Transserial Hardy fields

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 [AvdD02, AvdD01, AvdD04].

Other natural models of the theory of H-fields are fields of transseries [vdH97, Sch01, MMvdD97, MMvdD99, Kuh00, vdH06]. Contrary to Hardy fields, these models are purely formal, which makes them particularly useful for the automation of asymptotic calculus [vdH97]. Furthermore, the so called field of grid-based transseries

(for instance) satisfies several remarkable closure properties. Namely,

is differentially Henselian [vdH06, theorem 8.21] and it satisfies the differential intermediate value theorem [vdH06, theorem 9.33].

Now the purely formal nature of the theory of transseries is also a drawback, since it is not a priori clear how to associate a genuine real function to a transseries

, even in the case when

satisfies an algebraic differential equation over

. One approach to this problem is to develop Écalle's accelero-summation theory [Éca85, Éca87, Éca92, Éca93, Bra91, Bra92], 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 transserial Hardy field.

Roughly speaking, a transserial Hardy field is a truncation-closed differential subfield

, which is also a Hardy field. The main objectives of this paper are to show the following two things:

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 [AvdDvdH05, AvdDvdH] 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 [Har10, Har11]. The definition of a Hardy field and the possibility to add integrals, exponentials and algebraic functions is due to Bourbaki [Bou61]. More generally, Hardy fields can be extended by the solutions to Pfaffian first order differential equations [Sin75, Bos81] and solutions to certain second order differential equations [Bos87]. Further results on Hardy fields can be found in [Ros83a, Ros83b, Ros87, Bos82, Bos86]. 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 [vdH05]). For instance, consider a differential equations such as

The recursive replacement of the left-hand side by the right-hand side then yields a “convergent” expansion for

using iterated integrals

where we understand that each of the integrals in this expansion are taken from

In order to make this idea work, one has to make sure that the extension of

with a solution

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” (a notion which refines Ritt rank). In section 2, we will show how to put the equation in normal form

over

. In section 4, it will be show how to solve (1.1) using iterated integrals, using the fact that the equation

admits

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 3 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 (1.1) 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 5. Throughout the paper, we will freely use notations from [vdH06]. For the reader's convenience, some of the notations are recalled in section 2.1. 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 [vdDMM94]. A second step would be to consider postcompositions with operators

for sufficiently flat transseries

for which Taylor's formula holds:

This requires the existence of suitable analytic continuations of

in the complex domain. Typically, if

with

, then

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

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 [vdD98]. 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.

2.Preliminaries

2.1.Notations

Let

be the totally ordered field of grid-based transseries [vdH06]. Any transseries is an infinite linear combination

of transmonomials, with grid-based support

. Transmonomials

are systematically written using the fraktur font. Each transmonomial is an iterated logarithm

or the exponential of a transseries

with

for each

. The asymptotic relations

and

are defined by

It is convenient to use relations as superscripts in order to filter elements, as in

We denote the derivation on

w.r.t.

and the corresponding distinguished integration (with constant part zero) by

. The logarithmic derivative of

is denoted by

. The operations

and

of upward and downward shifting correspond to postcomposition with

resp.

. We finally write

if the transseries

is a truncation of

, i.e.

for all

2.2.Differential fields of transseries and cuts

By convention,

contains less than two elements. We also define the ultimate canonical span of

Consider a differential subfield

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

as soon as

. Notice also that we must have

has span

A transseries

is said to be a serial cut over

, if

for every

and

admits no minimal element for

. In that case, let

be maximal for

such that

. Then

and

are called the head and the tail of

. We say that

is a normal serial cut if

, which implies in particular that

Assuming that

has span

, any serial cut over

is necessarily in

. Conversely, any

with

is a serial cut over

. We will denote by

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

and differential subfields

. If

has span

, then the set

coincides with the set of all

which are either in

or serial cuts over

with

2.3.Complements on differential algebra

Let

be a differential field. We denote by

the ring of differential polynomials in

over

and by

its quotient field. Given

and

, we recall that

denotes the homogeneous part of degree

. We will denote by

the linear operator in

with

. Assuming that

, we also denote the order of

, the degree of

and the total degree of

. Thus, the Ritt rank of

is given by the pair

. The triple

will be called the complexity of

; likewise ranks, complexities are ordered lexicographically.

