Ilyashenko algebras based on transserial asymptotic expansions
aa r X i v : . [ m a t h . L O ] J a n ILYASHENKO ALGEBRAS BASED ON TRANSSERIALASYMPTOTIC EXPANSIONS
ZEINAB GALAL, TOBIAS KAISER AND PATRICK SPEISSEGGER
Abstract.
We construct a Hardy field that contains Ilyashenko’sclass of germs at + ∞ of almost regular functions found in [12] aswell as all log-exp-analytic germs. This implies non-oscillatory be-havior of almost regular germs with respect to all log-exp-analyticgerms. In addition, each germ in this Hardy field is uniquely char-acterized by an asymptotic expansion that is an LE-series as de-fined by van den Dries et al. [7]. As these series generally havesupport of order type larger than ω , the notion of asymptotic ex-pansion itself needs to be generalized. Introduction
The purpose of this paper is to extend Ilyashenko’s construction in[12] of the class of germs at + ∞ of almost regular functions to obtaina Hardy field containing them. In addition, each germ in this Hardyfield is uniquely characterized by an asymptotic expansion that is anLE-series as defined by van den Dries et al. [7] and a transseries asdefined by van der Hoeven [11]. As these series generally have supportof order type larger than ω , the notion of asymptotic expansion itselfneeds to be generalized. This can be done naturally in the context of aquasianalytic algebra, leading to our definition of quasianalytic asymp-totic algebra , or qaa algebra for short. Any qaa algebra constructedby generalizing Ilyashenko’s construction will be called an Ilyashenkoalgebra .The Hardy field H = H an , exp of all unary germs at + ∞ of unaryfunctions definable in the o-minimal structure R an , exp is an example ofan Ilyashenko field; see van den Dries and Miller [8] and van den Drieset al. [5]. The third author’s paper [25] contains a first attempt at Date : Wednesday 9 th January, 2019 at 03:09.1991
Mathematics Subject Classification.
Primary 26A12, 41A60, 30E15; Sec-ondary 37E35, 03C99.
Key words and phrases.
Transserial asymptotic expansions, quasianalyticity,Hardy fields, analysable functions.Supported by NSERC of Canada grant RGPIN 261961 and the Zukunftskollegof the University of Konstanz. constructing an Ilyashenko field F containing Ilyashenko’s almost reg-ular germs. The implied non-oscillatory properties of its germs wereused in Belotto et al.’s recent solution [3] of the strong Sard conjec-ture. However, this field F does not contain H ; the Ilyashenko fieldconstructed here is a Hardy field that contains both F and H , implyingnon-oscillatory behaviour with respect to all log-exp-analytic germs.Our main motivation for generalizing Ilyashenko’s construction inthis way is the conjecture that the class of almost regular germs gen-erates an o-minimal structure over the field of real numbers. Thisconjecture, in turn, might lead to locally uniform bounds on the num-ber of limit cycles in subanalytic families of real analytic planar vec-tor fields all of whose singularities are hyperbolic. Establishing suchuniform bounds for planar polynomial vector fields follows Roussarie’sapproach [23] to Hilbert’s 16th problem (part 2); see Ilyashenko [13] foran overview on the latter. Our conjecture implies a generic instanceof Roussarie’s finite cyclicity conjecture [22]; see the third author’spreprint [26] explaining this connection. In Kaiser et al. [17] we gavea positive answer to our conjecture in the special case where all singu-larities are, in addition, non-resonant. (For a different approach to thegeneral hyperbolic case, see Mourtada [20].)The almost regular germs also play a role in the description of Rie-mann maps and solutions of Dirichlet’s problem on semianalytic do-mains; see Kaiser [15, 16] for details. Finally, in the spirit of theconcluding remark of van den Dries et al. [6], this paper provides arigorous construction of a Hardy subfield of ´Ecalle’s field of “fonctionsanalysables” [10] that properly extends H , and we do so without the useof “acc´el´ero-sommation”; for more details on this, see the concludingremarks in Section 10.We plan to eventually settle our o-minimality conjecture by adaptingthe procedure in [17], which requires three main steps:(1) extend the class of almost regular germs into an Ilyashenko field;(2) construct corresponding algebras of germs of functions in sev-eral variables, such that the resulting system of algebras is sta-ble under various operations (such as blowings-up, say);(3) obtain o-minimality using a normalization procedure.While [25] contains a first successful attempt at Step (1), Step (2) posessome challenges. For instance, it is not immediately obvious what thenature of LE-series in several variables should be; they should at leastbe stable under all the operations required for Step (3). They should lyashenko algebras 3 also contain the series used in [20] to characterize parametric transi-tion maps in the hyperbolic case, which use so-called ´Ecalle-Roussariecompensators as monomials.Our approach to this problem is to enlarge the set of monomials usedin asymptotic expansions. A first candidate for such a set of monomi-als is the set of all (germs of) functions definable in the o-minimalstructure R an , exp (in any number of variables). This set of germs isobviously closed under the required operations, because the latter areall definable, and it contains the ´Ecalle-Roussarie compensators. How-ever, it is too large to be meaningful for use as monomials in asymptoticexpansions, as it is clearly not R -linearly independent (neither in theadditive nor the multiplicative sense) and contains many germs thathave “similar asymptotic behavior” such as, in the case of unary germs,belonging to the same archimedean class. More suitable would be tofind a minimal subclass L n of all definable n -variable germs such thatevery definable n -variable germ is piecewise given by a convergent Lau-rent series (or, if necessary, a convergent generalized Laurent series, seevan den Dries and Speissegger [9]) in a finite tuple of germs in L n .Thus, the purpose of this paper is to determine such a minimal setof monomials L = L contained in the set H of all unary germs at + ∞ definable in R an , exp , and to further adapt the construction in [25] tocorresponding generalized series in one variable. Recalling that H isa Hardy field, we can summarize the results of this paper (Theorems1.12 and 1.13 below) as follows: Main Theorem.
There is a multiplicative subgroup L of H such thatthe following hold: (1) no two germs in L belong to the same archimedean class; (2) every germ in H is given by composing a convergent Laurentseries with a tuple of germs in L ; (3) the construction in [25] generalizes, after replacing the finiteiterates of log with germs in L , to obtain a correspondingIlyashenko field K .The resulting Ilyashenko field K is a Hardy field extending H as wellas the Ilyashenko field F constructed in [25] . Remark.
As mentioned earlier, the Ilyashenko field F constructed in[25] does not extend H .We obtain this set L of monomials by giving an explicit description ofthe Hardy field H as the set of all convergent LE-series , as suggested in[7, Remark 6.31], with L being the corresponding set of convergent LE-monomials ; see Example 5.15 below. The proof that the construction Zeinab Galal, Tobias Kaiser and Patrick Speissegger in [25] generalizes to this set L relies heavily on our recent paper [18];indeed, our construction here was the main motivation for [18]. In thenext two sections, we give a more detailed overview of the definitionsand results of this paper (Section 1) and their proof (Section 2).1. Main definitions and results
We let C be the ring of all germs at + ∞ of continuous functions f : R −→ R . A germ f ∈ C is small if lim x → + ∞ f ( x ) = 0 and large if lim x → + ∞ | f ( x ) | = ∞ . To compare elements of C , we use thedominance relation ≺ as found in Aschenbrenner and van den Dries [1,Section 1], defined by f ≺ g if and only if g ( x ) is ultimately nonzero andlim x → + ∞ f ( x ) /g ( x ) = 0, or equivalently, if and only if g ( x ) is ultimatelynonzero and f ( x ) = o ( g ( x )) as x → + ∞ . Thus, f (cid:22) g if and only if f ( x ) = O ( g ( x )) as x → + ∞ , and we write f ≍ g if and only if f (cid:22) g and g (cid:22) f . Note that the relation ≍ is an equivalence relation on C , andthe corresponding equivalence classes are the archimedean classes of C ; we denote by Π ≍ : C −→ C / ≍ the corresponding projection map.We denote by H ⊆ C the Hardy field of all germs of unary func-tions definable in R an , exp . Below, we let K be a commutative ring ofcharacteristic 0 with unit 1.Recall from [9] that a generalized power series over K is a powerseries G = P α ∈ [0 , ∞ ) n a α X α , where X = ( X , . . . , X n ), each a α ∈ K and the support of G ,supp( G ) := { α ∈ [0 , ∞ ) n : a α = 0 } , is contained in a cartesian product of well-ordered subsets of R . More-over, we call the support of G natural (see Kaiser et al. [17]) if, forevery a >
0, the intersection [0 , a ) ∩ Π X i (supp( G )) is finite, whereΠ X i : R n −→ R denotes the projection on the i th coordinate.Throughout this paper, we work with the following series: we fix amultiplicative R -vector subspace M of H > := { h ∈ H : h > } . Definition 1.1. An M -generalized Laurent series (over K ) is aseries of the form n · G ( m , . . . , m k ), where k ∈ N , G ( X , . . . , X k ) isa generalized power series with natural support, m , . . . , m k ∈ M aresmall and n belongs to the R -multiplicative vector space h m , . . . , m k i × generated by m , . . . , m k . In this situation, we say that F has gener-ating monomials m , . . . , m k . lyashenko algebras 5 Example 1.2.
Every logarithmic generalized power series, as definedin [25, Introduction], is an L -generalized Laurent series, where L := h exp , x, log , log , . . . i × is the multiplicative R -vector space generated by { exp , x, log , log , . . . } and log i denotes the i -th compositional iterate of log.Every M -generalized Laurent series F belongs to the ring K (( M ))of generalized series , as defined for instance in [6]. Correspondingly,we write F = P m ∈ M a m m , and the support of F is the reverse-wellordered set supp( F ) := { m ∈ M : a m = 0 } . We show in Section 5 that the set K (( M )) ls of all M -generalized Laurentseries is a subring of K (( M )) in general, and is a subfield if K is a field. Remarks 1.3.
