Hardy type derivations on generalised series ﬁeldsmath.usask.ca/~skuhlman/KuhlmannMatuPart1-14.pdf · 2009. 8. 31. · of real valued functions in a Hardy ﬁeld. We provide a necessary

Hardy type derivations on generalised series fields

S KM̈M

Abstract: We consider the valued field K := R((Γ)) of generalised series (withreal coefficients and monomials in a totally ordered multiplicative group Γ ). Weinvestigate how to endow K with a series derivation, that is a derivation that sat-isfies some natural properties such as commuting with infinite sums (strong lin-earity) and (an infinite version of) Leibniz rule. We characterize when such aderivation is of Hardy type, that is, when it behaves like differentiation of germsof real valued functions in a Hardy field. We provide a necessary and sufficentcondition for a series derivation of Hardy type to be surjective.

2000 Mathematics Subject Classification 12J10, 12J15, 12L12, 13A18 (primary);03C60, 12F05, 12F10, 12F20 (secondary)

Keywords: generalised series fields, Hardy fields, derivation, valuation, asymp-totic integration

1 Introduction

In his seminal paper, I. Kaplansky established (see 11, Corollary, p. 318) that if avalued field (K, v) has the same characteristic as its residue field, then (K, v) is analyt-ically isomorphic to a subfield of a suitable field of generalised series (for definitionsand terminology, see Section 2). Fields of generalised series are thus universal do-mains for valued fields. In particular, real closed fields of generalised series providesuitable domains for the study of real algebra.

The work presented in the first part of this paper is motivated by the following query:are fields of generalised series suitable domains for the study of real differential al-gebra? We investigate in Section 3 how to endow a field of generalised series (ofcharacteristic 0) with a natural derivation d , namely a series derivation (see Definition3.2). Our investigation is based on the notion of fundamental monomials, which are infact representatives of the various comparability classes of series (see Section 2). Westart with a map d from these fundamental monomials to the field of series. The cen-tral object of investigation is to extend d first to the group of monomials (via a strong

http://www.ams.org/mathscinet/search/mscdoc.html?code=12J10, 12J15, 12L12, 13A18,(03C60, 12F05, 12F10, 12F20)http://www.ams.org/mathscinet/search/mscdoc.html?code=12J10, 12J15, 12L12, 13A18,(03C60, 12F05, 12F10, 12F20)

2 Salma Kuhlmann and Mickaël Matusinski

version of Leibniz rule) and then from the group of monomials to the field of series(via an infinite version of linearity) so that we obtain a series derivation. The mainchallenge in doing so is to keep control of the resulting supports and coefficients ofthe resulting series. The criterion that we obtain in Theorem 3.5 is rather abstract, butit easily provides more concrete sufficient (but not necessary) conditions (Corollaries3.8, 3.7, 3.10 and 3.11) to build such series derivations. These results are applied inSection 5 to obtain concrete examples.

Hardy fields (i.e. fields of germs of differentiable real functions at infinity) were intro-duced by G. H. Hardy (9) as the natural domain for the study of asymptotic analysis.They represent prime examples of valued differential fields. In a series of papers, M.Rosenlicht studied the valuation theoretic properties of these derivations. This alge-braic approach has been resumed and enhanced by M. Aschenbrenner and L. van denDries in the formal axiomatic setting of H-fields and asymptotic couples (see (1), (2)and (4)). The motivation for the second part of our paper is to understand the possibleconnection between generalised series fields and Hardy fields as differential valuedfields. Continuing our investigations in Section 4, we study derivations (on fields ofgeneralised series) that satisfy the valuative properties discovered by Rosenlicht forHardy fields (Definition 4.1). We obtain a necessary and sufficient condition on a se-ries derivation to be of Hardy type (Theorem 4.2). In the last section, we derive acriterion, Corollary 6.5, for a series derivation of Hardy type to be surjective.

We conclude with a few comments. For convenience, we have assumed throughoutthe paper that our field of generalised series has field of coefficients R and group ofmonomials a Hahn group (see Remark 2.9); however the results of the paper hold inmore generality (see the Final Remark).

Derivations on Logarithmic-Exponential series fields (see (5)(6)) and on fields of Transseries(see (10)) have been successfully introduced and studied in the literature. More pre-cisely, in the case of Transseries, it appears in (17) that one can lift a given (stronglylinear and compatible with the logarithm) derivation on some field of transseries tothe various transfinite exponential extensions of it. In a forthcoming paper, we extendour present investigations to study Hardy type derivations on Exponential-Logarithmicseries fields (see (13)).

2 Preliminary definitions

In this section, we introduce the required terminology and notations. For orderedset theory, we refer to (16). In particular, we will repeatedly use the following easy

Hardy type derivations on generalised series fields 3

corollary of Ramsey’s theorem (16, ex. 7.5 p. 112):

Lemma 2.1 Let Γ be a totally ordered set. Every infinite sequence (λn)n∈N ⊂ Γhas an infinite subsequence which is either constant, or strictly increasing, or strictlydecreasing.

Definition 2.2 Let (Φ,4) be a totally ordered set, that we call the set of fundamentalmonomials. We consider the set Γ of formal products γ ∈ Γ of the form

γ =∏φ∈Φ

φγφ

where γφ ∈ R, and the support of γ

supp γ := {φ ∈ Φ | γφ , 0}

is an anti-well-ordered subset of Φ.

Multiplication of formal products is defined pointwise: for α, β ∈ Γ

αβ =∏φ∈Φ

φ αφ+βφ

With this multiplication, Γ is an abelian group with identity 1 (the product with emptysupport). We endow Γ with the anti lexicographic ordering 4 which extends 4 of Φ:

γ � 1 if and only if γφ > 0 , for φ := max(supp γ) .

With this definition, we see that φ � 1 for all φ ∈ Φ. Thus, Γ is a totally orderedabelian group (8), that we call the Hahn group of generalised monic monomials .

For any γ , 1, we will refer to γφ as the exponent of φ, and the additive group (R,+)as the group of exponents of the fundamental monomials.

Definition 2.3 We define the leading fundamental monomial of 1 , γ ∈ Γ byLF(γ) := max(supp γ) . We set by convention LF (1) := 1. This map verifies theultrametric triangular inequality :

∀α, β ∈ Γ, LF (αβ) 4 max{ LF (α), LF (β)}

andLF (αβ) = max{ LF (α), LF (β)} if LF (α) , LF (β) .

We define the leading exponent of 1 , γ ∈ Γ to be the exponent of LF (γ), andwe denote it by LE (γ). For α ∈ Γ we set |α| := max(α, 1/α); and define sign(α)


accordingly. For any monomial α =∏

φ∈supp αφαφ ∈ Γ, any fundamental monomial

ψ ∈ Φ and any relation R ∈ {≺,4,�,


such that Supp a = {α ∈ Γ | aα , 0} is anti-well-ordered in Γ. As usual, we writethese maps a =

∑α∈Supp a

aαα, and denote by 0 the series with empty support.