As usual, we will denote the initial and separator of

resp.

and set

. Given

with

, Ritt reduction of

provides us with a relation

Let

be a differential field extension of

. An element

is said to be differentially algebraic over

if there exists an annihilator

with

. An annihilator

of minimal complexity

will then be called a minimal annihilator and

is also called the complexity of

over

. The order

of such a minimal annihilator

is called the order of

over

. We say that

is a differentially algebraic extension of

if each

is differentially algebraic over

We say that

is differentially closed in

, if

contains no elements which are differentially algebraic over

.. Given

(resp.

), we say that

is -differentially closed (resp. -differentially closed) in

(resp.

) for all

. We say that

is weakly differentially closed if every

admits a root in

. We say that

is weakly -differentially closed if every

of order

admits a root in

Given a differential polynomial

and

, we define the additive and multiplicative conjugates of

We also notice that additive and multiplicative conjugation are compatible with Ritt reduction: given

and assuming (2.3), we have

Remark 2.2. 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 a differential polynomial

may also be regarded as a series in

. Similarly, elements

of the fraction field

may be regarded as series with coefficients in

. Indeed, writing

and

, where

denotes the dominant term of

, we may expand

In the case when

for some transbasis

, then

and

may also be expanded lexicographically with respect to

2.4.Linear differential operators and factorization

Let

be a differential field and consider a linear differential operator

. We will denote the order of

. Given

, we define the multiplicative conjugate

and the twist

We notice that

is also obtained by substitution of

for

. We say that

splits over

, if it admits a complete factorization

We say that

is -linearly closed if any linear differential operator of order

splits over

Proof. The proof proceeds by induction over

. For

, we have nothing to prove, so assume that

and let

be of order

. Then the differential Riccati polynomial

has order

, so it admits a root

. Division of

yields a factorization

where

has order

. By the induction hypothesis,

splits over

, whence so does

Proof. Recall that greatest common divisors and least common multiples exist in the ring

. Given a splitting (2.4), consider the operators

We have

and

. Moreover, the orders of

and

(resp.

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 atomic real operator if

has either one of the forms

where each

is an atomic real operator. A splitting (2.4) over

is said to preserve realness, if it gives rise to a real splitting (2.5) for

and

Proof. Assuming that

, we claim that there exists an atomic real right factor

. Consider a splitting (2.4) over

. If

, then we may take

. Otherwise, we write

and take

to be the least common multiple of

and

. Since

, we indeed have

. Since

and

, we also have

. In particular, proposition 2.4 implies that

splits over

. Such a splitting is necessarily of the form

whence

is atomic. Having proved our claim, the proposition follows by induction over

. Indeed, let

be such that

. By proposition 2.4,

splits over

. By the induction hypothesis,

therefore admits a real splitting

over

. But then

is a real splitting of

2.5.Factorization at cuts

Let

be a differential subfield of

of span

. Given

and

, we say that

splits over

, if

and

have the same order

and

splits over

By the minimality hypothesis for

, we have

and

, so that

and

. Similarly, we have

. Consequently, when considering the linear part of the equation (2.7), we obtain

