QQUASIANALYTIC ILYASHENKO ALGEBRAS
PATRICK SPEISSEGGER
Abstract.
I construct a quasianalytic field F of germs at + ∞ of realfunctions with logarithmic generalized power series as asymptotic expan-sions, such that F is closed under differentiation and log-composition;in particular, F is a Hardy field. Moreover, the field F ◦ ( − log) of germsat 0 + contains all transition maps of hyperbolic saddles of planar realanalytic vector fields. Introduction
In his solution of Dulac’s problem, Ilyashenko [3] introduces the class A of germs at + ∞ of almost regular functions, and he shows that this classis quasianalytic and closed under log -composition , by which I mean thefollowing: given f, g ∈ A such that lim x → + ∞ g ( x ) = + ∞ , it follows that f ◦ ( − log) ◦ g ∈ A . As a consequence, A ◦ ( − log) is a quasianalytic class ofgerms at 0 + that is closed under composition. Ilyashenko also shows that if f is the germ at 0 + of a transition map near a hyperbolic saddle of a planarreal analytic vector field, then f belongs to A ◦ ( − log); from this, it followsthat limit cycles of a planar real analytic vector field ξ do not accumulate ona hyperbolic polycycle of ξ . (For a discussion of Dulac’s problem and relatedterminology used here, we refer the reader to Ilyashenko and Yakovenko [2,Section 24]. The class A also plays a role in the description of Riemann mapsand solutions of Dirichlet’s problem on semianalytic domains; see Kaiser[4, 5] for details.)That A is closed under log-composition is due to a rather peculiar as-sumption built into the definition of “almost regular”: by definition, a func-tion f : ( a, + ∞ ) −→ R is almost regular if there exist real numbers0 ≤ ν < ν < . . . such that lim i ν i = + ∞ , polynomials p i ∈ R [ X ] for each i and a standard quadratic domain Ω = Ω C := (cid:8) z + C √ z : Re z > (cid:9) ⊆ C , with C > , Date : Tuesday 25 th September, 2018 at 13:18.1991
Mathematics Subject Classification.
Primary 41A60, 30E15; Secondary 37D99,03C99.
Key words and phrases.
Generalized series expansions, quasianalyticity, transitionmaps.Supported by NSERC Discovery Grant a r X i v : . [ m a t h . C V ] M a y such that(i) f has a bounded holomorphic extension f : Ω −→ C ;(ii) p is a nonzero constant and, for each N ∈ N , f ( z ) − N (cid:88) i =0 p i ( z ) e − ν i z = o (cid:0) e − ν N z (cid:1) as | z | → + ∞ in Ω . Remark.
For an almost regular f as defined here, the function log ◦ f isalmost regular in the sense of [2, Definition 24.27].It is the assumption that p be a nonzero constant that makes the class A closed under log-composition. However, one drawback of this assumptionis that the class A is not closed under addition (because of possible cancel-lation of the leading terms), which makes it unamenable to study by manycommonly used algebraic-geometric methods.I show here that Ilyashenko’s construction of A can be adapted, usinghis notion of superexact asymptotic expansion [3, Section 0.5], to obtain aquasianalytic class F that is closed under addition and multiplication, con-tains exp and log and is closed under differentiation and log-composition.This construction comes at the cost of replacing the asymptotic expansionsabove by the following series: for k ∈ Z , we denote by log k the k th compo-sitional iterate of log. Recall from van den Dries and Speissegger [12] thata generalized power series is a power series F = (cid:80) α ∈ R k a α X α , where X = ( X , . . . , X k ), each a α ∈ R and the support of F ,supp( F ) := { α ∈ R n : a α (cid:54) = 0 } , is contained in a cartesian product of well-ordered subsets of R ; the set ofall generalized power series in X is denoted by R [[ X ∗ ]]. Moreover, we callthe support of F natural (see Kaiser et al. [6]) if, for every compact box B ⊆ R k , the intersection of B ∩ supp( F ) is finite. Definition 1.1. A logarithmic generalized power series is a series ofthe form F (cid:16) i , . . . , ik (cid:17) , where i , . . . , i k ≥ − F ∈ R [[ X ∗ ]] hasnatural support.I denote by L the divisible multiplicative group of all monomials of theform (log i ) r · · · (log i k ) r k , with − ≤ i < · · · < i k and r , . . . , r k ∈ R . Notethat L is linearly ordered by setting m ≤ n if and only if lim x → + ∞ m ( x ) n ( x ) ≤ L is a multiplicative subgroup of the Hardy field of all germs at+ ∞ of functions definable in the o-minimal structure R exp , see Wilkie [14].)Indeed, this ordering can be described as follows: identify each m ∈ L witha function m : {− } ∪ N −→ R in the obvious way. Then for m, n ∈ L we have m < n in L if and only if m < n in R {− }∪ N in the lexicographicordering.For a divisible subgroup L (cid:48) of L , I denote by R [[ L (cid:48) ]] the set of all logarith-mic generalized power series with support contained in L (cid:48) . Note that, by definition, every series in R [[ L (cid:48) ]] has support contained in L (cid:48) ∩ (cid:8) (log i ) r · · · (log i k ) r k : − ≤ i < · · · < i k and r , . . . , r k ≤ (cid:9) . It is straightforward to see that R [[ L (cid:48) ]] is an R -algebra under the usual addi-tion and multiplication of series, and I denote its fraction field by R (( L (cid:48) )). (Sothe general series in R (( L (cid:48) )) is of the form mF , where m ∈ L (cid:48) and F ∈ R [[ L (cid:48) ]].)This notation agrees with the usual notation for generalized series , see forinstance van den Dries et al [10]. To simplify notations, I sometimes write F ∈ R (( L (cid:48) )) as F = (cid:80) m ∈ L (cid:48) a m m as in [10]; in this situation, I call the setsupp( F ) := (cid:8) m ∈ L (cid:48) : a m (cid:54) = 0 (cid:9) the support of F . Note that, under the ordering on L (cid:48) , the set supp( F ) isa reverse well-ordered subset of L (cid:48) of order-type at most ω k for some k ∈ N .I call supp( F ) L (cid:48) -natural if supp( F ) ∩ ( m, + ∞ ) is finite for any m ∈ L (cid:48) .For F = (cid:80) m ∈ L (cid:48) a m m ∈ R (( L (cid:48) )) and n ∈ L , I denote by F n := (cid:88) m ≥ n a m m the truncation of F above n . A subset A ⊆ R (( L (cid:48) )) is truncation closed if, for every F ∈ A and n ∈ L , the truncation F n belongs to A .Since the support of a logarithmic generalized power series can have ordertype ω k for arbitrary k ∈ N , I need to make sense of what it means to havesuch a series as asymptotic expansion. I do this in the context of an algebraof functions: Definition 1.2.
