> <\body> <\hide-preamble> >>> >>> >>> >>> >>> \; |||<\author-address> de Mathématiques ( 425) Université Paris-Sud, CNRS 91405 Orsay Cedex France |>|>|<\doc-note> This work has partially been supported by the ANR Gecko project. > <\abstract> It is well known that Hardy fields can be extended with integrals, exponentials and solutions to Pfaffian first order differential equations =P(f)/Q(f)>. From the formal point of view, the theory of transseries allows for the resolution of more general algebraic differential equations. However, until now, this theory did not admit a satisfactory analytic counterpart. In this paper, we will introduce the notion of a transserial Hardy field. Such fields combine the advantages of Hardy fields and transseries. In particular, we will prove that the field of differentially algebraic transseries over {{x1>}}> carries a transserial Hardy field structure. Inversely, we will give a sufficient condition for the existence of a transserial Hardy field structure on a given Hardy field. A Hardy field is a field of infinitely differentiable germs of real functions near infinity. Since any non-zero element in a Hardy field > is invertible, it admits no zeros in a suitable neighbourhood of infinity, whence its sign remains constant. It follows that Hardy fields both carry a total ordering and a valuation. The ordering and valuation can be shown to satisfy several natural compatibility axioms with the differentiation, so that Hardy fields are models of the so called theory of H-fields. Other natural models of the theory of H-fields are fields of transseries . Contrary to Hardy fields, these models are purely formal, which makes them particularly useful for the automation of asymptotic calculus. Furthermore, the so called field of grid-based transseries > (for instance) satisfies several remarkable closure properties. Namely, > is differentially Henselian and it satisfies the differential intermediate value theorem 9.33>. Now the purely formal nature of the theory of transseries is also a drawback, since it is not clear how to associate a genuine real function to a transseries , even in the case when satisfies an algebraic differential equation over {{x1>}}>. One approach to this problem is to develop Écalle's accelero-summation theory , which constitutes a more or less canonical way to associate analytic functions to formal transseries with a ``natural origin''. In this paper, we will introduce another approach, based on the concept of a. Roughly speaking, a transserial Hardy field is a truncation-closed differential subfield> of >, which is also a Hardy field. The main objectives of this paper are to show the following two things: <\enumerate> The differentially algebraic closure in > of a transserial Hardy field can be given the structure of a transserial Hardy field. Any differentially algebraic Hardy field extension of a transserial Hardy field, which is both differentially Henselian and closed under exponentiation, admits a transserial Hardy field structure. We have chosen to limit ourselves to the context of grid-based transseries. More generally, an interesting question is which H-fields can be embedded in fields of well-based transseries and which differential fields of well-based transseries admit Hardy field representations. We hope that work in progress on the model theory of H-fields and asymptotic fields will enable us to answer these questions in the future. The theory of Hardy fields admits a long history. Hardy himself proved that the field of so called L-functions is a Hardy field . The definition of a Hardy field and the possibility to add integrals, exponentials and algebraic functions is due to Bourbaki. More generally, Hardy fields can be extended by the solutions to Pfaffian first order differential equations and solutions to certain second order differential equations . Further results on Hardy fields can be found in. The theory of transserial Hardy fields can be thought of as a systematic way to deal with differentially algebraic extensions of any order. The main idea behind the addition of solutions to higher order differential equations to a given transserial Hardy field > is to write such solutions in the form of ``integral series'' over > (see also ). For instance, consider a differential equations such as =\>+f,> for large 1>. Such an equation may typically be written in integral form \>+ f.> The recursive replacement of the left-hand side by the right-hand side then yields a``convergent'' expansion for using iterated integrals \>+ \>+2* \>* \>+\,> where we understand that each of the integrals in this expansion are taken from \>: <\equation*> (g)(x)=>g(t)*\ t. In order to make this idea work, one has to make sure that the extension of > with asolution of the above kind does not introduce any oscillatory behaviour. This is done using a combination of arguments from model theory and differential algebra. More precisely, whenever a transseries solution to an algebraic differential equation over > is not yet in >, then we may assume the equation to be of minimal ``complexity'' (anotion which refines Ritt rank). In section, we will show how to put the equation in normal form <\equation> L f=P(f), where \{F}> is ``small'' and \[\]> admits a factorization <\equation*> L=(\-\)*\*(\-\) over [\]>. In section, it will be show how to solve() using iterated integrals, using the fact that the equation -\) f=g> admits \>*\\>*g> as a solution. Special care will be taken to ensure that the constructed solution is again real and that the solution admits the same asymptotic expansion over > as the formal solution. Section contains some general results about transserial Hardy fields. In particular, we prove the basic extension lemma: given a transseries and a real germ> at infinity which behave similarly over > (both from the asymptotic and differentially algebraic points of view), there exists a transserial Hardy field extension of > in which and> may be identified. The differential equivalence of and > will be ensured by the fact that the equation () was chosen to be of minimal complexity. Using Zorn's lemma, it will finally be possible to close > under the resolution of real differentially algebraic equations. This will be the object of the last section. Throughout the paper, we will freely use notations from . For the reader's convenience, some of the notations are recalled in section. We also included a glossary at the end. It would be interesting to investigate whether the theory of transserial Hardy fields can be generalized so as to model some of the additional compositional structure on >. A first step would be to replace all differential polynomials by restricted analytic functions. A second step would be to consider postcompositions with operators > for sufficiently flat transseries for which Taylor's formula holds: <\equation*> f\(x+\)=f+f*\+*f*\+\. This requires the existence of suitable analytic continuations of in the complex domain. Typically, if \g>> with \,\>>, then g> should be defined on some sector at infinity (notice that this can be forced for the constructions in this paper). Finally, more violent difference equations, such as <\equation*> f(x)=>>>+f(x+1), generally give rise to quasi-analytic solutions. From the model theoretic point view, they can probably always be seen as convergent sums. Finally, one may wonder about the respective merits of the theory of accelero-summation and the theory of transserial Hardy fields. Without doubt, the first theory is more canonical and therefore has a better behaviour with respect to composition. In particular, we expect it to be easier to prove o-minimality results. On the other hand, many technical details still have to be worked out in full detail. This will require a certain effort, even though the resulting theory can be expected to have many other interesting applications. The advantage of the theory of transserial Hardy fields is that it is more direct (given the current state of art) and that it allows for the association of Hardy field elements to transseries which are not necessarily accelero-summable. Let =\=\>> be the totally ordered field of grid-based transseries. Any transseries is an infinite linear combination \\>f>*\> of transmonomials, with grid-based support \>. Transmonomials ,\,\> are systematically written using the fraktur font. Each transmonomial is an iterated logarithm x> of or the exponential of a transseries with \1> for each \supp g>. The asymptotic relations >,>,>,>,>,>,>> and >>g>| is dominated by >g>| is negligible >g>| is asymptotic to >g>| is asymptotically similar to >g>| is flatter than or as flat as >g>| is flatter than >g>| is as flat as >g>| and are similar modulo flatness> on > are defined by <\eqnarray*> g>|>|>|g>|>|>|g>|>|g\f>>|g>|>|g>>|g>|>|log \|g\|>>|g>|>|log \|g\|>>|g>|>|log \|g\|>>|g>|>|log \|g\|.