whence

divides

. Now

splits over

, whence so does

. By proposition 2.4, we infer that

splits over

. Since

, we also have

and we conclude that

splits over

Proof. Applying the lemma to

, we see that

splits over

. Now

, whence

and

also split over

Proof. Let

be as in the above corollary, so that

splits over

. Since

has minimal complexity and

, Ritt division of

yields

Consequently, the quotient of

and

has order at most

, whence it splits over

. It follows that

splits over

and

splits over

2.6.Normalization of linear operators

Let

be a differential subfield of

of span

. Recall from [vdH06, Section 7.7] that

with

admits a canonical fundamental system of oscillatory transseries solutions

with

. We will denote by

the set of dominant monomials of

. The neglection relation on

is extended to

if and only if

with

and

We say that

is normal, if we have

for each

. In that case, any quasi-linear equation of the form

with

admits

as its only solution in

. If

is a first order operator of the form

, then

is normal if and only if

for some

. In particular, we must have

and

Proof. Let

. For each

, the operator

admits

as solutions, which implies in particular that

. Now

for all

. Choosing

sufficiently large, it follows that

for all

with

, so that

is normal. Similarly, if

for some

with

, then

for all

Proof. We will call

normal, if

is normal. Let us first prove the following auxiliary result: given

and

such that

and

are normal and

, then

is also normal. If

, then

, whence

. In the other case, we have

. Now if

, then

, since

. If

, then

implies

, whence

. It again follows that

Let us now prove the proposition by induction over

. For

, we have nothing to do, so assume that

. Since

is normal, the induction hypothesis implies that

is normal for all

. Now let

be the unique element in

. Since

is normal,

is also normal for

, by the auxiliary result. We conclude that

is normal, since

Let

and

be as above. The smallest real number

with

for all

will be called the growth rate of

, and we denote

. For all

, we notice that

Proof. Assume for contradiction that

for some

and choose

maximal with this property. Setting

satisfies

, as well as

. But such an

cannot be a linear combination of the

with

Remark 2.14. 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

which splits over

for every

. In particular,

-linearly closed if and only if

-linearly closed over

2.7.Normalization of quasi-linear equations

Assume now that

is a differential subfield of

of span

. We say that

is normal if

is normal of order

and

. In that case, the equation

is quasi-linear and it admits a unique solution in

. Indeed, let

be the distinguished solution to (2.8). By proposition 2.12, the operator

is normal. If

were another solution to (2.8), then

would be in

, whence

, which is impossible.

Proof. Let

and

. Modulo a multiplicative conjugation by

for some

, we may assume without loss of generality that

. Modulo an additive conjugation by

, we may also assume that

. For any

and

, we have

Since

, we have

. By proposition 2.10, there exists a

for which

is normal. Now take

. Denoting

, proposition 2.12 and (2.9) imply that

is normal with

and

We say that

is split-normal, if

is normal and

can be decomposed

such that

splits over

and

. In that case, we may also decompose

for

with

. If

is monic, then we say that

is monic split-normal. Any split-normal equation (2.8) is clearly equivalent to a monic split-normal equation of the same form.

Proof. By proposition 2.15 and modulo a replacement of

, we may assume without loss of generality that

is normal. By lemma 2.9,

splits over

. Let

be such that

Then

and proposition 2.12 implies that

is normal, with

. Denoting

, we finally have

3.Transserial Hardy fields

3.1.Transserial Hardy fields

Let

be the field of grid-based transseries [vdH06] and

the set of infinitely differentiable germs at infinity. A transserial Hardy field is a differential subfield

, together with a monomorphism

of ordered differential

-algebras, such that

In what follows, we will always identify

with its image under

, which is necessarily a Hardy field in the classical sense. The integer

in TH3 is called the depth of

; if

for all

, then the depth is defined to be

. We always have

, since

is stable under differentiation. If

, then

is exponential for all

and

contains

. If

and

, then

contains

for all sufficiently large

Example 3.1. The field

is clearly a transserial Hardy field. As will follow from theorem 3.12 below, other examples are

Remark 3.2. Although the axioms TH4 and TH5 are not really necessary, TH4 allows for the simplification of several proofs, whereas it is natural to enforce TH5. Notice that TH5 automatically holds for

with

since

whence

for some

. Since both

and

are infinitesimal in

, we have

. Consequently, it suffices to check TH5 for monomials

with

, then

is called a plane transbasis and

is stable under differentiation. The incomplete transbasis theorem for

also holds for transserial Hardy fields:

Proof. The same proof as for [vdH06, Theorem 4.15] 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 transrank of

is the minimal size of a plane transbasis

with

. This notion may be extended to allow for differential polynomials

(modulo the replacement of

by its set of coefficients).

Remark 3.6. The span and ultimate span of

are not necessarily in

. Nevertheless, if

and

is a transbasis for

, then we do have

for some

(and similarly for the ultimate span of

3.2.Cuts in transserial Hardy fields

We say that

and

are asymptotically equivalent over

if for each

(or, equivalently, for each

), we have

Proof. Let

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

. Let

be such that

and let

be the dominant monomial of

. Modulo division of

and

, we may also assume that

is a normal serial cut. But then the equation

gives rise to the equation

for

. The complexity of

is clearly bounded by

As to their differential equivalence, let us first assume that

is differentially transcendent over

. Given

, let us denote

Assume now that

is differentially algebraic over

and let

be a minimal annihilator. Given

, Ritt reduction of

w.r.t.

gives

, this clearly implies

. Otherwise,

vanishes neither at

nor at

and the relations (3.1) and (3.2) again yield

and

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

and replacing

, we may finally assume that

For general

, we use Ritt reduction of

w.r.t.

and conclude in a similar way as in the proof of lemma 3.8.