Let K be an R -algebra of germs at + ∞ of functions f :( a, ∞ ) −→ R (with a depending on f ), let L (cid:48) be a divisible subgroup of L , and let T : K −→ R (( L (cid:48) )) be an R -algebra homomorphism. The triple( K , L (cid:48) , T ) is a quasianalytic asymptotic algebra (or qaa algebra forshort) if(i) T is injective;(ii) the image T ( K ) is truncation closed;(iii) for every f ∈ K and every n ∈ L (cid:48) , we have f − T − (( T f ) n ) = o ( n ) . In this situation, for f ∈ K , I call T ( f ) the K -asymptotic expansion of f . The result of this note can now be stated: Theorem 1.3. (1)
There exists a qaa field ( F , L, T ) that contains theclass A as well as exp and log . (2) The field F is closed under differentiation and log -composition. The remainder of this paper is divided into six sections: Section 2 dis-cusses some basic properties of standard quadratic domains, Section 3 intro-duces strong asymptotic expansions, Section 4 contains the construction of ( F , L, T ), Section 5 contains the proof of closure under differentiation andSection 6 that of closure under log-composition. Finally, Section 7 containssome remarks putting this paper in a wider context.In Section 6, I rely on the observation that R (( L )) is a subset of the set T of transseries as defined by van der Hoeven in [13]; I use, in particular, thefact that T is a group under composition.The construction of F is based on the following consequence of the Phrag-m´en-Lindel¨of principle [2, Theorem 24.36]: Fact 1.4 ( [2, Lemma 24.37] ). Let Ω ⊆ C be a standard quadratic domainand φ : Ω −→ C be holomorphic. If φ is bounded and, for each n ∈ N , | φ ( x ) | = o (cid:0) e − nx (cid:1) as x → + ∞ in R , then φ = 0 . Indeed, I use this consequence of the Phragm´en-Lindel¨of principle as ablack box. I suspect that other Phragm´en-Lindel¨of principles, such as theone found in Borichev and Volberg [1, Theorem 2.3], might be used in asimilar way to obtain other qaa algebras.2.
Standard quadratic domains
This section summarizes some elementary properties of standard qua-dratic domains and makes some related conventions. For a ∈ R , I set H ( a ) := { z ∈ C : Re z > a } , and I define φ C : H ( − −→ H ( −
1) by φ C ( z ) := z + C √ z. I denote by C the set of all germs at + ∞ of continuous functions f : R −→ R . For f, g ∈ C I write f ∼ g if f ( x ) /g ( x ) → x → + ∞ . Lemma 2.1.
Let
C > . (1) The map φ C is conformal with compositional inverse φ − C given by φ − C ( z ) = z + C − C (cid:114) z + C in particular, the boundary of Ω C is the set φ C ( i R ) . (2) We have Re φ C ( ix ) ∼ C √ √ x and Im φ C ( ix ) ∼ x . (3) There exists a continuous f C : [ C, + ∞ ) −→ (0 , + ∞ ) such that Im φ C ( ix ) = f C (Re φ C ( ix )) for x > and f C ( x ) ∼ x/C ) . Proof.
These observations are elementary and left to the reader. (cid:3)
Figure 2 shows a standard quadratic domain with its boundary φ C ( i R ).From now on, I denote by φ C the restriction of φ C to the closed right half-plane H (0). Figure 1.
A standard quadratic domain and its boundary φ C ( i R )Two domains Ω , ∆ ⊆ H (0) are equivalent if there exists R > ∩ D ( R ) = ∆ ∩ D ( R ), where D ( R ) := { z : | z | > R } . The correspondingequivalence classes of domains in H (0) are called germs at ∞ of domainsin H (0). If clear from context, we shall not explicitely distinguish betweena domain in H (0) and its germ at ∞ .For A ⊆ C and (cid:15) >
0, let T ( A, (cid:15) ) := { z ∈ C : d ( z, A ) < (cid:15) } be the (cid:15) -neighbourhood of A . Convention.
Given a standard quadratic domain Ω and a function g : R −→ R that has a holomorphic extension on Ω, I will usually denote thisextension by the corresponding boldface letter g . I also write exp and x for the holomorphic extensions on Ω of exp and the identity function x ,respectively, and log for the principal branch of log on Ω. Thus, every m ∈ L has a unique holomorphic extension m on Ω. (Strictly speaking,these extensions depend on Ω, but I do not indicate this dependence.) Lemma 2.2.
Let
C > . The following inclusions hold as germs at ∞ in H (0) : (1) for D > C and (cid:15) > , we have T (Ω D , (cid:15) ) ⊆ Ω C ; (2) for ν > , we have ν · Ω C ⊆ (cid:40) Ω νC if ν ≤ , Ω C if ν ≥ for any standard quadratic domain Ω , we have log (Ω C ) ⊆ Ω ; (4) we have Ω C + Ω C ⊆ Ω C . Proof. (1) follows from Lemma 2.1(3).(2) follows from Lemma 2.1(3) and the equality ν · ( x, x/C ) ) = ( νx, νx/ √ νC ) )in R .(3) Note that log ( H (0) ∩ {| z | > } ) = H (0) ∩ {| Im z | < π/ } .(4) Note first that, for a ∈ C with Re a ≥
0, the boundary of a + Ω C in { z ∈ C : Im z ≥ Im a } , viewed as a subset of R , is the graph of a function f a,C : [ C + Re a, + ∞ ) −→ [Im a, + ∞ ) such that f a,C ( x ) ∼ Im a + (cid:18) x − Re aC (cid:19) . In particular, if a ∈ ∂ Ω C , then a = b + if C ( b ) for some b ≥ C ; therefore, f a,C ( x ) ∼ b + ( x − b ) C < f C ( x )in C , which proves the claim. (cid:3) The following is the main reason for working with standard quadraticdomains.