Let F ∈ K (( M )) ls , and let m , . . . , m k ∈ M be small, n ∈ h m , . . . , m k i × and a generalized power series G with natural sup-port be such that F = n · G ( m , . . . , m k ).(1) The support of F is of reverse-order type at most ω k +1 (seeCorollary 5.5 below). For instance, the logarithmic generalizedpower series X m,n ∈ N x − m exp − n has reverse-order type ω .(2) The latter is not a unique representation of F as an M -generalizedLaurent series: taking, say, H ( X , . . . , X k ) := G (cid:0) X , X , . . . , X k (cid:1) , we have F = n · H (cid:0) √ m , m , . . . , m k (cid:1) as well.To justify using M -generalized Laurent series as asymptotic expan-sions, we need some further notations. Definition 1.4. (1) M is an asymptotic scale if m ≍ m = 1, for m ∈ M (or, equivalently, if every archimedean classof H > has at most one representative in M ).(2) A set S ⊆ M is called M -natural if, for all a ∈ M , the inter-section S ∩ ( a, + ∞ ) is finite. Examples 1.5. (1) For k ∈ N we set L k := h exp , x, log , . . . , log k − i × ⊂ L. It follows from basic calculus that L , and hence each L k , isan asymptotic scale. Every Dulac series (see Ilyashenko and
Zeinab Galal, Tobias Kaiser and Patrick Speissegger
Yakovenko [14, Section 24]) belongs to R (( L )) and has L -natural support.(2) Let L be the set of all principal monomials of H as definedin [18, Section 2]. Since every archimedean class of H > has aunique representative in L [18, Proposition 2.18(2)], the latteris a maximal asymptotic scale.(3) Two germs g, h ∈ C are comparable if there exist r, s > | f | r < | g | < | f | s (see Rosenlicht [21]). By Lemma5.4 below, if G ( X , . . . , X k ) is a generalized power series withnatural support and m , . . . , m k ∈ M are small and pairwisecomparable and n ∈ h m , . . . , m k i × , the M -generalized Laurentseries nG ( m , . . . , m k ) has h m , . . . , m k i × -natural support.We assume from now on that M is an asymptotic scale. Definition 1.6.
Let f ∈ C and F = P a m m ∈ R (( M )). We say that f has asymptotic expansion F (at + ∞ ) if supp( F ) is M -natural and( ∗ ) f − X m ≥ n a m m ≺ n for every n ∈ M . Example 1.7.
Every almost regular f ∈ C , in the sense defined in theintroduction of [25], has an asymptotic expansion in R (( L )) ls .We denote by C ( M ) the set of all f ∈ C that have an asymp-totic expansion in R (( M )). By Lemmas 6.3 and 6.4 below, C ( M ) isan R -algebra, every f ∈ C ( M ) has a unique asymptotic expansion in T M ( f ) ∈ R (( M )), and the map T M : C ( M ) −→ R (( M )) is an R -algebrahomomorphism. In this paper, we are interested in the following kindof subalgebras of C ( M ): Definition 1.8.
We call a subalgebra K of C ( M ) quasianalytic if therestriction of T M to K is injective.Note that, since R (( M )) is a field, every quasianalytic subalgebra of C ( M ) is an integral domain.We now want to extend the definition of asymptotic expansion to allseries in R (( M )), not just the ones with natural support. However, forthis generalization we cannot separate “asymptotic expansion” from“quasianalyticity”; both need to be defined simultaneously in the con-text of a ring of germs, in the spirit of [25, Definition 2].For F = P a m m ∈ R (( M )) and n ∈ M , we denote by F n := X m ≥ n a m m lyashenko algebras 7 the truncation of F above n . A subset S ⊆ R (( M )) is truncationclosed if, for every F ∈ S and n ∈ M , the truncation F n belongs to S . Example 1.9.
The set T M ( C ( M )) is truncation closed. Definition 1.10.
Let
K ⊆ C be an R -subalgebra and T : K −→ R (( M )) be an R -algebra homomorphism. The triple ( K , M, T ) is a quasianalytic asymptotic algebra (or qaa algebra for short) if(i) T is injective;(ii) the image T ( K ) is truncation closed;(iii) for every f ∈ K and every n ∈ M , we have f − T − (( T f ) n ) ≺ n. Example 1.11.
Let L be the set of all principal monomials of H asdefined in [18, Section 2]. We show in Corollary 5.20(2) below thatthere is a field homomorphism S L : H −→ R (( L )) such that ( H , L , S L )is a qaa field. The image of S L is what we call the set of all convergentLE-series (or convergent transweries), see Section 5.6.Let M ′ ⊂ H > be another asymptotic scale, and let ( K , M, T ) and( K ′ , M ′ , T ′ ) be two qaa algebras. We say that ( K , M, T ) extends ( K ′ , M ′ , T ′ ) if K ′ is a subalgebra of K , M ′ is a multiplicative R -vectorsubspace of M and T ↾ K ′ = T ′ . Theorem 1.12 (Construction).
Let h be a finite tuple of smallgerms in L . (1) There exists a qaa field ( K h , h h i × , T h ) such that h ⊆ K h . (2) If g is finite tuple of small germs in L and h ⊆ g , then the qaafield ( K g , h g i × , T g ) extends ( K h , h h i × , T h ) . Remark.
For general f ∈ K h , the series T h ( f ) is not convergent.In view of the Construction Theorem, we consider the set consistingof all qaa fields ( K h , h h i × , T h ), for finite tuple h of small germs in L ,partially ordered by the subset ordering on the tuples h , and we let( K , L , T ) be the direct limit of this partially ordered set. Theorem 1.13 (Closure). (1) ( K , L , T ) is a qaa field extendingeach ( K h , h h i × , T h ) . (2) ( K , L , T ) extends the qaa field ( F , L, T ) constructed in [25, The-orem 3] . (3) ( K , L , T ) extends the qaa field ( H , L , S L ) of Example 1.11 above. (4) K is closed under differentiation; in particular, K is a Hardyfield. Zeinab Galal, Tobias Kaiser and Patrick Speissegger Outline of proof and the Extension Theorem
The proof of the Construction Theorem proceeds by adapting theconstruction in [25] to the more general setting here. The role of stan-dard quadratic domain there is taken on by the following domains here:for a ∈ R , we set H ( a ) := { z ∈ C : Re z > a } . Definition 2.1. A standard power domain is a set U ǫC := φ ǫC ( H (0)) , where C > ǫ ∈ (0 ,
1) and φ ǫC : H (0) −→ U ǫC is the biholomorphicmap defined by φ ǫC ( z ) := z + C (1 + z ) ǫ , where ( · ) ǫ denotes the standard branch of the power function on H (0)(see Section 3 for details).Note that ǫ = corresponds to the standard quadratic domainsof [25]. We use the following consequence of the Phragm´en-Lindel¨ofprinciple [14, Theorem 24.36]: Uniqueness Principle.
Let U ⊆ C be a standard power domain and φ : U −→ C be holomorphic. If φ is bounded and φ ↾ R ≺ exp − n for each n ∈ N , then φ = 0 . The Uniqueness Principle follows from [14, Lemma 24.37], because x < φ ǫC ( x ) for x >
0. The reason for working with standard powerdomains in place of standard quadratic domains is technical; see theremark following Lemma 4.10 below for details.Recall that the construction in [25] is for the tuples (cid:18) , x , . . . , k (cid:19) = exp ◦ ( − x, − log , . . . , − log k +1 ) , and it proceeds by induction on k . To understand how we can gener-alize this construction to more general sequences h ∈ D k +1 , where D := (cid:8) h ∈ H > : h ≺ (cid:9) is the set of all positive small germs, we let I := (cid:8) h ∈ H > : h ≻ (cid:9) lyashenko algebras 9 be the set of all infinitely increasing (i.e., positive large) germs in H and write h = exp ◦ ( − f ) = 1exp ◦ f, where f = ( f , . . . , f k ) with each f i ∈ I . In this situation, we shall alsowrite M f for the multiplicative R -vector subspace h h i × .We first recall how the induction on k works in [25]: assuming theqaa field ( F k − , L, T k − ) has been constructed such that every germ in F k − has a complex analytic continuation on some standard quadraticdomain, we “right shift” by log, that is, we(i) [25] set F ′ k := F k − ◦ log and define T ′ k : F ′ k −→ L by T ′ k ( h ◦ log) :=( T k − h ) ◦ log.Note that, since log has a complex analytic continuation on any stan-dard quadratic domain with image contained in every standard qua-dratic domain [18, Example 3.13(2)], the tuple ( F ′ k , L, T ′ k ) is also aqaa field as defined in [25] such that every germ in F ′ k has a complexanalytic continuation on some standard quadratic domain. So we(ii) [25] let A k be the R -algebra of all germs h ∈ C that have a bounded,complex analytic continuation on some standard quadratic do-main U and an asymptotic expansion F = P h m m ∈ F ′ k (( L ))that holds not only in R , but in all of U , and we set T k h := X ( T ′ k h m ) m ∈ R (( L k )) . (Note that, in general, T k h is an L -series over R , but not an L -genera-lized Laurent series over R ; this observation was not explicitely men-tioned in [25].) The corresponding generalization of asymptotic ex-pansion ( ∗ ) to allowing coefficients in F ′ k works, because each germ in F ′ k is polynomially bounded, and the quasianalyticity follows from theUniqueness Principle. Finally, since R (( L k )) is a field, the ring A k isan integral domain, and we(iii) [25] let F k be the fraction field of A k and extend T k accordingly.We represent this construction by the schematic in Figure 1.Throughout this construction, the following property of L is used: Definition 2.2.
Let M be a multiplicative R -vector subspace of H > .We call M a strong asymptotic scale if(1) there is a basis { m , . . . , m k } of M consisting of pairwise in-comparable small germs;(2) every m ∈ M has a complex analytic continuation m : U −→ C on every standard power domain U ; R (UP) −−−→ (cid:20) F e − x ◦ ( x ) (cid:21) ւ ◦ log ւ (cid:20) F ′ e − x ◦ (log) (cid:21) (UP) −−−→ (cid:20) F e − x ◦ ( x, log) (cid:21) ւ ◦ log ւ ... (UP) −−−→ (cid:20) F k − e − x ◦ ( x, log , . . . , log k − ) (cid:21) ւ ◦ log ւ (cid:20) F ′ k e − x ◦ (log , . . . , log k ) (cid:21) (UP) −−−→ (cid:20) F k e − x ◦ ( x, log , . . . , log k ) (cid:21) Figure 1.