>>>> Given \1>, one also defines variants of >,>>, modulo flatness>g>|g> modulo elements flatter than >>>g>|g> modulo elements flatter than >>>>g>|g> modulo elements flatter than or as flat as >>>>g>|g> modulo elements flatter than or as flat as >>: <\eqnarray*> >g>|>|\\\,f\g*\>>|>g>|>|\\\,f\g*\>>|>>g>|>|\\\,f\g*\>>|>>g>|>|\\\,f\g*\.>>>> It is convenient to use relations as superscripts in order to filter elements, as in>>|shorthand for \:f\0}>>>>|shorthand for \:f\0}>>>>|shorthand for \:f\1}>> <\eqnarray*> >>||\:f\0}>>|>>||\:f\0}>>|>>||\:f\1}.>>>> Similarly, we use subscripts for filtering on the support:>>|infinite part of >\>>|part of which is flatter than >>>>|shorthand for >:f\\}>>\>>|shorthand for \>:f\\}>> <\eqnarray*> >>||\supp f,\\1>f>*\>>|\>>||\supp f,\\\>f>*\>>|>>||>:f\\}>>|\>>||\>:f\\}.>>>> We denote the derivation on > by >>|derivation with respect to > and the corresponding distinguished integration (with constant part zero) by >>|integration with respect to >. The logarithmic derivative of is denoted by>>>>|logarithmic derivative of >. The operations >>|upward shifting> and >>|downward shifting> of upward and downward shifting correspond to postcomposition with . We finally write g>g>| is a truncation of > if the transseries is a truncation of , \supp f> for all \supp(g-f)>. Given \>, we define the of by|canonical span of > <\equation> span f=max> {\\(log(\/\))>:\,\\supp f}. By convention, if contains less than two elements. We also define the of by|ultimate canonical span of > <\equation> uspan f=min> {span f\>:\\supp f}. We notice that 1> if and only if admits no minimal element for >>. <\example> We have <\eqnarray*> 1+x>|1-x1>>>||x>>>|1+x>|1-x1>>>||1>>>>> Consider a differential subfield > of > and let \\>>. We say that > has span >, if\> for all \> and \> for at least one \> (notice that we do not require \\\>>). Since > is stable under differentiation, we have \x1>> as soon as \\>. Notice also that we must have \\\>> if > has span >. A transseries \\\> is said to be a over>, if \\> for every \f> and admits no minimal element for >>. In that case, let \supp f> be maximal for >> such that 1>*supp f\>\span f>. Then =f\>> and =f\>> are called the and the of . We say that is a if \span f>>, which implies in particular that =0>. Assuming that > has span >, any serial cut over > is necessarily in \>>. Conversely, any \\>\\> with \> is a serial cut over>. We will denote by |^>>|^>>|completion of > with serial cuts> the set of all \\>> which are either in > or serial cuts over > with \>. Notice that |^>> is again a differential subfield of \>>. The above definitions naturally adapt to the complexifications [\]> and [\]> of > and differential subfields > of >. If > has span >, then the set |^>[\]> coincides with the set of all \\>[\]=\[\]\>> which are either in [\]> or serial cuts over [\]> with \>. Let > be a differential field. We denote by {F}>{F}>|ring of differential polynomials in over >> the ring of differential polynomials in over > and by \F\>\F\>|quotient field of {F}>> its quotient field. Given \{F}> and \>, we recall that > denotes the homogeneous part of degree of . We will denote by >>|linear part of as an operator> the linear operator in [\]> with F=P(F)>. Assuming that \>, we also denote the order of by>>|order of >, the degree of in )>> by >>|degree of in its leader> and the total degree of by >>|total degree of >. Thus, the Ritt rank of is given by the pair ,s)>. The triple =(r,s,t)>>|complexity of > will be called the of; likewise ranks, complexities are ordered lexicographically. As usual, we will denote the initial and separator of by >>|initial of > >>|separant of > and set =I*S>>|the product *S>>. Given \{F}> with \>, Ritt reduction of by provides us with a relation <\equation> H>*Q=\ P+R, where \\{F}[\]> is a linear differential operator, \\> and the remainder \{F}> satisfies \\>. Let > be a differential field extension of >. An element \> is said to be over > if there exists an annihilator \{F}\\> with . An annihilator of minimal complexity > will then be called a and =\>>|complexity of over >> is also called the of over >. The order =r>>|order of over >> of such a minimal annihilator is called the of over >. We say that > is a of > if each \> is differentially algebraic over >. We say that > is differentially closed in >, if \\> contains no elements which are differentially algebraic over >.. Given \\> ( \>), we say that > is >-differentially closed> ( -differentially closed>) in > if \\> ( \r>) for all \\\>. We say that > is if every \{F}\\> admits a root in >. We say that> is -differentially closed> if every \{F}\\> of order > r> admits arootin>. Given a differential polynomial \{F}> and \\>, we define the and of by >:>>|additive conjugation of by >>\>>|multiplicative conjugation of by >> <\eqnarray*> >(F)>||)>>|\>(F)>||*F).>>>> We have >,P\>\\{F}> and <\eqnarray*> >>>||>>|\>>>||>>|>>>||>>>|\>>>||\>>>|>>>||>>>|\>>>||\>>>>> We also notice that additive and multiplicative conjugation are compatible with Ritt reduction: given \\> and assuming(), we have <\eqnarray*> >>>*Q>>|| P>+R>>>|\>>>*Q\>>|| P\>+R\>,>>>> <\remark> The compatibility of Ritt's reduction theory with additive and multiplicative conjugation holds more generally for rings of differential polynomials in a finite number of commutative partial derivations (or with a finite dimensional Lie algebra of non-commutative derivations). Similar compatibility results hold for upward shiftings or changes of derivations (in the partial case, this requires the rankings to be order-preserving). In the case when > is a differential subfield of =\>>, we recall that adifferential polynomial \{F,\,F}> may also be regarded as a series in {F,\,F}>>. Similarly, elements of the fraction field \F,\,F\> of {F,\,F}> may be regarded as series with coefficients in \F,\,F\>. Indeed, writing *\+R> and *\+R>, where *\> denotes the dominant term of , we may expand <\equation*> =|D>\|\>\|D*\>|1+|D+\>> In the case when \;\;\>{F,\,F}> for some transbasis ={\,\,\}>, then and may also be expanded lexicographically with respect to ,\,\>. Let > be a differential field and consider a linear differential operator \[\]>>. We will denote the order of by >. Given \\>, we define the \>>\>>|multiplicative conjugate of by >> and the \>>\>>|twist of by >> by <\eqnarray*> \>>||>>|\>>||1>*L*\>>>> We notice that \>> is also obtained by substitution of +\>> for > in . We say that over >, if it admits a complete factorization <\equation> L=c*(\-\)*\*(\-\) with ,\,\\\>. In that case, each of the twists \>> of also splits: <\equation*> L\>=c*(\+\>-\)*\*(\+\>-\). We say that > is -linearly closed> if any linear differential operator of order > r> splits over>. <\proposition> If > is weakly -differentially closed, then > s -linearly closed. <\proof> The proof proceeds by induction over . For , we have nothing to prove, so assume that 0> and let \[\]> be of order . Then the differential Riccati polynomial> has order , so it admits a root \\>. Division of by -\> in [\]> yields a factorization *(\-\)> where \\[\]> has order . By the induction hypothesis, > splits over >, whence so does . <\proposition> Let \[\]>> be an operator which splits over > and let \[\]> be such that . Then and split over >. <\proof> Recall that greatest common divisors and least common multiples exist in the ring [\]>. Given a splitting(), consider the operators <\eqnarray*> >||-\)*\*(\-\))>>|>||-\)*\*(\-\))>>>> We have \|\\|\=A*B> and \|\\|\=B>. Moreover, the orders of > and > ( > and >) differ at most by one for each . It follows that and split over >. Assume now that > is a totally ordered differential field. A monic operator \[\]>> is said to be an if has either one of the forms <\equation*> |||||-\,>||\\>>|||-(\-\*\+\>))*(\-(\+\*\)),>||,\\\>>>>> A of an operator \[\]>> over> is afactorization of the form <\equation> L=K*\*K, where each > is an atomic real operator. A splitting () over [\]> is said to , if it gives rise to a real splitting() for =(\-\>)> or =(\-\>)*(\-\+1>)> and \\\i>. <\proposition> Let \[\]>> be an operator which splits over [\]>. Then admits areal splitting over >. <\proof> Assuming that \>, we claim that there exists an atomic real right factor \[\]> of . Consider a splitting () over [\]>. If \\>, then we may take -\>. Otherwise, we write <\equation*> L=>*(\-|\>)*\*(\-|\>) and take to be the least common multiple of -\> and -|\>> in [\]>. Since >>, we indeed have \[\]>. Since -\\|L> and -|\>\|L>, we also have . In particular, proposition implies that splits over [\]>. Such a splitting is necessarily of the form <\equation*> K=(\-(\-\*\+\>))*(\-(\+\*\)),\,\\\, whence is atomic. Having proved our claim, the proposition follows by induction over . Indeed, let \\[\]> be such that *K=L>. By proposition, > splits over [\]>. By the induction hypothesis, > therefore admits a real splitting =K*\*K> over >. But then *\*K*K> is a real splitting of . <\corollary> An operator \[\]>> is atomic if and only if is irreducible over > and splits over [\]>. Let > be a differential subfield of > of span >. Given \[\]{F}> and |^>[\]>, we say that over |^>[\]> at , if >> and have the same order and >> splits over |^>[\]>. <\lemma> Let > be a differential subfield of > of span >. Let \[\]{F}> be aminimal annihilator of a differentially algebraic cut |^>[\]> over [\]>, which splits over |^>[\]> at. Then any minimal annihilator \[\]\f\>> of >> over [\]\f\> splits over |^>[\]> at >>. <\proof> Since >(>)=0>, Ritt division of >> by yields <\equation> H>*>=\ Q for some \\> and \\[\]\f\{>}[\]>. Additive conjugation of () yields <\equation> H>>>>*>>>=\ Q>>. By the minimality hypothesis for , we have >>,r>=S(>)\0> and (>)\0>, so that >>=1> and >>>=0>. Similarly, we have >>>=1>. Consequently, when considering the linear part of the equation (), we obtain <\equation*> H>>,0>>*L>>>>=\ L>>>, whence >>>> divides >>>>> in [\]\f\[\]>. Now >> splits over |^>[\][\]>, whence so does>>>>>. By proposition , we infer that >>>> splits over |^>[\][\]>. Since (>)\0>, we also have >>>>=r> and we conclude that splits over |^>[\]> at >>. <\corollary> Let > be a differential subfield of > of span >. Let \[\]{F}> be aminimal annihilator of a differentially algebraic cut |^>[\]> over [\]>, which splits over|^>[\]> at. Then any minimal annihilator \[\]\f\G> of over [\]\f\> splits over |^>[\]> at . <\proof> Applying the lemma to >, we see that >>>> splits over |^>[\]>. Now >>=R>, whence >> and >=L,\2>> also split over |^>[\]>. <\lemma> Let > be a differential subfield of > of span >, such that |^>[\]> is -linearly closed. Let \[\]{F}> be aminimal annihilator of a differentially algebraic cut |^>[\]> over [\]>, such that has order . Assume that \> and let \{G}> be aminimal annihilator of over >. Then splits over |^>[\]> at . <\proof> Let be as in the above corollary, so that splits over |^>[\]> at . Since has minimal complexity and 0>, Ritt division of by yields <\equation*> H> S=\ R for some \\> and \\[\]\f\{G}[\]>. Additive conjugation and extraction of the linear part yields <\equation*> H,0>>*L>=\ L>, so >> divides >> in |^>[\][\]>. Since the separants of and don't vanish at , we have <\equation*> ||>>>||>||[\]\f,Re f\:\[\]\f\)>>|||||[\]\Re f,Im f\:\[\])-tr deg (\[\]\f\:\[\])>>|||||\Re f,Im f\:\)-tr deg (\[\]\f\:\[\])>>|>>>||>||\Re f\:\)>>|||||\Re f,Im f\:\)-tr deg (\\Re f,Im f\:\\Re f\)>>>>> and <\equation*> r-r=tr deg (\[\]\f\:\[\])-tr deg (\\Re f,Im f\:\\Re f\)\r. Consequently, the quotient of >> and >> has order at most , whence it splits over |^>[\]>. It follows that >> splits over |^>[\]> and splits over |^>[\]> at . Let > be a differential subfield of > of span \x>. Recall from that with \[\][\]> admits a canonical fundamental system of oscillatory transseries solutions ={h,\,h}\\> with ,\,log h\\\>[\]>. We will denote by >>|set of dominant monomials of solutions to > the set of dominant monomials of ,\,h>. The neglection relation on > is extended to > by 1> if and only if >*\*\>+\+f>*\*\>> with >,\,f>\\[\]>> and ,\,\\\>. We say that is , if we have \>1> or \log \> for each . In that case, any quasi-linear equation of the form <\equation*> L f=g,f\>1 with \\>[\]> admits 1> g> as its only solution in \>[\]>. If is a first order operator of the form -\>, then is normal if and only if \c*\>> for some 0> or \\>>. In particular, we must have \>1> and \\>>. <\proposition> Let \[\][\]\\[\]>. <\enumerate-alpha> There exists a \\> such that \>>> is normal. If is normal and \0>, then \>>> is normal. <\proof> Let ={h,\,h}>. For each \\>, the operator \>>> admits /\>,\,h/\>> as solutions, which implies in particular that \>>>=\\>*\>. Now /\>)\log \\Re log h\log \> for all . Choosing > sufficiently large, it follows that /\>\>1> for all with /\>)\log \>, so that \>>> is normal. Similarly, if \>1> for some with /\>)\log \>, then \>\>> for all \0>. <\proposition> Consider a normal operator \[\][\]>, which admits a splitting <\equation*> L=(\-\)*\*(\-\) with ,\,\\\[\]>. Then each -\> is a normal operator. <\proof> We will call \\>[\]*\*\\>>> normal, if -h>> is normal. Let us first prove the following auxiliary result: given \\[\]> and \\>[\]*\*\\>>> such that -\> and are normal and =\\\-\>>, then -\)*h> is also normal. If log \>, then (\-\) h\>>h>, whence -\) h=Re log h+O(log \)\log \>. In the other case, we have >1>. Now if >\\>, then -\) h\>1>, since \>1>. If >\\>, then \\-\>> implies \-\)h>>>, whence -h>\1/(x*log x*\)>. It again follows that -\)*h\>h/(x*log x*\)\>1>. Let us now prove the proposition by induction over . For , we have nothing todo, so assume that 1>. Since =(\-\)*\*(\-\)> is normal, the induction hypothesis implies that -\> is normal for all 2>. Now let be the unique element in \\>>. Since is normal, -\) \ (\-\) h> is also normal for ,2>, by the auxiliary result. We conclude that -\> is normal, since =( h)>>. Let and ={h,\,h}> be as above. The smallest real number \0> with \>\\>> for all will be called the of , and we denote =\>. For all \\>, we notice that \>>>=\>. <\proposition> Let \[\][\]> be operators of the same order with <\equation*> K=L+o>(\*\>*L). Then=\>. <\proof> Given \>, we have <\equation*> Kh>=Lh>+o>(Lh>), since >\>log h\\\>>. In particular, h,0>\>K>, whence \h>>> and \\>. <\proposition> Given a splitting <\equation*> L=(\-\)*\*(\-\) with ,\,\\\\>[\]>, we have \>\>> for all . <\proof> Assume for contradiction that \>\\>> for some and choose maximal with this property. Setting <\equation*> K=(\-\)*\*(\-\), the transseries <\equation*> h=K1>(\\>)\\\>[\]*\\> satisfies , as well as >\\>\\>>. But such an cannot be a linear combination of the > with \>\\>>. <\remark> It can be shown (although this will not be needed in what follows) that an operator \[\][\]> splits over |^>[\]> if and only if there exists an approximation \\[\][\]> with -L\>\>> which splits over [\]> for every\\>. In particular, |^>[\]> is -linearly closed if and only if [\]> is -linearly closed over |^>[\]>. Assume now that > is a differential subfield of > of span \x>. We say that is if > is normal of order > and 1>\>\*\>>*L>. In that case, the equation <\equation> P(f)=0,f\>1 is quasi-linear and it admits a unique solution in \>>. Indeed, let \\>> be the distinguished solutionto (). By proposition , the operator >> is normal. If \\\>> were another solution to(), then -f>> would be in >>, whence \1>, which is impossible. <\proposition> Let > be a differential subfield of > of span >. Let \[\]{F}> be aminimal annihilator of a differentially algebraic cut |^>[\]> over [\]>. Then there exists a truncation \f> and \\> such that ,\\>>> is normal. <\proof> Let =P> and =r>>*\>>>. Modulo a multiplicative conjugation by >> for some \0>, we may assume without loss of generality that \L>>. Modulo an additive conjugation by >1>>, we may also assume that >1>. For any ,\\0> and =f>\>>\f>, we have <\equation*> P>=-f>=+o>(\>*), whence <\equation> P,\\>>=\>>+O>(\>*)+o>(\>*). Since (f)\0>, we have \0>. By proposition , there exists a \\> for which ,\\>>> is normal. Now take =\+\>. Denoting ,\\>>>, proposition and () imply that > is normal with >=\> and 1>\>\>*\>>\\>*L>. We say that \[\]{F}> is , if is normal and > can be decomposed =L+K> such that splits over [\]> and >\*\>*L>. In that case, we may also decompose for 1>(F)+K F> with >\*\>*L>. If is monic, then we say that is . Any split-normal equation () is clearly equivalent to a monic split-normal equation of the same form. <\proposition> Let > be a differential subfield of > of span > such that |^>[\]> is -linearly closed. Let \[\]{F}> be aminimal annihilator of a differentially algebraic cut |^>[\]> of order over [\]>. Let \{F}> be a minimal annihilator of and assume that \r>. Then there exists a truncation \Re f> and \\> such that ,\\>>> is split-normal. <\proof> By proposition and modulo a replacement of by \>*(f-\)>, we may assume without loss of generality that is normal. By lemma , splits over |^>[\]> at . Let ,\,\\|^>[\]> be such that <\equation*> L>=c*(\-\)*\*(\-\). Setting =s*\>>, we notice that =L>+o>(\>*L)>. Now take <\equation*> L=c>\>*\>*(\-\>\>>)*\*(\-\>\>>)\\[\][\]. Then +o>(\>*L)> and proposition implies that is normal, with =\>=\>>>. Denoting , we finally have >\>*L>. Let =\=\>> be the field of grid-based transseries and >>|ring of infinitely differentiable germs at infinity> the set of infinitely differentiable germs at infinity. A field is a differential subfield> of >, together with a monomorphism :\\\> of ordered differential >-algebras, such that <\description> For every \>, we have \>. For every \>, we have >\\>. There exists an \>, such that \\+\*log x> for all \\\\>. The set \\> is stable under taking real powers. We have (log f)=log \(f)> for all \>> with \>. In what follows, we will always identify > with its image under >, which is necessarily aHardy field in the classical sense. The integer in is called the of >; if \\> for all \\\\>, then the depth is defined to be \>. We always have 0>, since> is stable under differentiation. If \>, then > is exponential for all \> and> contains x>. If > and \\>, then > contains x> for all sufficiently large . <\example> The field =\> is clearly a transserial Hardy field. As will follow from theorem below, other examples are <\eqnarray*> (x>)>||,\,\\\>\(x>,\,x>)>>|(\*x>)>||,\,\\\>\(\*x>,\,\*x>).