3.3.Elementary extensions

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

is closed under truncation and that

for all

with

(this implies in particular that

is increasing). Notice that

implies

Truncation closedness. Given

, let us prove by induction over the transrank

that

. So let

be a plane transbasis for

and

. Assume first that

. Writing

Preservation of dominant terms. Given

with

, let us prove by induction over the transrank

that

. Let

be a plane transbasis for

and

and assume first that

. Since

, there exists a maximal

with

, when considering

as a series in

. But then

by the induction hypothesis. If

, then there exists an

such that, for all sufficiently large truncations

, the Taylor series expansion of

yields

Proof. Assume that

and choose

of minimal complexity. By lemma 3.7, we may assume without loss of generality that

is a serial cut. Consider the monic minimal polynomial

. Since

, we have

for a sufficiently large truncation

(we refer to [vdH06, Section 8.3] for a definition of the Newton degrees

). But then

admits unique solutions

and

resp.

, by the implicit function theorem. It follows in particular that

. Let

and consider

with

. Then

Since

, we obtain

, whence

and

are asymptotically equivalent over

. By lemmas 3.9 and 3.10, it follows that

carries a transserial Hardy field structure which extends the one on

. Since (3.3) has a unique solution

, this structure is unique. We conclude by Zorn's lemma.

3.4.Exponential and logarithmic extensions

Proof. Each element in

is of the form

for

and

-linearly independent

. Given

, let

be a transbasis for

. We may write

with

(or the obvious adaptations if

). Modulo the substitution of

, we may assume without loss of generality that

, then we may regard

as a convergent grid-based series in

with coefficients in

. In particular,

Furthermore, if

admits

as its dominant exponent in

, then

holds both in

and in

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

holds both in

and in

This shows that

is truncation closed and that the extension of

is increasing. We also have

. In other words,

is a transserial Hardy field.

Proof. The proof is similar to the proof of theorem 3.12, when replacing

3.5.Complex transserial Hardy fields

Let

be a transserial Hardy field. Asymptotic and differential equivalence over

are defined in a similar way as over

Proof. Assume that

and

are asymptotically equivalent over

and let

. Consider

. We have

, so that

. Moreover,

, whence

and

. The relation

is proved similarly. Inversely, assume that

and

as well as

and

are asymptotically equivalent over

. Given

, we have

, whence there exist

with

and

. It follows that

, whence

Proof. Differential equivalence over

clearly implies differential equivalence over

. Assuming that

and

are differentially equivalent over

, we also have

Remark 3.16. Given

and

, it can happen that

and

are differentially equivalent over

, without

and

being differentially equivalent over

. This is for instance the case for

and

. Indeed, the differential ideals which annihilate

resp.

are both

Most results from the previous sections generalize to the complex setting in a straightforward way. In particular, lemmas 3.7, 3.8 and 3.9 also hold over

. However, the fundamental extension lemma 3.10 admits no direct analogue: when taking

and

such that the complexified conditions i, ii and iii hold, we cannot necessarily give

the structure of a transserial Hardy field. This explains why some results such as lemmas 2.9 and 2.16 have to be proved over

instead of

. Of course, theorem 3.11 does imply the following:

4.Analytic resolution of differential equations

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

An operator

(resp.

) is said to be continuous if there exists an

with

(resp.

) for all

. The smallest such

is called the norm of

and denoted by

(resp.

). The above definitions generalize in an obvious way to the complexifications

and

4.1.Continuous right-inverses of first order operators

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

for all

, whence (4.2). In the attractive case, the change of variables

leads in a similar way to the bound

4.2.Continuous right-inverses of higher order operators

Let

be a transserial Hardy field of span

. A monic operator

is said to be split-normal, if it is normal and if it admits a splitting

with

. In that case, proposition 2.11 implies that each

is a normal first order operator. For a sufficiently large

, it follows that

admits a continuous “factorwise” right-inverse

, where

. We have

By proposition 2.13 and induction over

, we have

for all

. Since

, it follows that

Proof. It clearly suffices to prove the proposition for an atomic real operator

. If

has order

, then the result is clear. Otherwise, we have

for certain

. In particular, we are in the same case (attractive or repulsive) for both factors of

. Setting

, let

be as in the previous section. Consider

and

. In the repulsive case, we have

In particular, we have

, whence

, since

satisfies the differential equation

of order

with real coefficients. In the attractive case, we have

so that

. Since

, the difference

satisfies

. Now

is the only solution with

to the equation

. This proves that

4.3.The fixed point theorem

Let

be a transserial Hardy field of span

and consider a monic split-normal quasi-linear equation

where

has order

and

has degree

. Of course, we understand that

is a monic split-normal operator with

. We will denote by

the valuation of

(i.e.

for

and

). We will show how to construct a solution to (4.5) using the fixed-point technique.

