). Consider the algebraic solution ( k, 0) of that polynomial. By the scheme (DL), there exists a K with a ( k, 0) which is a solution. By continuity of the field operations, we have that f(a) 0. Lemma Under the same hypotheses as in the previous Lemma, the subfield of the constants C K is dense in K. Proof: Let a be in K. Then, by Hypothesis (V), we can write a = σ 1 σ 2 for some σ 1, σ 2 V. We are going to show that given a neighbourhood W τ of 0, there is an element b C K such that a b W. Choose W 0, W 1 τ, 0 W 0 such that W 0.W 0 W 1, W 1 + W 1 + W 1 W and σ 1.W 0 W 1, σ 1 2.W 0 W 1. We apply the preceding lemma to σ 1 and σ 2. Choose ɛ 1, ɛ 2 in C K such that (ɛ 1 σ 1 ) W 0 and (ɛ 1 2 σ2 1 ) W 0. Thus we obtain ɛ 1 ɛ2 σ 1 σ 2 = (ɛ 1 σ 1 ).(ɛ 1 2 σ 1 2 )+σ 1.(ɛ 1 2 σ 1 2 )+σ 1 2.(ɛ 1 σ 1 ) W. Proposition Let K, τ be a differential topological L-field which satisfies the scheme (DL) and let f(x) := f (X,..., X (n) ) be a non zero differential polynomial of order n in K{X}. If there exists a 0,..., a n K such that f (a 0,..., a n ) = 0 and s f (a 0,..., a n ) 0 then, for all W V, there exists b K with f(b) = 0 and s f (b) 0 such that n (b (i) a i ) W. i=0 Proof: Let f (X,..., X (n) ) = (i 0,...,i n) c (i 0,...,i n).m (i0,...,i n)(x) K{X} such that the c (i0,...,i n) s are non-zero elements of K, M (i0,...,i n)(x) = X i 0... ( D (n) (X) ) i n and M = max ī { n j=0 i j}. Now we use that K satisfies Hypothesis (V) with respect to a neighbourhood V of zero. Let ī = (i 0,..., i n ), c ī := d 1,ī.d 1 with d 2,ī 1,ī, d 2,ī V, let d := ī d 2,ī V and so, c ī.d V. In the same way, we write a i K as a i0.a 1 i1 where a i0 V, a i1 V and we set d := i a i1 V, so a i.d V. By Lemma (3.10), there exists d C K sufficiently close to d.d (so d V ) such that for all i, ī we get d a i V and d c ī V. We obtain:

11 TOPOLOGICAL DIFFERENTIAL FIELDS 11 0 = d M+1 f (a 0,..., a n ) = Moreover, we have (i 0,...,i n) = f (da 0,..., da n ) where f V [X 0,..., X n ]. s f (d.a 0,..., d.a n ) = (i 0,...,i n) d c ī d (M P n j=0 ij) (da 0 ) i0 (da n ) in d c ī d (M P n j=0 ij) i n (da 0 ) i0 (da n ) in 1 = d M s f (a 0,..., a n ) K. Since, for all i {0,..., n}, da i V, we can apply the axiom (DL). We find, for all neighbourhoods of the form d.w with W V, an element b of V which satisfies f(b) = 0 and s f(b) 0 such that (b (i) da i ) d.w. Thus we have (d 1 ) M+1 f (b, b,..., b (n) ) = 0. It is equal to (d 1 ) M+1 (i 0,...,i d c n) (i 0,...,i n) d (M P n j=0 i j) b i0 b (n)in = f(d 1 b) = 0. In the same way, s f (d 1 b) 0 and (d 1.b (i) a i ) W. Now, since d 1 C K, we have that d 1.b (i) = (d 1.b) (i). We have the following corollary of the proof of Proposition 3.11 together with Lemma 3.9. Corollary Let f(x 1,, X n ) be a differential polynomial with coefficients in K vanishing on K, then f vanishes on K. Proof: We argue by the way of contradiction. Let k K be such that f ( k) 0. As in the proof of the preceding proposition, we find d V C K such that k.d V and d N.f ( k) = f ( k.d), where N is some non-zero natural number and f is a differential polynomial with coefficients in V. By Lemma 3.9, there is an element a K such that f(a) 0, which implies that f(a.d 1 ) 0, a contradiction. Definition Let φ(x 1,, x n ) be an open L -formula. We