Schematic of the construction in [25]: goinghorizontally from left to right represents one use of theUniqueness Principle (UP) and adds e − x to the generat-ing monomials on the left; going from the right to thenext lower left represents a right shift by log.(3) for every standard power domain U and every m, n ∈ M , wehave m ≺ n if and only if m ( z ) = o ( n ( z )) as | z | → ∞ in U . Remark 2.3. If M is a strong asymptotic scale, then M is an asymp-totic scale: let { m , . . . , m k } be a basis of M consisting of pairwiseincomparable small germs such that m < · · · < m k , and set m k +1 := 1.Let m ∈ M be such that m ≍
1, let α , . . . , α k ∈ R be such that m = m α · · · m α k k , and set α k +1 := 1. Since the m i are pairwise incomparable, m iscomparable to m j , where j := min { i = 0 , . . . , k + 1 : α i = 0 } ; hence m ≍ α = · · · = α k = 0, that is, m = 1.The use of strong aymptotic scales is to extend the notion of asymp-totic expansion to standard power domains, see strong asymptoticexpansions in Section 6. lyashenko algebras 11 For some of the examples below, we let U be the set of all purely infi-nite germs in H , as defined in [18, Section 2]. Recall that L = exp ◦ U ;in particular, two germs f, g ∈ U belong to the same archimedean classif and only if the germs exp ◦ f and exp ◦ g are comparable. Examples 2.4. (1) L is a strong asymptotic scale by [25, Lemma8].(2) L is not a strong asymptotic scale: the germ e − x ◦ x belongsto L and is bounded, but its complex analytic continuation onany standard power domain is unbounded.(3) Not every tuple from L is a basis consisting of pairwise incom-parable small germs: consider the germs f := x , f = x − logand f := log + log log in U . While { f , f , f } is additivelylinearly independent, we have f ≍ f . However, M f has thebasis e − x ◦ ( x, log , log )consisting of pairwise incomparable small germs, because x , logand log belong to distinct archimedean classes.(4) We show in Lemma 8.1 below that, if each f i belongs to U , thenthe additive R -vector space h f i generated by the f i has a basisconsisting of infinitely increasing germs belonging to pairwisedistinct archimedean classes; hence, M f has a basis consistingof pairwise incomparable small germs.The most straightforward generalization of the construction in [25]is to any sequence f of the form f = (cid:0) g ◦ , g ◦ , . . . , g ◦ k (cid:1) , where k ∈ N , g ∈ I belongs to a strictly smaller archimedean class than x , g ◦ i denotes the i -th compositional iterate of g and M f is an asymp-totic scale on standard power domains, and such that the followingholds:( † ) for every standard power domain V , the germ g has a complexanalytic continuation g on some standard power domain U suchthat g ( U ) ⊆ V .The additional assumption ( † ) means that we can compose on theright (“right shift”) with g in place of log, as in the construction in[25].In general, we assume that k > f > f > · · · > f k belongto I and that M f is an asymptotic scale with basis e − x ◦ f consistingof pairwise incomparable small germs; this implies, in particular, that f ≻ · · · ≻ f k . In this situation, we aim to adapt the construction in [25] as represented by the schematic pictured in Figure 2. The “rightshifts” are now by germs of the form f k − i +1 ◦ f − k − i —which still belongto H since they are definable—and the monomials at the i -th step are e − x ◦ f h i i , where we set f h i i := (cid:0) x, f k − i +1 ◦ f − k − i , . . . , f k ◦ f − k − i (cid:1) and f h i i ′ := (cid:0) f k − i +1 ◦ f − k − i , . . . , f k ◦ f − k − i (cid:1) . In particular, we have f h i = ( x ), so that the first step in the construc-tion yielding K = K f, is the same as the first step of the constructionin Figure 1, that is, K = F .To determine what additional conditions f has to satisfy in order forthis adaptation to go through at the i -th step, we assume that M f h i − i is a strong asymptotic scale with basis e − x ◦ f h i − i consisting of pairwiseincomparable germs, and that we have constructed K i − = K f,i − suchthat every h ∈ K i − has an analytic continuation h on some standardpower domain. (We shall omit the subscript “ f ” in K f,i if clear fromcontext.) Provided that( † ) for every standard power domain V , the germ f k − i +1 ◦ f − k − i hasa complex analytic continuation f i,i − on some standard powerdomain U such that f i,i − ( U ) ⊆ V ,the set M f h i i′ is also a strong asymptotic scale (because f − i,i − mapsstandard power domains into standard power domains) with basis e − x ◦ f h i i ′ consisting of pairwise incomparable small germs. Therefore, weright shift by f k − i +1 ◦ f − k − i , that is, we(i) set K ′ i = K ′ f,i := K i − ◦ (cid:0) f k − i +1 ◦ f − k − i (cid:1) and define T ′ i = T f,i ′ : K ′ i −→ R (cid:0)(cid:0) M f h i i′ (cid:1)(cid:1) by T ′ i (cid:0) h ◦ (cid:0) f k − i +1 ◦ f − k − i (cid:1)(cid:1) := ( T i − h ) ◦ (cid:0) f k − i +1 ◦ f − k − i (cid:1) . Again by assumption ( † ) , the triple (cid:0) K ′ i , M f h i i′ , T ′ i (cid:1) is a qaa field suchthat every germ in K ′ i has a complex analytic continuation on somestandard power domain. So we(ii) let A i be the set of all germs h ∈ C that have a bounded,complex analytic continuation on some standard power domain U and a strong asymptotic expansion H = X h m m ∈ K ′ i (( M x ))in U (that is, this asymptotic expansion holds as | z | → ∞ in U , see Section 6 for details), and we set T i ( h ) := X T ′ i ( h m ) m ∈ R (cid:0)(cid:0) M f h i i (cid:1)(cid:1) . lyashenko algebras 13 R (UP) −−−→ (cid:20) K f, e − x ◦ f h i (cid:21) ւ ◦ (cid:0) f k ◦ f − k − (cid:1) ւ (cid:20) K ′ f, e − x ◦ f h i ′ (cid:21) (UP) −−−→ (cid:20) K f, e − x ◦ f h i (cid:21) ւ ◦ (cid:0) f k − ◦ f − k − (cid:1) ւ ... (UP) −−−→ (cid:20) K f,i − e − x ◦ f h i − i (cid:21) ւ ◦ (cid:0) f k − i +1 ◦ f − k − i (cid:1) ւ (cid:20) K ′ f,i e − x ◦ f h i i ′ (cid:21) (UP) −−−→ (cid:20) K f,i e − x ◦ f h i i (cid:21) ... (UP) −−−→ (cid:20) K f,k − e − x ◦ f h k − i (cid:21) ւ ◦ (cid:0) f ◦ f − (cid:1) ւ (cid:20) K ′ f,k e − x ◦ f h k i ′ (cid:21) (UP) −−−→ (cid:20) K f,k e − x ◦ f h k i (cid:21) ւ ◦ f ւ (cid:20) K f e − x ◦ f (cid:21) Figure 2.
Schematic of the generalized construction:going horizontally from left to right represents one useof the Uniqueness Principle (UP) and adds e − x to thegenerating monomials on the left; going from the rightto the next lower left represents a “right shift” by f k − i +1 ◦ f − k − i . One final right shift by f yields the desired qaafield K f . As in [25], the corresponding generalization of asymptotic expansion( ∗ ) in Definition 1.6 to allowing coefficients in K ′ i works, because eachgerm in K ′ i has comparability class strictly smaller than that of e x andstrictly larger than that of e − x , and the quasianalyticity follows fromthe Uniqueness Principle. Finally, we(iii) let K i be the fraction field of A i , and we extend T i accordingly.Iterating this construction leads to the schematic pictured in Figure2. The final step, a right shift by f , leads to the desired qaa field( K f , M f , T f ). Note that, for this last step, we do not need any analyticcontinuation assumptions and, consequently, we do not expect analyticcontinuation of the germs in K f on standard power domains.The crucial additional assumption we need to make this work is( † ) above, which we need for each i . Requiring this condition to beinherited by all subtuples of f , we shall consider the following strongerassumption:( † ) for 0 ≤ j < i ≤ k and every standard power domain V , the germ f i ◦ f − j has a complex analytic continuation on some standardpower domain U with image contained in V .This leads us to the following condition on general tuples f : Definition 2.5.
We call the tuple f admissible if ( † ) holds and M f ◦ f − is a strong asymptotic scale with basis e − x ◦ (cid:0) f ◦ f − (cid:1) con-sisting of pairwise incomparable small germs.Note that if f is admissible and g is a subtuple of f , then g isadmissible as well and, in this situation, the above construction shows(see Proposition 7.12 below) that ( K f , M f , T f ) extends ( K g , M g , T g ).Since not every germ in I satisfies ( † ) , not every tuple f is admissi-ble. To figure out what tuples f are admissible, recall from [18] that agerm f ∈ H is simple if eh( f ) = level( f ), where eh( f ) is the exponen-tial height of f as defined in [18] and level( f ) is the level of f as foundin [19]. In Section 4, we use Application 1.3 and Corollary 1.6 of [18]to establish the following: Theorem 2.6 (Admissibility).
Assume the f i are simple and havepairwise distinct archimedean classes. Then f is admissible. Note that the Admissibility Theorem fails for non-simple germs ingeneral:
Example 2.7.
Consider the tuple f = ( f , f ) := (cid:16) x, x + e − x (cid:17) : while f has a bounded complex analytic continuation on every standardpower domain, the germ f does not have a bounded complex analytic lyashenko algebras 15 continuation on any standard power domain. In fact, M f is not a strongasymptotic scale.Since every germ in U is simple [18, Example 8.7], we obtain thefollowing from the Admissibility Theorem 2.6: Corollary 2.8 (Admissibility).
If each f i belongs to U , then everysubtuple of h f i consisting of infinitely increasing germs belonging topairwise disjoint archimedean classes is admissible. (cid:3) Therefore, if each f i ∈ U , we obtain the qaa field ( K f , M f , T f ) asfollows: by Example 2.4(4), h f i has a basis g consisting of infinitelyincreasing germs belonging to pairwise disjoint archimedean classes.By the Admissibility Corollary 2.8, our construction then produces theqaa field ( K g , M g , T g ), and we set K f := K g and T f := T g . The resulting K f is independent of the chosen basis g , see Proposition8.6.Finally, we show in Section 9 that the direct limit ( K , L , T ) is max-imal in the following sense: if each f i belongs to U , the qaa field( K f , M f , T f ) constructed here is extended by ( K , L , T ); this implies,in particular, parts (1) and (2) of the Closure Theorem. For part (3)of the latter, it suffices to verify that every germ given by a convergent L -generalized power series in R (( M f )) belongs to K f . The proof of part(4) of the Closure Theorem is adapted from the proof of [25, Theorem3(2)]. 3. Standard power domains
This section summarizes some elementary properties of standardpower domains and makes some related conventions. Given two germs f, g ∈ C , we set f ∼ g if g ( x ) = 0 for sufficiently large x > f ( x ) /g ( x ) → x → + ∞ . Lemma 3.1.