>>>> <\remark> Although the axioms and are not really necessary, allows for the simplification of several proofs, whereas it is natural to enforce . Notice that automatically holds for \>> with 1> since <\equation*> \(log f)=\((log f))=\(f/f)=\(f)/\(f)=(log \(f)), whence (log f)=log \(f)+c> for some \>. Since both (log f)-log f>> and (f)-log f>> are infinitesimal in >, we have . Consequently, it suffices to check for monomials \\\> with \>. <\proposition> Let > be a transserial Hardy field with \>. Then the upward shift \> of > carries a natural transserial Hardy field structure with (f\)=\(f)\\>. <\proof> The field \> is stable under differentiation, since =(x*f)\> forall\>. <\corollary> If > has depth \>, then \> is a transserial Hardy field of depth . We recall that a > is a finite set of transmonomials ,\,\}> with <\description> ,\,\\1> and \\\\>. =log x> for some \>. \\;\;\>> for all i\n>. If , then > is called a transbasis and ;\;\>> is stable under differentiation. The incomplete transbasis theorem for> also holds for transserial Hardy fields: <\proposition> Let \\> be a transbasis and \>. Then there exists an supertransbasis |^>\\> of > with \>>>. Moreover, if > is plane and is exponential, then |^>> may be taken to be plane. <\proof> The same proof as for may be used, since all field operations, logarithms and truncations used in the proof can be carried out in >. Given a set > of exponential transseries in >, the of > is the minimal size of a plane transbasis ={\,\,\}> with \\;\;\>>. This notion may be extended to allow for differential polynomials in > (modulo the replacement of by its set of coefficients). <\remark> The span and ultimate span of \> are not necessarily in >. Nevertheless, if 1> and ={\,\,\}\\> is a transbasis for , then we do have \> for some (and similarly for the ultimate span of ). Let > be a transserial Hardy field. Given \> and \\>, we write >>| is asymptotically similar to > over >> if there exists a\\> with <\equation*> f\>\\>. We say that and > are over > if for each \\> (or, equivalently, for each \f>), we have <\equation*> f-\\-\. We say that and> are over > if <\equation*> P(f)=0\P()=0 for all \{F}>. <\lemma> Let > be a transserial Hardy field and let \\\> be differentially algebraic over >. Let \supp f> be maximal for >>, such that =f\>\\>. Then > is differentially algebraic over > and >\\>. <\proof> Let \{F}> be a minimal annihilator of . Modulo upward shifting, we may assume without loss of generality that and are exponential. Since \|^>>, all monomials in > are in >, whence there exists a plane transbasis ,\,\}\\> for and >. Modulo subtraction of >> from and >, we may assume without loss of generality that >=0>. Let be such that \\> and let >*\*\>> be the dominant monomial of>. Modulo division of and > by >*\*\>>, we may also assume that > is a normal serial cut. But then the equation gives rise to the equation \>(\)=0> for =f\>>. The complexity of \>> is clearly bounded by =\>. <\lemma> Let > be a transserial Hardy field and \\\\>>. Let \>|^>> and \\> be such that and > are both asymptotically and differentially equivalent over \>>. Then and > are both asymptotically and differentially equivalent over >. <\proof> Given \\>, we either have \>>1> and <\equation*> f-\\>\\\>-\ or \>>1>, in which case <\equation*> f-\\>f-\>>1>\-\>>1>\>-\. This proves that and > are asymptotically equivalent over >. As to their differential equivalence, let us first assume that is differentially transcendent over \>>. Given \{F}>>, let us denote <\equation*> D=\1>*Q>>\>\\\>. We have (f)\0>, ()\0> and <\eqnarray*> |>>>|(f)*\>>|)>|>>>|()*\,>>>> whence 0> and )\0>. Assume now that is differentially algebraic over \>> and let \\>{F}> be a minimal annihilator. Given \{F}>, Ritt reduction of gives <\equation*> H*Q=\ P+R, where \\{F}[\]> and \{F}> is such that \\>. Since >\\> and \\\>>, we both have (f)\0> and ()\0>, whence <\eqnarray*> ||(f)>>>|)>||)|H()>.>>>> If , this clearly implies )=0>. Otherwise, > vanishes neither at nor at> and the relations() and() again yield 0> and )\0>. <\lemma> Let > be a transserial Hardy field and let |^>\\> be a differentially algebraic cut over > with minimal annihilator . Let \\> be a root of such that and> are asymptotically equivalent over >. Then and > are differentially equivalent over >. <\proof> Let \\> be such that \>. Modulo some upward shiftings, we may assume without loss of generality that and are exponential. Modulo an additive conjugation by > and a multiplicative conjugation by >, we may also assume that is a normal cut. Modulo a division of by > and replacing by \>>, we may finally assume that \\>{F}>. Now consider \\>{F}>> with \\>. Since 0>, there exists a \f> with \>1> and ,\0>\>Q(\)>. But then <\equation*> Q()=Q(\)+Q,\0>(-\)\Q(\)\0. For general \{F}>, we use Ritt reduction of and conclude in a similar way as in the proof of lemma . <\lemma> Let \\\> and \\\\> be such that <\enumerate-roman> is a serial cut over >. and > are asymptotically equivalent over >. and > are differentially equivalent over >. Then \f\> carries the structure of a transserial Hardy field for the unique differential morphism :\\f\\\> over > with (f)=>. <\proof> Modulo upward shifting, an additive conjugation by > and a multiplicative conjugation by >, we may assume without loss of generality that is an exponential normal serial cut. Let \\> be such that \>. We have to show that \f\> is closed under truncation and that P()> for all \{F}> with 0> (this implies in particular that > is increasing). Notice that \> implies \f\\\=\\\>. Given \\F\>, let us prove by induction over the transrank of that >\\\f\>. So let ,\,\}> be a plane transbasis for and . Assume first that \\>. Writing <\equation*> R=\\>R>*\>\\;\;\>\F\>, the sum <\equation*> R>>=\0>R>*\> is finite, whence <\equation*> R(f)>>=R>>(f)=\0>R>(f)*\>\\\f\. By the induction hypothesis, we also have (f)>\\\f\> and >\\\f\>. If\\>, then <\equation*> R(f)>=R(\)> for a sufficiently large truncation \f>, whence >\\>. Given \{F}> with 0>, let us prove by induction over the transrank of that P()>. Let ,\,\}> be aplane transbasis for and and assume first that \\>. Since 0>, there exists amaximal> with >(f)\0>, when considering \\>P>*\>>> as a series in >. But then <\equation*> P(f)\P>(f)*\>\P>()*\>\P(), by the induction hypothesis. If \\>, then there exists an \\> such that, for all sufficiently large truncations \f>, the Taylor series expansion of +(f-\))> yields <\eqnarray*> ||)+O>((f-\)*\>)>>|)>||)+O>((-\)*\>).