associate with φ( x) an open L-formula φ ( x ) by substituting to each term x (i) j, 1 j n, i ω, appearing in φ a new variable x j,i. We will call m(j) the formal order x j in φ if m(j) is the maximum of the natural numbers i such that x (i) j occurs in φ. Lemma Let C be a class of topological L-fields. Suppose that the first-order theory of C is model-complete. Let K, τ be a differential topological L-field satisfying Hypothesis (I) with respect to C and the scheme (DL). Then, given any open L - formula φ(x 1,, x n ) such that the corresponding L-formula φ defines an open set in K N, for some natural number N, if x φ ( x ) is satisfied in a topological L-extension of K belonging to C, then x φ( x) is satisfied in K. Proof: Let m(j) be the formal order x j in φ and consider the formula φ( x) n j=1 x j (m(j)+1) = 0. By model-completeness of C, there exists a tuple b in K satisfying φ. Applying Proposition (3.11) to the differential polynomial x j (m(j)+1) = 0, 1 j n, and to the tuple b j := (b j,1,, b j,m(j), 0), we get a tuple

12 12 NICOLAS GUZY AND FRANÇOISE POINT (a j, a (1) j,, a (m(j)) j, 0) of K close to b j for the product topology. Since φ defines an open set O in K N, where N = n j=1 (m(j) + 1), we may require that ( (aj, a (1) j,, a (m(j)) j ) ) n belongs to O and so we get a solution to our formula j=1 φ. 4. Model-completion In this section, we will revert to first-order languages and so in order to apply the results of the previous section, we will put on our topological structures the extra-hypothesis that the topology is first-order definable (with parameters). Definition 4.1. A topological L-field K satisfies Hypothesis (D) if there is a formula φ(x, ȳ) such that in any K elementarily equivalent to K, the set {x K : K = φ(x, ā), ā K } can be chosen as a basis of neighbourhoods of 0. Note that our notion of topological L-fields satisfying Hypothesis (D) is equivalent to the notion of topological structures used by A. Pillay in [15]. Further, note that if we had only required the above condition on K, it would have held in any topological L-field L t -elementarily equivalent to K. Let L be the language L fields {R i ; i I} {c j ; j J} where the c j s are constants and the R i are predicates of arity, say n i. Let T be an universal L-theory of topological L fields -fields of characteristic zero which satisfies Hypothesis (D) and such that every relation R i and its complement, with i I, is interpreted in any model K of T, as an union of an open set and an algebraic set of K n i. So, on one hand we weaken the topological L-field assumption on K, but on the other hand we add a topological condition on the complement of the relations. Note that these conditions on the language are much alike the ones used by L. Mathews (in [10]) and already appeared in the special case of real-closed rings (see [7]). We will denote the subset R i K n i by O Ri { x K n i : k r i,k( x) = 0} and the subset R i K n i by O Ri { x K n i : l s i,l( x) = 0}, where r i,k, s i,l K[X 1,, X ni ]. Assume that the theory T admits a model-completion T c and that the class of the models of T c satisfies Hypothesis (I). Let T D (respectively T c,d ) be the L -theory T (respectively T c ) together with the axioms ( ) stating that D is