Proof. Since

for all

, the number

from (4.8) tends to

for

. When constructing

using proposition 4.1, the number

from (4.9) decreases as a function of

. Taking

sufficiently large so that

, we obtain (4.10). By induction over

, it follows that

Now let

be the space of

times continuously differentiable functions

, such that

are bounded. This space is complete, whence

converges to a limit

. Since this limit satisfies the equation (4.5), the function

is actually infinitely differentiable, i.e.

4.4.Asymptotic analysis

With the notations from the previous section, assume now that

-differentially closed in

, i.e. any solution

to an equation

with

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

so the sequence

also admits a formal limit

. In order to show that the fixed point

from proposition 4.6 and

are asymptotically equivalent over

, we need some further notations. Given

and

, let us write

, i.e.

for all

. We also write

whence

for some fixed

. This proves (4.11). More generally,

additional applications of (4.12) yield

for some

. Since

is normal, we either have

(in which case

for all

) or

. In both cases, we get

, so that

Proof. With the above notations, let

and

be the limits in

resp.

of the sequences

resp.

. Given

, there exists an

with

Proof. In view of propositions 2.5 and 4.4, we may assume that

and

preserve realness in all results from sections 4.3 and 4.4. In particular, the solutions

and

in the conclusion of theorem 4.9 are both real.

5.Differentially algebraic Hardy fields

5.1.First order extensions

Proof. With the notations of section 4.1, let

. Given a truncation

, we claim that

In the attractive case,

implies

. In the repulsive case, we have

and again

. By proposition 4.8, we also have

Since

, it follows that

, whence

and

are asymptotically equivalent over

. Furthermore,

is a minimal annihilator of

over

, since

is transcendental over

. Lemma 3.9 therefore implies that

and

are differentially equivalent over

Proof. By theorems 3.11, 3.12 and 3.13, we may assume that

is closed under the resolution of real algebraic equations, exponentiation and logarithm. Assume that

and let

be of minimal complexity

, such that

for some

. Without loss of generality, we may make the following assumptions:

By Zorn's lemma, it suffices to show that

carries the structure of a transserial Hardy field, which extends the structure of

, then lemma 5.1 implies the existence of an

such that

and

are both asymptotically and differentially equivalent over

. Hence, the result follows from lemmas 3.8 and 3.10.

, then

and

are

-differentially closed in

resp.

. Now

, since

is exponential. Therefore, theorem 4.10 provides us with an

with

, such that

and

are asymptotically equivalent over

. We conclude by lemmas 3.9, 3.8 and 3.10.

5.2.Higher order extensions

Proof. The fact that

and

are asymptotically equivalent over

is proved in a similar way as for lemma 5.1. 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 3.9.

Proof. By theorems 3.12, 3.13 and 5.2, we may assume that

is closed under exponentiation, logarithm and the resolution of first order differential equations. Assume that

and let

be of minimal complexity

, such that

for some

with

. Let

be a minimal annihilator of

and notice that

, since

. Without loss of generality, we may make the following assumptions:

By Zorn's lemma, it now suffices to show that

carries the structure of a transserial Hardy field, which extends the structure of

, then lemma 5.3 and the fact that

-differentially closed imply the existence of an

such that

and

are both asymptotically and differentially equivalent over

. The result follows by lemmas 3.8 and 3.10.

, then

and

are

-differentially closed in

resp.

. Now

, since

is exponential. Therefore, theorem 4.10 provides us with a

with

, such that

and

are asymptotically equivalent over

. We conclude by lemmas 3.9, 3.8 and 3.10.

Proof. Take

and endow it with a transserial Hardy field structure. Let

and

with

be such that

. By [vdH06, Theorem 9.33], there exists a

with

and

. But

implies

Proof. Take

. By a straightforward adaptation of [vdH06, Chapter 8] (see also [vdH01, theorem 9.3]), it can be shown that any differential equation

of degree

with

admits

distinguished solutions in

when counting with multiplicities. Let

be such a solution. Since

, both

and

are differentially algebraic over

, whence

5.3.Differential Newton polynomials for Hardy fields

Let

be a differentially algebraic Hardy field extension of a transserial Hardy field

Proof. The functional inverse

satisfies an algebraic differential equation

over

. Let

be the leading term of

for its logarithmic decomposition. As in [vdH06, Section 8.1.4]. there exists an

with

for all

. It follows that

and

Given a differential polynomial

, we define its dominant part to be the unique monic

such that

for some

and

. Here

is said to be monic if its leading coefficient w.r.t.