By (8), this set provided with component-wise sum and the following convolutionproduct

(∑

α∈Supp aaαα ) (

∑β∈Supp b

bββ ) =∑

γ∈(Supp a∪Supp b)(∑αβ=γ

aαbβ) γ

is a field.

For any series 0 , a, we define its leading monomial: LM (a) := max(Supp a

) ∈ Γwith the usual convention that LM (0) := 0 ≺ γ , for all γ ∈ Γ. The map

LM : K \ {0} → Γis a field valuation; it verifies the following properties :

∀a, b ∈ K : LM (a.b) = LM (a). LM (b) and LM (a + b) 4 max{ LM (a), LM (b)} ,(ultrametric triangular inequality), with

LM (a + b) = max{ LM (a), LM (b)} if LM (a) , LM (b) .We define the leading coefficient of a series to be LC (a) := a LM (a) ∈ R (with theconvention that LC (0) = 0) and use it to define a total ordering on K as follows:∀a ∈ K, a ≤ 0⇔ LC (a) ≤ 0 . For nonzero a ∈ K, the term LC (a) LM (a) is calledthe leading term of a, that we denote LT(a).

We use the leading monomial to extend the ordering 4 on Γ to a dominance relationalso denoted 4 on K :

∀a, b ∈ K, a 4 b⇔ LM (a) 4 LM (b)That is 4 verifies the following definition :

Definition 2.6 Let (K,≤) be an ordered field. A dominance relation on K is a binaryrelation 4 on K such that for all a, b, c ∈ K :(DR1) 0 ≺ 1(DR2) a 4 a(DR3) a 4 b and b 4 c ⇒ a 4 c(DR4) a 4 b or b 4 a(DR5) a 4 b ⇒ ac 4 bc(DR6) a 4 c and b 4 c ⇒ a − b 4 c(DR7) 0 ≤ a ≤ b ⇒ a 4 b


Given a and b non zero elements of K, we define the corresponding equivalencerelations thus :

a and b are asymptotic ⇔ a � b ⇔ a 4 b and b 4 aa and b are equivalent ⇔ a ∼ b ⇔ a − b ≺ b

Definition 2.7 We denote by K41 := {a ∈ K | a 4 1} the valuation ring of K.Similarly, we denote by K≺1 := {a ∈ K | a ≺ 1} the maximal ideal of K41 . We haveK41 = R ⊕ K≺1 . We denote by K�1 := R

((Γ�1

)), the subring of purely infinite

series. This is an additive complement group of K41 in K, i.e. K = K41 ⊕K�1 .

Finally, we extend the notion of leading fundamental monomial to K\{0}:LF : K \ {0} → Φ ∪ {1}

a 7→ LF(a) = LF ( LM (a)) .

We use it to define the notion of comparability of two series:

Definition 2.8 Let a � 1, b � 1 be two elements of K. a and b are comparable ifand only if LF (a) = LF (b).

It is straightforward to verify that comparability is an equivalence relation on K.

Remark 2.9 The group of generalised monic monomials Γ is a multiplicative copyof the classical additive Hahn group H(Φ,R) =

←−RΦ which consists of formal sums:

{ h =∑

φ∈supp hhφ1φ | support h ⊆ Φ is anti-well-ordered }

where 1φ denotes the characteristic function of {φ}. We endow←−RΦ with point-wise

addition and anti-lexicographic ordering. The natural isomorphism ν : Γ → ←−RΦis defined by: ν(γ) = ν(

∏φ∈supp γ

φγφ) :=∑

φ∈supp γγφ1φ . Hahn’s embedding theorem

(8) states that every ordered abelian group embeds in such a Hahn group. Thus Hahngroups are universal domains for ordered abelian groups.

3 Defining derivations on generalised series fields

To state the main result of this section, we need some definitions. throughout, we fixK the generalised series field with real coefficients and monomials in the Hahn groupΓ defined by a chain of fundamental monomials Φ.


Definition 3.1 Let I be an index set and F = (ai)i∈I be a family of series in K.Then F is said to be summable if the two following properties hold (7, Part II, Ch.8,Sect.5):

(SF1) Supp F :=⋃i∈I

Supp ai (the support of the family) is an anti-well-ordered

subset of Γ.

(SF2) For any α ∈ Supp F , the set Sα := {i ∈ I | α ∈ Supp ai} ⊆ I is finite.

Write ai =∑α∈Γ

ai,αα, and assume that F = (ai)i∈I is summable. Then∑i∈I

ai :=∑

α∈Supp F(∑i∈Sα

ai,α )α ∈ K

is a well defined element of K that we call the sum of F .

Definition 3.2 LetdΦ : Φ → K\{0}

φ 7→ φ′

be a map.1) We say that dΦ extends to a series derivation on Γ if the following property holds:

(SD1) For any anti-well-ordered subset E ⊂ Φ, the family(φ′

φ

)φ∈E

is summable.

Then the series derivation dΓ on Γ (extending dΦ ) is defined to be the map

dΓ : Γ→ K

obtained through the following axioms:

(D0) 1′ = 0

(D1) Strong Leibniz rule: ∀α =∏

φ∈supp αφαφ ∈ Γ, (α)′ = α

∑φ∈supp α

αφφ′

φ.

2) We say that a series derivation dΓ on Γ extends to a series derivation on K if thefollowing property holds:

(SD2) For any anti-well-ordered subset E ⊂ Γ, the family (α′)α∈E is summable.

Then the series derivation d on K (extending dΓ ) is defined to be the map

d : K→ K

obtained through the following axiom:


(D2) Strong linearity: ∀a =∑

α∈Supp aaαα ∈ K, a′ =

∑α∈Supp a

aαα′ .

Remark 3.3 A series derivation is a derivation in the usual sense, i. e. :

(1) d is linear : ∀a, b ∈ K, ∀K,L ∈ R, (K.a + L.b)′ = K.a′ + L.b′ .(2) d verifies the Leibniz rule : ∀a, b ∈ K, (ab)′ = a′b + ab′ .

The main result of this section, Theorem 3.5, consists in giving a necessary and suffi-cient condition on the map dΦ so that properties (SD1) and (SD2) hold. (In the sequel,we drop the subscripts Φ and Γ of dΦ and dΓ to relax the notation). We first study theproblem of extending d to Γ.

Definition 3.4 Let φ, ψ ∈ Φ. Since by definition Supp φ′

φand Supp

ψ′

ψare anti-well-

ordered subsets of Γ, we may assume, without loss of generality (by interchangingthe role of φ and ψ if necessary) that there exists an embedding of ordered sets Iφ,ψ :

Suppφ′

φ→ Supp ψ

′

ψ, mapping LM (

φ′

φ) to LM (

ψ′

ψ) . Thus Iφ,ψ : Supp

φ′

φ→

Iφ,ψ(Suppφ′

φ) ⊂ Supp ψ

′

ψis an isomorphism, and we denote Iψ,φ its reciprocal. We

shall say that Iφ,ψ is a left shift if Iφ,ψ(γ) ≺ γ for any γ in the domain of Iφ,ψ .Given any φ ∈ Φ, since Supp φ

′

φis anti-well-ordered, we can enumerate its elements

in the decreasing direction τ0 � τ1 � · · · � τλ � · · · where λ is an ordinal called theposition of τλ in Supp

φ′

φ. Thus, denoting ON the proper class of all ordinals (16),

we define the following function :

p :⊔φ∈Φ

Suppφ′

φ→ ON

which maps any element τλ ∈ Suppφ′

φto its position λ in Supp

φ′

φ.