a derivation (see Definition 3.5). Note that since the models of T satisfies Hypothesis (D), the scheme of axioms (DL) is now first-order. Theorem 4.2. Under the above hypotheses on T and T c, we have that the L -theory T c,d consisting of T c,d together with the scheme (DL) is the model-completion of T D. Proof: We will apply Blum s criterion for the existence of a model-completion (see [24] Theorem 17.2). First, let us check that any model K 0 of T D embeds into a model of T c,d. Since T is inductive, we embed K 0 into a model K 1 of T c. Moreover, we may always assume that K 1 is a differential extension of K 0 by extending the derivation D defined on K 0 ; first, to the relative algebraic closure K 0 of K 0 and then, on a transcendence

13 TOPOLOGICAL DIFFERENTIAL FIELDS 13 basis of K 1 over K 0 and finally, again to the relative algebraic closure. Then, using Proposition (3.8), we embed K 1 into a model K 2 of T c,d which satisfies the scheme (DL). Then, given K 0 a model of T D, and two L -extensions; on one hand a 1-extension K 0 c which is a model of T D and on the other hand K a K 0 + -saturated model of T c,d, we have to find a L -embedding of K 0 c into K over K 0. Since T c is the model-completion of T, we can embed K 0 c in K over K 0, as an L-substructure. In order to obtain an L -embedding, we have to show that any set of open L - formulas belonging to the 1-type t(c, K 0 ) of c over K 0 is finitely satisfiable in K (we use that K is K 0 + -saturated). Since T c admits quantifier elimination (as the modelcompletion of an universal theory ([24] Theorem 13.2), we only need to consider conjunctions of basic L -formulas of t(c, K 0 ); namely the formula ψ(x) = ψ 1 (x) ψ 2 (x), where ψ 1 (x) := i f i (x, x (1),, x (n) ) = 0 ψ 2 (x) := g (x, x (1),..., x (n) ) 0 ( R j p 1,j (x, x (1),..., x (n) ),..., p n j,j(x, x (1),..., x (n) ) ) j ( R k q 1,k (x, x (1),..., x (n) ),..., qm k,k(x, x (1),..., x (n) ) ), k such that g, f i K 0 {X} and p r,k, q s,l belong to the field K 0 X of differential rational functions. We associate to the L -formula ψ(x), the L-formula, say, ψ (x, x 1,, x n ) as in Lemma (3.14) (by replacing in x (i) by x i ). We divide the basic sub-formulas into two sorts: (1) the first sort concerns the differential equations and inequations which determine the differential field structure of K 0 c, (2) the second sort concerns the formulas which contain the predicates R i, i I. (1) Either c is differentially algebraic in which case we let f(x,, X (m) ) be its generic polynomial of order, say, m and degree d, or c is differentially transcendental, in which case we will apply Lemma (3.14). Using the same calculation that we performed in the proof of Lemma (3.7), if f(x,, X (m) ) = d l=0 f l(x,, X (m 1) ).X (m)l then f(x,, X (m) ) (1) = d l=0 f l(x, X (m 1) ) (1).X (m)l + d l>0 f l(x,, X (m 1) ).l.x (m)l 1.X (m+1). Now, if we evaluate this polynomial at c, we get that c (m+1) = ( ld f l (c,, c (m 1) ) (1).c (m)l ).s f (c) 1. We can iterate this process to express differential polynomials of order higher than m evaluated at c as polynomial over K 0 in c,, c (m) and s f (c) 1. Since f is the generic polynomial of c, the separant s f (c) is non-zero, being of degree less than or equal to d 1.