Lemma 2.3.
Let
C > and set K := C √ . There exists k ∈ (0 , dependingon C such that k exp (cid:16) K (cid:112) | z | (cid:17) ≤ | exp ( z ) | ≤ exp( | z | ) for z ∈ Ω C . Proof.
Let
C > C and, for r >
0, denote by C r thecircle with center 0 and radius r . Since | exp ( x + iy ) | = exp x , the point inΩ ∩ C r where | exp z | is maximal is z = r . On the other hand, the point z ( r ) = x ( r ) + iy ( r ) in Ω ∩ C r where | exp z | is smallest lies on the boundaryof Ω r , so that y ( r ) = f C ( x ( r )). It follows from Lemma 2.1(3) that r = (cid:112) x ( r ) + f C ( x ( r )) ∼ x ( r ) (cid:115) x ( r ) + 4 C . Hence x ( r ) ≥ K √ r for all sufficiently large r ∈ R , as required. (cid:3) Convention.
Given an unbounded domain Ω ⊆ H (0) and holomorphic φ, ψ : Ω −→ C , I write ψ = o ( φ ) in Ωif | ψ ( z ) /φ ( z ) | → | z | → ∞ in Ω.The reason why the notion of qaa algebra makes sense for the set ofmonomials L is that, for m, n ∈ L , we have m < n if and only if m = o ( n ).This equivalence remains true on standard quadratic domains: Lemma 2.4.
Let m, n ∈ L be such that m < n , and let Ω be a standardquadratic domain. Then m = o ( n ) in Ω . Remark.
While exp − < x − in L , we have exp − (cid:54) = o ( x − ) in H (0) (orindeed in any right half-plane). Proof.
First, let z ∈ H (0) with | z | ≥ e . Then1 ≤ log | z | = Re( log z ) ≤ | log z | and, since Im( log z ) ∈ ( − π , π ), we also have | log z | ≤ | z | . Second, define e := 1 and, for k >
0, we set e k := e e k − . It follows from(1), by induction on k ∈ N , that if z ∈ H (0) with | z | ≥ e k , there exists C = C ( k ) > ≤ log k | z | ≤ | log k z | ≤ C log k | z | . The previous two observations, together with Lemma 2.3 and the charac-terization of the ordering of L given in the introduction, imply that if m ∈ L is such that m <
1, then m = o ( ) in Ω. Since L is a multiplicative group,the lemma follows. (cid:3) Strong asymptotic expansions
Set E := { exp r : r ∈ R } . Note that E is co-initial in L ; in particular,a series F ∈ R (( E )) has E -natural support if and only if it has L -naturalsupport. Definition 3.1.
Let f ∈ C and F = (cid:80) f r exp − r ∈ C (( E )). The germ f has strong asymptotic expansion F (at ∞ ) if(i) F has E -natural support;(ii) f has a holomorphic extension f on some standard quadratic domainΩ;(iii) each f r has a holomorphic extension f r on Ω such that f r = o ( exp s )in Ω, for each s > r ∈ R , we have( ∗ f,r ) f − (cid:88) s ≤ r f s exp − s = o (cid:0) exp − r (cid:1) in Ω . In this situation, Ω is called a strong asymptotic expansion domain of f . Example 3.2.
Let f ∈ C be almost regular with asymptotic expansion F := (cid:80) ∞ n =0 p n exp − ν n as defined in the introduction. Then F is a strongasymptotic expansion of f .To see this, let r ∈ R ; Condition ( ∗ f,r ) holds by definition if r = ν N forsome N ∈ N , so assume that ν N − < r < ν N for some N (setting ν − := −∞ to make sense of all cases). The definition of “almost regular” implies that f − (cid:88) ν n ≤ r p n exp − ν n − p N exp − ν N = o (cid:0) exp − ν N (cid:1) in Ω . Condition ( ∗ f,r ) now follows, because | z | → ∞ in Ω implies Re z → + ∞ , sothat q exp − ν N = o ( exp − r ) in Ω, for every polynomial q . Remark.
Let f ∈ C have strong asymptotic expansion F ∈ C (( E )), and let s ∈ R . Then f · exp s has strong asymptotic expansion F · exp s . Lemma 3.3.
Let f, g ∈ C have strong asymptotic expansions (cid:80) a s exp − s and (cid:80) b s exp − s , respectively, in a standard quadratic domain Ω . Then (1) f + g has strong asymptotic expansion (cid:80) ( a s + b s ) exp − s in Ω ; (2) f g has strong asymptotic expansion ( (cid:80) a s exp − s ) ( (cid:80) b s exp − s ) in Ω ; (3) if f = 0 and s := min { s ∈ R : a s (cid:54) = 0 } , then there exists r > suchthat a s = o ( exp − r ) in Ω . Proof.
Fix r ≥
0. Then in Ω, f + g − (cid:88) s ≤ r ( a s + b s ) exp − s = f − (cid:88) s ≤ r a s exp − s + g − (cid:88) s ≤ r b s exp − s = o (cid:0) exp − r (cid:1) , which proves (1). For (2), write (cid:88) c s exp − s = (cid:16)(cid:88) a s exp − s (cid:17) (cid:16)(cid:88) b s exp − s (cid:17) , so that c s = (cid:80) s + s = s a s b s . By the remark before this lemma, after re-placing f and g by f exp s and g exp s for some s ≤
0, I may assume that a s = b s = 0 for s ≤
0; then f and g , as well as a s exp − s and b s exp − s for each s , are bounded in Ω. Since fg − (cid:88) s ≤ r c s exp − s = f − (cid:88) s ≤ r a s exp − s g ++ (cid:88) s ≤ r a s exp − s g − (cid:88) s ≤ r b s exp − s ++ (cid:88) s ≤ r a s exp − s (cid:88) s ≤ r b s exp − s − (cid:88) s ≤ r c s exp − s , it follows that the first and second of these four summands are o ( exp − r ) inΩ. As to the third and fourth summands, (cid:88) s ≤ r a s exp − s (cid:88) s ≤ r b s exp − s − (cid:88) s ≤ r c s exp − s = (cid:88) s ≤ r a s exp − s (cid:88) s ≤ r b s exp − s − (cid:88) s + s ≤ r a s b s exp − s − s = (cid:88) s ,s ≤ rs s >r a s b s exp − s − s , which is o ( exp − rx ) in Ω, because the latter sum is finite.For (3) set s := min { s > s : a s (cid:54) = 0 } > s . Then Condition ( ∗ f,r ),with r := ( s + s ), implies that a s exp − s = o ( exp − r ) in Ω, so that a s = o (cid:0) exp − ( r − s ) (cid:1) . (cid:3) For F = (cid:80) r ∈ R f r exp − r ∈ C (( E )), I setord( F ) := min { r ∈ R : f r (cid:54) = 0 } . Recall that, given series F n ∈ C (( E )) for n ∈ N such that ord( F n ) → + ∞ as n → ∞ , the infinite sum (cid:80) n F n defines a series in C (( E )). The next criterionis useful for obtaining strong asymptotic expansions. Lemma 3.4.