Note that, given any φ, ψ ∈ Φ and any τ(φ), τ(ψ) in the domain of Iφ,ψ , respectively Iψ,φ ,we have p(τ(φ)) = p(τ(ψ)) if and only if Iφ,ψ(τ(φ)) = τ(ψ) (if and only if Iψ,φ(τ(ψ)) =τ(φ) ).

The main challenge in using (D1) (and (D2)) to extend d to Γ (and then to K) is tofind a criterion on dΦ such that the corresponding families in (SD1) (respectively in(SD2)) are summable. We isolate the following two crucial ”bad” hypotheses:


(H1) there exists an infinite decreasing sequence (φn)n∈N ⊂ Φ and an infinite se-quence (τ(n))n∈N ⊂ Γ such that for any n, τ(n) ∈ Supp

φ′nφn

and τ(n) 4 τ(n+1) 4

Iφn,φn+1(τ(n))

(H2) there exist infinite increasing sequences (φn)n∈N ⊂ Φ and (τ(n))n∈N ⊂ Γ such

that for any n, τ(n) ∈ Supp φ′n

φnand LF

(τ(n+1)

τ(n)

)< φn+1 .

Theorem 3.5 A map d : Φ→ K\{0} extends to a series derivation on K if and onlyboth hypothesis (H1) and (H2) fail.

To emphasize the role of each hypothesis, we divide the proof of the theorem into thestatement and the proof of the two following lemmas 3.6, 3.9.

Lemma 3.6 .A map d : Φ→ K\{0} extends to a series derivation on Γ if and only if (H1) fails.

Proof. 1 Suppose that (H1) holds. There exists a decreasing sequence (φn)n∈N such

that for all n, there are τ(n) ∈ Supp φ′n

φnwith τ(n) 4 τ(n+1) . Applying Lemma 2.1 to the

sequence (τ(n))n∈N , there exists a sub-sequence which is either increasing or constant.

Thus the family(φ′nφn

)n∈N

is not summable.

Conversely, suppose that (SD1) does not hold. So there exists an anti-well-ordered

subset E ⊂ Φ such that the family(φ′

φ

)φ∈E

fails to be summable. That is, the support

of this family contains an infinite strictly increasing sequence or an infinite constant

sequence (τ(n))n∈N , say τ(n) ∈ Suppφ′nφn

for some φn ∈ E , with the τ(n) ’s determining

distinct φn ’s in the case where (τ(n))n∈N is constant. We claim that also in the casewhere (τ(n))n∈N is strictly increasing, we may assume without loss of generality that

the φn ’s are distinct. Indeed since for any φ ∈ E , Supp (φ′

φ) is anti-well-ordered in

Γ, we have {τ(n) | n ∈ N} ∩ Supp (φ′

φ) is finite. In other words, the map

{τ(n) | n ∈ N} → {φn | n ∈ N}τ(n) 7→ φn


has infinitely many finite fibers. Choosing a complete set of representatives for the setof fibers, we may extract an infinite increasing subsequence of (τ(n))n∈N which satisfiesthe assertion of the claim (and which we continue to denote by (τ(n))n∈N below).

We now apply Lemma 2.1 to the sequence S = (φn)n∈N :Since S ⊆ E and E is anti-well-ordered in Φ, S cannot have an infinite strictly increas-ing subsequence. So S has an infinite decreasing subsequence which we continue todenote by (φn)n∈N for convenience.

Consider the following restriction of p (see Definition 3.4) :

p : {τ(n) | n ∈ N} → {λn | n ∈ N} ⊂ ONτ(n) 7→ λn := f (τ(n))

Suppose p has an infinite fiber, say p−1(λ) = {τ(nj) | j ∈ N}. Note that for any ordinal λin the image of p and any elements τ(k), τ(l) (k , l) in p−1(λ), we have Iφk ,φl(τ

(k)) = τ(l)

(see the final remark in Definition 3.4). Thus we have τ(nj) 4 τ(nj+1) = Iφnj ,φnj+1 (τ(nj))

for any j, satisfying conditions of (H1). Otherwise, all the fibers of p are finite, whichimplies that the image of p is infinite.

We shall define by induction an increasing subsequence (λ̃i)i∈N of (λn)n∈N , togetherwith the desired sequence (τ(nj))j∈N . Set λ̃0 := min{λn | n ∈ N} and let τ(n0) 4 τ(n1) 4· · · 4 τ(nj0 ) be the elements of its fiber p−1(λ̃0). Now suppose that we have built a finitesequence τ(n0) 4 τ(n1) 4 · · · 4 τ(nj0 ) 4 · · · 4 τ(nji ) together with an ordinal λ̃i for somei ≥ 0. Then, set λ̃i+1 := min{λn | n > nji} and let τ(nji+1) 4 · · · 4 τ

(nji+1) be the elements

of p−1(λ̃i+1) ∩ {τ(n) | n > nji}. By construction, {τ(n) | n > nji} ∩ i⋃

l=0

p−1(λ̃l)

= ∅. Soλ̃i+1 > λ̃i . Moreover, τ(nji+1) ≺ Iφnji ,φnji+1 (τ

(nji )). Thus we obtain as required an infinitesequence (τ(nj))j∈N such that τ(nj) 4 τ(nj+1) ≺ Iφnj ,φnj+1 (τ

(nj)) if j = ji for some i ≥ 0,and τ(nj) 4 τ(nj+1) = Iφnj ,φnj+1 (τ

(nj)) otherwise. �

We deduce from the preceding lemma a more explicit sufficient condition (but notnecessary: see Example Figure 2) such that (SD1) holds.

Corollary 3.7 Let a map d : Φ → K\{0} be given. Then d extends to a seriesderivation on Γ if the following property holds :(H1’) the set E1 = {φ ∈ Φ | ∃ψ � φ, Iψ,φ is not a left shift} is well ordered in Φ.

Proof. 2 For any decreasing sequence S = (φn)n∈N , since E1 ⊂ Φ is well-ordered,E∩S is finite. So all but finitely many couples (φn, φn+1) are such that Iφn,φn+1 is a leftshift. It implies that we can not obtain a sequence as in (H1). �


To visualize (H1’), we illustrate in Figure 1 the supports Suppφ′

φfor some φ ∈ Φ.

The ordered sets Φ and Γ are represented as linear orderings.