>>>> Taking \f> such that )*\>\>P(f)>, we obtain <\equation*> P(f)\P(\)\P(). This completes the proof. <\theorem> Let > be a transserial Hardy field. Then its real closure >>|real closure of >> admits aunique transserial Hardy field structure which extends the one of >. <\proof> Assume that \\> and choose \\\> of minimal complexity. By lemma, we may assume without loss of generality that is a serial cut. Consider the monic minimal polynomial \[F]> of . Since (f)\0>, we have <\equation*> degf-\> P>=1 for a sufficiently large truncation \f> of (we refer to 8.3> for a definition of the Newton degrees \> P>\> P>|Newton degree of modulo )>>). But then <\equation> P>(g)=0,g\f-\ admits unique solutions and > in > >, by the implicit function theorem. It follows in particular that +g>. Let =\+> and consider > with \\\f>. Then <\eqnarray*> )>|>|,1>*(f-\)>>|)-P(\)>|>|,1>*(-\)>>>> Since )=0>, we obtain \-\>, whence and > are asymptotically equivalent over >. By lemmas and, it follows that \f\> carries a transserial Hardy field structure which extends the one on >. Since () has a unique solution > in >, this structure is unique. We conclude by Zorn's lemma. <\theorem> Let > be a transserial Hardy field and let \\>> be such that >\\>. Then the set (\*\>)> carries the structure of a transserial Hardy field for the unique differential morphism :\(\*\>)\\> over > with (\*\>)=\*\(\)>> for all \\>. <\proof> Each element in (\*\>)> is of the form *\>,\,\*\>)> for \(F,\,F)> and >-linearly independent ,\,\\\>. Given \(F,\,F)>, let ,\,\}> be atransbasis for . We may write <\equation*> \>=\|~>>*\>*\*\> with \\|~>>\\> (or the obvious adaptations if or ). Modulo the substitution of > by *log \+\+\*log \+|~>>, we may assume without loss of generality that =\=\=0>. If \\>>, then we may regard \\>f>*\*\>> as a convergent grid-based series in>> with coefficients in \\;\;\>>. In particular, <\equation*> f>=*sign \\0>f>*\*\>+f>\\(\*\>). Furthermore, if admits > as its dominant exponent in >>, then f>*\*\>> holds both in > and in >. If >\\>, then we may consider as a series <\equation*> R\\\(\\\;\;\>)(F,\,F);\;\> in ,\,\>. Since > is closed under truncation, both >>> and >>> lie in >, whence <\equation*> f>=R>>(\*\>,\,\*\>)+R>>(\*\>,\,\*\>)>\\(\*\>), by what precedes. Similarly, if ,\,\>*\>*\*\>> is the dominant term of as a series in ,\,\> and *\>> is the dominant term of ,\,\>(\*\>,\,\*\>)> as a series in >> (with \\\;\;\>>), then c*\*\>*\>*\*\>> holds both in > and in >. This shows that (\*\>)> is truncation closed and that the extension of > to (\*\>)> is increasing. We also have (\*\>)\\=(\\\)*\*\>>. In other words, (\*\>)> is a transserial Hardy field. <\theorem> Let > be a transserial Hardy field of depth \>. Then ((log x)>)> carries the structure of a transserial Hardy field for the unique differential morphism :\((log x)>)\\> over > with ((log x)>)=(log x)>> for all \\>. <\proof> The proof is similar to the proof of theorem , when replacing >> by x>. Let > be a transserial Hardy field. Asymptotic and differential equivalence over [\]> are defined in a similar way as over >. <\proposition> Let > be a transserial Hardy field. Let \[\]> be a serial cut over [\]> and \\[\]>. Then and > are asymptotically equivalent over [\]> if and only if and > as well as and > are asymptotically equivalent over >. <\proof> Assume that and > are asymptotically equivalent over [\]> and let \Re f>. Consider =(Im f)Re f-\>\Im f>. We have +\*\\f>, so that -\*\\-\-\*\>. Moreover, -\*\\Re f-\>, whence \Re -\> and Re >. The relation Im > is proved similarly. Inversely, assume that and > as well as and > are asymptotically equivalent over >. Given \f>, we have ,Im \\\>, whence there exist \> with \g\Re -Re \> and \h\Im -Im \>. It follows that \g+h*\\-\>, whence >. <\proposition> Let > be a transserial Hardy field, \> and \\>. Then and > are differentially equivalent over [\]> if and only if they are differentially equivalent over >. <\proof> Differential equivalence over [\]> clearly implies differential equivalence over >. Assuming that and > are differentially equivalent over >, we also have <\equation*> P(f)=0\(Re P)(f)=0\(Im P)(f)=0\(Re P)()=0\(Im P)()=0\P()=0 for every \[\]{F}>. <\remark> Given \> and \\>, it can happen that and > are differentially equivalent over [\]>, without and > being differentially equivalent over >. This is for instance the case for =\(x>)>, > and =\*\>. Indeed, the differential ideals which annihilate > are both -F>. Most results from the previous sections generalize to the complex setting in a straightforward way. In particular, lemmas, and also hold over [\]>. However, the fundamental extension lemma admits no direct analogue: when taking \[\]\\[\]> and \\[\]\\[\]> such that the complexified conditions , and hold, we cannot necessarily give \Re f\> the structure of a transserial Hardy field. This explains why some results such as lemmas and have to be proved over > instead of [\]>. Of course, theorem does imply the following: <\theorem> Let > be a transserial Hardy field. Then there exists a unique algebraic transserial Hardy field extension > of > such that [\]> is algebraically closed. Recall that > stands for the differential algebra of infinitely differentiable germs of real functions at \>. Given \\>, we will denote by >> the differential subalgebra of infinitely differentiable functions on ,\)>. We define a norm on >>={f\\>:f\1}>f\<\|\|\>>>|norm of for x>> by <\equation*> \<\|\|\>f\<\|\|\>>=supx> \|f(x)\| Given \>, we also denote ;r>>={f\\>:f,\,f\1}>;r>>>|shorthand for \>:f,\,f\1}>> and define a norm on ;r>>> byf\<\|\|\>;r>>|norm of and its first derivatives for x>> <\equation*> \<\|\|\>f\<\|\|\>;r>=max {\<\|\|\>f\<\|\|\>>,\,\<\|\|\>f\<\|\|\>>}. Notice that <\equation*> \<\|\|\>f*g\<\|\|\>;r>\2*\<\|\|\>f\<\|\|\>;r>*\<\|\|\>g\<\|\|\>;r>. An operator >\\>> ( >\\;r>>) is said to be if there exists an \> with K f\<\|\|\>>\M*\<\|\|\>f\<\|\|\>>> ( K f\<\|\|\>;r>\M*\<\|\|\>f\<\|\|\>>>) for all \>>. The smallest such is called the of and denoted by K\>>K\>>|operator norm for >\\>>> ( K\;r>>K\;r>>|operator norm for >\\;r>>>). The above definitions generalize in an obvious way to the complexifications >>[\]> and ;r>>[\]>. Let > be a transserial Hardy field of span \\>. Consider a normal operator -\> with \\[\]> and let > be sufficiently large such that > does not change sign on ,\)>. We define a primitive \\> of > by <\equation*> \(x)=>\(t)*\ t>>|\\>>|>\(t)*\ t>>|>>>>> Decomposing =\+\*\>, we are either in one of the following two cases: <\enumerate> The repulsive case when >\>1>. The attractive case when both >\>1> and >\\>. Notice that the hypothesis \\> implies =Re \\\>\1>. <\proposition> The operator -\)1>>>, defined by <\equation> (J f)(x)=||(x)>*>\\(t)>*f(t)*\ t>>||>>|(x)>*>\\(t)>*f(t)*\ t>>||>>>>> is a continuous right-inverse of -\> on >[\]>, with <\equation> \J\>\>>. <\proof> In the repulsive case, the change of variables (t)=u> yields <\equation*> (J f)(x)=\(x)>*>(x)>\u-\(\(u))*\>*(u))|\(\(u))>*\ u*. It follows that <\equation*> \|(J f)(x)\|\\(x)>*>(x)>\u>*\<\|\|\>f\<\|\|\>>*>*\ u=\<\|\|\>f\<\|\|\>*> for all x>, whence (). In the attractive case, the change of variables \(t)=u> leads in a similar way to the bound <\eqnarray*> |>|(x)>*\(x)>\(x)>\*\<\|\|\>f\<\|\|\>>*>>*\ u>>|||1-\(x)-\(x)>*\<\|\|\>f\<\|\|\>>*>>>>||>|f\<\|\|\>>*>>,>>>> for all x>, using the monotonicity of >. Again, we have (). <\corollary> In the attractive case, the operator <\equation*> J>:f\(J f)(x)+\*\(x)>*\<\|\|\>f\<\|\|\>> is a continuous right-inverse of on >[\]>, for any \\>. Let > be a transserial Hardy field of span \\>. A monic operator \[\][\]> is said to be , if it is normal and if it admits a splitting <\equation> L=(\-\)*\*(\-\) with ,\,\\\[\]>. In that case, proposition implies that each -\> is a normal first order operator. For a sufficiently large >, it follows that admits a continuous ``factorwise'' right-inverse *\*J> on [\]>>, where =(\-\)1>>>. We have <\equation*> \J*\*J\>\\J\>*\*\J\>. <\proposition> >*J*\*J:\>>[\]\\;r>>[\]> is a continuous operator for every \r*\>. <\proof> Given \>[\]>, the the first derivatives of >*J*\*J) f> satisfy <\equation*> [(\>*J*\*J) f]=c*(\>*J*\*J) f, with <\eqnarray*> >||>|>||+\*\>*c+\*c+>>*c.>>>> By proposition and induction over , we have \>\i*\>> for all . Since \r*\>, itfollows that <\equation> \<\|\|\>[(\>*J*\*J) f]\<\|\|\>>\C*\<\|\|\>f\<\|\|\>>, for all \>[\]> and , where <\equation*> C=\<\|\|\>\>*c\<\|\|\>>*\J\>*\*\J\>. We conclude that <\equation*> \\>*J*\*J\;r>\max {C,\,C}. <\proposition> If \[\]> and the splitting )> preserves realness, then *\*J> preserves realness in the sense that it maps >>> into itself. <\proof> It clearly suffices to prove the proposition for an atomic real operator . If has order , then the result is clear. Otherwise, we have <\equation*> L=(\-(a-b*\+b>))*(\-(a+b*\)) for certain \>. In particular, we are in the same case (attractive or repulsive) for both factors of . Setting =a+b*\>, let =\+\*\> be as in the previous section. Consider \>>> and J f>. In the repulsive case, we have <\equation*> g(x)=b(x)*\|\>(x)>*>*\(t)>|b(t)>*>\\(u)>*f(u)*\ u*\ t. In particular, we have )=g(x)=0>, whence \>>>, since satisfies the differential equation of order with real coefficients. In the attractive case, we have <\equation*> g(x)=b(x)*\|\>(x)>*>*\(t)>|b(t)>*>\\(u)>*f(u)*\ u*\ t, so that \>1>. Since >=f>, the difference >-g> satisfies >-g)=0>. Now is the only solution with >1> to the equation . This proves that >=g>. Let > be a transserial Hardy field of span \\> and consider a monic split-normal quasi-linear equation <\equation> L f=P(f),f\1, where \[\][\]> has order and \[\]{F}> has degree . Of course, we understand that is a monic split-normal operator with >\>>. We will denote by \r*\> the valuation of in > ( >\>> for 0> and =\>). We will show how to construct asolution to () using the fixed-point technique. <\proposition> Given > with \\\v>, let \>>*\*J\>>> be a continuous factorwise right-inverse of \>>> beyond> and consider the operator <\equation> \:f\(J*\*J)(P(f)) on >;r>>. Then there exists a constant >> with <\equation> \<\|\|\>\(f+\)-\(f)\<\|\|\>;r>\C>*(1+\+\<\|\|\>f\<\|\|\>;r>)*(\<\|\|\>\\<\|\|\>;r>+\+\<\|\|\>\\<\|\|\>;r>), for all \\>;r>>. <\proof> Consider the Taylor series expansion <\eqnarray*> )>||>P)>(f)*\)>>>|||>>P)>>*f)>*\)>>>>> Since )>>\>\>> for all > and >, we may define >> by <\equation> A>=,\>\\>*P)>>> and obtain <\equation*> \\>*(P(f+\)-P(f))>\A>*(1+\+\<\|\|\>f\<\|\|\>;r>)*(\<\|\|\>\\<\|\|\>;r>+\+\<\|\|\>\\<\|\|\>;r>). On the other hand, for each \>> with \>>, we have <\equation*> \<\|\|\>(J*\*J)(g)\<\|\|\>;r>=\<\|\|\>(\>*J\>>*\*J\>>)(\\>*g)\<\|\|\>;r>\B>*\<\|\|\>\\>*g\<\|\|\>>, where <\equation> B>=\\>*J\>>*\*J\>>\;r> Consequently, the proposition holds for >=A>*B>>. <\theorem> Let )> be a monic split-normal equation and let > be such that \\\v>. Then for any sufficiently large >, there exists a continuous factorwise right-inverse \>>*\*J\>>> of \>>>, such that the operator )> satisfies <\equation> \<\|\|\>\(f+\)-\(f)\<\|\|\>;r>\*\<\|\|\>\\<\|\|\>;r> for all <\equation*> f,\\\\>;r>,>=f\\>;r>:\<\|\|\>f\<\|\|\>;r>\>. Moreover, taking > such thatP\<\|\|\>;r>\>, the sequence (0)> tends to a unique fixed point \(\>;r>,)> for the operator >. <\proof> Since \>*P)>>\1> for all , the number >> from () tends to for \\>. When constructing \>>,\,>\>>> using proposition, the number >> from() decreases as a function of >. Taking > sufficiently large so that >=A>*B>\>, we obtain(). By induction over, it follows that <\eqnarray*> \(0)-\(0)\<\|\|\>;r>>|>|>>>|\(0)\<\|\|\>;r>>|>|->.>>>> Now let |^>>;r>> be the space of times continuously differentiable functions on ,\)>, such that ,f> are bounded. This space is complete, whence (0)> converges to alimit \(|^>>;r>,)>. Since this limit satisfies the equation (), the function is actually infinitely differentiable, \(\>;r>,)>. With the notations from the previous section, assume now that [\]> is -differentially closed in [\]\>>, any solution \[\]\>> to an equation -\) f=g> with ,g\\[\]> is already in [\]>. Each > is the right-inverse of an operator -\> with \\[\]>. Now -\> also admits a formal distinguished right-inverse >. Consequently, the operator > also admits a formal counterpart <\equation*> |~>:f\(*\*)(P(f)). For each \>, we have <\equation*> |~>>(0)-|~>>(0)\>|~>>(0) so the sequence |~>>(0)> also admits a formal limit > in |^>[\]>. In order to show that the fixed point from proposition and > are asymptotically equivalent over [\]>, we need some further notations. Given \>[\]> and \\[\]>, let us write > if \\>>, \\>> for all \\>. We also write > if ,\,f\>. <\proposition> For \>[\]>, ,\\[\]> and \>, we have <\eqnarray*> \g\>|>|+>>|\g\>|>|*>>|>|>|\>>>> <\proof> Trivial. <\proposition> For \>[\]>, \\[\]> and \> with \>\>>, we have <\equation*> f\\J f\ . <\proof> Let us first show that <\equation> f\0\J f\0. Given \\> with \>>, we have \>> (\\>*f)\1>, whence f\\>>. Moreover, <\equation> (J f)=\1>*f+\*(J f), whence \>\(J f)\\+\>> for some fixed >. This proves(). More generally, additional applications of() yield <\equation*> f\0\J f\0. Now assume that > and write <\equation*> J f- =J (f-)+(J-) (). By what precedes, we have (f-)\0>. On the other hand, <\equation*> (J-)()=c*\\> for some \>. Since -\> is normal, we either have \>\\>> (in which case \>)\\>> for all \>) or . In both cases, we get -)()\0>, so that f\ >. <\theorem> Let > be a transserial Hardy field of span \\> such that [\]> is >-differentially closed in \>>. Consider a monic split-normal quasi-linear equation )> without solutions in >. Then there exist solutions \[\]> and \|^>[\]> to )>, such that and > are asymptotically equivalent over [\]>. <\proof> With the above notations, let and > be the limits in [\]> |^>[\]> of the sequences (0)> |~>>(0)>. Given \[\]>, there exists an with <\equation*> \(0)-\(0)\>g. At that point, we have <\equation*> f-g\\(0)-g\|~>>(0)-g\-g In other words, and > are asymptotically equivalent over [\]>. <\theorem> Let > be a transserial Hardy field of span \\>. Consider a monic split-normal quasi-linear equation )> without solutions in > such that and have coefficients in>. Assume that one of the following conditions holds: <\enumerate-alpha> > is -differentially closed in \>> and =r=1>. [\]> is -differentially closed in [\]\>>. Then there exist solutions \> and \|^>> to )>, such that and > are asymptotically equivalent over>. <\proof> In view of propositions and, we may assume that *\*J> and > preserve realness in all results from sections and. In particular, the solutions and > in the conclusion of theorem are both real. <\lemma> Let > be a transserial Hardy field of span \\>. Let -\\\[\]> be anormal operator. Let \|^>>> and \>> be such that > is transcendental over > and =g>. Then there exists an \>> with , such that and > are both differentially and asymptotically equivalent over >. <\proof> With the notations of section , let . Given a truncation \>, we claim that <\equation*> f-\\J(g-(\-\*\)). Indeed, consider <\equation*> \=\-J(\-\*\)\\*\>. In the attractive case, \>\>> implies =0>. In the repulsive case, we have >\>>1> and again \0>. By proposition, we also have <\equation*> -\=(g-\+\*\)\J(g-\+\*\). Since -\*\\g>, it follows that -\\f-\>, whence and > are asymptotically equivalent over >. Furthermore, is a minimal annihilator of > over >, since > is transcendental over >. Lemma therefore implies that and > are differentially equivalent over >. <\theorem> Let > be a transserial Hardy field. Let \\>>|first order differential closure of > in >> be the smallest differential subfield of >, such that for any \{F}>> with \1> and \> we have f\\>. Then the transserial Hardy field structure of > can be extended to >. <\proof> By theorems , and , we may assume that > is closed under the resolution of real algebraic equations, exponentiation and logarithm. Assume that \\> and let \{F}>> be of minimal complexity =(1,s,t)>, such that for some \>. Without loss of generality, we may make the following assumptions: <\itemize> and are exponential (modulo upward shifting). is a serial cut (by lemma ). is a normal cut (modulo additive and multiplicative conjugations by > >). \[\]\>{F}>, where \\\\> satisfies \> (modulo replacing by \>>). is monic split-normal (modulo proposition, additive and multiplicative conjugations, and division by >). By Zorn's lemma, it suffices to show that \f\> carries the structure of a transserial Hardy field, which extends the structure of >. If , then lemma implies the existence of an \\>> such that and > are both asymptotically and differentially equivalent over \>>. Hence, the result follows from lemmas and. If 1>, then > and \>> are -differentially closed in > \>>. Now \\>, since is exponential. Therefore, theorem provides us with an \\>> with )=0>, such that and > are asymptotically equivalent over \>>. We conclude by lemmas, and. <\lemma> Let > be a transserial Hardy field of span \\>. Let -\\\[\][\]> be anormal operator. Let \|^>[\]>> and \[\]>> be such that > has order over > and =g>. Then there exists an \>[\]> with , such that and > are both differentially and asymptotically equivalent over >. <\proof> The fact that and > are asymptotically equivalent over > is proved in a similar way as for lemma. It follows in particular that and > are asymptotically equivalent. Since >)> annihilates , >>, > and |\>>, it also annihilates both and >. The fact that > has complexity over > now guarantees that >)> is a minimal annihilator of >. We conclude by lemma. <\theorem> Let > be a transserial Hardy field. Let \\>>|differentially algebraic closure of > in >> be the smallest differential subfield of >, such that for any \{F}>> and \> we have f\\>. Then the transserial Hardy field structure of > can be extended to >. <\proof> By theorems , and , we may assume that > is closed under exponentiation, logarithm and the resolution of first order differential equations. Assume that \\> and let \[\]{F}>> be of minimal complexity =(r,s,t)>, such that for some \[\]> with \>. Let \{F}> be a minimal annihilator of and notice that \r>, since \>. Without loss of generality, we may make the following assumptions: <\itemize> , and are exponential (modulo upward shifting). is a serial cut (by the complexified version of lemma ). is a normal cut (modulo additive and multiplicative conjugations by > >). \[\]\>{F}> and \\>{F}>, where \\\\> satisfies \> (modulo the replacement of and by \>> \>>). is monic split-normal (modulo proposition, additive and multiplicative conjugations, and division by >). By Zorn's lemma, it now suffices to show that \Re f\> carries