> <\body> |~>>|~>>|~>|~>>|~>>>algebraic differential equations>Université Paris-Sud91405 Orsay CEDEXFrance>||> <\abstract> In our PhD. we have given an algorithm for the algebraic resolution of algebraic differential equations with real transseries coefficients. Unfortunately, not all equations do admit solutions in this strongly monotonic setting, even though we recently proved an intermediate value theorem. In this paper we show that the algorithm from our PhD. generalizes to the setting of weakly oscillatory or complex transseries. Modulo a finite number of case separations, we show how to determine the solutions of an arbitrary algebraic differential equation over the complex transseries. We will show that such equations always admit complex transseries solutions. However, the field of complex transseries is not differentially algebraically closed. In , we have studied the asymptotic behaviour of solutions to algebraic differential equations in the setting of strongly monotonic or real transseries. We have given a theoretical algorithm to find all such solutions, which is actually effective for suitable subclasses of transseries. More recently, we have proved the following ``differential intermediate value theorem''. > Let be the real field of grid-based transseries in and let be a differential polynomial with coefficients in . Then, given transseries g\> with 0> and 0>, there exists a > with h\g> and .> This theorem implies in particular that any algebraic differential equation of odd degree, such as +e>*f*f+\(log x+1)*f+log (e+\(\(x))=0,> has at least one real transseries solution. This theorem is striking in the sense that it suggests the existence of theories of ordered and/or valuated differential algebra.\ However, a main drawback of the setting of real transseries, is that not every algebraic differential equation can be solved; actually, even an equation like +1=0> has no solutions. In order to get a better understanding of the asymptotic behaviour of solutions to algebraic differential equations, it is therefore necessary to search for a complex analogue of the theory of real transseries. This paper is a first contribution in this direction. The first problem is to actually define complex transseries. The difficulty is that it is not clear whether an expression like >> should be seen as an infinitely large or an infinitely small transmonomial. Several approaches can be followed. A first approach, based on pointwise algebras, was already described in chapter 6 of . However, this approach has the drawback that it is not easy to compute with complex transseries. A second more computational approach is described in section . Roughly speaking, it is based on the observation that all computations with complex transseries can be done in a similar way as in the real setting, except for testing whether a monomial like >> is infinitely large or small. Now whenever we have to make such a choice, we will actually consider both cases, by applying the automatic case separation strategy (see ). We implicitly reject the case when >> is bounded, which is ``degenerate'', but which deserves to be studied later. The last approach, which is described in section , is more structural and really allows us to define a complex transseries in a not too difficult way. The underlying idea is analogue to the concept of a maximal ideal. Intuitively speaking, we assume the existence of some ``god'', who has decided for us which monomials like >> are infinitely large and which ones are infinitely small. It turns out that all possible choices lead to isomorphic fields of transseries. However, the geometric significance of these fields is hard to grasp. In section , we introduce parameterized complex transseries, which are necessary to express generic solutions to differential equations. Indeed, such solutions may involve integration constants. As usual, our approach is based on the automatic case separation strategy. The remaining sections deal with the resolution of asymptotic algebraic differential equations with complex transseries coefficients. Our approach is similar to the one followed in , but we have made a few simplifications and we corrected an error (see section ). Our main results are stated in sections and . We show that there exists a theoretical algorithm to express the generic solution to an algebraic differential equation by means of parameterized complex transseries and we give a bound for the logarithmic depth of the generic solution. We also show that an algebraic differential equation of degree admits at least complex transseries solutions when counting with multiplicities. As a consequence, each linear differential equation admits a full system of solutions. However, our fields of complex transseries are not differentially algebraically closed and several interesting problems still need to be solved (see section ). The reader should be aware of a few changes in notations w.r.t.> , which are summarized in the following table: >|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>>>>>> In all what follows, let be a and its complexification. This means that has the structure of a totally ordered field and functions R>, which are compatible with this ordering. More precisely, we assume that admits an inverse function with domain >> and that the function restricted to /2,\/2[> admits a totally defined inverse. Here , where /2)> and =4*arctan 1>. Furthermore, ||>|||>>>> for all R>. Finally, for each > resp.>>> and R>, we require that |>|>*x+\+>*x;>>||>|>*x+\>*x->*x.>>>>> form a real trigonometric field.> <\proof> The functional equations are classical. The inequality for was first proved in . As to the inequality for , we have )|(2*k)!>*x=n>|(4*k)!>*1-|(4*k+1)*(4*k+2)>\0,> if 4*n>. Otherwise, )|(2*k)!>*x=1-\ |2>+|(4*k)!