14 14 NICOLAS GUZY AND FRANÇOISE POINT We have that K 0 c = ψ (c,..., c (n) ) (n m). Since f i (c, c (1),, c (n) ) = 0, we get that s ki f f i < f > for some positive integers k i. Define φ(x) to be the L -formula f(x) = 0 s f (x) 0 ψ 2 (x). Since K 0 c embeds in K as an L-substructure, we can find (d 0,..., d n ) K such that K = φ (d 0,..., d n ). Using the scheme (DL), we will show that we can find an element z K such that K = φ(z). Note that for such element z, the formula ψ 1 (z) holds too. We apply this scheme (DL) to either the generic polynomial f of c in case c is differentially algebraic over K 0 or to the polynomial X (n+1) in case c is differentially transcendental. Note that, in both cases, the separant s f is non-zero at c. In the differential algebraic case, since f has order m n, a priori we only get that for any neighbourhood W m+1 of zero in K m+1, there exists z such that (z, z (1),, z (m) ) (d 0,, d m ) W m+1. But note that d m+1 is equal to ( l=0d g l (d 0,, d m 1 ) (1).d l m ).s f (d) 1. So, since K is a topological L fields -field, using the continuity of the field operations (for the inverse operation, excluding 0), we get that if the tuple (z, z (1),, z (m) ) is close to the tuple (d 0, d 1,, d m ) then so are the tuples (z, z (1),, z (m+1) ), (d 0, d 1,, d m+1 ). (2) Now, we have to determine suitable neighbourhoods of (d 0,, d n ) which will enforce that any element in these neighbourhoods also satisfies the second sort of formulas. Given any relation R i or R j, we have to distinguish two cases: either, in case of R i (the other case is similar) r i,k (c, c (1),, c (li,k) ) = 0 holds in which case k r i,k (X, X (1),, X (li,k) ) belongs to I(c, K 0 ); otherwise the tuple (p i1( d),, p in i ( d)) belongs to O Ri. In this last case, again, we have to use the continuity of the field operations to find an open subset O i of d in K0 n such that its image by the rational map x (p 1( x),, p n i ( x)) in K n i 0 goes into the open subset O Ri. 5. Applications to topological fields with an absolute value. The examples we are going to consider will be either non-archimedean fields with an order topology or a valuation topology. In both cases, the topology will be firstorder definable, and so Hypothesis (D) will hold. In the first case, we will add an order to the field language and in the second case, instead of adding a valuation map, we will use a linear divisibility relation, that we introduce below. Definition 5.1. (see [11, Section (4.2)] ) Let A be a commutative domain. Then a linear divisibility relation (l.d. relation) on A is a binary relation D(.,.) on A such that: D is transitive, D(0, 1), compatible with + and., namely D(a, b) and D(a, c) implies that D(a, b + c), and for all c 0, we have D(a, b) implies D(a.c, b.c), and either D(a, b) or D(b, a). A l.d. relation D on the domain A induces a valuation ring O A of the fraction field F := F rac(a) of A: O A = { a b : a, b A, b 0, D(b, a)}

15 TOPOLOGICAL DIFFERENTIAL FIELDS 15 The corresponding valuation v D on F rac(a) is defined by: for any a, b A, v D (a) v D (b) D(a, b). We have a bijection between the set of l.d. relations and the set of valuation rings of F rac(a). So any field with a l.d. relation can be endowed with a valuation topology. Let L:=L fields {D} {d} and let V F be the L-theory of non-trivially valued fields (i.e. D(d, 1) belongs to V F and so, V F is a universal L-theory). In a nontrivially valued field K, the l.d. relation D(x, y) defines a set in K 2 which is the union of an open set and {(0, 0)}. Then ACV F is the L-theory of algebraically closed non-trivially valued fields and it is model-complete (see [22]), moreover using prime extensions, it is easy to see that it admits quantifier elimination in L D (see [11, p. 83]). In this case, Hypothesis (I) is trivially satisfied (see Examples 2.2 (2)), a model of ACV F being Henselian. So we get the following corollary to Theorem (4.2). We use the subscript 0 to indicate that our fields are now of characteristic 0. Corollary 5.2. The L -theory ACV F0,D is the model-completion of the L -theory V F 0,D. Note that a model of ACV F D is in particular a differentially closed field. Recall that a p-valued field of p-rank d (d ω) with p a prime number, is a valued field of characteristic 0, residue field of characteristic p and denoting the valuation ring by O the dimension of O/(p) over the prime field F p is equal to d. Let L := L fields {D} {P n ; n ω {0, 1}} {c 2,, c d }. The theory pcf d of p- adically closed fields of p-rank d admits quantifier elimination in L and so is the modelcompletion of its universal part (pcf d ). This last theory has been axiomatized by L. Bélair who denoted the set of axioms T d (see [4, Theorem (2.4)]). In order to apply our results, we need to check that the predicate P n defines an open subset union an algebraic set (in this case a finite set) as well as its negation in any model K of pcf d. Let us denote by P n the set of non-zero n-th powers and K the multiplicative group of the field K. Then, one can find an integer r n such that x 0 y 0 [ ] (x P n v(x y) > v(x) + r n.v(x)) y P n (see [4, Lemma (1.10)] and also [11]). So, P n is open in K and since it has finite index in K, its complement is also open in K. Thus, we get the following corollary to Theorem (4.2). Corollary 5.3. The L -theory (pcf d ) D is the model-completion of (T d) D. Let L := L fields {D} { } {d} and let OV F be the theory of non-trivially valued ordered fields, namely the L fields { }-theory of ordered fields together with the theory of fields with a l.d. relation D, and the following compatibility condition between the valuation topology and the order topology: ( ) a b 0 a b D(b, a). Let RV F be the L-theory of real closed valued fields, namely the theory OV F together with axioms of real closed fields. Note that an L-substructure of a model of RV F is a model of OV F.