Let
C > and ǫ ∈ (0 , . (1) The map φ ǫC is biholomorphic onto its image. (2) We have Re φ ǫC ( ir ) ∼ C cos (cid:0) ǫ π (cid:1) r ǫ and Im φ ǫC ( ir ) ∼ r , as r → + ∞ in R . (3) There exists a continuous f ǫC : [ C, + ∞ ) −→ (0 , + ∞ ) such that Im φ ǫC ( ir ) = f ǫC (Re φ ǫC ( ir )) for r > and f ǫC ( r ) ∼ K ǫC r /ǫ as r → + ∞ in R , where K ǫC is the constant (cid:0) C cos (cid:0) ǫ π (cid:1)(cid:1) − /ǫ . Proof. (1) Note that z + C (1+ z ) ǫ − ( w + C (1+ w ) ǫ ) = (1+ z )+ C (1+ z ) ǫ − ((1+ w )+ C (1+ w ) ǫ );so it suffices to show that the map ψ = ψ ǫC : H (0) −→ C defined by ψ ( z ) := z + Cz ǫ is injective. Note also that ψ maps the first quadrant H (0) + into itself, the real line into the real line and the fourth quadrant H (0) − into itself. So we let z, w ∈ H (0) + be distinct and show that ψ ( z ) = ψ ( w ); the other cases are similar and left to the reader.We may assume that Re w ≥ Re z , and we let Γ ⊆ H (0) + be thestraight segment connecting z to w and parametrized by the corre-sponding affine linear curve γ : [0 , −→ H (0) + ; note that γ ′ = a + ib with a, b ∈ R such that a ≥ ξ ∈ H (0) + , we have ψ ′ ( ξ ) = 1 + Cǫξ − ǫ , and since ξ − ǫ ∈ S (cid:0) (1 − ǫ ) π (cid:1) ∩ H (0) + , it follows thatRe ψ ′ ( ξ ) > ψ ′ ( ξ ) < . By the Fundamental Theorem of Calculus for holomorphic functionswe have ψ ( w ) − ψ ( z ) = Z S ψ ′ ( ξ ) dξ = Z ( a Re ψ ′ ( γ ( t )) − b Im ψ ′ ( γ ( t ))) dt + i Z ( b Re ψ ′ ( γ ( t )) + a Im ψ ′ ( γ ( t ))) dt. Thus, if b ≥ a and b are zero, it follows thatRe( ψ ( w ) − ψ ( z )) >
0. Arguing similarly if b ≤
0, we obtain thatIm( ψ ( w ) − ψ ( z )) < r >
1, that φ ǫC ( ir ) = ir + C ( ir ) ǫ ∞ X k =0 (cid:18) ǫk (cid:19) ( ir ) − k . Taking real and imaginary parts givesRe φ ǫC ( ir ) = Cr ǫ (cid:18) c ǫ K ǫ (cid:18) r (cid:19) − s ǫ L ǫ (cid:18) r (cid:19)(cid:19) , lyashenko algebras 17 where c ǫ := cos (cid:0) ǫ π (cid:1) , s ǫ := sin (cid:0) ǫ π (cid:1) , K ǫ ( X ) := ∞ X k =0 (cid:18) ǫ k (cid:19) ( − k X k and L ǫ ( r ) := ∞ X k =0 (cid:18) ǫ k − (cid:19) ( − k X k +1 , and Im φ ǫC ( ir ) = r + Cr ǫ (cid:18) c ǫ L ǫ (cid:18) r (cid:19) + s ǫ K ǫ (cid:18) r (cid:19)(cid:19) . Since the series K ǫ and L ǫ converge, part (2) follows.(3) Arguing as in part (1), we get Re ( φ ǫC ) ′ ( ξ ) > ξ ∈ H ( − ξ >
0. Thus, the map h : (0 , + ∞ ) −→ ( C, + ∞ ) definedby h ( r ) := Re φ ǫC ( ir ) is injective and has a compositional inverse g :( C, + ∞ ) −→ (0 , + ∞ ), and we define f ǫC : [ C, + ∞ ) −→ [0 , + ∞ ) by f ǫC ( t ) := ( Im φ ǫC ( ig ( t )) if t > C, t = C. Now note that p − ◦ h ◦ p − ( t ) = H ( t ), where H ∈ R [[ X ∗ ]] is aconvergent generalized power series as in [9] with leading monomiallm( H ) = X ǫ /Cc ǫ . By [9, Lemma 9.9], the series H has a compositionalinverse G ∈ R [[ X ∗ ]]; it follows that g = p − ◦ G ◦ p − . Since lm( H ◦ G ) =lm( H ) ◦ lm( G ), and since Im φ ǫC ( ir ) = p − ◦ I ◦ p − ( r ) for some convergent I ∈ R [[ X ∗ ]] with leading monomial X , it follows that f ǫC ( r ) ∼ K ǫC r /ǫ as r → + ∞ . (cid:3) From now on, we denote by φ ǫC the restriction of φ ǫC to the closedright half-plane H (0). Convention.
Given a standard power domain U and a function g : R −→ R that has a complex analytic continuation on U , we shalldenote this extension by the corresponding boldface letter g .For A ⊆ C and ǫ >
0, let T ( A, ǫ ) := { z ∈ C : d ( z, A ) < ǫ } be the ǫ -neighbourhood of A . Lemma 3.2.
Let
C > and ǫ ∈ (0 , . The following inclusions holdas germs at ∞ in H (0) : (1) for D > , ǫ ′ ∈ ( ǫ, and δ > , we have T (cid:0) U ǫ ′ D , δ (cid:1) ⊆ U ǫC ; (2) for D > C and δ > , we have T ( U ǫD , δ ) ⊆ U ǫC ; (3) for ν > , we have ν · U ǫC ⊆ ( U ǫνC if ν ≤ ,U ǫC if ν ≥ we have U ǫC + U ǫC ⊆ U ǫC/ ; (5) for any standard power domain U , there exists a > such that log ( U ǫC ) ∩ H ( a ) ⊆ U ∩ H ( a ) . Proof. (1) and (2) follow from Lemma 3.1(3).(3) follows from Lemma 3.1(3) and the equality ν · x, xC cos (cid:0) ǫ π (cid:1) ! /ǫ = νx, νxν − ǫ C cos (cid:0) ǫ π (cid:1) ! /ǫ in R .(4) Note that, for a ∈ C with Re a ≥
0, the boundary of a + U ǫC in { z ∈ C : Im z ≥ Im a } , viewed as a subset of R , is the graph of afunction f ǫa,C : [ C + Re a, + ∞ ) −→ [Im a, + ∞ ) such that f ǫa,C ( x ) ∼ Im a + K ǫC ( x − Re a ) /ǫ ≺ f ǫC/ ( x ) , which proves the claim.(5) Note that log ( { Re z > } ) = { z ∈ H (0) : | arg z | < π/ } . (cid:3) The following is the main reason for working with standard powerdomains.
Lemma 3.3.
Let
C > , ǫ ∈ (0 , and set K := C cos ( ǫ π ) ǫ/ . Thereexists k ∈ (0 , depending on C and ǫ such that k exp ( K | z | ǫ ) ≤ | exp ( z ) | ≤ exp( | z | ) for z ∈ U ǫC . Proof.
For r >
0, denote by C r the circle with center 0 and radius r . Since | exp ( x + iy ) | = exp x , the point in U ∩ C r where | exp z | ismaximal is z = r . On the other hand, the point z ( r ) = x ( r ) + iy ( r ) in U ǫC ∩ C r where | exp z | is smallest lies on the boundary of U ǫC , so that y ( r ) = f ǫC ( x ( r )). It follows from Lemma 3.1(3) that r = q x ( r ) + f ǫC ( x ( r )) ∼ x ( r ) /ǫ q x ( r ) − /ǫ + ( K ǫC ) . Hence x ( r ) ≥ Kr ǫ for all sufficiently large r ∈ R , as required. (cid:3) lyashenko algebras 19 Monomials on standard power domains
We now use Corollary 1.6 and Application 1.3 of [18] to find outwhich tuples f of germs in I are admissible. Example 4.1.
The restriction of 1 / exp to any right half-plane H ( a )with a ∈ R is bounded. Hence exp ◦ ( − x ) has a bounded complexanalytic continuation on every standard power domain.Below, we denote by arg the standard argument on C \ ( −∞ , α ∈ (0 , π ], we set S ( α ) := { z ∈ C : | arg z | < α ) } . We will need to work, for f ∈ H , with both the exponential height eh( f ) and the level level( f ). The former measures the logarithmic-exponential complexity of f ; roughly speaking, if f is unbounded, theneh(exp ◦ f ) = eh( f ) + 1, while if f is bounded, then eh(exp ◦ f ) = eh( f )(see [18, Section 2] for details). The latter measures the exponentialorder of growth of the germ f ; we refer the reader to Marker and Miller[19] for details. The level extends to all log-exp-analytic germs in anobvious manner, see [18, Section 3]. Remarks. (1) The two quantities are not equal in general: we havelevel( x + e − x ) = 0 = 1 = eh( x + e − x ).(2) The map level : ( H , ◦ ) −→ ( Z , +) is a group homomorphism;in particular, for f ∈ H and g ∈ I , we have level( f ◦ g − ) =level( f ) − level( g ). In contrast, the map eh : H −→ Z is not agroup homomorphism, and the definition of eh gives no boundson eh( f ◦ g − ) in terms of eh( f ) and eh( g ).As in [18, Section 2], we set H ≤ := { f ∈ H : eh( f ) ≤ } . Fact 4.2 (Corollary 2.16 in [18] ). (1) Let f ∈ H . Then eh( f ◦ exp) = eh( f ) + 1 and eh( f ◦ log) = eh( f ) − . (2) Let f ∈ H be infinitely increasing. Then level( f ) ≤ eh( f ) . (3) The set H ≤ is a differential subfield of H . The second fact we need gives an upper bound for exponential heightin the situation of the second remark above:
Fact 4.3 (Application 1.3 in [18] ). Let f ∈ I and g ∈ H . Then eh (cid:0) g ◦ f − (cid:1) ≤ max { eh( g ) + eh( f ) − f ) , eh( f ) − level( f ) } . The third fact summarizes analytic continuation properties of germsin H of low enough exponential height and level. Given a domain Ω ⊆ C and a map f : Ω −→ C , we call f half-bounded if f or 1 /f is bounded.We denote by arg( z ) the standard argument of z ∈ C \ ( −∞ , Fact 4.4 (Corollary 1.6 in [18] ). Let f ∈ H be such that eh( f ) ≤ . (1) There are a ≥ and a half-bounded complex analytic continu-ation f : H ( a ) −→ C of f . (2) Assume in addition that f ∈ I . Then (a) | f ( z ) | → ∞ as | z | → ∞ , for z ∈ H ( a ) ; (b) if f ≺ x , then f ( H ( a )) ⊆ C \ ( −∞ , , f : H ( a ) −→ f ( H ( a )) is biholomorphic and we have sgn(arg f ( z )) = sgn(arg z ) = sgn(Im z ) = sgn(Im f ( z )) for z ∈ H ( a ) ; (c) if eh( f ) < then, for every α > , there exists b ≥ a suchthat f ( H ( a )) ∩ H ( b ) ⊆ S ( α ) . (cid:3) From Fact 4.4, we immediately get the following:
Corollary 4.5.