>*1-|(4*k+1)*(4*k+2)>\-1\c\ os x,> since 1>. and for expansions at order instead of .> and may naturally be extended to , numbers in may naturally be written in polar form, etc.> and are no longer required to be total and the functional equations resp. inequalities are only required to hold, whenever they make sense. For instance, if dom exp>, then we require that dom exp> and .> Let > be a totally ordered monomial group (or set) with -powers. Then we recall that the field >> of grid-based power series is naturally totally ordered by 0\c\0>, for all 0>. This ordering is compatible with the multiplication: 0\g\0\<\ Rightarrow\>f*g\0>. More generally, if > is only partially ordered, then we define an ordering on >> to be with the asymptotic ordering on >, if g\g\0\f\0> for all R>>. In what follows, we are rather interested in the complexification >> of >>. Obviously, this -algebra can not be given an ordering which is compatible with the multiplication. Nevertheless, it is interesting to consider orderings on >> which are only compatible with the -algebra structure of >>. Such an ordering is again said to be compatible with the asymptotic ordering on >, if () holds for all R>>\ . Assuming that such orderings on > and >> are total, the condition () implies that =\\sign f=sign g> for all non zero C>>. Consequently, the ordering on >> is totally determined by the sets >={c\C\|c*\0},> where > runs over >. Each >> is actually the set of strictly positive elements of a total ordering on , which is compatible with the -module structure of . Therefore, each >> is characterized by an angle >\[\ >,\)> and a direction >\{,1}>, via >={c\C\|(Re (c*e>>)\0)\(Re (c*e>>)=0\Im(\>*c*e>>)\0)}.> This situation is illustrated in figure . |Shape of the region >> for >=1> resp. >=>.> In is also possible to consider complex powers of monomials: a is a monomial group > with -powers, with an asymptotic ordering > which is compatible with the expo-linear -vector space structure of >. For instance the formal group >*z>> is a monomial group with -powers for the ordering >*z>\1\(Re \\0)\(Re \=0\Re \\0)>. This group is not totally ordered, since > and are incomparable. We may make the ordering total by deciding that >*i>\(log z)>*i>\1>. Now consider a totally ordered grid-based algebra of the form =C>>, where > is a totally ordered complex monomial group, and where the ordering on is assumed to be compatible with the asymptotic ordering on >. Assume that we also have a partial logarithmic function on , such that <\description> coincides with the usual logarithm on >>; If dom log>, then dom log> and . We say that is a if the following conditions are satisfied: <\description> >\|c\R>}>; \>>, for all \>; For all \>>, we have )=l\\>, where >)|k>\C[[z]]>, as well as the following conditions for the logarithm: <\description> >=\*log > for all \> and \C>; \\log \log > for all ,\>; \> for all \\\{1}>. In , we write g>, if and only if \log > for all >>. In view of , this means that g\f>\g>> for all ,\\C>>. until replace the requirement that should be a logarithmic function in the definition of real fields of transseries. It can indeed be checked that our conditions are to the usual conditions on in this case.> <\remark> The domain of the logarithm on may be further extended, by setting +log (f/c)> for all >>, where > is defined using the function on . Of course, such an extension of the logarithm to involves a choice of a principal determination. Furthermore, such an extension cannot satisfy both the properties and . On the other hand, the partial inverse of may be extended canonically in such a way that the equation admits a solution for each >>. Indeed, it suffices to extend via exp(f)*exp(c)> for all dom exp> and C>. In what follows, we will always assume that the partial inverse of has been extended in this way. > Consider the formal -vector space =C*log z\C*log log z\\> generated by the formal symbols >. Given angles ,\,\\[-\,\)> and directions ,\,\\{,1}>, we define a total ordering on > as explained in section . Then the formal exponential =z*(log z)*\> of > is a complex monomial group for the asymptotic ordering > defined by \\log \log >, where >*(log z)>*\)=\*log z+\*log log z+\>. In order to avoid confusion, we will sometimes write >,>>> instead of >. Assume from now on that > and > were chosen such that >*log z\R>*log log z\\\0>. Given a non-zero grid-based series =C>> with \R>>, we define its logarithm by +log +l\ (1+\).> We may extend the total ordering on \> to in a similar way as in section , by extending the angle and direction families > resp. > into larger families |^>> resp. |^>>. It is easily verified that the field |^>>,|^>>>\> with this ordering is a pre-field of complex transseries. Actually, the structure of |^>>,|^>>>> does not really depend on the choices of >, >, |^>> and |^>>, modulo rotations and conjugations. Indeed, assume that > and > are a second family of angles and directions with indices in }>. Then we define an increasing isomorphism > between >,>>> and >,>>> by :\{log z, log log z,\}>f>*\\{log z, log log z,\}>*e>>\ *\>*\>><\ left|(>f>*e>>*,> where (z)>||\ >|(z)>||>\ ,>>>>> for all C>. We infer that |\>:>,>>\>,>>; exp f\exp \(f)> is an isomorphism of complex monomial groups. Now if |^>> and |^>> are families of angles resp. directions with indices in >, and which extend > and >, then we define an increasing isomorphism |^>> between |^>>,|^>>>> and |^>>,|^>>>> by |^>:\>,>>>f>*\\>,>>>*e>>*\>*\>>f>*e>>*|\>().> We notice that |^>> extends > if and only if |\>=Id>, which is again equivalent to the condition that for each \{log z,log log z,\}> we have >=\>\ \ \>*\>=\>*\>.