Let f ∈ C and f n ∈ C , for n ∈ N , and let Ω be a standardquadratic domain. Assume that each f n has strong asymptotic expansion F n ∈ C (( E )) in Ω such that ord( F n ) → + ∞ for n ∈ N , and assume that f has a holomorphic extension f on Ω such that f − n (cid:88) i =0 f i = o ( f n ) in Ω , for each n. Then the series (cid:80) n F n is a strong asymptotic expansion of f in Ω . Proof.
Let r ∈ R , and choose N ∈ N such that ord( F n ) > r for all n ≥ N .Then f n = o ( exp − r ) in Ω, for n ≥ N , so f − n (cid:88) i =0 f i = o ( exp − r ) in Ω . Increasing N if necessary, we may assume that (cid:32) ∞ (cid:88) i =0 F i (cid:33) exp − r = N (cid:88) i =0 ( F i ) exp − r . Therefore, with h r the holomorphic extension of ( (cid:80) F i ) exp − r on Ω and h i,r the holomorphic extension of ( F i ) exp − r on Ω, we get f − h r = f − N (cid:88) i =0 h i,r = (cid:32) f − N (cid:88) i =0 f i (cid:33) + N (cid:88) i =0 ( f i − h i,r )= o ( exp − r ) in Ω , as required. (cid:3) To extend the notion of strong asymptotic expansion to series in R (( L )),I proceed as in Definition 1.2: Definition 3.5.
Let
K ⊆ C be an R -algebra, let L (cid:48) be a divisible subgroupof L , and let T : K −→ R (( L (cid:48) )) be an R -algebra homomorphism. We saythat the triple ( K , L (cid:48) , T ) is a strong qaa algebra if(i) T is injective;(ii) the image T ( K ) is truncation closed;(iii) for every f ∈ K , there exists a standard quadratic domain Ω suchthat f and each g n := T − (( T f ) n ), for n ∈ L (cid:48) , have holomorphicextensions f and g n on Ω, respectively, that satisfy(3.1) f − g n = o ( n ) in Ω . In this situation, I call T ( f ) the strong K -asymptotic expansion of f and Ω a strong K -asymptotic expansion domain of f . Lemma 3.6.
Let ( K , L (cid:48) , T ) be a strong qaa algebra, with L (cid:48) a divisiblesubgroup of L . Then ( K , L, T ) is a strong qaa algebra. Proof.
Let f ∈ K and n ∈ L ; if n ∈ L (cid:48) , then the asymptotic relation(3.1) holds by assumption, so assume n / ∈ L (cid:48) . If n ≤ supp( T f ), then T − (( T f ) n ) = f , so the asymptotic relation (3.1) holds trivially. So as-sume also that n (cid:54)≤ supp( T f ) and choose the maximal p ∈ supp( T f ) such that p < n (which exists because supp( T f ) is reverse well-ordered). By as-sumption, writing g p and g n for the holomorphic extensions of T − (( T f ) p )and T − (( T f ) n ), respectively, o ( p ) = f − g p = f − g n − a p , for some nonzero a ∈ R . Since p = o ( n ) in Ω by Lemma 2.4, the asymptoticrelation (3.1) follows. (cid:3) The construction
The initial Ilyashenko algebra.
In view of Fact 1.4 and in the spiritof [2, Section 24], I define A to be the set of all f ∈ C that have a strongasymptotic expansion F = (cid:80) r ≥ a r exp − r ∈ R (( E )). Note that the conditionsupp( F ) ⊆ [0 , + ∞ ) implies that f has a bounded holomorphic extension tosome standard quadratic domain. Lemma 4.1. (1) A is an R -algebra. (2) Each f ∈ A has a unique strong asymptotic expansion T f ∈ R (( E )) . (3) The map T : A −→ R (( E )) is an injective R -algebra homomor-phism. Proof.
Part (1) follows from Lemma 3.3(1,2). For part (2), assume for a con-tradiction that 0 has a nonzero strong asymptotic expansion (cid:80) a r exp − r ∈ R (( E )) of order s . Then by Lemma 3.3(3), we have a s = o (exp − r ) for some r >
0; since a s ∈ R , it follows that a s = 0, a contradiction. For part (3),the map T is a homomorphism by Lemma 3.3(1,2), and its kernel is trivialby Fact 1.4. (cid:3) Corollary 4.2.
The triple ( A , L, T ) is a strong qaa algebra. Proof.
By Lemma 3.6, it suffices to show that ( A , E, T ) is a strong qaaalgebra. For r ≥ − r has a bounded holomorphic extensionon H (0), so it belongs to A with T exp − r = exp − r . Since the supportof T f , for f ∈ A , is E -natural, every truncation of T f is an R -linearcombination of exp − r , for various r ≥
0, and therefore belongs to A aswell. (cid:3) Examples 4.3. (1) Let p ∈ R [[ X ∗ ]] be convergent with natural support[12, 6]. Then p ◦ exp − ∈ A .(2) The algebra A ◦ ( − log) is the class A = A , considered in [6,Definition 5.4]. In particular, for f ∈ A the series T ( f ) ◦ ( − log) ∈ R [[ X ∗ ]]has natural support and, for r ≥ g r := T − ( T ( f )) exp − r , wehave f ( − log x ) − g r ( − log x ) = o ( x r ) as x → + . The initial Ilyashenko field.