Let f ∈ I be such that eh( f ) ≤ − . Then there exists a ≥ and a complex analytic continuation f : H ( a ) −→ C of f suchthat, for all standard power domains U, V ⊆ H (0) , there exists b ≥ a with f ( U ∩ H ( b )) ⊆ V . In particular, exp ◦ ( − f ) is a bounded on everystandard power domain. Proof.
By Fact 4.2(2), we have level( f ) ≤ eh( f ) ≤ −
1, while level( x ) =0; hence f ≺ x . So by Fact 4.4(2b), there are a ≥ f : H ( a ) −→ C such that f ( H ( a )) ⊆ C \ ( −∞ ,
0] and f : H ( a ) −→ f ( H ( a )) is biholomorphic. Let V be a standard powerdomain, and let c ≥ S ( π/ ∩ H ( c ) ⊆ V . By Facts4.4(2ac), there is d ≥ c such that f ( H ( a )) ∩ H ( d ) ⊆ S ( π/ ∩ H ( c ).By Fact 4.4(2a) again, there is a b ≥ a such that f ( H ( b )) ⊆ f ( H ( a )) ∩ H ( d ). Hence f ( U ∩ H ( b )) ⊆ V for any standard power domain U , asclaimed. (cid:3) However, for f ∈ I with eh( f ) = 0, things are still not clear: while f = x works by Example 4.1, the germ f = x does not: if U ′ ⊆ H (0)is a standard power domain, the set of squares of elements of U ′ is notcontained in any right half-plane H ( a ) with a ∈ R , so the complexanalytic continuation exp ◦ ( − z ) on U ′ is unbounded. Arguing simi-larly, we see that the germ exp ◦ ( − x r ) has a bounded complex analyticcontinuation on some standard power domain if and only if r ≤ lyashenko algebras 21 in this case, it has a bounded complex analytic continuation on allstandard power domains).What about the general f ∈ I with eh( f ) = 0? While we do notfully characterize all such f for which exp ◦ ( − f ) has a bounded com-plex analytic continuation to some standard power domain, we do givea sufficient condition in Corollary 4.12 below that suffices for our pur-poses.To determine which of these germs satisfy ( † ) , we also need a notionfor studying asymptotic behavior on standard power domains: given U ⊆ H (0) and φ, ψ : U −→ C , we write φ (cid:22) U ψ iff (cid:12)(cid:12)(cid:12)(cid:12) φ ( z ) ψ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) is bounded in U and φ ≺ U ψ iff lim | z |→∞ ,z ∈ U φ ( z ) ψ ( z ) = 0 . Correspondingly, we write φ ≍ U ψ if both φ (cid:22) U ψ and ψ (cid:22) U φ .We start with an easy case where dominance is preserved on theright half-plane: Lemma 4.6.
Let f, g ∈ H be such that eh( f ) , eh( g ) ≤ , and let f and g be corresponding complex analytic continuations obtained fromFact 4.4. If f ≺ g , then there exists a ≥ such that f ≺ H ( a ) g . Proof.
By Fact 4.2(3), the germ h := gf satisfies eh ( h ) ≤
0; let h be acomplex analytic continuation of h obtained from Fact 4.4. Now choose a ≥ f , g and h are defined on H ( a ). By assumption, either h or − h belongs to I ; hence | h ( z ) | → ∞ as | z | → ∞ in H ( a ) by Fact4.4(2a). Since f / g = 1 / h in H ( a ) by the holomorphic identity theorem,the lemma is proved. (cid:3) Corollary 4.7.
Let f ∈ H be such that eh( f ) ≤ , and let f bea complex analytic continuation of f obtained from Fact 4.4. If wehave lim x → + ∞ f ( x ) = c ∈ R , then there exists a ≥ such that lim | z |→∞ f ( z ) = c in H ( a ) . Proof.
Set g := f − c ; then eh( g ) ≤ g ≺
1, so that g ≺ H ( a ), for some a ≥
0, by Lemma 4.6. (cid:3)
Recall that, for α ∈ [0 , π ] we denote by S ( α ) = { z ∈ C : | arg z | < α } the sector of opening α and bisecting line (0 , + ∞ ). Lemma 4.8.
Let f ∈ I be such that f > and eh( f ) ≤ , and let f bea complex analytic continuation of f obtained from Fact 4.4. Assumethere exists ǫ ∈ (0 , such that and f ≺ x ǫ . Then f ( H ( a )) ⊆ S ( ǫ · π/ for some a ≥ . Proof.
The assumptions imply that h := x ǫ f ≤ x ǫ ≺ x belongs to I ,and eh( h ) ≤ a ≥ z ∈ H ( a ) with arg z >
0, wehave 0 < arg z ǫ f ( z ) = ǫ arg z − arg f ( z ) , so that arg f ( z ) < ǫ π , as claimed. We argue similarly if arg z < (cid:3) Corollary 4.9.
Let f ∈ I be such that f > and eh( f ) ≤ , andlet f be a complex analytic continuation of f obtained from Fact 4.4.Assume there exists ǫ ∈ (0 , such that and f ≺ x ǫ . Then there exists a ≥ such that (1) for z ∈ H ( a ) we have | f ( z ) | ≤ | z | ǫ and Re f ( z ) ≥ cos (cid:16) ǫ · π (cid:17) | f ( z ) | ;(2) if f ∈ I and U, V are standard power domains, there exists b ≥ a such that f ( U ∩ H ( b )) ⊆ V . Proof.
From Lemma 4.6 with g = x ǫ we get f ≺ H ( a ) p ǫ , for some a ≥ f ( z )) ≥ cos (cid:0) ǫ · π (cid:1) · | f ( z ) | , for some a ≥ z ∈ H ( a ); this proves part (1). Part (2) follows fromLemma 4.8 and Fact 4.4(2a), first choosing c ≥ w ∈ V forall w ∈ S ( ǫ · π/
2) with | w | > c , then choosing b ≥ a such that | f ( z ) | ≥ c for z ∈ H ( b ). (cid:3) Lemma 4.10.
Let f ∈ I be such that f > and eh( f ) ≤ , and let f be a complex analytic continuation of f obtained from Fact 4.4. As-sume that f ≺ x but f ≻ x ǫ for all ǫ ∈ (0 , . Then for every standardpower domain V , there exist a ≥ and standard power domains U and U such that f ( U ∩ H ( a )) ⊆ V and f ( V ∩ H ( a )) ⊆ U . Remark.
We do not know if this lemma remains true with “quadratic”in place of “power”; this is the technical reason for working with stan-dard power domains instead of standard quadratic domains.
Proof.
Let U = U δC ⊆ C be a standard power domain, with δ ∈ (0 , C >
0, and let δ ′ ∈ (0 , δ ); we claim that there exists a ≥ lyashenko algebras 23 that f ( U ∩ H ( a )) ⊆ U δ ′ D , where (cid:0) D cos (cid:0) δ ′ π (cid:1)(cid:1) δ ′ = cos ( ( δ − δ ′ ) π ) C cos ( δ π ) , whichthen proves the lemma.Set g := fx ; by assumption and Fact 4.2(3), 1 /g ∈ I , eh(1 /g ) ≤ /g ≺ x ǫ for all ǫ ∈ (0 , g has acomplex analytic continuation g : H ( a ) −→ C , for some a ≥
0, suchthat Im g ( z ) < z ∈ U with Im z >
0. By Lemma 3.1(2) andafter shrinking C a bit if necessary, we have Re z ≥ C cos (cid:0) δ π (cid:1) | z | δ forsufficiently large z ∈ U . By Corollary 4.9(1) with ǫ = δ − δ ′ , we alsohave for sufficiently large z ∈ U with Im z >
0, that | g ( z ) | ≥ | z | δ ′ − δ and, because Re g = g Re g , that Re g ( z ) ≥ cos (cid:0) ( δ − δ ′ ) π (cid:1) | g ( z ) | .Since f = xg and Im g ( z ) <
0, it follows thatRe f ( z ) = (Re z )(Re g ( z )) − (Im z )(Im g ( z )) > (Re z )(Re g ( z )) ≥ (Re z ) · cos (cid:16) ( δ − δ ′ ) π (cid:17) | g ( z ) |≥ C cos (cid:16) δ π (cid:17) cos (cid:16) ( δ − δ ′ ) π (cid:17) | z | δ ′ . On the other hand, since g ( z ) → | z | → ∞ in U , we have forsufficiently large z ∈ U with Im z > < Im f ( z ) = (Re z )(Im g ( z )) + (Im z )(Re g ( z )) ≤ (Im z )(Re g ( z )) ≤ | z | ;arguing similarly for Im z <
0, we get | Im f ( z ) | ≤ | z | for all sufficientlylarge z ∈ U . Hence | Im f ( z ) | < (cid:18) Re f ( z ) C cos ( δ π ) cos ( ( δ − δ ′ ) π ) (cid:19) /δ ′ for such z ∈ U , and the claim is proved by Lemma 3.1(2). (cid:3) Lemma 4.11.