> In this case, we say that >,>)> and >,>)> are . We say that >,>)> and >,>)> are if the relation holds for all =log z> with sufficiently large >. > Assume now that we are given a complex field of transseries =C>>, which is not stable under exponentiation (modulo the extension of the exponentiation as described in remark ). Let > and > be the associated families of angles and directions. Now consider the formal complex monomial group =exp >,> whose asymptotic ordering is given by exp g\f\g>, for all >>. Given extensions >> and >> of > and > to families indexed by monomials in >, we may totally order =C\ >> as explained in section . It is easily verified that > is a pre-field of complex transseries, which we call the of , relative to > and >. In cases of confusion, we will write >,>>> instead of . Notice that the exponential of any series in is defined in >. Again, the structure of >,>>> does not really depend on the choice of >,>)>. Indeed, if |^>>,|^>>)> and |~>>,|~>>)> are two different such choices, then :\>f>*\\>*e>>*\>*\>>f>*e>>*> is an increasing isomorphism between |^>>,|^>>>> and |~>>,|~>>>>. Starting with from the previous section, we may now consider the iterated exponential extensions =C>=>, =C>=>, =C>=> of . The union =C=\\\\=C\\\\>> of these fields is called a field of complex transseries in . Of course, the construction of depends on the successive choices of angles >> and directions >> for >, with indices in >. The angles > and directions > for coincide with these choices on each >. We will write >,>>> instead of whenever confusion may arise. We claim that >,>>> and >,>>> are isomorphic as soon as the restrictions of >,>)> and >,>)> to }> are compatible. We have already shown (see formulas () and ()) that there exist isomorphisms :>,>>\>,>>> for each . Now let > be such that z>=\ z>> for each l>. Then we observe that (log z)=\(log z)> for all > and i\l-l>. By induction over l>, it then follows that for ()=\()> for all \>> and i>. Given \>, this shows that the value of ()> does not depend on the choice of , for sufficiently large . In other words, the > can be glued together into an isomorphism between >,>>> and >,>>>. It is possible to slightly generalize the construction of pre-fields of complex transseries in , when starting with =C*log z\C*log z\\> instead of =C*log z\log z\\>. Notice that is not necessarily a monomial when adopting this generalization.> Actually, in our construction of pre-fields of complex transseries in , it is reasonable to require that require that x>=0> for all sufficiently large >, thereby eliminating all ambiguity (up to isomorphism) in the construction of . More generally, a pre-field of complex transseries is a , if it satisfies the following axiom: <\description> For each \>, there exists an \>, such that for all i> we have <\itemize> (log )=log (log )>. (log )>=0>. Then up to isomorphism, we have constructed field of grid-based complex transseries in . Actually, the same procedure of exponential extensions and direct limits can be used to close any field of complex transseries under exponentiation. Again, this closure is unique up to isomorphism. <\remark> In this paper we restrict our attention to grid-based complex transseries. Nevertheless, the results of this sections can easily be generalized to the case of Noetherian complex transseries. In this case, we recommend to replace the axiom by the following more complicated but better axiom: <\description> Let )\ >> be a sequence of monomials in >, such that \supp log >. Then there exists an \>, such that for all i\ > we have <\itemize> \> for all \log supp >. >=0>. This axiom allows the resolution of certain functional equations like +f(log z)>,> which admits natural solutions of the form +e+e>>>,> which are called . > Consider a tuple =(,\,<\ value|mb>)> of non zero complex transseries in with \\\>. We call > a if the following conditions are satisfied: <\description> =log z> for some >, which is called the of >. \C;\;>> for each 1>. \\\> (i.e. >\\\\ >>). Such a transbasis generates a >. We say that C> can be expanded w.r.t. > if C>>. If , then we say that > (and any C>>) is . The following is proved in a similar way as in the case of real transseries: > be a transbasis and C> a complex transseries. Then can be expanded w.r.t. a super-transbasis |^>\> of >.> We define a strong derivation w.r.t. on =C> in the usual way: we take >*\*(log z)>)=|z>+\<\ cdots\>+|z*log z*\*log z>z>*\*(log z)>> for all monomials >*\*(log<\ rsub|l> z)>\>. This yields a derivation on > through extension by strong linearity. Given a derivation on >, we define =f*exp f,> for all monomials exp ()>=>. This again yields a derivation on > through extension by strong linearity. By induction over , we thus obtain a derivation on . We recall that a derivation on is said to be resp. if the following conditions are satisfied: <\itemize> g\f\g>, for all > with 1>; 1\(f\0\f\0)>, for all >. Contrary to the case of real transseries, our derivation on cannot be strictly positive. Indeed, either \1> or \1>, say \1>. Then we have )=-e>, so either )\sign e> or )\sign (e)>. On the other hand, the following may be proved in the usual way: > is strictly valuated.