equals

because

would imply

. Applying [vdH06, Lemma 8.5] to (5.2), and similarly for

, we get

By proposition 5.8 and (5.3), we have

and

for some

. Modulo upward shiftings, we may thus assume without loss of generality that

. More generally, assume that

for some

. By (5.3), this implies

for all

and

for all

. Without loss of generality, we may assume that

and

are in

. Then proposition 5.8 implies

and even

(by integrating from

when possible). Again by proposition 5.8, it follows that

and

for some

. But then (5.4) yields

which contradicts the fact that

. The relations (5.5) and (5.4) imply the existence of an

with

. By induction over

and modulo upward shiftings, we may thus ensure that

for all

The polynomial

in theorem 5.9 is called the differential Newton polynomial of

. The generalization of this concept to

allows us to mimic a lot of the theory from [vdH06, chapter 8] in

. In what follows, we will mainly need a few more definitions. The Newton degree of an equation

We say that

is a solution to (5.6) modulo

. We say that

is differentially Henselian, if every quasi-linear equation over

admits a solution. Given a solution

to (5.6), we say that

has algebraic type if

is not homogeneous and differential type in the other case. The following proposition is proved along the same lines as [vdH06, proposition 8.16]:

Remark 5.11. 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 [AvdDvdH].

5.4.Transserial models of differentially algebraic Hardy fields

Proof. By theorems 3.11, 3.12 and 5.2, we may assume that

is closed under the resolution of real algebraic equations, exponentiation and integration. Assume that

and choose

of minimal complexity

, such that either

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

which extends the structure on

Let

be the set of

such that

. The set

is totally ordered for

, so there exists a minimal well-based transseries

with

for all

. We call

the initializer of

over

. Assume first that

. Then we may assume without loss of generality that

, modulo an additive conjugation by

. Now

is of differential type, since

for no

. Let

be such that

modulo

. Since

has lower complexity than

, there exists a

with

modulo

. Since

is truncation closed we may take

. But then

. This contradiction proves that we cannot have

Let us now consider the case when

. Since

, there exists a root

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 C2, we notice that

, whence

, since

. Consequently, we always have

We claim that the cuts

and

are differentially equivalent over

. Assume the contrary and let

be a minimal annihilator of

. By lemma 2.15 and modulo an additive and multiplicative conjugation, we may assume without loss of generality that

and that

is normal. Since

is differentially Henselian, it follows that

admits a root

. Now

in case C1 and

in case C2, so this root is already in

, by the induction hypothesis. But

admits at most one solution in

, whence

. This contradiction completes the proof of our claim. By lemma 3.10, we conclude that

carries the structure of a transserial Hardy field extension of

Proof. The fact that

admits no non-trivial algebraically differential Hardy field extensions implies that

is stable under exponentiation. By theorem 5.12, we may give

the structure of a transserial Hardy field. By theorem 5.4, we also have

. We conclude in a similar way as in the proof of corollary 5.5.

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 [Bos87, Proposition 3.7]:

Glossary

is dominated by 4

is negligible w.r.t. 4

is asymptotic to 4

is asymptotically similar to 4

is flatter than or as flat as 4

is flatter than 4

is as flat as 4

and are similar modulo flatness 4

modulo elements flatter than 4

modulo elements flatter than or as flat as 4

shorthand for 4

infinite part of 4

part of which is flatter than 4

shorthand for 4

derivation with respect to 4

integration with respect to 4

logarithmic derivative of 4

upward shifting 4

downward shifting 4

is a truncation of 4

canonical span of 4

ultimate canonical span of 4

completion of with serial cuts 5

ring of differential polynomials in over 5

quotient field of 5

linear part of as an operator 5

order of 5

degree of in its leader 5

total degree of 5

complexity of 5

initial of 5

separant of 5

the product 5

complexity of over 5

order of over 5

additive conjugation of by 6

multiplicative conjugation of by 6

multiplicative conjugate of by 6

twist of by 6

set of dominant monomials of solutions to 9

ring of infinitely differentiable germs at infinity 12

is asymptotically similar to over 13

real closure of 15

Newton degree of modulo 15

norm of for 17

shorthand for 17

norm of and its first derivatives for 17

operator norm for 18

first order differential closure of in 23

differentially algebraic closure of in 24

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			(5.1)
			(5.2)
			(5.3)

1.Introduction

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

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

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

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

Glossary

Bibliography