Let f, g ∈ I be such that eh( f ) , eh( g ) ≤ , and let f and g be corresponding complex analytic continuations obtained fromFact 4.4. Assume that f ≍ g ≺ x . Let also U, V ⊆ C be standardpower domains and set s := f U, Im and t := g V, Im so that U = U s and V = U t , and assume that g ( U ) ⊆ V . Then there exist a, b, c > suchthat f ( U ∩ H ( c )) ⊆ U m a ◦ t ◦ m b . Proof.
Let c > f ∼ cg . If f = cg , then the conclusionfollows directly from Lemmas 3.2(3) and 3.1(3), so we assume f = cg .We distinguish two cases: Case 1: cg − f is bounded. Then there exists d ∈ R such that ǫ := cg − d − f ∈ D or − ǫ ∈ D . Hence either 1 /ǫ or − /ǫ is infinitely increasing; by Fact 4.2(3) both are of exponential height at most 0. ByFact 4.4(2a), it follows that | ǫ ( z ) | → | z | → ∞ in U s .Since f = cg − d − ǫ , it follows that, for large enough z ∈ U s withIm z >
0, we haveIm f ( z ) = c Im g ( z ) − Im ǫ ( z ) ≤ c Im g ( z ) + 1and Re f ( z ) = c Re g ( z ) − d − Re ǫ ( z ) ≥ c Re g ( z ) − ( d + 1) . Since | g ( z ) | → ∞ as | z | → ∞ by Fact 4.4(2a) and since, for z ∈ U t , wehave Re z → + ∞ as | z | → ∞ , it follows that Re f ( z ) ≥ c Re g ( z ) → + ∞ as | z | → ∞ in U s . A similar inequality as the first one above holdsif Im z <
0, so that | Im f ( z ) | ≤ c | Im g ( z ) | + 1for sufficiently large z ∈ U s . By assumption | Im g ( z ) | < t (Re g ( z ))for z ∈ U s . Since t is infinitely increasing, we get for sufficiently large z ∈ U s that | Im f ( z ) | ≤ c | Im g ( z ) | + 1 ≤ c · t (Re g ( z )) ≤ c · t (cid:18) c Re f ( z ) (cid:19) , so we can take a := 2 c and b := c . Case 2: cg − f is unbounded. We assume here that cg − f >
0; thecase f − cg > cg − f ) ≤ cg − f ≺ f ≺ x , wehave for large enough z ∈ U s with Im z > f ( z )) > c g ( z ) − f ( z )) > . Since Im( c g ( z )) = Im f ( z ) + Im( c g ( z ) − f ( z )), it follows thatIm f ( z ) < c Im g ( z )for such z . Arguing similarly for z ∈ U s with Im z <
0, we concludethat | Im f ( z ) | ≤ c | Im g ( z ) | for sufficiently large z ∈ U s (the inequality not being strict to accountfor the real z ).On the other hand, the germ 1 /δ belongs to I as well, where δ := cg − fcg has exponential height at most 0 by Fact 4.2(3). Since 1 /δ ≤ cg ≺ x ,by Fact 4.4(2b) again, we have Im δ ( z ) < z ∈ U s with Im z >
0. Since f = cg (1 − δ ) we have, for sufficiently large z ∈ U s with Im z > f ( z ) = c Re g ( z )(1 − Re δ ( z )) − c Im g ( z ) Im δ ( z ) > c g ( z ) . lyashenko algebras 25 We now conclude as in Case 1 with a := c and b := c . (cid:3) Corollary 4.12.
Let f ∈ I be such that eh( f ) ≤ and f (cid:22) x ,and let f be a complex analytic continuation of f obtained from Fact4.4. Then for every standard power domain V , there exist a ≥ and standard power domains U , U such that f ( U ∩ H ( a )) ⊆ V and f ( V ∩ H ( a )) ⊆ U . In particular, the restriction of exp ◦ ( − f ) to anystandard power domain is bounded. Proof. If f ≍ x the conclusion follows from Lemmas 4.11 and 3.1(2),so we assume that f ≺ x . If f ≺ x ǫ for some ǫ ∈ (0 , (cid:3) Next, dominance is preserved by all pairs of germs covered by theprevious corollary:
Proposition 4.13.
Let f, g ∈ H be such that eh( f ) , eh( g ) ≤ and f, g (cid:22) x , and let a ≥ and f , g : H ( a ) −→ C be corresponding complexanalytic continuations obtained from Fact 4.4. Then, for any standardpower domain U ⊆ H ( a ) , we have exp ◦ ( − f ) (cid:22) exp ◦ ( − g ) if and only if exp ◦ ( − f ) (cid:22) U exp ◦ ( − g ) . Proof.
By hypothesis and Fact 4.2(3), we have eh( f − g ) ≤
0. Ifexp ◦ ( − f ) ≍ exp ◦ ( − g ), then lim x → + ∞ ( f − g )( x ) = c ∈ R so, by Corol-lary 4.7, we have lim | z |→∞ ( f − g )( z ) = c in H ( a ), so that exp ◦ f ≍ H ( a ) exp ◦ g . So we assume from now on that exp ◦ ( − f ) ≺ exp ◦ ( − g ); then f − g ∈ I .Fix a standard power domain U ⊆ H ( a ). Since f − g (cid:22) x byassumption, it follows from Corollary 4.12 that ( f − g )( U ) ⊆ V forsome standard power domain V . Since V is a standard power domain,we have for w ∈ V that Re w → + ∞ as | w | → ∞ . Since f − g : U −→ ( f − g )( U ) is a biholomorphism, we have for z ∈ U that | ( f − g )( z ) | → ∞ as | z | → ∞ . Hence | exp ◦ ( f − g )( z ) | = exp(Re( f − g )( z )) → ∞ as | z | → ∞ in U , which shows that exp ◦ ( − f ) ≺ U exp ◦ ( − g ). (cid:3) We set H x ≤ := { f ∈ H ≤ : f (cid:22) x } and M x := (cid:8) exp ◦ ( − f ) : f ∈ H x ≤ (cid:9) . Recall also that, for h ∈ ( H > ) k , we denote by h h i × the multiplicative R -vector space generated by h . Corollary 4.14.
The set M x is a multiplicative R -vector subspaceof H > , and if m ∈ M x is bounded, then m has a complex analyticcontinuation m : H ( a ) −→ C , for some a > , such that m is boundedon every standard power domain U ⊆ H ( a ) . Proof.
That M x is a multiplicative R -vector subspace of H > followsfrom the observation that H x ≤ is an additive R -vector subspace of H ≤ .Let m ∈ M x be bounded and f ∈ H x ≤ be such that m = exp ◦ ( − f ).Let also a > f : H ( a ) −→ C be a complex analytic continuationof f obtained from Fact 4.4, and set m := exp ◦ ( − f ). Since 0 ∈ H x ≤ and m (cid:22) ◦ ( − m (cid:22) U ,as required. (cid:3) Proof of the Admissibility Theorem.
Let k ∈ N and f , . . . , f k ∈ I besimple such that f > · · · > f k , and assume that the f i have pairwisedistinct archimedean classes. Fix 0 ≤ i < j ≤ k . Since f i > f j andboth are simple, we have λ i := level( f i ) ≥ λ j := level( f j ) = eh( f j ), sothat λ j − λ i ≤
0. So by Fact 4.3 with f j and f i in place of f and g there,we get f j ◦ f − i ∈ H x ≤ . It follows from Corollary 4.12 that ( † ) holds for f := ( f , . . . , f k ). Moreover, by Corollary 4.14, every bounded m ∈ M f has a bounded complex analytic continuation on U ∩ H ( a ), for some a ≥ U . Since the f i have pairwisedistinct archimedean classes, if follows that f is admissible. (cid:3) M -generalized power series We work in the setting of Section 1; in particular, we let M be amultiplicative R -vector subspace of H > and K be a commutative ringof characteristic 0 with unit 1.An M -generalized power series over K is a series of the form F = G ( m , . . . , m k ), where G ∈ K [[ X ∗ , . . . , X ∗ k ]] has natural supportand m , . . . , m k ∈ M are small; in this situation, we refer to the m i asthe generating monomials of F . Remark.
In general, in a representation of an M -generalized powerseries F of the form G ( m , . . . , m k ) as above, the generalized series G is not uniquely determined by m , . . . , m k . However, if M = L and wechoose the latter carefully (see Lemma 8.1 below), G is indeed uniquelydetermined by m , . . . , m k . Lemma 5.1.
Let G ∈ K [[ X ∗ , . . . , X ∗ k ]] , m , . . . , m k ∈ M be small and n ∈ M . Then the set S Gn := { α ∈ supp( G ) : m α = n } is finite. lyashenko algebras 27 Proof.
We show that Π X i (cid:0) S Gn (cid:1) is finite for each i , where Π X i : R k +1 −→ R denotes the projection on the i th coordinate. Assume, for a contra-diction, that Π X i (cid:0) S Gn (cid:1) is infinite, where i ∈ { , . . . , k } ; without lossof generality, we may assume that i = k .Since Π X k (cid:0) S Gn (cid:1) ⊆ Π X k (supp( G )) and the latter is well ordered, thereis a strictly increasing sequence 0 ≤ α k < α k < · · · of elements ofΠ X k (cid:0) S Gn (cid:1) . For l ∈ N , choose α l , . . . , α lk − ≥ α l := ( α l , . . . , α lk ) ∈ S Gn . We claim that there exists i ( l ) ∈ { , . . . , k − } such that α l +1 i ( l ) < α li ( l ) :otherwise, we have m α l +1 ≤ m α l · · · m α lk − k − m α l +1 k k < m α l = n, a contradiction. By the claim, there exist i ∈ { , . . . , k − } and asequence l p ∈ N , for p ∈ N , such that α l i > α l i > · · · , which contradictsthat Π X i (cid:0) S Gn (cid:1) is well ordered. (cid:3) Lemma 5.2.
Every M -generalized power series is a generalized seriesin K (( M )) . Proof.
Let F = G ( m , . . . , m k ), where G = P α ∈ [0 , + ∞ ) k a α X α ∈ K [[ X ∗ , . . . , X ∗ k ]] has natural support and m , . . . , m k ∈ M are small.By Lemma 5.1, there are unique F n ∈ K , for n ∈ M , such that F = P n ∈ M F n n ; we need to show that the set T := { n ∈ M : F n = 0 } is anti-well ordered. So let T ′ ⊆ T be nonempty and define S GT ′ := [ n ∈ T ′ S Gn , where S Gn is defined as in Lemma 5.1. Then Π X i (cid:0) S GT ′ (cid:1) is well ordered foreach i , so there exist a nonzero l ∈ N and α , . . . , α l ∈ S GT ′ such that, forevery α ∈ S GT ′ , one of the X α j divides X α . Hence max n m α , . . . , m α l o is the maximal element of T ′ , which proves the lemma. (cid:3) We denote by K [[ M ]] ps the subring of K (( M )) of all M -generalizedpower series. Lemma 5.3.