Γ

ψ ∈ (Φ\E1)Iφ,ψ

Φ

φ ∈ (Φ\E1)

φ̃ ∈ E1

Figure 1. Illustration of (H1′)

Under an additional hypothesis, we deduce from Lemma 3.6 a necessary and sufficientcondition for d to extend to a series derivation:

Corollary 3.8 Let a map d : Φ → K\{0} be given. We suppose that there existsN ∈ N such that, for any φ ∈ Φ, Card

(Supp

φ′

φ

)≤ N . Then d extends to a series

derivation on Γ if and only if the following property holds :(H1”) for any decreasing sequence (φn)n∈N ⊂ Φ, there exists a pair of integers m < nsuch that Iφm,φn is a left shift.

Proof. 3 Suppose that (SD1) does not hold. As in the proof of Lemma 3.6, thisimplies that there exist a decreasing sequence (φn)n∈N and an increasing or constant

sequence τ(0) 4 τ(1) 4 · · · with τ(n) ∈ Supp φ′n

φnfor any n. We set kn := p(τ(n)) ∈

{1, . . . ,N}, n ∈ N (see Definition 3.4). Applying Lemma 2.1 to the sequence (kn)n∈N ,there exists a constant subsequence (kni = k)i∈N . Hence, for any i < j, τ

(ni) 4 τ(nj) =


Iφni ,φnj (τ(ni)) (see the final remark in Definition 3.4). The sequence (φni)i∈N is such that

the corresponding isomorphisms Iφni ,φnj for any i < j fail to be left shifts.

Conversely, suppose that there exists a decreasing sequence (φn)n∈N for which theIφm,φn ’s, m < n, are not left shifts. That is, given m, for any n > m, there exists

τ(m) ∈ Supp φ′m

φmsuch that τ(m) 4 Iφm,φn(τ

(m)). We set lm,n := p(τ(m)) ∈ {1, . . . ,N}for any n. By Lemma 2.1, there exists a constant subsequence (lm,ni = lm)i∈N , thatis we have τ(m) 4 Iφm,φni (τ

(m)) for any i ≥ 0. Now, consider the sequence (lm)m∈N .Again by Lemma 2.1 and since for any m lm ∈ {1, . . . ,N}, there exists a constantsubsequence, say (lmj = l)j∈N for some l ∈ {1, . . . ,N}. Hence for any j ∈ N, τ(mj) 4Iφmj ,φmj+1 (τ

(mj)) = τ(mj+1) . This implies that (SD1) does not hold for the family {φmj | j ∈N}. �

Example: If we drop the assumption that the cardinality of Suppφ′

φis uniformly

bounded, (SD1) may still hold even if (H1′′) fails, as illustrated by the followingFigure 2.The dashed lines indicate changes of comparability classes (for instance, take τ0,k =φ1 ≺ Iφ0,φk (τ0,k) = φ

1/k0 for any k ∈ N

∗ ). The lines connect τk,l and Iφk ,φl(τk,l) forwhich the isomorphism Iφk ,φl fails to be a left shift.We observe that even if there is an infinite decreasing sequence (φn)n∈N for whichthe Iφn,φn+1 ’s are not left shifts, (SD1) holds (we obviously can not build any infiniteincreasing or constant sequence of τ’s).


Γ

φ0

φ1

φ2

Figure 2. Counter-example when Card(Supp φ′

φ )is not uniformely bounded.

Φ

Now we prove the second lemma that completes the proof of Theorem 3.5.

Lemma 3.9 Let d a series derivation on Γ be given. Then d extends to a seriesderivation on K if and only if (H2) fails.

Proof. 4 First, we suppose that (H2) holds. For any n ∈ N, set τ(n+1)

τ(n)= ψηn+1n+1 γ

(n+1)

where ψn+1 = LF(τ(n+1)

τ(n)

), ηn+1 = LE

(τ(n+1)

τ(n)

)and γ(n+1) ∈ Γ. Then ψn+1 <

φn+1 , ηn+1 > 0 (the sequence (τ(n))n is increasing) and LF(γ(n+1)

)≺ ψn+1 . Consider

now the sequence (α(n))n∈N where α(0) = φ−�00 for some �0 > 0, α

(n+1) = φ−�n+1n+1 forsome �n+1 > 0 if ψn+1 � φn+1 , and α(n+1) = (φηn+1n+1 γn+1)

−1 = (α(n))−1 if ψn+1 =φn+1 . This sequence is decreasing since the sequence (φn)n∈N is increasing. Moreover,setting β(n) := α(n)τ(n) , we have β(n) ∈ Supp (αn)′ for any n (see (D1)). Then it is

routine to prove thatβ(n+1)

β(n)=

α(n+1)τ(n+1)

α(n)τ(n)� 1, meaning the sequence (β(n))n∈N


is increasing. It implies that the family ((αn)′)n∈N is not summable, witnessing that(SD2) does not hold.

Conversely, suppose that (SD2) does not hold. There exists an anti-well-ordered set ofmonomials E ⊂ Γ such that the set

⋃α∈E

Supp α′ contains either an increasing in-

finite sequence or a constant one (coming from distinct α’s in the constant case),say β(0) 4 β(1) 4 β(2) 4 · · · . Set (α(n))n∈N the corresponding sequence in E suchthat β(n) ∈ Supp (α(n))′ for any n. From (D1), we note that for any α, Supp α′ ⊂α.

⋃φ∈supp α

Suppφ′

φ. Hence, for any n we set β(n) = α(n)τ(n) for some τ(n) ∈ Supp φ

′n

φn

with φn ∈ supp α(n) . We claim that in the case where (β(n))n∈N is strictly increasing,we may assume without loss of generality that the α(n) ’s are distinct. Indeed since forany α, Supp α′ is anti-well-ordered in Γ, we have {β(n) | n ∈ N} ∩ Supp α′ is finite. Inother words, the map

{β(n) | n ∈ N} → {α(n) | n ∈ N}β(n) 7→ α(n)

has infinitely many infinite fibers. Choosing a complete set of representatives for theset of fibers, we may extract an infinite increasing subsequence of (β(n))n∈N whichsatisfy the assertion of the claim (and which we continue to denote by (β(n))n∈N below).

We now apply Lemma 2.1 to the sequence S = (α(n))n∈N . Since S ⊆ E and E isanti-well-ordered in Γ, S cannot have an infinite strictly increasing subsequence. SoS has an infinite decreasing subsequence which we continue to denote (α(n))n∈N forconvinience.

Since for any k < l ∈ N, β(k) = α(k)τ(k) 4 β(l) = α(l)τ(l) , we have :

(1) ∀k < l ∈ N, 1 ≺ α(k)

α(l)4τ(l)

τ(k)

The sequence (τ(n))n∈N is strictly increasing.

Now consider the corresponding sequence (φn)n∈N (for which τ(n) ∈ Suppφ′nφn

for any

n). Considering the map

{τ(n) | n ∈ N} → {φn | n ∈ N}τ(n) 7→ φ(n)

we may assume without loss of generality as above that the φn ’s are distinct.