> Actually, the proof involves upward shiftings of transseries: given C>, its upward (resp. downward) shifting is defined by =f\exp> (resp. =f\log>). Contrary to the case of real transseries, this transseries does not necessarily live in the same field of transseries as : if C>,>>>, then we have \C>\,>\>>, where \\>=\<\ rsub|>> and \\>=\>> for all transmonomials \>. In the case of downward shiftings, one may have to consider the generalized fields of complex transseries in from remark . It is more difficult to extend functional composition from the real to the complex setting due to possible incompatibility between the angles and directions. For instance, if =0>, then the transseries +e+\> can not be composed on the right with . In general, right composition with a given transseries is only defined on a certain subfield of >. Contrary to the case of real transseries, certain functional equations like +f(\*z),> with \C> seem to fall outside the scope of the theory of complex transseries, unless someone comes up with some really new ideas to incorporate the solutions to such equations inside this theory. One of the main ideas behind the construction of fields of complex transseries is that we do not longer require the ordering on the constant field to be compatible with the multiplication. Indeed, we just need the compatibility with the addition (or multiplication with reals), in order to obtain ordered monomial groups via exponentiation. The above idea may be used to generalize the results from this section to other circumstances. Consider for instance the set > of -adic complex numbers, where 2>. Then it is classical that there exists a partial logarithm on , which is defined for all C> with z=0>. By Zorn's lemma, there exists a total ordering on the -vector space . The theory of this section may now be adapted in order to construct the field > of complex -adic transseries. A first change concerns the condition , which should now become >\|val c=0}={f\>\|c\dom log}.> Furthermore, it is not as easy as before to characterize the total orderings on , which are compatible with the -vector space structure. Consequently, there is no natural analogue to the condition and we have to satisfy ourselves with the construction of pre-fields of complex transseries. Also, the exponentiation on > is not total. Notice that it seems to be possible to take itself for the indeterminate in the construction of >. This would yield a field of transseries which contains > and such that the logarithm is defined for all non zero elements. In practical computations with complex transseries the angles > and directions > are not known in advance and we have to choose them (or more precisely, to put constraints on them) as the computation progresses. This can be done by introducing a closed interval >\R/(2*\*)> for each transmonomial >, which corresponds to the constraint \\\>:\\ |\-\>\|\\/2> on >>. Given such sets >>, we will work with which are in the ``intersection'' of all >,>>> such that > and > satisfy the above constraints. Actually, it is convenient to always work w.r.t. generic complex transbases, which we will introduce now. Let =(,\,)> be an -tuple of symbols. Assume that each > comes with closed interval \/2*\*i> modulo >, such that \(\+\)=\>. Then we may order the monomial group =*\*> by >*\*>\1\arg \\\,> for each non zero monomial >*\*>> with \0>. We call > a of the scale >. Such a basis is called a , if <\description> =log z> for some >, which is called the of >, and \>. > is a regular, infinitely large transseries in ;\\ ;>> for each 1>. >\\\>>. An important question is whether the asymptotic constraints on the > determine a non empty region of the complex transplane (see chapter 6 of ). This question will be addressed in a forthcoming paper. ,e/(1-z)>)> is a transbasis, for the constraints z\e\e/(1-z)>>. Computations with respect to this transbasis are valid in regions of , where /(1-z)>\|\\|e\|\\ \|z\|\1>. This is for instance the case for , such that +\> in a region where +\\>\k+-\> for some small \(0,)> and >.> A is an element of ;\;>> for some complex transbasis ,\,)>. It can be shown that two transbases which have a non empty region of definition in common can be merged together. In the remainder of the paper we will follow an easier approach, which consists of working with respect to a , which may be enlarged and on which we may impose additional asymptotic constraints during computations with complex transseries. By construction, all ring operations can already be carried out in an algebra of the form ;\;>>. In order to invert a complex transseries, we first have to be able to compute its dominant monomial. In principle, both or > might be ``the'' dominant monomial of a transseries like >. Nevertheless, given a transseries C;\;>> with dominant monomials ,\,>, then we may always separate cases \\\<\ value|md>\\\\>>|\\\\\\\>>|>>|\\\\\\\\ >>>>>,> in each of which has only one dominant monomial. This case separation technique is explained in detail in . In the present context, the imposition of a constraint >*\*>\1> with \0> reduces to the insertion of > in the interval >>. If the length of the new interval exceeds \>, then () can not be satisfied, so that the corresponding case does not need to be considered. <\remark> In order to be really complete, we should also consider the cases when several dominant monomials are asymptotic. For instance, in the case of the series >, we should consider the cases e> and e>, but also e>. However, in the present paper, we argue that the situation when e> is ``degenerate'' in the sense that it corresponds to a single ``direction'' 0[\]> among a continuous number if possibilities. As a consequence, we notice that the process of ``regularization'' of a complex transseries is much easier than in the case of multivariate transseries studied in . Indeed, in the case when one has to consider the possibility that e>, one also has to consider the possibility of cancellation =0> or \\ 1>. This would necessitate refinements of the coordinates and rewriting of the series in ;\;>>. <\example> Modulo cases separations, we may thus carry out all field operations. For instance, the inverse of > is either given by >=1-e+e+\,e\1,> or >=e-e+e+\,e\1,> Consider a non zero complex transseries C;\;\ >>. Modulo case separations, we may assume that is regular, so that we can write >*\*\ >*(1+\),> with ,\,\\C\ > and \1>. Consequently, *log +\+\*log +c+log(1+\).