Assume that M is an asymptotic scale, and let F ∈ K [[ M ]] ps be nonzero. Then there are a nonzero a ∈ K and an E ∈ K [[ M ]] ps such that lm( E ) is small and F = a lm( F )(1 − E ) . Proof.
Let G ∈ K [[ X ∗ ]] have natural support and m , . . . , m k ∈ M besmall such that F = G ( m ). Set p := lm( G ( m )); changing G if neces-sary, by Lemma 5.1, we may assume that there is a unique minimalelement β ∈ supp( G ) such that p = m β . Since M is an asymptoticscale this implies, in particular, that if α ∈ supp( G ) is minimal and α = β , we have that m α /p is small.Let now S be the finite set of minimal elements of supp( G ) (see [9,Lemma 4.2(1)]), so that β ∈ S . Then there are U α ∈ K [[ X ∗ ]] withnatural support (see [9, Concluding Remark 2]) and nonzero a α ∈ K ,for α ∈ S , such that U α (0) = 1 for each α and G = X α ∈ S a α X α U α = a β X β U β + X β = α ∈ S a α X α U α . Let Y = ( Y α : β = α ∈ S ) be a tuple of new indeterminates and set U ( X, Y ) := U β + X β = α ∈ S a α a β Y α U α , a generalized power series with natural support satisfying U (0) = 1.Hence H ( X, Y ) := 1 − U ( X, Y )is a generalized power series with natural support satisfying ord( H ) >
0. On the other hand, since m α /p is small for β = α ∈ S , the series E := H ( m, m ′ ) is an M -generalized power series, where m ′ := ( m α /p : β = α ∈ S ), and we have G ( m ) = a β pH ( m, m ′ ) = a β p (1 − E ) . Since ord( H ) >
0, the leading monomial of E is small, so the lemma isproved. (cid:3) Order type of support.
Recall that m, n ∈ M are comparable if there exist a, b > n a < m < n b . It is straightforward tosee that the comparability relation is an equivalence relation on M . If C ⊆ M consists of pairwise comparable germs, we say that n ∈ M is comparable to C if n is comparable to any germ in C . Lemma 5.4.
Let F ∈ K [[ M ]] ps have generating monomials m , . . . ,m k , and assume that the m i are pairwise comparable. Then F has h m , . . . , m k i × -natural support. Proof.
Let G ∈ K [[ X ∗ , . . . , X ∗ k ]] with natural support be such that F = G ( m , . . . , m k ). Let p ∈ h m , . . . , m k i × ; we claim that n ≤ p for all but finitely many n ∈ supp( F ). Since the m i are pairwise com-parable, each h m i i × is coinitial in h m , . . . , m k i × ; in particular, there lyashenko algebras 29 exist r , . . . , r k ∈ R such that m ri < p for each r > r i , for i = 0 , . . . , k .Since the box B := [0 , r ] × · · · × [0 , r k ] is compact, the set B ∩ supp( G )is finite. But by definition of B , if n ∈ supp( F ) satisfies n ≥ p , then n = m α for some α ∈ B , which proves the claim. (cid:3) Lemma 5.5.
Let F ∈ K [[ M ]] ps have generating monomials m < · · · Let G ( X ) ∈ R [[ X ∗ ]] with natural support be such that F = G ( m , . . . , m k ). If l = 0, the corollary follows from Lemma 5.4, so weassume l > l .We let j ∈ { , . . . , k } be such that m i is comparable to C if and onlyif i ≤ j , and we set M ′ := h m j +1 , . . . , m k i × and M ′′ := h m , . . . , m j i × .Identifying K (( M ′′ M ′ )) with a subring of K (( M ′′ )) (( M ′ )), we write F = e G ( m j +1 , . . . , m k ) , where e G ∈ K [[ M ′′ ]] ps (cid:2)(cid:2) ( X ∗ j +1 , . . . , X ∗ k ) (cid:3)(cid:3) has natural support. Applyingthe inductive hypothesis with K [[ M ′′ ]] ps in place of K , we obtain thatthe support of e G ( m j +1 , . . . , m k ) has reverse-order type at most ω l . Onthe other hand, by Lemma 5.4, every coefficient of e G has M ′′ -naturalsupport, that is, support of reverse-order type at most ω ; it followsthat the support of F has reverse-order type at most ω l +1 . (cid:3) Infinite sums. Let F ν ∈ K (( M )) for ν ∈ N . Recall that, if thesequence (lm( F ν ) : ν ∈ N ) is decreasing and coinitial in M , then theinfinite sum P ν F ν defines a series in K (( M )). In this general context, M -naturality is preserved: Lemma 5.6. Assume that each F ν has M -natural support and thesequence (lm( F ν ) : ν ∈ N ) is decreasing and coinitial in M . Then P ν F ν has M -natural support. Proof. Let n ∈ M ; it suffices to show that the set [ ν supp( F ν ) ! ∩ ( n, + ∞ )is finite. By hypothesis, we have lm( F ν ) → M as ν → ∞ . So wechoose N ∈ N such that lm( F ν ) ≤ n for ν > N ; then [ ν supp( F ν ) ! ∩ ( n, + ∞ ) = N [ ν =0 supp( F ν ) ! ∩ ( n, + ∞ ) , and the latter is finite since each F ν has natural support. (cid:3) Assume now that each F ν is an M -generalized power series of theform G ν ( m , . . . , m k ) with generating monomials m < · · · < m k and G ν ∈ K [[ X ∗ ]] of natural support. In this situation, we want a generalcriterion for when P ν F ν defines a series in K [[ M ]] ps . The fact that M may contain more than one comparability class plays a role here:let l ∈ N be such that { m , . . . , m k } has l + 1 comparability classes C < · · · < C l , and let i := 0 < i < · · · < i l ≤ k < i l +1 := k + 1 besuch that C j = { m i j , . . . , m i j +1 − } for 0 ≤ j ≤ l , and set M j := h C j i × . We denote by Π j : R k +1 −→ R i j +1 − i j the projection on the coordinates( x i j , . . . , x i j +1 − ), and we let Π M j : h m , . . . , m k i × −→ M j be the mapdefined by Π M j ( m α ) := ( m i j , . . . , m i j +1 − ) Π j ( α ) . Note that, since each G ν has natural support, each set Π M j (supp( F ν ))is an anti-well ordered subset of M j ; so we set m ν,j := max Π M j (supp( F ν )) . Lemma 5.7. Assume that S ν supp( G ν ) is natural and there exists j ∈ { , . . . , l } such that the sequence ( m ν,j : ν ∈ N ) is coinitial in M j . Then P ν F ν is an M -generalized power series with generatingmonomials m , . . . , m k . Remark. Note that the assumption that S ν supp( G ν ) be natural isnecessary: if M = h , x i × and F ν = exp − ν x − /ν , then M = h i × and m ν, = exp − ν → M as ν → ∞ , while G ν = X ν X /ν impliesthat S ν supp( G ν ) is not natural. Proof. We claim that ord( G ν ) → ∞ as ν → ∞ . Assuming this claim,it follows from [9, Paragraph 4.6] that G := P ν G ν belongs to K [[ X ∗ ]].Since G has natural support by hypothesis, it follows that P ν F ν = F := G ( m , . . . , m k ) ∈ K [[ M ]] ps , as required.To see the claim, let j be as in the hypothesis of the lemma. Foreach ν , choose β ν ∈ supp( G ν ) such that ord( G ν ) = | β ν | . Since( m i j , . . . , m i j +1 − ) Π j ( β ν ) = Π M j (cid:0) m β ν (cid:1) ≤ m ν,j → M j as ν → ∞ , we must have | Π j ( β ν ) | → ∞ as ν → ∞ . Since β ν only has nonnegative coordinates, it follows that ord( G ν ) = | β ν | → ∞ as ν → ∞ , as claimed. (cid:3) lyashenko algebras 31 Composition with power series. For A ⊆ [0 , ∞ ) and ν ∈ N ,we set + ν A := { a + · · · + a ν : a i ∈ A } and B ( A ) := [ ν ≥ (+ ν A ) . Lemma 5.8. Let A ⊆ [0 , ∞ ) be natural. Then B ( A ) is natural. Proof. Assume first that 0 / ∈ A , and set a := min A > 0. Thenmin (+ ν A ) = νa for all ν ≥ 1. So, given b > 0, choose N ≥ ba ;then [0 , b ] ∩ B ( A ) = [0 , b ] ∩ S N − ν =1 (+ ν A ), which is finite. This provesthe lemma in this case.In general, we have A ⊆ { } ∪ A ′ , where A ′ ⊆ (0 , ∞ ) is natural. Bythe previous case, it now suffices to show that B ( A ) = { } ∪ B ( A ′ ),which follows if we show that( ∗ ) ν + ν A ⊆ { } ∪ B ( A ′ )for each ν ≥ 1. We show ( ∗ ) ν by induction on ν : ( ∗ ) follows by choiceof A ′ . So assume ν > ∗ ) η holds for η < ν . Then+ ν A = A + (cid:0) + ν − A (cid:1) ⊆ ( { } ∪ B ( A ′ )) + ( { } ∪ B ( A ′ ))= { } ∪ B ( A ′ ) ∪ B ( A ′ ) ∪ ( B ( A ′ ) + B ( A ′ )) ⊆ { } ∪ B ( A ′ ) , where the second line follows from the inductive hypothesis and thefourth line follows from the observation that B ( A ′ ) + B ( A ′ ) ⊆ B ( A ′ ).This proves ( ∗ ) ν and hence the lemma. (cid:3) Let F ∈ K [[ M ]] ps with generating monomials m , . . . , m k be suchthat lm( F ) is small, and let P = P ν ∈ N a ν T ν ∈ K [[ T ]] be a powerseries in the single indeterminate T . Let also G ∈ K [[ X ∗ ]] have naturalsupport such that F = G ( m ). Since lm( F ) is small, we have ord( G ) > 0, so by [9, Paragraph 4.6], the sum P ◦ G := P ν a ν G ν belongs to K [[ X ∗ ]]. We therefore define P ◦ F := ( P ◦ G ) ( m ) . This composition does not depend on the particular series G chosen(see for instance [7]), and it is associative in the following sense: if P ∈ K [[ T ]] has positive order and Q ∈ K [[ T ]], then Q ◦ ( P ◦ G ) = ( Q ◦ P ) ◦ G .We will therefore simply write Q ◦ P ◦ G for these compositions. Proposition 5.9. Let F ∈ K [[ M ]] ps with generating monomials m ,. . . , m k be such that lm( F ) is small, and let P ∈ K [[ T ]] . (1) P ◦ F is an M -generalized power series with generating mono-mials m , . . . , m k . (2) Assume in addition that supp( F ) is M -natural and the sequence (lm( F ) ν : ν ∈ N ) is coinitial in M . Then P ◦ F has M -naturalsupport. Proof. (1) Let G ∈ K [[ X ∗ ]] with natural support and small m , . . . , m k ∈ M be such that F = G ( m ); we need to show that P ◦ G has naturalsupport. Since Π X i (supp( P ◦ G )) ⊆ S ν Π X i (supp( G ν )), and sinceΠ X i (supp( G ν )) = + ν Π X i (supp( G )) , Lemma 5.8 implies that supp( P ◦ G ) is natural, as required.