We apply Lemma 2.1 to the sequence S̃ = (φn)n∈N . Suppose that it has an infinite de-creasing subsequence, say Ŝ = (φni)i∈N . This anti-well-ordered subset Ŝ ⊂ Φ would besuch that the corresponding subsequence (τ(ni))i∈N is increasing, contradicting (SD1).So S̃ has an infinite increasing subsequence which we continue to denote (φn)n∈N forconvinience.

We shall define by induction increasing subsequences (φni)i∈N of (φn)n∈N and (τ(ni))i∈N

of (τ(n))n∈N as in the statement of (H2). Set n0 = 0 and recall that for any n,φn ∈ supp α(n) . Suppose that we have subsequences φn0 ≺ φn1 ≺ · · · ≺ φni andτ(n0) ≺ τ(n1) ≺ · · · ≺ τ(ni) for some i ≥ 0. Since the sequence (φn)n∈N is increas-ing and supp α(ni) is anti-well-ordered in Φ, there exists a lowest index ni+1 > ni

such that φni+1 < supp α(ni) . But φni+1 ∈ supp α(ni+1) . So φni+1 ∈ supp

α(ni)

α(ni+1)and

LF(α(ni)

α(ni+1)

)< φni+1 . Moreover by (1) we have LF

(τ(ni+1)

τ(ni)

)< LF

(α(ni)

α(ni+1)

). So

LF(τ(ni+1)

τ(ni)

)< φni+1 as required. �

We deduce a positive version of (H2), i.e. a necessary and sufficient condition suchthat (SD2) holds :

Corollary 3.10 Let a series derivation d on Γ be given. Then d extends to a seriesderivation on K if and only if the following property holds :(H2’) for any infinite increasing sequences (φn)n∈N ⊂ Φ and (τ(n))n∈N ⊂ Γ such that

for any n, τ(n) ∈ Supp φ′n

φn, the set S =

{n ∈ N | LF

(τ(n+1)

τ(n)

)< φn+1

}is finite.

Proof. 5 Hypothesis (H2′) implies that (H2) does not hold, which means that (SD2)holds. Conversely, suppose that there exist infinite increasing sequences (φn)n∈N ⊂ Φand (τ(n))n∈N ⊂ Γ as in (H2’), for which S is infinite. Denote S = {ni | i ∈ N} withni < ni+1 for all i, and set mi := ni + 1, i ∈ N. We notice that

τ(mi+1)

τ(mi)=τ(ni+1+1)

τ(ni+1)=

τ(ni+1+1)

τ(ni+1)τ(ni+1)

τ(ni+1−1)· · · τ

(ni+2)

τ(ni+1). Moreover we have LF

(τ(ni+1+1)

τ(ni+1)

)< φni+1+1 and for

any n such that ni < n < ni+1 , LF(τ(n+1)

τ(n)

)≺ φn+1 . So applying the ultrametric

inequality for LF , we have LF(τ(jk+1)

τ(jk)

)= LF

(τ(ni+1+1)

τ(ni+1)

)< φni+1+1 . Thus the

increasing sequences (φmi)i∈N and (τ(mi))i∈N verify (H2). �


From Lemma 3.9 and Corollary 3.7 we deduce a more explicit sufficient (but not nec-essary) condition such that a map d : Φ → K\{0} extends to a series derivation onK:

Corollary 3.11 Assume that d : Φ → K\{0} satisfies (H1’), then d extends to aseries derivation on K if the following property holds :(H2”) the set

E2 : = {ψ ∈ Φ |∃φ ≺ ψ, ∃τ(φ) ∈ Suppφ′

φ, ∃τ(ψ) ∈ Supp ψ

′

ψs.t. LF (

τ(φ)

τ(ψ)) < ψ }

is anti-well-ordered in Φ.

Proof. 6 By Corollary 3.7, d extends to a series derivation on Γ. From Lemma 3.9,(SD2) does not hold if and only if there exist infinite increasing sequences (φn)n∈N ⊂ Φ

and (τ(n))n∈N ⊂ Γ such that for any n, τ(n) ∈ Suppφ′nφn

and LF(τ(n+1)

τ(n)

)< φn+1 . But

from (H2′′), for any increasing sequence S = (φn)n∈N , since E2 ⊂ Φ is anti-well-

ordered, E2 ∩ S is finite. So for all but finitely many n, we have LF(τ(n+1)

τ(n)

)≺ φn+1

for any τ(n) ∈ Supp φ′n

φnand any τ(n+1) ∈ Supp

φ′n+1φn+1

. This contradicts (H2). �

Example: If we omit the assumption that the sequence (τ(n))n∈N is increasing in (H2)(or (H2′)), the condition is not anymore necessary, even if we restrict to the case that

the supports ofφ′

φare finite and uniformly bounded as in Corollary 3.8. Indeed we

have the following example :given an infinite increasing sequence (φn)n∈N , suppose that there exists ψ ∈ Φ such

that ψ � φn for any n. Then defineφ′0φ0

= τ(0)1 + τ(0)2 = 1 + ψ

−1 and for any n ∈ N∗ ,φ′nφn

= τ(n)1 + τ(n)2 = φn−1 + ψ

−1φn−1 .

We observe that any infinite increasing sequence of τ’s contains either infinitely many

τ1 ’s, or infinitely τ2 ’s. Moreover for any k < l, LF

τ(k)1τ(l)1

= LFτ(k)2τ(l)2

= φl−1 ≺ φl .So (SD2) holds, even if for any n ∈ N, LF

τ(n+1)2τ(n)1

= LF τ(n+1)1τ(n)2

= ψ � φn+1 .


4 Hardy type derivations.

Definition 4.1 Let (K,4,C) be a field endowed with a dominance relation (cf. Defi-nition 2.6), which contains a sub-field C isomorphic to its residue field K41/K≺1 (soK41 = C ⊕ K≺1 ). A derivation d : K → K is a Hardy type derivation if :

(1) the sub-field of constants of K is C : ∀a ∈ K, a′ = 0⇔ a ∈ C.

(2) d verifies l’Hospital’s rule : ∀a, b ∈ K\{0} with a, b - 1 we havea 4 b⇔ a′ 4 b′ .

(3) d is compatible with the logarithmic derivative (in the sense of Hardy fields):

∀a, b ∈ K with |a| � |b| � 1,we have a′

a<

b′

b. Moreover,

a′

a� b

′

bif and only if

a and b are comparable.

Axioms 1. and 2. are exactly those which define a differential valuation ((14, Defi-nition p. 303); see Theorem 1 and Corollary 1 for the various versions of l’Hospital’srule that hold in this context). Axiom 3. is a slightly stronger version of the propertiesobtained in (15, Propositions 3 and 4) and (14, Principle (*) p. 314) in the context ofHardy fields.

Below we prove the following criterion for a series derivation to be of Hardy type. Setθ(φ) := LM (φ′/φ).