For f ∈ A , I setord( f ) := ord ( T ( f )) . Below, I call f ∈ C infinitely increasing if f ( x ) → + ∞ , small if f ( x ) → unit if f ( x ) →
1, as x → + ∞ .Similarly, let G ∈ R (( L )), and let g ∈ L be the leading monomial of G ; sothere are nonzero a ∈ R and (cid:15) ∈ R (( L )) such that G = ag (1 + (cid:15) ). Note thatthe leading monomial of (cid:15) is small. I call G small if g is small, and I call G infinitely increasing if both g is infinitely increasing and a > Remark and Definition 4.4.
Let G ∈ R (( L )), and let g ∈ L be the leadingmonomial of G ; so there are nonzero a ∈ R and small (cid:15) ∈ R (( L )) such that G = ag (1 + (cid:15) ). Let also k ∈ {− } ∪ N and F ∈ R ((( X − , . . . , X k ) ∗ )) be suchthat F has natural support and G = F (cid:18) , , . . . , k (cid:19) . Let α = ( α − , . . . , α k ) ∈ R k be the minimum of the support of F withrespect to the lexicographic ordering on R k , so that g = exp − α − log − α · · · log − α k k . Case 1: Let P ∈ R [[ X ∗ ]] be of natural support, and assume that G is small.Then α > (0 , . . . ,
0) in the lexicographic ordering of R k .Case 2: Let P ∈ R [[ (cid:0) X (cid:1) ∗ ]] be of natural support, and assume that G is infin-itely increasing. Then α < (0 , . . . ,
0) in the lexicographic orderingof R k .In both cases, P ◦ F belongs to R ((( X − , . . . , X k ) ∗ )) and has natural supportas well. I therefore define P ◦ G := ( P ◦ F ) (cid:18) , , . . . , k (cid:19) . This composition is associative in the following sense: whenever P ∈ R [[ X ∗ ]]is small and of natural support and Q ∈ R [[ X ∗ ]] is of natural support, then Q ◦ ( P ◦ G ) = ( Q ◦ P ) ◦ G . A similar statement holds in Case 2; as usual, Iwill therefore simply write Q ◦ P ◦ G for these compositions. Lemma 4.5.
Let f, g ∈ A , and set d := ord( g ) ≥ . (1) There exist unique nonzero g d ∈ R and (cid:15) ∈ A such that g = g d exp − d (1 − (cid:15) ) and ord( (cid:15) ) > . (2) Assume that g is small with strong asymptotic expansion domain Ω , and let P ∈ R [[ X ]] be convergent. Then P ◦ g belongs to A ,has strong asymptotic expansion domain Ω and satisfies T ( P ◦ g ) = P ◦ T ( g ) . Remark.
In the situation of Part (1) above, the germ gg d exp − d is a unitbelonging to A . Proof. (1) Say T ( g ) = (cid:80) r ≥ d g r exp − r ; then take (cid:15) := − g − g d exp − d g d exp − d , which belongs to A by Lemma 3.3(2).(2) By Condition ( ∗ g, ), the function P ◦ g is a bounded, holomorphicextension of P ◦ g on Ω. Moreover, say P ( X ) = (cid:80) a n X n ∈ R [[ X ]]; since P ( z ) − (cid:80) ni =0 a i z i = O ( z n ) at 0 in C by absolute convergence, it follows that P ◦ g − n (cid:88) i =0 g i = o ( g n ) in Ω . From Lemma 3.3, it follows that a n g n ∈ A has strong asymptotic expansiondomain Ω and satisfies T ( a n g n ) = a n T ( g ) n , for each n . Since g is small, we have d >
0, so we also get ord ( g n ) = ns → ∞ as n → ∞ . Part (2) now follows from Lemma 3.4. (cid:3) Let F be the fraction field of A and extend T to an R -algebra homo-morphism T : F −→ R (( E )) in the obvious way (also denoted by T ). Notethat the functions in F do not all have bounded holomorphic extensions tostandard quadratic domains; hence the need for first defining A . Remark.
Let K be a subfield of C . Let F, G ∈ K (( E )), let g be the leadingterm of G and set (cid:15) := − G − gg . Recall that FG = Fg · (G eom ◦ (cid:15) ) , where G eom = (cid:80) ∞ n =0 X n is the geometric series. Corollary 4.6. (1)
Let f ∈ F . Then f has strong asymptotic expan-sion T ( f ) , and there exist unique d, f d ∈ R and (cid:15) ∈ A such that f = f d exp − d (1 + (cid:15) ) and ord( (cid:15) ) > . (2) ( F , L, T ) is a strong qaa field. Proof. (1) Say f = g/h , for some g, h ∈ A with h (cid:54) = 0 of order s ≥
0. ByLemma 4.5(1) there are h s ∈ R \{ } and (cid:15) ∈ A such that h = h s exp − s (1 − (cid:15) )and ord( (cid:15) ) >
0. In particular, (cid:15) is small, so that f = gh s exp − s (1 − (cid:15) ) = exp s h s g G eom ( (cid:15) ) . Part (1) now follows from Lemmas 3.3 and 4.5(2). Since the series in T ( F ) have E -natural support and each monomial in E belongs to F , the triple ( F , E, T ) is a qaa field. Part (2) now followsfrom Lemma 3.6. (cid:3) Iteration.
I construct strong qaa fields ( F k , L, T k ), for nonzero k ∈ N , suchthat F k − is a subfield of F k and T k extends T k − , which I summarize bysaying that ( F k , L, T k ) extends ( F k − , L, T k − ). As in the initial stage ofthe construction, I will obtain F k as the fraction field of a strong qaa algebra( A k , L, T k ) such that(i) each f ∈ A k has a bounded, holomorphic extension to some standardquadratic domain;(ii) for each f ∈ F k , there exists s ∈ R such that f exp s belongs to A k .Note that, by Lemma 4.5(1), conditions (i) and (ii) hold for k = 0, providedI set A − = F − := R .The construction proceeds by induction on k ; the case k = 0 is done above.So assume k > A i , L, T i ) and ( F i , L, T i ) have been constructed,for i = 0 , . . . , k −
1. First, I set F (cid:48) k := F k − ◦ logand define T (cid:48) k : F (cid:48) k −→ R (( L (cid:48) )) by T (cid:48) k ( f ◦ log) := ( T k − f ) ◦ log , where L (cid:48) := { m ∈ L : m ( −
1) = 0 } . Corollary 4.7. ( F (cid:48) k , L, T (cid:48) k ) is a strong qaa field. Proof.
