Making differential closure more explicit Joris van der Hoeven Laboratoire CNRS d'informatique (LIX, UMR 7161) École polytechnique, Institut Polytechnique de Paris 1, rue Honoré d'Estienne d'Orves Bâtiment Alan Turing, CS35003 91120 Palaiseau, France

. This work has been supported by the ODELIX project (number 101142171).

Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

1.Introduction

Several well-known first order theories admit natural explicit models. For instance, the reals are the archetype of a real closed field and the complex numbers are a natural example of an algebraically closed field of characteristic zero. More recently, it was shown that both the field of transseries and maximal Hardy fields are models of the theory of closed -fields [1, 2]. (Here an -field is a differential field with an ordering that satisfies suitable compatibility conditions with the derivation.)

The first order theories behind the above three examples have excellent model theoretic properties: they are all complete, model complete, and they admit quantifier elimination for appropriate languages. In contrast, even though the theory of differentially closed fields of characteristic zero also enjoys these properties [14, 12, 3], no simple explicit models are known in this case. The goal of this paper is to construct such models.

We recall from [11, 7, 8] that a differential ring is a ring together with a derivation . The set is called its ring of constants. Given a differential ring , we define the set of differential polynomials in over by . Then is a differential ring for the unique derivation that extends the derivation on and such that for all . Given , the minimal with is called its order and we denote it by .

A differential field is a differential ring that is also a field. Let be a differential field of characteristic zero. We say that is differentially closed if it is algebraically closed and for any with , there exists a with and . This notion was first introduced by Robinson [12] and subsequently elaborated by Blum [3]. The definition that we use here is due to Blum.

Although it is known that any differential field of characteristic zero can be embedded into a differentially closed field , the proof is abstract and does not provide us with an explicit description of . A more natural—but still fairly lengthy—construction has recently been proposed in [ 18 ]. Other ideas were raised in a recent discussion on Mathoverflow

There are also partial constructions that go a long way, but fall short of delivering a full differential closure. For instance, Seidenberg's embedding theorem allows for the realization of certain countable differential fields as fields of meromorphic functions [15, 16, 9]. Fields of complex transseries from [4] have the property that any has a root in . In particular, such fields are closed under the resolution of linear differential equations. See also [6] for an analytic counterpart.

In this note, we give a short explicit construction for embedding an arbitrary differential field of characteristic zero into a differentially closed field . The key observation is that the ring of formal power series is very close to providing a differential closure: at a non-singular point, formal power series solutions are in sufficient supply to force the required inequality . Through the introduction of generic shifts we will reduce the general case to this favorable situation.

However, the generic shifts also introduce many new constants in our field. An interesting question remains: how to single out a differentially closed subfield of which has the same constant field as ?

Acknowledgment. We are grateful to Gleb Pogudin and François Ollivier for helpful comments and suggestions.

2.Construction

Let be an algebraically closed field of characteristic zero with the trivial derivation that turns every element of into a constant. We will write for the algebraically closed field of formal Puiseux series in with coefficients in . For , let

We view as a differential field with constant field for the derivation . Setting

we define a differential field embedding

Note that

(1)

and that the evaluation at of is non-zero for any non-zero .

Regarding as being naturally included in via the embedding , we have an infinite chain of inclusions

Now consider the differential field

(2)

Proof. The field is algebraically closed as an increasing union of algebraically closed fields.

Consider two non-zero differential polynomials of orders and , respectively. We have to prove that there exists a with and . Since and have finitely many coefficients, we have for some . Without loss of generality, we may assume that is square-free as a polynomial in , where . So is non-zero, where denotes the separant of . Let , where is the initial of . Hence is a non-zero element of .

In view of (1), we may regard , , , and as elements of . Let be their evaluations at . Note that , so . Similarly, . Consider with . Since is algebraically closed and , there exists some with . By construction, , since otherwise would vanish.

Now consider the differential equation over with initial conditions . Since , this equation admits a unique formal power series solution . By construction, the valuation of in is zero, so . This concludes our proof.

Remark 2. The field is still far away from being “the” differential closure of or . First of all, is not d-algebraic over . This problem can be addressed by constructing a sequence of differential subfields of as follows. We take . Assuming that with constant field has been constructed, let be the algebraic closure of in which we substituted for . Then we let be the field of that are d-algebraic over . The same argument as in the above proof shows that is differentially closed, but also d-algebraic over .

A second problem is that the constant fields of and of are much larger than . By construction, we actually have a replica of inside , since is a replica of inside for each . This suggests that the second problem may be related to the non-minimality of differential closures [13, 17]. Even though and are much larger than , they still admit a very explicit nature. In particular, elements of can recursively be described as solutions of suitable differential equations in .

3.Embedding arbitrary differential fields

For the theorem from the previous section, we worked over an algebraically closed constant field of characteristic zero. In fact, there is a simple way [10, §44.3] to embed an arbitrary differential field of characteristic zero into a differential field of Puiseux series whose coefficients are constants:

Lemma 3. Let be a differential field of characteristic zero and consider the map

We define a derivation on by . Then is an embedding of differential fields.

Proof. For , one verifies that

Proof. Without loss of generality, we may assume that is algebraically closed, since any derivation on a field of characteristic zero canonically extends to its algebraic closure. Using the lemma, we may embed into , where becomes a field of constants when regarded as a differential subfield of . This allows us to pursue the construction of the previous section and successively embed . We conclude by Theorem 1.

Remark 5. Lemma 3 also has the potential of replacing the embeddings from the previous section by alternative ones. For instance, by generalizing the construction from [4], we expect it to be possible to construct fields of “complex transseries” in over . Such a field would contain and be closed under the resolution of linear differential equations. Thanks to Lemma 3, we might then let play the role of in the construction of the previous section.

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