16 16 NICOLAS GUZY AND FRANÇOISE POINT The theory RV F is model-complete. Indeed, a real closed valued field is an Henselian valued ordered field with a real closed residue field and divisible ordered group (see [6, Theorem (3)]). Since the theory of real closed field RCF and the theory of divisible ordered groups are model-complete and complete, the L-theory RV F is model-complete and complete by Ax-Kochen-Ershov Theorems (see [6, Theorems (A) and (B)]) (note that the order in a real closed field is existentially definable). Then, we show that any L-substructure of a model of RV F has a prime extension and so RV F is the model-completion of OV F ([24]). Let K be a model of OV F and let O be its valuation ring and M its maximal ideal. Let K be the real-closure of K and let A be the convex hull of O in K. Then, A is a valuation ring of K and its maximal ideal M A is such that M A O = M (see [3]Lemma 1.1, Lemma 1.8 and the proof) and so K is an L-extension of K, satisfying RV F (see [6, Theorems (1) and (2)]). Therefore, we get the following corollary of Theorem (4.2). Corollary 5.4. The L -theory (RV F ) D is the model-completion of the L -theory (OV F ) D. 6. Hilbert s Seventeenth Problem for differential p-adically closed fields Let p be a prime number, let d, f be positive natural numbers. Let K, v be a differential p-valued field of p-rank d where v : K v(k ) { } is a p-valuation of p-rank d and [k K : F p ] = f. Let π belonging to K be such that v(π) is the least positive element of the value group. Set q = p f and let γ(x) := 1 [ X q X ] be the π-adic Kochen operator. π (X q X) 2 1 We have that γ(k) O K ([20] Lemma 6.1); if K is p-adically closed, then O K = γ(k) ([20] Theorem 6.15). Denote by K X := K X 1,..., X n the field of differential rational functions in n indeterminates. Assume now that K, v is a differential p-valued field of p-rank d with valuation v and derivation D. Then, we can extend the valuation and the derivation on K X in such a way it becomes a differential p-valued extension of K of p-rank d. Note that to check this statement it suffices to do it for the field of differential rational functions in one indeterminate. We extend D on K X as usual: define D( i a im i ( X)) := i D(a i).m i ( X) (1), where m i ( X) is a monomial in X 1,, X n and m i ( X) (1) stands for the formal derivative of this monomial. To extend v a p-valuation w of p-rank d, it suffices to check it for the ordinary polynomial ring K[X] ([20] Example 2.2), then we go to the fraction field and then we iterate considering first the polynomial ring K(X)[X (1) ]. Finally, we will get that w : K X Z ω v(k ) where Z ω is the set of sequences of elements in Z, with finite support, indexed by ω and that w is a p-valuation of the same p-rank as v. Now assume that K, v is a differential p-adically closed field of p-rank d. Before recalling the analogue of the Hilbert s Seventeenth Problem for p-adically closed fields of p-rank d, we need to introduce the following notation. Let L, v be a p-valued extension of K, v.

