> <\body> <\hide-preamble> >> >> >> >> >> >> > >> >> >> >> >> >> |]>|]>>> |)>|)>>> |]>>>> \; ||<\author-note> This work has been supported by the ODELIX project (number 101142171). <\wide-tabular> ||||||| ||LOGO_ERC-FLAG_FP.png>|61.62pt|28.51pt||>>>>> |<\author-affiliation> Laboratoire CNRS d'informatique (LIX, UMR 7161) �cole polytechnique, Institut Polytechnique de Paris 1, rue Honor� d'Estienne d'Orves B�timent Alan Turing, CS35003 91120 Palaiseau, France >>||<\doc-note> > Given a differential field of characteristic zero, we give a short explicit construction of adifferentially closed field > that contains . > 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>. (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, no simple explicit models are known in this case. The goal of this paper is to construct such models. We recall from> that a is a ring together with a derivation :R\R;y\y>>. The set \R\y=0|}>> is called its . Given a differential ring , we define the set of in over by \\>R,Y>|]>>>. Then > is adifferential ring for the unique derivation that extends the derivation on and such that Y>=Y>> for all . Given R>, the minimal \> with ,Y>|]>>> is called its and we denote it by r>. A is a differential ring that is also a field. Let be a differential field of characteristic zero. We say that is if it is algebraically closed and for any |P,Q|\>\K> with ord Q>, there exists a K> with =0> and \0>. This notion was first introduced by Robinson and subsequently elaborated by Blum. 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\Vbut still fairly lengthy\Vconstruction has recently been proposed in. Other ideas were raised in a recent discussion on Mathoverflow<\footnote> > . There are also partial constructions that go a long way, but fall short of delivering afull differential closure. For instance, Seidenberg's embedding theorem allows for the realization of certain countable differential fields as fields of meromorphic functions>. Fields of complex transseries from have the property that any K\K>> has aroot in . In particular, such fields are closed under the resolution of linear differential equations. See also 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 anon-singular point, formal power series solutions are in sufficient supply to force the required inequality \0>. 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. Aninteresting question remains: how to single out adifferentially closed subfield of > which has the same constant field as ? 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 =0,1,2,\>>,let <\eqnarray*> >>|>|

>|)>|)> \ >>|)>|)>>>|>>|>|>>>|)>|)>.>>>> We view >> as a differential field with constant field >> for the derivation >\\/\ z>>. Setting <\eqnarray*> >i>y*z>|)>>|>|i>y*+1>+z+1>|)>>>|||i>y*\+1>*0>*|k!>*+1>|\+1>>>>|||0>i>*|k!>*y*\+1>|)>*z+1>>>>> we define a differential field embedding <\equation*> L>|\>>L+1>. Note that <\equation> im \>\K+1>+1>|]>|]> and that the evaluation at +1>=0> of >> is non-zero for any non-zero L>>. Regarding >> as being naturally included in +1>> via the embedding>>, we have an infinite chain ofinclusions <\equation*> L\L\L\\. Now consider the differential field <\equation> L>\\0>L>. <\theorem> The field >> is differentially closed. <\proof> The field >> is algebraically closed as an increasing union of algebraically closedfields. Consider two non-zero differential polynomials |P,Q|\>\L>> of orders ord P> and r>, respectively. We have to prove that there exists a L>> with =0> and \0>>. Since and have finitely many coefficients, we have |P,Q|\>\L>> for some\0>>. Without loss of generality, we may assume that is square-free as a polynomial *>|)>+\+P>> in >>, where \0>. So >>|)>>> is non-zero, where \ P/\ Y>>> denotes the separantof. Let Q*I*Res>>|)>>, where \P> is the initial of . Hence is a non-zero element of >,Y>|]>>. In view of(), we may regard , >, >, and as elements of +1>+1>|]>|]>>. Let >,>,>,>\K+1>>> be their evaluations at +1>=0>. Note that >=ord P=r>, so >\+1>,Y>|]>>>>. Similarly, >\K+1>,Y>|]>>. Consider ,\,\

|)>\K+1>> with >,\,\

|)>\0>>. Since +1>> is algebraically closed and >,\,\

|)>\0>, there exists some \K+1>> with >,\,\|)>=0>>. By construction, >,\,\|)>\0>, since otherwise >