the structure of a transserial Hardy field, which extends the structure of >. If , then lemma and the fact that > is -differentially closed imply the existence of an \\>[\]> such that and > are both asymptotically and differentially equivalent over \>>. The result follows by lemmas and. If \(1,1,1)>, then [\]> and [\]\>> are -differentially closed in [\]> [\]\>>. Now \\>, since is exponential. Therefore, theorem provides us with a\>> with , such that and are asymptotically equivalent over \>>. We conclude by lemmas, and. <\corollary> There exists a transserial Hardy field >, such that for any \{F}> and \> with g> and 0>, there exists a \> with h\g> and . <\proof> Take =\(x>)> and endow it with a transserial Hardy field structure. Let \{F}> and \> with g> be such that 0>. By , there exists a \> with h\g> and . But implies \>. <\corollary> There exists a transserial Hardy field >, such that [\]> is weakly differentially closed. <\proof> Take =\>. By a straightforward adaptation of (see also ), it can be shown that any differential equation of degree with \[\]{F}> admits distinguished solutions in [\]> when counting with multiplicities. Let be such a solution. Since >(>)=0>, both and are differentially algebraic over >, whence \[\]>. <\corollary> There exists a differentially Henselian transserial Hardy field >, , such that any quasi-linear differential equation over > admits a solution in >. Let > be a differentially algebraic Hardy field extension of a transserial Hardy field >. <\proposition> Given \\>>, there exists an \> with \(log x)1>>. <\proof> The functional inverse 1>\|> of 1>\|> satisfies an algebraic differential equation 1>\|)=0> over >. Let \\>*f\\>> be the leading term of for its logarithmic decomposition. As in . there exists an \> with P\\>*f\\>> for all exp x> . It follows that 1>\|\exp x> and \(log x)1>>. Given a differential polynomial \{F}>>, we define its to be the unique monic \\{F}> such that *(D+E)> for some \\> and \\{F}>>. Here > is said to be monic if its leading coefficient )>,\,F> equals . <\theorem> Given \{F}>>, there exists a polynomial \\[F]*(F)>> with <\eqnarray*> >>||>>|>>||>(1)>>>> for all sufficiently large \>. <\proof> As in the proof of , we have <\equation*> wt D\wv D\wt D>\wv D>\\, so we may assume without loss of generality that >=wv D>=w> is constant for all\>. Now <\eqnarray*> >||>*(D>+E>)>>|||\*(D\+E\)>>|||\*(\w*x>*D>+E\),>>>> whence <\eqnarray*> >>||\*\w*x>>>|>>||\>>>|>>||\*\.>>>> Indeed, we must have <\equation*> E\*\=(Ew]>>\+Ew]>>\)*\\1, because w]>>\*\\1> would imply >\w>. Applying to(), and similarly for ,P\\,\>, we get <\equation*> D>=D\\[F]*(F) for all \>. By proposition and (), we have v]>\ x>1> and ,[\v]>\>1> for some \>. Modulo upward shiftings, we may thus assume without loss of generality that v]>\>1>. More generally, assume that v]>\>1> for some w>. By(), this implies ,[\v]>\>1> for all \> and <\eqnarray*> ,[\]>>||\]>+Ev]>\]>)*\>>|||*(E]>\+o>(1)),>>>> for all > of weight . We claim that there exists an \> with <\equation> E\[(log1> x)]. Assume the contrary and consider a coefficient ]>> of weight with <\equation*> \=]>|w-v>\(log1> x) for all \>. Without loss of generality, we may assume that> and \> are in >. Then proposition implies \\1> and even \\1> (by integrating from \> when possible). Again by proposition, it follows that \\log x> and \(log x)> for some \>. But then () yields <\equation*> E,[\]>=[(log x)]\*(E]>\+o>(1))\1, which contradicts the fact that >\1>. The relations () and () imply the existence of an \> with ,[v]>\>1>. By induction over ,0> and modulo upward shiftings, we may thus ensure that v]>\>1> for all w>. The polynomial > in theorem is called the of . The generalization of this concept to > allows us to mimic a lot of the theory from in >. In what follows, we will mainly need a few more definitions. The of an equation <\equation> P(f)=0,f\\ with \{F}> and \\>> is defined by \> P=deg N\>>>. Setting <\equation*> |^>= x*\> we also define <\equation*> deg|^>> P=min\|^>> deg\> P. We say that \> is a solution to() modulo),\\\\{|^>}> if \> P\0>. We say that > is , if every quasi-linear equation over > admits a solution. Given a solution to(), we say that has if f>>> is not homogeneous and in the other case. The following proposition is proved along the same lines as : <\proposition> Let be a solution to )> of differential type and let be the degree off>>>. Then >> is a solution modulo |^>)> of >>. <\remark> In this section, we assumed that > is a differentially algebraic Hardy field extension of a transserial Hardy field >. We expect that the theory can be adapted to even more general H-field. This is one of the objectives of a current collaboration with Lou van den Dries and Matthias Aschenbrenner . <\theorem> Let > be a transserial Hardy field and > a differentially algebraic Hardy field extension of >, such that > is differentially Henselian and stable under exponentiation. Then there exists a transserial Hardy field structure on > which extends the structure on >. <\proof> By theorems , and , we may assume that > is closed under the resolution of real algebraic equations, exponentiation and integration. Assume that \\> and choose \{F}> of minimal complexity =(r,s,t)>, such that either <\description> for some \>. modulo *|^>)> for some \>, \\\\> and admits no roots in > modulo *|^>)>. Moreover, > is >-differentially closed in >. Modulo upward shifting, we may assume without loss of generality that is exponential. In view of Zorn's lemma, it suffices to show that there exists a transserial Hardy field structure on\f\> which extends the structure on >. Let > be the set of \\> such that \supp >. The set > is totally ordered for>>, so there exists a minimal well-based transseries > with \> for all \\>. We call > the of over>. Assume first that \\>. Then we may assume without loss of generality that =0>, modulo an additive conjugation by >. Now is of differential type, since \> for no \\\\>. Let \> be such that >(f>)=0> modulo |^>)>. Since>> has lower complexity than , there exists a \> with >(g)=0> modulo |^>)>. Since > is truncation closed we may take \|^>*>>. But then \g>\\\\>. This contradiction proves that we cannot have \\>. Let us now consider the case when \\>. Since supp > P>\0>, there exists a root \> of in the set of well-based transseries with complex coefficients. But admits only grid-based solutions, whence \\>. By construction, and > are asymptotically equivalent over >. Let \\\\> be such that \\>. Modulo an additive and a multiplicative conjugation we may assume without loss of generality that > is a normal cut. In case , we notice that \\*|^>>, whence \>>1>, since =\>. Consequently, we always have \>()=0>. We claim that the cuts and > are differentially equivalent over >. Assume the contrary and let \\>{F}> be a minimal annihilator of >. By lemma and modulo an additive and multiplicative conjugation, we may assume without loss of generality that \>1> and that is normal. Since > is differentially Henselian, it follows that admits a root >1> in >. Now \\> in case and \\> in case , so this root is already in >, by the induction hypothesis. But admits at most one solution in \>>, whence =g\>\\>. This contradiction completes the proof of our claim. By lemma, we conclude that \f\> carries the structure of a transserial Hardy field extension of >. <\corollary> Let > be a transserial Hardy field and > a differentially algebraic Hardy field extension of >, such that > is differentially Henselian. Assume that > admits no non-trivial algebraically differential Hardy field extensions. Then > satisfies the differential intermediate value property. <\proof> The fact that > admits no non-trivial algebraically differential Hardy field extensions implies that > is stable under exponentiation. By theorem , we may give > the structure of a transserial Hardy field. By theorem, we also have =\>. We conclude in a similar way as in the proof of corollary. It is quite possible that there exist maximal Hardy fields whose differentially algebraic parts are not differentially Henselian, although we have not searched hard for such examples yet. The differentially algebraic part of the intersection of all maximal Hardy fields is definitely not differentially Henselian (and therefore does not satisfy the differential intermediate value property), due to the following result : <\theorem> Any solution of the equation <\equation*> f+f=\> is contained in a Hardy field. However, none of these solutions is contained in the intersection of all maximal Hardy fields. <\the-glossary|gly> g>| is dominated by |> g>| is negligible |> g>| is asymptotic to |> g>| is asymptotically similar to |> g>| is flatter than or as flat as |> g>| is flatter than |> g>| is as flat as |> g>| and are similar modulo flatness|> >g>|g> modulo elements flatter than >|> >g>|g> modulo elements flatter than >|> >>g>|g> modulo elements flatter than or as flat as >|> >>g>|g> modulo elements flatter than or as flat as >|> >>|shorthand for \:f\0}>|> >>|shorthand for \:f\0}>|> >>|shorthand for \:f\1}>|> >>|infinite part of |> \>>|part of which is flatter than >|> >>|shorthand for >:f\\}>|> \>>|shorthand for \>:f\\}>|> >|derivation with respect to |> >>>|integration with respect to |> >>|logarithmic derivative of |> >|upward shifting|> >|downward shifting|> g>| is a truncation of |> |canonical span of |> |ultimate canonical span of |> |^>>|completion of > with serial cuts|> {F}>|ring of differential polynomials in over >|> \F\>|quotient field of {F}>|> >|linear part of as an operator|> >|order of |> >|degree of in its leader|> >|total degree of |> >|complexity of |> >|initial of |> >|separant of |> >|the product *S>|> >|complexity of over >|> >|order of over >|> >>|additive conjugation of by >|> \>>|multiplicative conjugation of by >|> \>>|multiplicative conjugate of by >|> \>>|twist of by >|> >|set of dominant monomials of solutions to |> >|ring of infinitely differentiable germs at infinity|> >| is asymptotically similar to > over >|> >|real closure of >|> \> P>|Newton degree of modulo )>|> f\<\|\|\>>>|norm of for x>|> ;r>>>|shorthand for \>:f,\,f\1}>|> f\<\|\|\>;r>>|norm of and its first derivatives for x>|> K\>>|operator norm for >\\>>|> K\;r>>|operator norm for >\\;r>>|> >|first order differential closure of > in >|> >|differentially algebraic closure of > in >|> <\bibliography|bib|alpha|all> <\bib-list|AvdDvdH05> M.Aschenbrenner and L.vanden Dries. Liouville closed -fields. , 2001. to appear. M.Aschenbrenner and L.vanden Dries. -fields and their Liouville extensions. , 242:543--588, 2002. M.Aschenbrenner and L.vanden Dries. Asymptotic differential algebra. In . Amer. Math. Soc., Providence, RI, 2004. To appear. M.Aschenbrenner, L.vanden Dries, and J.vander Hoeven. Linear differential equations over H-fields. In preparation. Matthias Aschenbrenner, Lou vanden Dries, and J.vander Hoeven. Differentially algebraic gaps. , 11(2):247--280, 2005. M.Boshernitzan. An extension of Hardy's class l of 'orders of infinity'. , 39:235--255, 1981. M.Boshernitzan. New 'orders of infinity'. , 41:130--167, 1982. M.Boshernitzan. Hardy fields, existence of transexponential functions. , 30:258--280, 1986. M.Boshernitzan. Second-order differential equations over Hardy fields. , 35:109--120, 1987. N.Bourbaki. . Éléments de Mathématiques (Chap. 5. Hermann, 2-nd edition, 1961. B.L.J. Braaksma. Multisummability and Stokes multipliers of linear meromorphic differential equations. , 92:45--75, 1991. B.L.J. Braaksma. Multisummability of formal power series solutions to nonlinear meromorphic differential equations. , 42:517--540, 1992. J.Écalle. . Publ. Math. d'Orsay 1981 and 1985, 1985. J.Écalle. L'accélération des fonctions résurgentes (survol). Unpublished manuscript, 1987. J.Écalle. . Hermann, collection: Actualités mathématiques, 1992. J.Écalle. Six lectures on transseries, analysable functions and the constructive proof of Dulac's conjecture. In D.Schlomiuk, editor, , pages 75--184. Kluwer, 1993. G.H. Hardy. . Cambridge Univ. Press, 1910. G.H. Hardy. Properties of logarithmico-exponential functions. , 10(2):54--90, 1911. S.Kuhlmann. . Fields Institute Monographs. Am. Math. Soc., 2000. A.Macintyre, D.Marker, and L.vanden Dries. Logarithmic-exponential power series. , 56(2):417--434, 1997. A.Macintyre, D.Marker, and L.vanden Dries. Logarithmic exponential series. , 1999. To appear. M.Rosenlicht. Hardy fields. , 93:297--311, 1983. M.Rosenlicht. The rank of a Hardy field. , 280(2):659--671, 1983. M.Rosenlicht. Growth properties of functions in Hardy fields. , 299(1):261--272, 1987. M.C. Schmeling. . PhD thesis, Université Paris-VII, 2001. M.F. Singer. Asymptotic behavior of solutions of differential equations and Hardy fields: Preliminary report. , 1975. L.vanden Dries. , volume 248 of . Cambridge university press, 1998. L.vanden Dries, A.Macintyre, and D.Marker. The elementary theory of restricted analytic fields with exponentiation. , 140:183--205, 1994. J.vander Hoeven. . PhD thesis, École polytechnique, Palaiseau, France, 1997. J.vander Hoeven. Complex transseries solutions to algebraic differential equations. Technical Report 2001-34, Univ. d'Orsay, 2001. J.vander Hoeven. Integral transseries. Technical Report 2005-15, Université Paris-Sud, Orsay, France, 2005. J.vander Hoeven. , volume 1888 of . Springer-Verlag, 2006. <\initial> <\collection> <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> AvdD99 AvdD01 AvdD04 vdH:phd Schm01 MMV97 MMV99 Kuhl00 vdH:ln vdH:phd vdH:ln vdH:ln Ec85 Ec87 Ec92 Ec93 Br91 Br92 vdH:dagap vdH:hf Har10 Har11 Bour61 Sing75 Bos81 Bos87 Ros83 Ros83b Ros87 Bos82 Bos86 vdH:itr vdH:ln vdDMM94 vdD98 vdH:ln vdH:ln vdH:ln vdH:ln vdH:ln vdH:ln vdH:ln vdH:osc vdH:ln vdH:ln vdH:ln vdH:ln vdH:ln vdH:hf Bos87 <\associate|gly> |f\g>||f> is dominated by |g>|> |f\g>||f> is negligible |g>|> |f\g>||f> is asymptotic to |g>|> |f\g>||f> is asymptotically similar to |g>|> |f\g>||f> is flatter than or as flat as |g>|> |f\g>||f> is flatter than |g>|> |f\g>||f> is as flat as |g>|> |f\g>||f> and |g> are similar modulo flatness|> |f\>g>||f\g> modulo elements flatter than |\>|> |f\>g>||f\g> modulo elements flatter than |\>|> |f\>>g>||f\g> modulo elements flatter than or as flat as |\>|> |f\>>g>||f\g> modulo elements flatter than or as flat as |\>|> |\>>|shorthand for |{f\\:f\0}>|> |\>>|shorthand for |{f\\:f\0}>|> |\>>|shorthand for |{f\\:f\1}>|> |f>>|infinite part of |f>|> |f\>>|part of |f> which is flatter than |\>|> |\>>|shorthand for |{f>:f\\}>|> |\\>>|shorthand for |{f\>:f\\}>|> |\>|derivation with respect to |x>|> |||>>>|integration with respect to |x>|> |f>>|logarithmic derivative of |f>|> |\>|upward shifting|> |\>|downward shifting|> |f\g>||f> is a truncation of |g>|> |span f>|canonical span of |f>|> |uspan f>|ultimate canonical span of |f>|> ||^>>|completion of |\> with serial cuts|> |\{F}>|ring of differential polynomials in |F> over |\>|> |\\F\>|quotient field of |\{F}>|> |L>|linear part of |P> as an operator|> |r>|order of |P>|> |s>|degree of |P> in its leader|> |t>|total degree of |P>|> |\>|complexity of |P>|> |I>|initial of |P>|> |S>|separant of |P>|> |H>|the product |I*S>|> |\>|complexity of |f> over |\>|> |r>|order of |f> over |\>|> |P>>|additive conjugation of |P> by |\>|> |P\>>|multiplicative conjugation of |P> by |\>|> |L\>>|multiplicative conjugate of |L> by |\>|> |L\>>|twist of |L> by |\>|> |\>|set of dominant monomials of solutions to |L h=0>|> |\>|ring of infinitely differentiable germs at infinity|> |f\>||f> is asymptotically similar to |> over |\>|> |\>|real closure of |\>|> |deg\> P>|Newton degree of |P> modulo |O(\)>|> |\<\|\|\>f\<\|\|\>>>|norm of |f> for |x\x>|> |\;r>>>|shorthand for |{f\\>:f,\,f\1}>|> |\<\|\|\>f\<\|\|\>;r>>|norm of |f> and its first |r> derivatives for |x\x>|> |\K\>>|operator norm for |K:\>\\>>|> |\K\;r>>|operator norm for |K:\>\\;r>>|> |\>|first order differential closure of |\> in |\>|> |\>|differentially algebraic closure of |\> in |\>|> <\associate|toc> |math-font-series||font-shape||1.Introduction> |.>>>>|> |math-font-series||font-shape||2.Preliminaries> |.>>>>|> |2.1.Notations |.>>>>|> > |2.2.Differential fields of transseries and cuts |.>>>>|> > |2.3.Complements on differential algebra |.>>>>|> > |2.4.Linear differential operators and factorization |.>>>>|> > |2.5.Factorization at cuts |.>>>>|> > |2.6.Normalization of linear operators |.>>>>|> > |2.7.Normalization of quasi-linear equations |.>>>>|> > |math-font-series||font-shape||3.Transserial Hardy fields> |.>>>>|> |3.1.Transserial Hardy fields |.>>>>|> > |3.2.Cuts in transserial Hardy fields |.>>>>|> > |3.3.Elementary extensions |.>>>>|> > |3.4.Exponential and logarithmic extensions |.>>>>|> > |3.5.Complex transserial Hardy fields |.>>>>|> > |math-font-series||font-shape||4.Analytic resolution of differential equations> |.>>>>|> |4.1.Continuous right-inverses of first order operators |.>>>>|> > |4.2.Continuous right-inverses of higher order operators |.>>>>|> > |4.3.The fixed point theorem |.>>>>|> > |4.4.Asymptotic analysis |.>>>>|> > |math-font-series||font-shape||5.Differentially algebraic Hardy fields> |.>>>>|> |5.1.First order extensions |.>>>>|> > |5.2.Higher order extensions |.>>>>|> > |5.3.Differential Newton polynomials for Hardy fields |.>>>>|> > |5.4.Transserial models of differentially algebraic Hardy fields |.>>>>|> > |math-font-series||font-shape||Glossary> |.>>>>|> |math-font-series||font-shape||Bibliography> |.>>>>|>