> If =0>, then this series is already in ;\;>>. Otherwise, it still is, modulo the insertion of a new element =log =log z> in front of the transbasis, subject to the constraint >. Since > is a new symbol, this constraint is non contradictory with the existing expo-linear constraints on the >. The relation \>> is automatically verified, since >\z>. Consider a complex transseries C;\;>>. Modulo case separations, we may assume that is regular. In order to compute the exponential of , we distinguish three cases: is bounded. We may write >, with C> and \1>. Hence, =e*e>>, with C> and >\C\ ;\;>>. \\log > for some i\n> (where we understand that the left resp. right hand side relation is verified if resp. ). We decompose +f>, where =[*\*] f\C;\;>> and \1>. Inserting >> into > by \(,\,,e>,,\,)>, we then have =e>*e>\C;\;;e>;;\;>>. \log > for some . We may write *log +g>, with \C>> and f>. Then =>*e> and we compute > using the same algorithm. The computation of > cannot give rise to infinite loops, since the transbasis > would remain invariant in such a loop, while the index would strictly decrease. Consider the complex ``exp-log function'' e+i*z>+e>> and let us show how to expand it generically with respect to a generic complex transbasis. We start with \(z)> and recursively expand all subexpressions of . >.>In order to expand >, we fall into the second case of the exponentiation algorithm, since z> and 1>. Consequently, we insert > into > using \(z,e>), so that > expands as >. and +i*z>.>Since >> is a ring, we immediately have +i*z\C<\ gb|z;e>>. Since the expansions of sums and products do not present any problems, we will omit them in what follows. +i*z>>.>In order to expand +i*z>>, we first have to determine the dominant of +i*z>. Two cases need to be distinguished for this, namely >\ ={0}>, which corresponds to \1>, and >={\}>, which corresponds to \1>. In the first case, \i*z\log e>, so that +i*z>> needs to be inserted into >. In the second case, \i*z>, so we rewrite +i*z>=e*e>=e+e+*e+\\C>>. >>.>In the case when >={0}>, we have \log e+i*z>>, so we rewrite >=(e+i*z>)\ *e\C;e+i*z>>>. In the other case, when >={\}>, the argument > is bounded, so that >=1+i*e-*e+\<\ cdots\>\C>>. .>We first have to determine the dominant monomial of +i*z>+e<\ rsup|i*e>>. If >={0}>, then we separate the cases +i*z>>\ ={-|4>}> in which +i*z>\\ e>>, and +i\ *z>>={|4>}> in which +i*z>\e>>. In the first case, we obtain ||+i*\ z+log (1+e-i*z>)>>|||+i*z+e-i*z>+>*e-2*i*z>+\\C;e+i*z>>.>>>>> In the second case, we get ||+\ log (1+e+i*z>)>>|||+e+i*z>+>*e+2*i*z>+\\C;e+i*z>>.>>>>> If >={\}>, then +i*z>+e>=e+1+e+i*e+\>, so we separate the cases >=[|2>,\]> in which \1>, and >=[\,|2>]> in which \1>. If >=[|2>,\]>, then ||>-1)+e*e>)>>||<\ cell|=>|+e+(i-1)*e->*e+\\C>.>\ >>>> Otherwise, we obtain ||>-1)+e*e>)>>|||-e+(1+i)*e->*e+\\C>.>>>>\ > |A plot of the function e+i*z>+e>>, which illustrates the four possible asymptotic behaviours of on non degenerate regions. The ``rows'' of singularities correspond to the borders between regions of different types.> In order to deal with integration constants when solving differential equations, we need to consider parameterized transseries. As in the case of generic transseries, if will often be necessary to distinguish several cases as a function of the values of the parameters. Again, this can be done by putting constraints on the parameters. Let =(\,\,\>)> be a >-tuple of complex parameters. We call a subset > of \>> a , if > is the set of solutions of a system of polynomial equations or inequations )>||>|)>|>|>>>> where C[\,\,\>]>, and ``rational function inequalities on the real parts'' (\)|c(\)>\0,> where ,c\C(\,\,\>)> and > does not vanish on >. Notice that > may be seen as a special kind of semi-algebraic set, under the isomorphism \>\R>>. The polynomial algebra ,\,\>]> will also be called the or . Given a non empty region \C\>>, let =(,\,)> be an -tuple of symbols. Assume that each > comes with a finite set =\>={\\ ,\,\>}\P> of , such that > does not vanish on > for all i\r>, j\d>, and such that \\Cp>Re (\)|\>(\)>\0\\,> for all i\r> and j,j\d> with \j>. In the case when =0>, the directions > correspond to the extremal angles in the intervals > from the previous section. For each i\r>, there exists a natural partial ordering > on the -vector space , which is generated by the relations \0> for all . Indeed, the constraints () in an arbitrary point \\> guarantee the absence of relations *\+\+\\ >*\>=0,> with ,\,\>\ )\(R)>\\(0,\,0)>. Consequently, we may define a natural neglection relation > on the asymptotic scale =*\*> by >*\*>\1\\\ 0,> for each non zero monomial >*\*>> with \0>. We say that > is a , if <\description> =log z> for some >, which is called the of >, and \>. > is a regular, infinitely large transseries in ;\\ ;>> for each 1>. >\\\>>. A parameterized transseries is an element of ;\;>> for some transbasis ,\,)>. A regular parameterized transseries P>\ > is said to be , if either , or >(\)\0> for all \\>. In this section we prove that any parameterized transseries P>> can be uniformly regularized modulo case separations. We notice that a uniformly regular parameterized transseries on a region > remains uniformly regular on any subregion of >. Let \*\*> be a monomial. Then, modulo case separations, we may assume that either \1>, =1> or \1>.> <\proof> Write =>*\*>>, with ,\,\\P> and separate the following cases : <\itemize> For some i\n>, we have \0,\=0,\,\=0>; For some i\n>, we have \0,\=0,\,\=0>; =\=\=0>. In the cases , we have \1>. In the cases we have \1>. In case , we have =1>. Notice that the imposition of the constraints of the form =0> or \0> may involve a reduction of the region > and/or the insertion of new directions into >. Indeed, =0> is an additional algebraic constraint on >. In order to impose \0>, we first impose the constraints \0> and |\>\0> on >, for all j\d>. Next we insert > into >. Let ,\,> be infinitesimal monomials in an arbitrary monomial group > with -powers, such that =>*\*>>, for certain ,\,\\>. Then there exist infinitesimal monomials ,\,\>, such that \{,\,}>> for all i\k+1>.