(2) Arguing along the lines of Lemma 5.8, we see that each F ν has M -natural support (we leave the details to the reader). Since lm( F ν ) =lm( F ) ν , part (2) follows from part (1) and Lemma 5.6. (cid:3) M -generalized Laurent series. An M -generalized Laurentseries is a series of the form nF , where F is an M -generalized powerseries with generating monomials m , . . . , m k and n is a (possibly large)element of h m , . . . , m k i × . We denote by K (( M )) ls the subset of K (( M ))of all M -generalized Laurent series. Lemma 5.10. K (( M )) ls is subring of K (( M )) . Proof. It is easy to see that the set K (( M )) ls is closed under multipli-cation. As to closure under addition, let G , G ∈ K [[ X ∗ ]] have naturalsupport with X = ( X , . . . , X k ), let m , . . . , m k ∈ M be small and n , n ∈ h m , . . . , m k i × . Let α , α ∈ R k +1 be such that n i = m α i for i = 1 , 2, where m := ( m , . . . , m k ), and define β = ( β , . . . , β k ) := (cid:0) min (cid:8) α , α (cid:9) , . . . , min (cid:8) α k , α k (cid:9)(cid:1) and n := m β . Note that H i := X α i − β G i ( X )belongs to K [[ X ∗ ]] and has natural support, for i = 1 , 2. Then n G ( m ) + n G ( m ) = n ( H ( m ) + H ( m )) , which shows that K (( M )) ls is closed under addition. (cid:3) Proposition 5.11. Assume that K is a field and M is an asymptoticscale. lyashenko algebras 33 (1) Let F ∈ K (( M )) ls be nonzero. Then there are a nonzero a ∈ K and an E ∈ K [[ M ]] ls such that lm( E ) is small and F = a lm( F )(1 − E ) . (2) K (( M )) ls is the fraction field of K [[ M ]] ps in K (( M )) . Proof. (1) Let G ∈ K [[ X ∗ ]] have natural support, m , . . . , m k ∈ M besmall and n ∈ h m , . . . , m k i × be such that F = nG ( m ). Part(1) nowfollows from Lemma 5.3, since lm( F ) = n lm( G ( m )).(2) Let F ∈ K (( M )) ls be nonzero, and let a and E be for F as in part(1). We get from Proposition 5.9(1) that P ◦ E is an M -generalizedpower series with generating monomials m and m ′ , where P ∈ K [[ T ]]is the geometric series P ( T ) = P ν T ν . It follows from the binomialformula that 1 F = 1 a lm( F ) ( P ◦ E ) , which is an M -generalized Laurent series. (cid:3) M -series. Finally, the series we eventually obtain in the Con-struction Theorem all belong to R (( L )), but they are more generalthan L -generalized Laurent series. Once again, this has to do withthe possibility that M may contain more than one comparability class:let l ∈ N be such that { m , . . . , m k } has l + 1 comparability classes C < · · · < C l , and let i := 0 < i < · · · < i l ≤ k < i l +1 := k + 1 besuch that C j = { m i j , . . . , m i j +1 − } for 0 ≤ j ≤ l , and set M := h m , . . . , m i − i × and M := h m i , . . . , m k i × . Note that m j < M < m j for j = 0 , . . . , i − Definition 5.12. The set K (( M )) s of all M -series over K is definedby induction on l : K (( M )) s := ( K (( M )) ls if l = 0 ,K (( M )) s (( M )) ls if l > . Example 5.13. The series P ν ∈ N x ν exp − ν is an h x − , exp − i × -seriesover R that is not an h x − , exp − i × -generalized Laurent series over R . Lemma 5.14. If K is a field and M is an asymptotic scale, then K (( M )) s is a subfield of K (( M )) containing K (( M )) ls . Proof. By induction on l ; the case l = 0 follows from Proposition5.11(3), so we assume l > l . But then K (( M )) s is a subfield of K (( M )) containing K (( M )) ls ,by the inductive hypothesis, so K (( M )) s is a subfield of K (( M )) s (( M ))containing K (( M )) ls (( M )) ls , and the latter contains K (( M )) ls . There-fore, it remains to show that K (( M )) s is contained in K (( M )).To see this, let F ∈ K (( M )) s ; as an element of K (( M )) s (( M )), wecan write F = P m ∈ M F m m , with F m ∈ K (( M )) s , and by Lemma 5.4,the set supp M ( F ) := { m ∈ M : F m = 0 } is M -natural. Set S := (cid:26) m ∈ supp M ( F ) : m < M < m (cid:27) and S := supp M ( F ) \ S ; then S is finite, so the series F := P m ∈ S F m m belongs to K (( M )). On the other hand, since every m ′ ∈ M has strictly slower comparability class than every m ∈ S , the series F := P m ∈ S F m m also belongs to K (( M )), so the claim follows. (cid:3) Convergent M -generalized Laurent series. In this subsec-tion, we assume that K ⊆ C . Recall [9, Section 5] that if α is acountable ordinal and r β ≥ β < α , then the sum P β<α r β (re-spectively, the product Q β<α r β ) converges to a ∈ R if, for every ǫ > 0, there exists a finite set I ǫ ⊆ α such that (cid:12)(cid:12) P β ∈ I r β − a (cid:12)(cid:12) < ǫ (respectively, (cid:12)(cid:12) Q β ∈ I r β − a (cid:12)(cid:12) < ǫ ) for every finite set I ⊆ α containing I ǫ . It follows from the continuity and the morphism property of expthat P β<α r β converges to a if and only if Q β<α exp( r β ) converges toexp( a ). Correspondingly, a generalized power series G = X α ∈ [0 , + ∞ ) k +1 a α X α ∈ K [[ X ∗ ]] converges (absolutely) if P | a α ( x ) || x α | converges uniformly for allsufficiently small x ∈ [0 , ∞ ) k +1 . In this situation, there exist ǫ > g : [0 , ǫ ) k +1 −→ R such that G ( x )converges to g ( x ) for x ∈ [0 , ǫ ) k +1 . Therefore, if supp( G ) is natu-ral, m , . . . , m k ∈ M are small and n ∈ h m , . . . , m k i × , then the M -generalized Laurent series F := n · G ( m , . . . , m k ) converges to thegerm S K,M ( F ) := n · g ( m , . . . , m k ) ∈ C . We denote by K (( M )) conv the set of all convergent M -generalized Lau-rent series; arguing as in the proof of Proposition 5.11(1), we see lyashenko algebras 35 that K (( M )) conv is an R -subalgebra of K (( M )). Moreover, the map S K,M : K (( M )) conv −→ C is an R -algebra homomorphism; it followsthat the set C ( K, M ) conv := {S K,M ( F ) : F ∈ K (( M )) conv } is an R -subalgebra of C .The next example is central to this paper, and it provides a wayto make precise the notion of “convergent LE-series” hinted at in [7,Remark 6.31]. Recall from [18, Section 2] that E is the set of all germsin H defined by L an , exp -terms (that is, without the use of log) and M is the set of all germs in L defined by L an , exp -terms. We then have, inparticular, that H = [ k ∈ N E ◦ log k and L = [ k ∈ N M ◦ log k . Example 5.15. We have H ⊆ C ( R , L ) conv : to see this, let h ∈ H , andlet k ∈ N and f ∈ E be such that h = f ◦ log k . By definition of E , wehave f = S R , M ( F ) for some F ∈ R (( M )) conv , so h = S R , M◦ log k ( F ◦ log k ).Since M ◦ log k ⊆ L , we have F ◦ log k ∈ R (( L )) conv , so that H ⊆C ( R , L ) conv .Note in fact that H consists, by [18, Section 2], of all germs S R , L ( F )such that F = G ( m , . . . , m k ) for some small m , . . . , m k ∈ L anda convergent Laurent series G with support contained in Z k +1 . Inparticular, we have C ( R , L ) conv * H . Definition 5.16. In view of the previous example, we call a series F ∈ R (( L )) conv a convergent LE-series (or convergent transseries )if F = G ( m , . . . , m k ) for some small m , . . . , m k ∈ L and a convergentLaurent series G with support contained in Z k +1 . Correspondingly,we refer to the germs in L as the convergent LE-monomials (or convergent transmonomials ). In accordance with [7], we denote by R (( x − )) LE, conv the set of all convergent LE-series. Lemma 5.17. Let F = P a m m ∈ K (( M )) conv , let A ⊆ supp( F ) andset F A := P m ∈ A a m m . Then F A ∈ K (( M )) conv . In particular, the set C ( K, M ) conv is truncation closed. Proof. Let m , . . . , m k ∈ M be small, n ∈ h m , . . . , m k i × and a conver-gent G ( X , . . . , X k ) = P α ∈ [0 , ∞ ) k +1 b α X α ∈ K [[ X ∗ ]] with natural sup-port be such that F = nG ( m , . . . , m k ). Note that a p = P α ∈ S Gp/n b α ,for p ∈ M , since the set S Gp/n is finite by Lemma 5.1. Setting S A := S p ∈ A S Gp/n , the convergence of G implies that G A := P α ∈ S A b α X α isalso convergent; but F A = nG A ( m , . . . , m k ) , as required. (cid:3) Definition 5.18.