Since log maps H (0) into any standard quadratic domain, the triple( F (cid:48) k , L (cid:48) , T (cid:48) k ) is a strong qaa field. Since L (cid:48) is a divisible subgroup of L , thecorollary follows from Lemma 3.6. (cid:3) Remark 4.8.
Let g ∈ F (cid:48) k . There exists, by condition (ii) above, an s ∈ R such that g/x s has a bounded holomorphic extension on some standardquadratic domain Ω. Thus g = o (exp r ) for every r > F (cid:48) k is afield, it follows that g = o (exp − r ) for some r > g = 0.Now let A k be the set of all f ∈ C that have a bounded, holomorphicextension on some standard quadratic domain Ω and a strong asymptoticexpansion (cid:80) r ≥ f r exp − r ∈ F (cid:48) k (( E )) in Ω. (The boundedness assumption isincluded here, because not all f ∈ F (cid:48) k are bounded if k ≥ A k is an R -algebra,each f ∈ A k has a unique strong asymptotic expansion τ k f := (cid:88) r ≥ f r exp − r ∈ F (cid:48) k (( E )) , and the map τ k : A k −→ F (cid:48) k (( E )) is an R -algebra homomorphism. Moreover,it follows from Fact 1.4 that this map is injective. For f ∈ A k with τ k f = (cid:80) f r exp − r , I now define T k f := (cid:88) r ≥ ( T (cid:48) k f r ) exp − r . For completeness’ sake, I also set τ := T . Proposition 4.9.
The triple ( A k , L, T k ) is a strong qaa algebra that extends ( A k − , L, T k − ) . Proof.
The map σ : F (cid:48) k (( E )) −→ R (( L )) defined by σ (cid:16)(cid:88) f r exp − r (cid:17) := (cid:88) ( T (cid:48) k f r ) exp − r is an R -algebra homomorphism, and it is injective because T (cid:48) k is injective.Since T k = σ ◦ τ k , it follows that T k is an injective R -algebra homomorphism.Let now f ∈ A k be such that T k f = (cid:88) m ∈ L a m m and τ k f = (cid:88) r ≥ f r exp − r , and let n ∈ L ; we show there exists g ∈ A k such that T k g = ( T k f ) n .Considering n as a function n : {− } ∪ N −→ R , set r := − n ( −
1) and n (cid:48) := (cid:81) ∞ i =0 log n ( i ) i ∈ L (cid:48) , so that n = n (cid:48) exp − r and( T k f ) n = (cid:88) m ( − >n ( − a m m + (cid:0) T (cid:48) k f r (cid:1) n (cid:48) exp − r , and let Ω be a strong asymptotic expansion domain of f . Note that each f s exp − s has a bounded holomorphic extension on Ω. Since σ − (cid:88) m ( − >n ( − a m m = (cid:88) s
0. Then there exists F ∈ F (cid:48) k [[ X ∗ ]] such that F has natural support,ord( F ) > G = F (cid:0) exp − (cid:1) . Hence P ◦ F belongs to F (cid:48) k [[ X ∗ ]] and hasnatural support as well. We therefore define P ◦ G := ( P ◦ F ) (cid:0) exp − (cid:1) , which belongs to F (cid:48) k (( E )). Similar to the situation in Remark and Definition4.4, this composition is associative: if ord( P ) > Q ∈ R [[ X ∗ ]] hasnatural support, then ( Q ◦ P ) ◦ F = Q ◦ ( P ◦ F ). Lemma 4.10.
Let g ∈ A k , and set d := ord( g ) ≥ . (1) There exist unique g d ∈ F (cid:48) k and (cid:15) ∈ A k such that g = g d exp − d (1 + (cid:15) ) and ord( (cid:15) ) > . (2) Assume ord( g ) > , and let P ∈ R [[ X ]] be convergent. Then P ◦ g ∈A k , and we have τ k ( P ◦ g ) = P ◦ τ k ( g ) and T k ( P ◦ g ) = P ◦ T k ( g ) . Proof.
Replacing T by τ k throughout, the proof of Lemma 4.5(1,2) giveseverything except the statement T k ( P ◦ g ) = P ◦ T k ( g ). However, in thethe situation of part (2) with the notations from the proof of Lemma 4.5(2),since for each r ≥ N r ∈ N such that( P ◦ τ k ( f )) exp − r = N r (cid:88) n =0 a n ( τ k ( f ) n ) exp − r , it follows that σ ( P ◦ τ k ( f )) = P ◦ σ ( τ k ( f )). (cid:3) As in the construction of F , I now let F k be the fraction field of A k andextend τ k and T k correspondingly. Corollary 4.11. (1)
Let f ∈ F k . Then f has strong asymptotic expan-sion τ k ( f ) , and there exist unique d ∈ R , f d ∈ F (cid:48) k and (cid:15) ∈ A k suchthat f = f d exp − d (1 + (cid:15) ) and ord( (cid:15) ) > . In particular, f ∈ A k if and only if f is bounded. (2) The triple ( F k , L, T k ) is a strong qaa field. Proof. (1) Say f = g/h , for some g, h ∈ A k with h (cid:54) = 0 of order s ≥ h s ∈ F (cid:48) k and (cid:15) ∈ A k such that h = h s exp − s (1 − (cid:15) ) and ord( (cid:15) ) >
0. In particular, (cid:15) is small, so that f = gh s exp − s (1 − (cid:15) ) = exp s h s g G eom ( (cid:15) ) . Part (1) now follows from Lemmas 3.3 and 4.10(2).(2) The map T k is injective, because the restriction of T k to A k is. Also,by part (1), each f ∈ F k is of the form f = exp r g with g ∈ A k and r ∈ R .Since ( A k , L, T k ) is a strong qaa algebra, it follows that ( F k , L, T k ) is a strongqaa field. (cid:3) Remark 4.12.
Since A contains all polynomials in exp, the algebra A contains the class A of almost regular maps.In view of Proposition 4.9 and Corollary 4.11, we set A := (cid:91) k A k and F := (cid:91) k F k , and we let T be the common extension of all T k to F ; we denote the re-striction of T to A by T as well. It follows that ( A , L, T ) is a strong qaaalgebra and ( F , L, T ) is a strong qaa field such that F is the fraction fieldof A . This finishes the proof of Theorem 1.3(1).5. Closure under differentiation
The next lemma is a version of L’Hˆopital’s rule for holomorphic maps onstandard quadratic domains.