> <\proof> Since \1>, we may assume without loss of generality that \0\ >, modulo a permutation of indices. We will prove the lemma by induction over . For the lemma is trivial. So assume that 1> and let =>*\*>>. Then we have either \1>, =1> or \1>. If \1>, then there exist ,\,\>, such that ,\,,\{,\,}>>, by the induction hypothesis. Consequently, ,\,\{,\,,}>>. If =1>, then =>>, whence ,\,\{<\ value|mm>,\,}>>. If \1>, then there exist ,\,\>, such that >,\,>,\ >\{,\,}>>, by the induction hypothesis. Hence ,\,\{<\ value|mv>,\,,*>}>>. The lemma follows by induction. Any P>> can be uniformly regularized modulo case separations.> <\proof> Let \1,\,\1,,\,\*\*> be such that {,\,}>*{,\,}>. By lemma , we may assume without loss of generality that either \1>, =1> or \1> for each , modulo some case separations. Without loss of generality, we may therefore assume that admits a Cartesian representation in ,\,\ >, i.e. f\{,\,}>*>*\*>> for certain ,\,\\>. Choosing minimal, we will prove the theorem by induction over . If , then , and we have nothing to prove. So assume that 0>. We will first show how to regularize modulo case separations. So let ,\,> be the set of dominant monomials of . By repeated application of lemma , and modulo reordering, we may assume that \\\>. If all these inequalities are strict, then we are done, since > will be the only dominant monomial. Otherwise, we have => for certain j>, which yields a non trivial relation >*\*>=1> for certain ,\,\\>. Then lemma implies that we may find a Cartesian representation for in variables only, and we are done again, by the induction hypothesis. In order to make uniformly regular modulo case separations, we use the following algorithm: <\itemize> Regularize modulo case separations and let > be its dominant monomial (if 0>). If , or >(\)\0> for all \\>, then we are done. Separate the cases when >=0> and >\0> and go back to step 1. We have to show that this algorithm terminates. Assume the contrary and let ,,\> be the successive dominant monomials of in step 1 on smaller and smaller subregions \\\\> of >. Ultimately, for each , there exists a \\\ > with >(\)=0> in step 2, and the next region is given by ={\\\\\ |f>(\)=0}> in step 3. Now the numerators of all coefficients >,f>,\> belong to the Noetherian polynomial ring ,\,\>]>. Consequently, the increasing chain of ideals >)\(f>,f>)\\> is stationary and so is the decreasing chain \\\\> of subregions of >: contradiction. Using the tool of uniform regularization, we may compute with parameterized complex transseries in a similar way as explained in sections , and . Of course, it may happen that we need to exponentiate or to take logarithms of parameterized constants in . Nevertheless, this can only happen a finite number of times, so that we may see these exponentials resp. logarithms as new parameters. Furthermore, we will show that it is never necessary to exponentiate or take logarithms of parameterized constants during the resolution of algebraic differential equations. <\example> Consider the expansion of the function +\*z>+e*e>).> \1>.>We insert into >> and get +\*z>+e*e>\ =e*z>+e+1)*z>+\+1+\*e+\.> We thus have to determine whether *z>\1> and *z>\1>, which leads to the following cases and expansions for : |+|2>>*e+-+|4>-|8>>**e+\>|>|=0,\>={});>>|*z>+\+\*e+\>||\0,\>={,>});>>|*z+e+\+e*z>+\>||\0,\ \>={,\}).>>>>>> \1>.>We insert into >> and next need to determine whether +\*z>\e\ *e+\*\*z>> or +\*z>\e*e+\*\*z>>. This leads to the following cases and expansions for : |+\*z+e*\*z>*e-1)*(e+\*z)>+\>|>|\\1,\>={},\+\*z>>={1-\\ };>>|*e+e*\\ *z>*e)*(e+\*z)>+\>||\0,\>=\ {},\+\\ *z>>={\-1}).>>>>>> In the last exceptional case when =1>, we get +\*z>*(1+e*z>)),> so that we need to determine whether e*z>> or e*z>>. This leads to the following final cases and expansions for : |+\*z+log 2>|>|=0,\=1,\>={1});>>|+e*z>->*e*z>+\>||\0,\=1,\>={1,-\});>>\ |+\*z+e*z>+\>||\0,\=1,\>={1,\}).>>>>>> In the remainder of this paper, we will be concerned with the resolution of asymptotic algebraic differential equations like (f\),> where [f,f,\,f]> is a differential polynomial with transseries coefficients and \> a transmonomial. In this section, we describe the differential Newton polygon method, which enables us to compute the successive terms of solutions one by one. In the next sections, we will be concerned with the transformation of this transfinite process into a finite algorithm. In sections , and the transseries in are assumed to be as in section . In section , we will consider parameterized transseries solutions. Except for the usual asymptotic relations ,\,\,\,\,\,\> and >, we will also need the flattened relations ,\,\> and there variants >,\>,\>>, where is an infinitely large or small transseries. These relations are defined by g>|>|\\h:f*\\ \g;>>|g>|>|\\h:f*\\g;>>|g>|>|g\g\\ f;>>|>g>|>|\\log h:f*\\g;>>|<\ rsup|\>g>|>|\\log h:f*\\g;>>|>g>|>|>g\g\>f.>>>>> Notice that >g\f\ \g>, >g\ \f\g> and >g\f\g>. > The differential polynomial is most naturally decomposed as P(f)= > P>*f>> Here we use vector notation for tuples =(i,\,i)> and =(j,\,j)> of integers: <\expand|eqnarray*> \|>||>|\<\|\|\>>||+\+i;>>|<\ row|\>|>|\j\\\i\j;>>|>>||i>*(f)>*\*(\ f)>;>>||>>|||i>*\*|i>.>>>> The -th of is defined by =\<\|\|\>=i>P>*f>,> so that P.> along orders> Another very useful decomposition of is its : P(f)= >> P>]> f>]>> In this notation, > runs through tuples >=(\,\,\)> of integers in ,r}> of length deg P>, and >]>=P(1)>,\,\(l)>]>> for all permutations of integers. We again use vector notation for such tuples <\expand|eqnarray*> >\|>||>|>\<\|\|\>>||+\ *\*+\>\|>;>>|>\>>|>|>\|=\|>\|\\\\\\\\>\|>\\>\|>;>>|>]>>||)>*\*f>\|>)>;>>|>|>>>|||\>*\*<\ rsub|\|>\|>|\>\|>>.