Theorem 4.2 A series derivation d on K verifies l’Hospital rule and is compatiblewith the logarithmic derivative if and only if the following condition holds:

(H3’) : ∀φ ≺ ψ ∈ Φ, θ(φ) ≺ θ(ψ) and LF(θ(φ)

θ(ψ)

)≺ ψ.

Proof. 7 We suppose that (H3′) holds. To prove l’Hospital’s rule on K, it sufficesto prove it for the monomials. Let α =

∏φ∈supp α

φαφ and β =∏

φ∈supp βφβφ be arbitrary

monomials. Then α′ = α∑φ

αφφ′

φ� αθ(φ0) and β′ = β

∑φ

βφφ′

φ� βθ(φ1) where

φ0 = LF (α) and φ1 = LF (β).

If φ0 = φ1 , then θ(φ0) = θ(φ1) . Soα′

β′� αβ

.

If φ0 , φ1 , for instance φ0 ≺ φ1 , then LF(α

β

)= φ1 . But

α′

β′� α

β

θ(φ0)

θ(φ1), and


LF(θ(φ0)

θ(φ1)

)≺ φ1 . Applying the ultrametric inequality for LF, we obtain LF

(α′

β′

)=

φ1 and LE(α′

β′

)= LE

(α

β

). Thus

α′

β′and

α

βhave same sign.

To prove the compatibility with the logarithmic derivative, take a, b ∈ K with |a| �|b| � 1 and denote α = LM (a), β = LM (b), φ0 = LF (a) = LF (α) and φ1 =

LF (b) = LF (β). So we have LM(a′

a

)=

(α′

α

)= LM

(φ′0φ0

)= θ(φ0) and similarly

LM(b′

b

)= θ(φ1) (Lemma 2.4). Since |a| � |b| � 1, we have φ0 < φ1 . So θ(φ0) < θ(φ1)

by (H3′).Moreover, a and b are comparable if and only if φ0 = φ1 , which means that θ(φ0) =θ(φ1) .

Conversely, for φ, ψ ∈ Φ with φ ≺ ψ we have φ′

φ≺ ψ

′

ψ, since d is assumed to

be compatible with the logarithmic derivative (recall that φ � 1 for any φ ∈ Φ by

construction). Thus LM (φ′

φ) ≺ LM (ψ

′

ψ), that is θ(φ) ≺ θ(ψ) . In particular 1 ≺ θ

(ψ)

θ(φ).

Now consider any two real exponents r < 0 and s , 0. If φ, ψ ∈ Φ with φ ≺ ψ wehave ψr ≺ φs . Differentiating both sides of this inequality and applying l’Hospital’srule we obtain 1 ≺ ψ

′/ψ

φ′/φ≺ φsψ−r . Now LF (φsψ−r) = ψ and LE (φsψ−r) = −r > 0.

Thus LF (θ(φ)

θ(ψ)) ≺ ψ . �

Corollary 4.3 A series derivation d on K which verifies l’Hospital rule and is com-patible with the logarithmic derivative is a Hardy type derivation.

Proof. 8 By construction the field of coefficients R is included in the field of con-stants (see (D0), (D2)). Conversely, consider a non-constant series a =

∑α∈Supp a

aαα ∈

K\{0} such that a′ = 0. By (D1), we have a′ =∑

α∈Supp aaα(α)′ . Set α(0) =

max((Supp a)\{1}). By l’Hospital’s rule, we have (α(0))′ � α′ for any α ∈ ((Supp a)\{α(0)}).Thus we would have (α(0))′ = 0.But, setting φ0 = LF (α(0)), by (D1) and (H3′) we obtain (α(0))′ � α(0)θ(φ0) whichis non zero. Thus (α(0))′ cannot be zero, neither do a′ : this contradicts the initialassumption. �


Remark 4.4 Note that the first part of (H3′) is the particular case for fundamentalmonomials of the property that holds in Hardy fields (see Propositions 3 and 4 in (15)).This property also holds in pre-H-fields (see Lemma 3.5 in (3)).

If we want to get even closer to the functional case, that is if we want to interpret theφ’s as, for instance, germs of functions in a Hardy field, we need to assume that for any

φ ∈ Φ, we have φ′

φ> 0 since the logarithmic derivative of an infinite increasing germ

is always positive. This last assumption is equivalent to requiring that LC(φ′

φ

)> 0.

Then it is easy to prove that for any a ∈ K�1 , we have a′

a> 0, so (K, d) is an H-field

(see (3)).

5 Examples.

5.1 The monomial case.

Definition 5.1 A series derivation on K is monomial if for any φ ∈ Φφ′

φ= Tφθ(φ) ∈ R.Γ .

In this case, Corollary 3.8 applies with N = 1. Moreover by Theorem 4.2 it sufficesto define θ(φ) verifying Hypothesis (H3′) (with arbitrary non zero coefficients Tφ ) toobtain by (D1), (D2) and (D3) a series derivation of Hardy type.

1. For an elementary example, take the following chain of infinitely increasing realgerms at infinity (applying the usual comparison relations of germs) :

Φ := {expn(x) ; n ∈ Z}where expn denotes for positive n, the n’th iteration of the real exponential function,for negative n, the |n|’s iteration of the logarithmic function, and for n = 0 the identi-cal map. We denote K the generalized series field built with this chain Φ as chain offundamental monomials. Then applying the usual derivation on real germs, we obtain:

(expn(x))′

expn(x)= Πn−1k=1 expk(x) if n ≥ 2

(exp(x))′

exp(x)= 1

(expn(x))′

expn(x)= Πnk=0

1expk(x)

if n ≤ 0


So for any integers m < n, we have:

• expm(x) ≺ expn(x) and they are not comparable;

•(expm(x))

′

expm(x)≺ (expn(x))

′

expn(x);

•(expn(x))

′/ expn(x)(expm(x))′/ expm(x)

≺ expn(x).

The map expn(x) 7→ (expn(x))′ extends to a series derivation of Hardy type on K.

2. Let (Φ,4) be a totally ordered set that we suppose endowed with an endomorphisms : Φ→ Φ such that :a. if Φ has no least element, for all φ ∈ Φ, s(φ) ≺ φ

b. if Φ has a least element, say φm , then s(φ) ≺ φ for all φ , φm and s(φm) = φm .

Claim : if we define for all φ ∈ Φ, θ(φ) = s(φ) (respectively ∀φ , φm , θ(φ) = s(φ) andθ(φm) = 1), then these monomials verifies Hypothesis (H3′).

Indeed, for any φ1 , φ2 , with φ1,2 � φm in the case b, we have φ1 ≺ φ2 ⇔ θ(φ1) =

s(φ1) ≺ θ(φ2) = s(φ2). Moreover, LF(θ(φ1)

θ(φ2)

)= LF

(s(φ1)s(φ2)

)= max{s(φ1), s(φ2)} ≺

max{φ1, φ2}. That is we have (H3′) in case a.In case b. ∀φ � φm , s(φ) ≥ s(φm) = φm , and so θ(φ) = s(φ) � θ(φm) = 1. Moreover,

LF(θ(φ)

θ(φm)

)= LF (s(φ)) = s(φ) ≺ φ. Thus we have also (H3′) in this case.