Lemma 5.1.
Let < C < D and φ : Ω C −→ C be holomorphic. (1) Let r ∈ R be such that φ = o (exp − r ) in Ω C . Then φ (cid:48) = o (exp − r ) in Ω D . (2) If φ is bounded in Ω C , then φ (cid:48) is bounded in Ω D . Proof. (1) By Lemma 2.2(1), there is
R > D ( z, ⊆ Ω C forevery z ∈ Ω D with | z | > R . Let z ∈ Ω D be such that | z | > R , and let w z ∈ { w : | w − z | = 1 } be such that | φ ( w z ) | = max | w − z | =1 | φ ( w ) | ; then, byCauchy’s formula, we have | φ (cid:48) ( z ) | ≤ | φ ( w z ) | . On the other hand, | e − rz | = e − r Re z ≥ (cid:40) e − r (Re w z − = e r e − rw z if r ≤ ,e − r (Re w z +2) = e − r e − rw z if r ≥ . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) φ (cid:48) ( z ) e − rz (cid:12)(cid:12)(cid:12)(cid:12) ≤ e | r | (cid:12)(cid:12)(cid:12)(cid:12) φ ( w z ) e − rw z (cid:12)(cid:12)(cid:12)(cid:12) . Since | w z | ∼ | z | and φ = o (exp − r ) in Ω C , the conclusion follows.The proof of (2) is similar and left to the reader. (cid:3) I now set D := { f ∈ C : f is differentiable } and for F = (cid:80) f r exp − r ∈ D (( E )), I define F (cid:48) := (cid:88) ( f (cid:48) r − rf r ) exp − r ∈ C (( E )) . Proposition 5.2.
Let k ∈ N and f ∈ F k . Then f (cid:48) ∈ F k and τ k ( f (cid:48) ) = ( τ k f ) (cid:48) . Proof.
By induction on k ; let τ k ( f ) = (cid:80) f r exp − r . If k = 0, then ( τ k f ) (cid:48) ∈F (cid:48) k (( E )) because the coefficients of τ k f are real numbers. If, on the otherhand, k >
0, then f r = g r ◦ log for some g r ∈ F k − , so that f (cid:48) r = g (cid:48) r ◦ log x = g (cid:48) r exp ◦ log ∈ F (cid:48) k by the inductive hypothesis, so that again ( τ k f ) (cid:48) ∈ F (cid:48) k (( E )).To finish the proof of the proposition, we may assume (by the quotientformula for derivatives) that f ∈ A k . Let C > C is a domainof strong asymptotic expansion of f , and let D > C . By Lemma 5.1(2), themap f (cid:48) : Ω D −→ C is a bounded, holomorphic extension of f (cid:48) . Moreover, if r ≥
0, then f (cid:48) − (cid:88) s ≤ r ( f (cid:48) s − s f s ) exp − s = f − (cid:88) s ≤ r f s exp − s (cid:48) = o ( exp − r ) in Ω D , by Lemma 5.1(1) and Condition ( ∗ f,r ), so that f (cid:48) ∈ A k . (cid:3) Finally note that, for m ∈ L , the derivative m (cid:48) is a linear combination ofelements of L such that max supp( m (cid:48) ) → m → L . Therefore, for F = (cid:80) a m m ∈ R (( L )), I define F (cid:48) := (cid:88) a m m (cid:48) , and I note that the map F (cid:55)→ F (cid:48) is a derivation on R (( L )). Corollary 5.3. F is closed under differentiation and for f ∈ F , we have T ( f (cid:48) ) = ( T f ) (cid:48) . Proof.
Let k ∈ N and f ∈ F k ; I proceed by induction on k to show that T ( f (cid:48) ) = ( T f ) (cid:48) . If k = 0, then T ( f ) = τ ( f ) and ( T f ) (cid:48) = ( τ f ) (cid:48) , so the claimfollows from Proposition 5.2 in this case. So I assume k > k .Say τ k ( f ) = (cid:80) f r exp − r ; then T ( f ) = (cid:80) T ( f r ) exp − r by definition, while τ k ( f (cid:48) ) = ( τ k f ) (cid:48) = (cid:80) ( f (cid:48) r − rf r ) exp − r . It follows from the inductive hypothesisthat T ( f (cid:48) ) = (cid:88) T ( f (cid:48) r − rf r ) exp − r = (cid:88) (cid:0) T ( f (cid:48) r ) − rT ( f r ) (cid:1) exp − r = (cid:88) (cid:0) ( T f r ) (cid:48) − rT ( f r ) (cid:1) exp − r = ( T f ) (cid:48) , as claimed. (cid:3) Closure under log -composition
Note that, since F is a field, it is closed under log-composition if and onlyif for all f, g ∈ F such that lim x → + ∞ g ( x ) = + ∞ , the composition f ◦ log ◦ g belongs to F . First I show that, for infinitely increasing g ∈ F , the maplog ◦ g always has a holomorphic extension that maps standard quadraticdomains into standard quadratic domains. Lemma 6.1.
Let g ∈ F and Ω g be a strong F -asymptotic expansion domainof g , and assume that g is infinitely increasing. Then, for some standardquadratic domain Ω (cid:48) g ⊆ Ω g , the function log ◦ g has a holomorphic extension l g on Ω (cid:48) g such that, for every standard quadratic domain Ω , there exists astandard quadratic domain ∆ ⊆ Ω (cid:48) g with ( l g )(∆) ⊆ Ω . Proof.