17 TOPOLOGICAL DIFFERENTIAL FIELDS 17 Definition 6.1. ([20] chapter 6, section 2). The γ-kochen ring R L of L over K is the subring defined by: t R L = { 1 + π s : t, s O K[γ(L)] and 1 + π s 0}. The quotient field of R L is the field generated by K and γ(l) \ { } and by Merckel s Lemma ([20] Appendix), K(γ(L)) = L (see [20] Lemma 6.6). Theorem 6.2. ([20, Theorem (7.12)]) Let K be a model of pcf d. If f, g K[X 1,..., X n ] and f/g is integral definite (i.e. g(ā) 0 implies f(ā) g(ā) O K for all ā K n ), then there are t, t O[γ(K(X 1,..., X n ))] such that f g = t 1 + π t. Now, let us state and prove the differential case using the technology of Section 3 and the following result on holomorphy rings. Theorem 6.3. ([20, Theorem (6.14)]) The γ-kochen ring R L of L is the holomorphy ring v Γ O v, where Γ is the (non-empty) set of all p-valuations of p-rank d of L which extend the p-valuation of p-rank d of K. Theorem 6.4. Let K, D, v be a model of (pcf d ) D. Let us consider the ring K{X} of differential polynomials in n differential indeterminates over K. Let f, g be two differential polynomials in K{X} such that f is integral definite (i.e g(ā) 0 implies g f(ā) O g(ā) K for all ā K n ). Then f belongs to the γ-kochen ring R g K X of K X over K. Proof: Let us assume that f / R g K X. Then, by Theorem (6.3), there exists one p-valuation w of p-rank d of K X extending v over K such that w( f ) < 0. g We have: K X, w = φ := ȳ [w( f (ȳ) g (ȳ) ) < 0 g (ȳ) 0] where f, g are the usual polynomials corresponding to f and g. Then, using the fact that pcf d is the model-completion of T d, we embed K X in a differential p- adically closed field of p-rank d. So, since K satisfies the scheme (DL), we use the model-completeness of pcf d and apply Lemma (3.14). So, we get a contradiction with f integral definite. g 7. A theorem of differential transfer In this section, we will prove a differential analogue of Ax-Kochen-Ershov Theorem for Henselian valued fields which are in addition differential fields satisfying the scheme (DL) for the valuation topology.

18 18 NICOLAS GUZY AND FRANÇOISE POINT We shall use the following existentially closed (e.c.) version of this theorem, which can be found in [19, Section 1]. Let K 1, v 1 be a Henselian valued field and K 2, v 2 a valued field extension of K 1, v 1. Assume that (1) k K1 ec k K2 in the language of fields, (2) the characteristic of k K1 is zero and (3) v(k 1 ) ec v(k 2 ) in the language of totally ordered groups. Then K 1, v 1 ec K 2, v 2 in L := L fields {D} the language of fields with an l.d. relation (see Definition 5.1). First we translate Lemma (3.10) for topological fields endowed with a valuation topology: Lemma 7.1. Let K, v be a differential valued field which satisfies the scheme (DL). Then we have v(k ) = v(c K ) i.e. the value group is the set of values of the constant field C K. Proof: Let α be in v(k ) and let c be an element of K of value α. We apply the scheme (DL) to the differential polynomial X (1) (the separant is 1) at the point (c, 0) for the neighbourhood {x K v(x) > α} of 0. So we get a constant d C K such that v(c d) > α which implies that v(d) = α. Remark 1. Note that the hypothesis v ( C K) = v(k ) is actually used in [25] (where there is a stronger interaction between the derivation and the valuation). Now we are ready to prove the differential version of the Ax-Kochen-Ershov Theorem for differential valued fields satisfying the scheme (DL). Theorem 7.2. Let K 1, v 1 be a differential Henselian valued field which satisfies the scheme (DL) and let K 2, v 2 be a differential valued extension of K 1, v 1 such that (1) v ( C K 2 ) = v(k 2 ), (2) char(k K1 ) = 0, (3) k K1 e.c. k K2, (4) v(k 1 ) e.c. v(k 2 ), then K 1, v 1 e.c. K 2, v 2 in the language L of differential valued fields. Proof: We closely follow the proof of the Ax-Kochen-Ershov Theorem given in the Appendix of [19]. For convenience of the reader, we reproduce the proof here. As in [19], we can reduce to the following situation: (1) K, v is an L -elementary K 2 + -saturated extension of K 1, v 1, (2) k K1 ec k K2 k bk, (3) v(k 1 ) v(k 2 ) v( K ) and v(k 2 )/v(k 1 ) is torsion-free. Using the existential version of Frayne s Theorem, K 1 ec K 2 is equivalent to show that K 2, v 2 L -embeds into K, v over K 1. We then proceed in three steps. Note that, without loss of generality, we may assume that K 2, v 2 is also Henselian. Otherwise we take the Henselization of K 2 inside K and the hypotheses are still met since the Henselization is an immediate extension.