So for any φ ∈ Φ, φ � φm if φm exists, we can define φ′ = Tφθ(φ)φ = Tφs(φ)φ, andφ′m = Tφmφm , with arbitrary Tφ ∈ R∗ . Thus Theorem 3.5 applies.

As a remark, we could have put θ(φ) = s(φ)αφ for any αφ > 0 fixed.

As an illustration by germs of real functions, take φ0 = eαx and φ1 = eβeαx

for anyfixed reals α > 0 and β > 0 (x is supposed close to +∞). Then, using the usualderivation, we have φ′0 = αe

αx and φ′1 = αβeαxeβe

αx= αβφ0φ1 . With our notations, it

means that we have Tφ0 = α, Tφ1 = αβ and θ(φ0) = 1, θ(φ1) = φ0 .

3.We deal with a generalization of the preceding example in case a or in case b whenΦ\{φm} has no least element. For any n ∈ N, denote sn the nth iterate of s. Given

N ∈ N ∪ {+∞}, one can set θ(φ) =N∏

n=1

sn(φ) for any φ ∈ Φ or φ ∈ Φ\{φm}, and

θ(φm) = 1.


We could also define θ(φ) =N∏

n=1

sn(φ)αφ,n with αφ1 > 0 and for all k ≥ 2, αφ,n ∈ R.

The corresponding Tφ can be chosen arbitrarily.

4. Assume that Φ is isomorphic to a subset of R with least element φm , writing f thisisomorphism, we can put for any φ ∈ Φ, θ(φ) = φf (φ)+βm where β is some fixed real.

In order to illustrate this, we take Φ = {φα = exα

; α > 0} ∪ {φ0 = x} which isisomorphic to R+ , and arbitrary Tφα ∈ R∗ . With the usual derivation, we have ∀α >0, φ′α = αx

α−1exα

= αφα−10 φα and φ′0 = 1. Thus, θ

(φα) = φα−10 and Tφα = α.

5. Assume that Φ begins with an anti-well-ordered subset Φ̃, we can :• fix some ψ0 ∈ Φ̃, write Φ0 = {φ ∈ Φ | φ ≺ ψ0}, and set s : Φ\Φ0 → Φ\Φ0 and thecorresponding θ(φ) as in first example. In particular, θ(ψ0) = 1• set φ0 := max Φ0 , and for all φ ∈ Φ0 , s(φ) equal to the predecessor of φ in Φ0 . Fixalso for any φ ∈ Φ0 some αφ ∈ R, with in particular αphi0 < 0. Then one can definefor any φ ∈ Φ0 , θ(φ) =

∏ϕ≺φ

ϕαϕ,φ∏

φ0


Proposition 5.2 Let (Φ,4) be a totally ordered set without least element and K bethe corresponding field of generalised series. Let γ be some fixed monomial in Γ. Weconsider an endomorphism of ordered set : s : Φ→ Φ with for any φ ∈ Φ, s(φ) ≺ φ.Then the map ds,Φ : φ 7→ γφ

∑τ∈S

TτΠn∈N(sn+1(φ))τn is well-defined with values in K

(where sn+1 denotes the (n + 1)th iterate of s). Moreover, this map extends to a seriesderivation of Hardy type on K.

Proof. 9 We prove that conditions (H1’) and (H2”) from Corollary 3.11 and (H3’) of

Theorem 4.2 hold. We note that for any φ ∈ Φ, we have φ′

φ=

∑τ∈S

TτγΠn∈N(sn+1(φ))τn .

For any φ � ψ in Φ, the ordered sets Supp φ′

φand Supp

ψ′

ψare isomorphic by con-

struction. Moreover, consider some τ(φ) ∈ Supp φ′

φ, say τ(φ) = γΠn∈N(sn+1(φ))τn for

some real τn ’s, n ∈ N. Then we have Iφ,ψ(τ(φ)) = τ(ψ) where τ(ψ) = γΠn∈N(sn+1(ψ))τn .

Moreover,τ(φ)

τ(ψ)= Πn∈N

(sn+1(φ)

)τn (sn+1(ψ))−τn with for all n, sn+1(φ) � sn+1(ψ)(since s is an embedding). Thus LF

(τ(φ)

τ(ψ)

)= sn0+1(φ) for some n0 ∈ N. Moreover

LE(τ(φ)

τ(ψ)

)= τn0 which is positive (since d ∈ L�1 ). Hence we obtain that:

•τ(φ)

τ(ψ)� 1, which means that Iφ,ψ is decreasing. Condition (H1’) holds (the set

E1 is empty).

• LF(τ(φ)

τ(ψ)

)= sn0+1(φ) ≺ φ since s is a decreasing emdomorphism of Φ. Con-

dition (H2”) holds (the set E2 is empty).

• the same properties hold in particular for the leading monomials θ(φ) and θ(ψ)

ofφ′

φand

ψ′

ψ. The condition (H3’) holds.

�

6 Asymptotic integration and integration

Definition 6.1 Let (K, d,4) be a differentiable field endowed with a dominance rela-tion 4, and let a be one of its elements.


We say that a admits an asymptotic integral b if there exists b ∈ K \ {0} such thatb′ − a ≺ a.We say that a admits an integral b if there exists b ∈ K \ {0} such that b′ = a.

The following main result about asymptotic integration in fields endowed with a Hardytype derivation is an adaptation of (15, Proposition 2 and Theorem 1).

Theorem 6.2 (Rosenlicht) Let (K,4,C, d) be a field endowed with a Hardy typederivation d . Let a ∈ K\{0}, then a admits an asymptotic integral if and only ifa - g.l.b.4

{b′

b; b ∈ K\{0}, b - 1

}(if it exists). Moreover, for any such a, there

exists u0 ∈ K\{0} with u0 - 1 such that for any u ∈ K\{0} such that u±10 < u±1 � 1,then (

a.au/u′

(au/u′)′

)′∼ a

Proof. 10 Our statement is a straightforward combination of Proposition 2 and Theo-rem 1 in (15). It suffices to observe that the corresponding proofs in (15) only rely onthe fact that the canonical valuation of a Hardy field is a differential valuation and thatthe derivation is compatible with the logarithmic derivative (15, Proposition 3). �

In (15, Lemma 1), Rosenlicht provides a method to compute u0 :

• since a - g.l.b.4

{b′

b; b ∈ K\{0}, b - 1

}, we assume w.l.o.g. that

a � g.l.b.4{

b′

b; b ∈ K\{0}, b - 1

}(if not, take a−1 instead of a);

• take u1 � 1 such that a �u′1u1

• take any u0 such that u±10 � min{

u1,a

u′1/u1

}.

• as a remark, u0 verifies 1 � u±10 <(

au′0/u0

)±1. So LF (u0) 4 LF

(a

u′0/u0

).