>>>> We call >>\|\| the of > and P\<\|\|\>=max>\|P>]>\0>\<\|\|\>>\<\|\|\>> the of . > It is convenient to denote the successive logarithmic derivatives of by >>||/f;>>|i\>>||\\\> times)>.>>>>> Then each > can be rewritten as a polynomial in >,\,fi\>\ >: ||>|>||>*f;>>|>||>)+f\>*f\ >)*f;>>|>||>)+3*f\>*(f>)+(f\ \>)*f>+f\\\ >*f\>*f>)*f;>>||>|>>>> We define the of by =(i,\,i)>P\>*f\>,> where \>=f>*(f>)>*\*(fr\>)>.> Now consider the lexicographical ordering > on >, defined by \>|>|\j)\>>|||=j\i\j)\>>|||\>>|||=j\\\<\ wedge\>i=j\i\j).>>>>> This ordering is total, so there exists a maximal for > with \>\0>, assuming that 0>. For this , we have )\P\>*f\>> for all , whose dominant monomial is sufficiently large. Given a differential polynomial and a transseries it is useful to define the and > and h>> of w.r.t. and the > of as being the unique differential polynomials, such that for all , we have (f)>||>|h>(f)>||>|(f\)>||.>>>>> The coefficients of > are explicitly given by P>= \> |>*h->*P>.> The coefficients of h>> are more easily expressed using decompositions along orders: Ph,[>]>= >\>>>|>>*h>- >]>*P>]>.> The coefficients of the upward shifting (or compositional conjugation by >) are given by (P\)>]>=>\>>s>,>>*e>\<\|\|\>*z>*(P>]>\),> where the >,>>> are generalized Stirling numbers of the first kind: |||||>,>>>|=>|,\>*\*s\|>,\\|>>;>>|>|| s*x*f(log z).>>>>>> Given a differential polynomial with transseries coefficients, its > is defined by =max,\> >>.> and its (or coefficient) \C[c,c,\,c]> by =>P,>*c<\ rsup|>.> The following theorem shows how > looks like after sufficiently many upward shiftings: Let be a differential polynomial with purely exponential coefficients. Then there exists a polynomial C[c]> and an integer >, such that for all \<\|\|\>P\<\|\|\>>, we have >=Q*(c)>>.\ > <\proof> Let > be minimal, such that there exists an > with >\<\|\|\>=\> and \)>]>\0>. Then we have (D\)=e*z>> and >(c)=>\<\|\|\>=\>>\>>s>,>>*D>]>*c>]>,> by formula (). Since >\0>, we must have \\<\|\|\>D\<\|\|\>>. Consequently, D\<\|\|\>\\\ =\<\|\|\>D>\<\|\|\>\\<\|\|\>D\>\<\|\|\>\\>. Hence, for some >P\<\|\|\>>, we have D>\<\|\|\>=\<\|\|\>D<\ rsub|P\>\<\|\|\>>. But then () applied on > instead of yields >=D\ >>. This shows that >> is independent of , for \<\|\|\>P\<\|\|\>>. In order to prove the proposition, it now suffices to show that >=D> implies >=Q*(c)>> for some polynomial C[c]>. For all differential polynomials of homogeneous weight >, let >=([c*(c)>]R)*c*(c)>.> Since >>=D>>, it suffices to show that if and only if >=0>. Now >=0> implies that (z)=0>. Furthermore, () yields \=e*z>*D.> Consequently, we also have (e)=e*z>*(D\)(e)=e*z>*(D(z))\ \=0>. By induction, it follows that (exp z)=0> for any iterated exponential of . We conclude that =P=0>, by (). Given an arbitrary differential polynomial , the above proposition implies that there exists a polynomial C[c]> and an integer >, such that >=Q*(c)>> for all sufficiently large . We call =Q*(c)>> the of . More generally, given a monomial >, we call >>> the of >. Returning to the asymptotic differential equation (), we call \> a , if >>> admits a non trivial root (C)>>, where > stands for the algebraic closure of . If C\>\ >, then the corresponding term > is called a . The of (and of >) is the differential valuation of >,+c>>, i.e. the least such that >,+c,i\ >\0>. The of () is the largest possible degree of >>> for monomials \>. >| is a regular, non-zero transseries solution to )>. Then > is a potential dominant term.>> A potential dominant monomial > is said to be if >>> is non homogeneous, and if >>\C[c]>. A potential dominant monomial, which is both algebraic and differential, is said to be . Notice that () implies (P>)\>>*(P),> if the coefficients of and > are purely exponential. The algebraic potential dominant monomials correspond to the slopes of the Newton polygon in a non differential setting. However, they can not be determined directly as a function of the dominant monomials of the >>, because there may be some cancellation of terms in the different homogeneous parts during multiplicative conjugations. Instead, the algebraic potential dominant monomials are determined by successive approximation: <\proposition> Let j> be such that \0> and \0>. <\expand|enumerate-alpha> If is purely exponential, then there exists a unique purely exponential monomial >, such that (P>)=(P>)>. Denoting by > the monomial > in )>, there exists an integer \<\|\|\>P\<\|\|\>>, such that for all k> we have ,i,j>=,i,j>\>. There exists a unique monomial >, such that +P)>>> is non homogeneous. <\proof> In (), let =(,\,)> be a purely exponential transbasis for the coefficients of . We prove the existence of > by induction over the least possible , such that we may write (P)/(P)=>*\*\ >>. If , then we have =1>. Otherwise, let >> with =/(j-i\ )>>. Then (Q)\>\ (P)*\\ >(P)*\>(Q),> so that (Q)/(Q)=>*\*>> for some k> and ,\,\>. By the induction hypothesis, there exists a purely exponential monomial >, such that (Q>)=(Q>)>. Hence we may take =*>. As to the uniqueness of >, assume that =*>*\*>> with \0>. Then (P>)\>(P>)*<\ rsub|k>>\>(P>)*>\\ >(P>\ ).