19 TOPOLOGICAL DIFFERENTIAL FIELDS 19 Step 1 : In this step we extend the embedding of K 1, v 1 into K, v (which is the identity) to a differential valued subfield of K 2, v 2 which has residue field k K2. Suppose that K, v is a maximal differential valued subfield of K 2, v 2 having value group v(k ) = v(k 1 ) such that K, v can be L -embedded into K, v. We identify K, v with its image and K is Henselian. If k K k K2, we let 0 x k K2 \ k K, for some x K 2, and consider two cases, both leading to a contradiction to the maximality of K, v. case 1: x is algebraic over k K. This case is similar to Case 1 in [19]. In addition, we use that the derivation extends in a unique way to an algebraic extension since we are in characteristic zero. case 2: x is transcendental over k K. By hypothesis and Lemma (7.1), we can choose x C K2 and x C bk preimages of x respectively. We conclude as in [19] to obtain a contradiction. Step 2 : In this step we extend the above embedding further to a differential valued subfield of K 2, v 2 with value group v(k 2 ). Let now K, v be a maximal differential valued subfield of K 2, v 2 which embeds into K, v such that k K = k K2 and v(k 2 )/v(k ) is torsionfree. Such differential subfield exists by Zorn s Lemma and is Henselian. We identify K, v with its image. Assume that v(k 2 ) \ v(k ) contains an element α. By the assumption on v(k ), we have v(k ) Zα = {0}. Since v ( ) C K 2 = v(k 2 ), we can take x C K2 having value α (and x is transcendental over K) (*). Assigning the value α to x defines a unique extension of O K to the differential rational function field K x which coincide with the rational function field K(x). From K(x), we now pass to a valued algebraic extension (and so, differentially algebraic extension) K inside K 2 such that v(k 2 )/v(k ) is torsionfree where v(k ) is the value group corresponding to the valuation ring O of K. This can be done in the following way. If some δ v(k 2 ) \ (v(k ) Zα) satisfies q.δ v(k ) Zα for some prime q N, we choose y K 2 having value δ and a K x having value q.δ. Then y q.a 1 is a unit in O K2. Since k K2 = k K x, we find a unit e K x such that y q.a 1.e 1 is the unity of k K2. Since char(k K2 ) = 0, Hensel s Lemma gives us a q-th root of y q a 1.e 1 in K 2. Thus, a.e = z q for some z K 2. Thus, the value group of the differential valued field K x, z contains δ. Transfinite repetitions of this procedure (or simply an application of Zorn s lemma) yield an algebraic extension (so differential extension) K, v of K x of the desired nature. It remains to find a differential valued embedding of K, v into K, v. Since K, v is K 2 + -saturated, it is sufficient to find an embedding into K, v for every differential subfield of K, v finitely differentially generated. It follows that K, v admits an embedding into K, v. Therefore, we may assume that K, v itself is a finitely differentially generated field extension of K, v. We may assume that K is of the following form K x; x i : i < n where x C K2 and is algebraically transcendental over K and the x i s are algebraic over K x. Since v(k )/v(k ) is torsionfree, we have that v(k ) = v(k ) + Zβ for some β v(k ). We choose y in K such that v(y) = β. As in the proof in [19], we can embed K as an L-structure. We are