Our contribution here is to deduce explicit formulas for asymptotic integrals for ourfield of generalised series K = R((Γ)) endowed with a Hardy type derivation. Notethat this is equivalent (by l’Hospital’s rule) to provide formulas for asymptotic integrals


of monomials. We recall also that for any b ∈ K\{0}, b - 1, we have b′

b� φ

′

φ� θ(φ)

where φ = LF (b). So g.l.b.4

{b′

b; b ∈ K\{0}, b - 1

}= g.l.b.4

{φ′

φ; φ ∈ Φ

}=

g.l.b.4{θ(φ); φ ∈ Φ

}.

Notation 6.3 For any monomial α ∈ Γ, α - 1, for any ψ ∈ supp α, we denote αψthe exponent of ψ in α. In particular, for any φ ∈ Φ, for any ψ ∈ supp θ(φ) , we denoteθ

(φ)ψ the exponent of ψ in θ

(φ) . We also set Fφ := LC(φ′

φ

).

Corollary 6.4 Let α ∈ Γ be some monomial such that α - g.l.b.4{θ(φ); φ ∈ Φ

}. If

α - 1, set φ0 := LF (α). It implies that LT(α′

α

)= αφ0 LT

(φ′0φ0

)= αφ0Fφ0θ

(φ0) .

Then we have :

• if LF (θ(φ0)) 4 φ0 � LF(α

θ(φ0)

), then

1Fφ0(αφ0 − θ(φ0)φ0 )α

θ(φ0)

′

∼ α;

• if LF (θ(φ0)) = φ1 � φ0 , then

1−Fφ1θ(φ0)φ1α

θ(φ1)

′

∼ α (note that θ(φ0)φ1 = θ(φ1)φ1

);

• if LF (θ(φ0)) � φ0 � LF(α

θ(φ0)

)or if α = 1, then

1Fφ1(αφ1 − θ(φ1)φ1 )α

θ(φ1)

′

∼ α

where φ1 is the element of Φ such that LF(α

θ(φ1)

)= φ1 ≺ φ0 .

Proof. 11 For the first case, it suffices to observe that LF(α

θ(φ0)

)= φ0 with exponent

αφ0 − θ(φ0)φ0

. So we have :(α

θ(φ0)

)′∼ αθ(φ0)

(αφ0 − θ(φ0)φ0

)φ′0φ0∼ αθ(φ0)

(αφ0 − θ(φ0)φ0

)Fφ0θ(φ0) .

For the second one, since LF (θ(φ1)) = φ1 � φ0 , from (H3) we deduce that LF (θ(φ0)) =φ1 with the same exponent θ

(φ0)φ1

. So LF(α

θ(φ1)

)= φ1 with exponent −θ(φ0)φ1 , and then(

α

θ(φ1)

)′∼ αθ(φ1)

(−θ(φ0)φ1 )φ′1φ1∼ αθ(φ1)

(−θ(φ0)φ1 Fφ1)θ(φ1) .

For the third one, firstly we have to show that there exists φ1 as in the statement of thecorollary. We define u0 corresponding to α as in the preceding theorem and we denote

φ̂0 = LF (u0) and φ1 = LF(

α

u′0/u0

). So we have LM

(u′0u0

= β0θ(φ̂0)). Moreover by

Rosenlicht’s computation of u0 , we note that φ̂0 4 φ1 . Thus we obtain by (H3) that


LFθ(φ̂0)θ(φ1)

≺ φ1 . and as desired:φ1 = LF

(α

θ(φ̂0)

)= LF

αθ(φ̂0)

.θ(φ̂0)

θ(φ1)

= LF

(α

θ(φ1)

)Secondly, we compute :(

α

θ(φ1)

)′∼ α

θ(φ1)(αφ1 − θ

(φ1)φ1

)φ′1φ1

∼ (αφ1 − θ(φ1)φ1

)α

θ(φ1)Fφ1θ

(φ1)

= Fφ1(αφ1 − θ(φ1)φ1

)α

.

�

Concerning integration, we apply to our context (12, Theorem 55) (recall that fields ofgeneralised series are pseudo-complete (see e.g. 11, Theorem 4, p. 309)).

Corollary 6.5 Assume that K is endowed with a series derivation of Hardy typederivation d . Set θ̃ = g.l.b.4

{θ(φ) φ ∈ Φ

}(if it exists). Then, any element a ∈ K with

a ≺ θ̃ admits an integral in K. Moreover K is closed under integration if and only ifθ̃ < Γ.

Proof. 12 Firstly, since d is a Hardy type derivation we notice that for any b ∈K\{0}, b - 1, then LM

(b′

b

)= LM

(β′

β

)where β = LM (b). But for any

β ∈ Γ\{1}, LM(β′

β

)= LM

(φ′

φ

)where φ = LF (β) = LF(b). So LM

(b′

b

)=

LM(φ′

φ

)and thus

g.l.b.4

{b′

b; b ∈ K\{0}, b - 1

}= g.l.b.4

{θ(φ) = LM

(φ′

φ

); φ ∈ Φ

}.

Secondly, given a ∈ K with a ≺ θ̃ , then there exists a monomial γ ∈ Γ that is anasymptotic integral of a. That is γ′ � a. But since d verifies l’Hospital’s rule, itimplies that for any γ̃ ∈ Supp γ′ , γ̃ ≺ θ̃ . So it admits itself an asymptotic integral. theresult now follows from (12, Theorem 55). �

Examples 2 and 3 in the case when Φ has no least element and the one of Proposition5.2 are closed under integration.


Final Remark: The results in this paper hold for an arbitrary field of coefficients Cand an arbitrary ordered group of exponents (Definition 2.2) G , with the followingrequirement due to the strong Leibniz rule (Axiom (D1) in Section 3): we requirethe existence of a group embedding (G,+) → (C,+). Moreover, given a chain offundamental monomials Φ ( 2.2), we can choose a particular group of exponents Gφfor each fundamental monomial φ ∈ Φ, provided that each of these groups embedsinto the field of coefficients C.

Acknowledgements: The last versions of this paper were written while both authorswere participants of the Thematic Program on o-minimal Structures and Real AnalyticGeometry at the Fields Institute, Toronto, Canada. We wish to thank the Institute forits hospitality.

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Research Center for Algebra, Logic and Computation, University of Saskatchewan, S7N 5E6,Canada.

I.M.B., Université de Bourgogne, , avenue Savary, B.P. 47870, 21078 Dijon cedex, France

[email protected], [email protected]

http://math.usask.ca/˜skuhlman/, http://sites.google.com/site/mickaelmatusinski/

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1 Introduction2 Preliminary definitions3 Defining derivations on generalised series fields4 Hardy type derivations.5 Examples.5.1 The monomial case.5.2 A general example.

6 Asymptotic integration and integration

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Hardy type derivations on generalised series ﬁeldsmath.usask.ca/~skuhlman/KuhlmannMatuPart1-14.pdf · 2009. 8. 31. · of real valued functions in a Hardy ﬁeld. We provide a necessary