> This proves (). With the notations from proposition , we have already shown that D\>\<\|\|\>\<\ leqslant\>\<\|\|\>D>\<\|\|\>> and that equality occurs if and only if >=cD>\<\|\|\>>*(c)D>\<\|\|\>>>. Because of (), we also notice that D,\*e*z>>\<\ \|\|\>=\<\|\|\>D>\<\|\|\>> for all >. It follows that D,\>\<\|\|\>\\<\|\|\>D\,\,i,j>>\<\|\|\>\\> and similarly for > instead of >, since we necessarily have \,i,j>=,i,j>\*e*z>> for some >. We finally notice that D,\>\ \<\|\|\>=\<\|\|\>D\,\,i,j>>\<\|\|\>> and D,\>\<\|\|\>=\<\|\|\>D\\ ,\,i,j>>\<\|\|\>> imply that ,i,j>=\\ >, since D>*(c)>)ex>>>\<\|\|\>=0\\=\<\|\|\>D>*(c)\ >>\<\|\|\>> whenever \0> and \0>. Consequently, D\,\,i,j>>\<\|\|\>> and D\,\,i,j>>\<\|\|\>> stabilize for k> with \<\|\|\\ >P\<\|\|\>>. For this , we have (). With the notations from (), ,i,j>\> is actually the unique monomial > such that +P)>\\ >=D>\>+D>\>> is non homogeneous for all sufficiently large . Now +P)>>=D\ +P)>\>\ > for sufficiently large . This proves () for purely exponential differential polynomials , and also for general differential polynomials, after sufficiently many upward shiftings. The unique monomial > from part () of the above proposition is called an or the -equalizer for . An algebraic potential dominant monomial is necessarily an equalizer. Consequently, there are only a finite number of algebraic potential dominant monomials and they can be found as described in the proof of proposition . Notice that, given a transbasis =(,\,\ )> for the coefficients of , all equalizers for belong to P\<\|\|\>> )*\*(log )*>. In order to find the differential potential dominant monomials, it suffices to consider the homogeneous parts > of , since \ >,i>=N>>>, if \|N>>> and >,i>\0>. Now we may rewrite > as \ i>> times a differential polynomial > of order 1> in >>. We call > the -th Ricatti equation associated to . Since solving (f)=0> is equivalent to solving (f>)=0>, we are entitled to expect that finding the potential dominant monomials of w.r.t.> is equivalent to solving (f>)=0> ``up to a certain extent''. <\proposition> The monomial \\ > is a potential dominant monomial of w.r.t. (f)=0> if and only if the equation >>(f>)=0f>\>>\ > has strictly positive Newton degree. <\proof> We first notice that ,i>=(R\\ )e>> for all and . We claim that the equivalence of the proposition holds for and > if and only if it holds for > and \>. Indeed, > is potential dominant monomial w.r.t. (), if and only if > is a potential dominant monomial w.r.t. \(f\)=0> and () has strictly positive Newton degree if and only if >>\(f>\)=0f>\\*z*log z**\>>> has strictly positive Newton degree. Now the latter is the case if and only if >>\)e>(f\>)=0f\>\>>> has strictly positive Newton degree. But >>\)e>=(R\)>\,\e>=(R\)e,+\>>=R,i,+\>>.> This proves our claim. Now assume that > is a potential dominant monomial w.r.t.)>. In view of our claim, we may assume without loss of generality that and > are purely exponential and that >>=D>>>. Since > is homogeneous, we have >>=\*(c)> for some \C\>> and >>>=\*ci>.> Since >>> is purely exponential, it follows that >,\z>>> has degree , so that the Newton degree of () is at least . Similarly, if > is not a potential dominant monomial w.r.t. (), then >>=\*c\ > and >>>=\*> for some \C\>>. Consequently, >,\>>=\> for any infinitesimal monomial >, and the Newton degree of () vanishes. Now we know how to determine potential dominant terms of solutions to (), let us show how to obtain more terms. A is a change of variables together with an asymptotic constraint +(\\ ),> where \\>. Such a refinement transforms () into >()=0(\<\ apply|tmv>).> We call the refinement , if () has strictly positive Newton degree. Let > be the dominant term of > and assume that =>. Then the Newton degree of )> is equal to the multiplicity > of as a root of >>>.> <\proof> Let us first show that ,\>>\> \ for any monomial \>. Modulo replacing by >> we may assume without loss of generality that =1>. Modulo a sufficient number of upward shiftings, we may also assume that =D>, that ,\>>=D,\>>>, and that , > and > are purely exponential. The differential valuation of =D>> being >, we have in particular (P,>)=(P>)>. Hence, (P,\,i>)\<\ asymp\>>(P,\\ ,i>)*\>(P,\>)*>\>(P,\,>)> for all >. We infer that ,\>>\>. At a second stage, we have to show that ,\>>\>. Without loss of generality, we may again assume that =1>, that =D>, and that and > are purely exponential. The differential valuation of =D>> being >, we have (P,i>)\(P>)> for all >. Taking =z>, we thus get (P,\,i>)\<\ asymp\>>(P,i>)\>(P>)=(P,>)\>(P,\<\ value|mn>,>)> for all >. We conclude that ,\>>\>. Consider the algebraic differential equation P(f)=f+f*f-(f)=0.> Let us start by computing the potential dominant monomials of . We first have to find the -equalizer relative to (). Since >\c>*(c)>>, we cannot have >=P>, so we have to compute =f+e*(-f*f+f*f-(f)).> In order to ``equalize'' > and >, we have to conjugate multiplicatively with >: e>=e*(f-2*\ f-f*f+f*f-(f)).> At this point, we observe that e>\>=c-2*c\C[c]>, so we have found the -equalizer, which is =e\=z>. Since >>=c-2*c>, the corresponding algebraic potential dominant term of is =*z>. As to the differential potential dominant monomials, we have >||>|>||>.>>>>> Clearly, > has no roots and (f>)=0> has all constants \C> as its solutions modulo )>. Consequently, *z>> is a potential dominant monomial of for all \C>, such that *z>\1>. The corresponding differential potential dominant terms are of the form ,\>=\*e*z>\ >, with \0> and *z>\1>. In order to find more terms of the solution to (), we have to refine the equation. First of all, consider the refinement +(\\),> which transforms () into 2*-2*z*+>*z*+*<\ wide|f|~>-()=0(\z).> Since =0>, we first observe that *z> is actually a solution to (). On the other hand, since > is a potential dominant term of multiplicity 1 of , the Newton degree of () is one. The only potential dominant monomials of > therefore necessarily correspond to solutions modulo )> of the Ricatti equation >+>*z*((>)+f'>)=0.> These solutions are of the form >=\ +\> and >=+\>, which leads to the potential dominant monomials and >, from which we remove >, since \z>. Expanding one term further, we see that the generic solution to () is =\*z+|2>>,> with \C> and where the case =0> recovers the previous solution. In other words, >*z+\*z+|2>>> is the first type of generic solution to (). As to the second case, we consider the refinement ,\>+(\ \\,\>),> which transforms () into \*e*z>+(\*f-2*\*f+f)*\*e*z>+f+<\ wide|f|~>*-()