Let a > m ∈ L be the leading monomial of F and small (cid:15) ∈ F be such that g = am (1 + (cid:15) ). Shrinking Ω g if necessary, I may assume thatΩ g is also a strong F -asymptotic expansion domain of (cid:15) with correspondingholomorphic extension e : Ω g −→ C . Then by the asymptotic relation (3.1),we have g = a m (1 + e ) with e = o ( ) in Ω g ;in particular, after shrinking Ω g again if necessary, the function log a +log(1 + (cid:15) ) has holomorphic extension log a + log (1 + e ) on Ω g such that log (1 + e ) = o ( ) in Ω g . Sincelog ◦ g = log a + log ◦ m + log(1 + (cid:15) ) , I may therefore assume by Lemma 2.2 that g = m ∈ L . However log ◦ m isan R -linear combination of log i , for various i ∈ N . Let i be the smallest i such that log i appears in this R -linear combination. Since m is infinitelyincreasing, the coefficient of log i in this R -linear combination must be pos-itive. Since log i = o (log i ) in H (0), for i > i , it follows as above that I may even assume that m = log i . But this last case follows from Lemma2.2(3). (cid:3) Formal log -composition in R (( L )) . Let G ∈ R (( L )), and let g ∈ L be theleading monomial of G ; so there are nonzero a ∈ R and small (cid:15) ∈ R (( L ))such that G = ag (1 + (cid:15) ).(L1) Assume that a >
0. Note that log ◦ g is an R -linear combination ofelements of the set { log k : k ∈ N } . Therefore, with F log ∈ R [[ X ]] theTaylor series at 0 of log(1 + x ), I definelog ◦ G := log a + log ◦ g + ( F log ◦ (cid:15) ) . Note that if G is small and G >
0, then − log ◦ G = log ◦ G ; and if G is infinitely increasing, then so is log ◦ G . Thus, for G infinitelyincreasing and nonzero i ∈ N , I definelog i ◦ G := log ◦ (log i − ◦ G )by induction on i .(L2) Recall that L (cid:48) = { m ∈ L : m ( −
1) = 0 } , and let F ∈ R (( L (cid:48) )). Sothere are l ∈ N and P ∈ R ((( X , . . . , X l ) ∗ )) with natural supportsuch that F = P (cid:18) , . . . , l (cid:19) ;i.e., the support of F contains no exponential monomials. Assumethat G is infinitely increasing. Then, by (L1) above, there exist k i ∈ N and Q i ∈ R ((( X − , . . . , X k i ) ∗ )) with natural support suchthat 1log i ◦ G = Q i (cid:18) , , . . . , k i (cid:19) , for i ∈ N . Since G is infinitely increasing, each i ◦ G is small, and it followsthat P ( Q , . . . , Q l ) ∈ R ((( X , . . . , X k ) ∗ )), where k = max { k , . . . , k l } .Therefore, I set F ◦ G := P ( Q , . . . , Q l ) (cid:18) , , . . . , k (cid:19) ∈ R (( L )) . (L3) Let F ∈ R (( L )), and let l ∈ N and P ∈ R ((( X − , . . . , X l ) ∗ )) withnatural support be such that F = P (cid:18) , , . . . , l (cid:19) . Then I set F ◦ log := P (cid:18) , . . . , l +1 (cid:19) ;note that F ◦ log ∈ R (( L (cid:48) )). Lemma 6.2.
Let F ∈ R (( L (cid:48) )) and G ∈ R (( L )) be such that G is infinitelyincreasing. Then ( F ◦ log) ◦ G = F ◦ (log ◦ G ) . Proof.
Let Q i be for i ◦ G be as in (L2). Then for i ∈ N , I have by (L1)that 1log i ◦ (log ◦ G ) = 1log i +1 ◦ G = Q i +1 (cid:32) , , . . . , k i +1 (cid:33) . On the other hand, let l ∈ N and P ∈ R ((( X , . . . , X l ) ∗ )) with naturalsupport be such that F = P (cid:18) , . . . , l (cid:19) . Then by (L2), I have F ◦ (log ◦ G ) = P ( Q , . . . , Q l +1 ) (cid:18) , , . . . , k (cid:19) , where k := max { k , . . . , k l +1 } . On the other hand, by (L3), I have F ◦ log = P (cid:18) , . . . , kl +1 (cid:19) , so again by (L2), I get( F ◦ log) ◦ G = P ( Q , . . . , Q l +1 ) (cid:32) , , . . . , k i +1 (cid:33) , and the lemma is proved. (cid:3) I continue working in the setting of (L1)–(L3) above.(L4) For r ∈ R , I let P r ∈ R [[ X ]] be the Taylor series at 0 of (1 + x ) r , andI define G r := a r g r · ( P r ◦ (cid:15) ) . Note that, if G is infinitely increasing, then so is G r .(L5) For r ∈ R , I let F exp r be the Taylor series at 0 of the function x (cid:55)→ exp( rx ), and I setexp r ◦ (log ◦ G ) := a r g r ( F exp r ◦ ( F log ◦ (cid:15) ) . Note that this series has order r · ord( g ); thus, for F = (cid:80) f r exp − r ∈ R (( L )) with f r ∈ R (( L (cid:48) )) I set F ◦ (log ◦ G ) := (cid:88) ( f r ◦ (log ◦ G )) · G − r . Corollary 6.3.
Let
F, G ∈ R (( L )) be such that G is infinitely increasing.Then ( F ◦ log) ◦ G = F ◦ (log ◦ G ) . Proof.
Note that P r ( x ) = (1 + x ) r = exp( r log(1 + x )) = ( F exp r ◦ F log )( x )for r ∈ R and small x ∈ R , so that P r ◦ (cid:15) = F exp r ◦ F log ◦ (cid:15) . It follows from(L3), (L4) and Lemma 6.2 that F ◦ (log ◦ G ) = ( F ◦ log) ◦ G . (cid:3) In the situation of the previous corollary, I write F ◦ log ◦ G for the com-position F ◦ (log ◦ G ) = ( F ◦ log) ◦ G , called the log -composition of F with G . Closure under log -composition.
First I show that F is closed underlog-composition. Lemma 6.4.
Let f, g ∈ F and assume that g is infinitely increasing. Then f ◦ log ◦ g ∈ F and T ( f ◦ log ◦ g ) = T ( f ) ◦ log ◦ T ( g ) . Proof.
It suffices to prove the lemma for f ∈ A . Let Ω and ∆ be strong as-ymptotic expansion domains for f and g , respectively. (Recall that “‘strongasymptotic expansion” and “strong F -asymptotic expansion” mean the samething for h ∈ F .) By Lemma 6.1, after shrinking Ω if necessary, the germlog ◦ g has a holomorphic extension l g on Ω such that ( l g ) (Ω) ⊆ ∆. There-fore, the function h := f ◦ log ◦ g has bounded, holomorphic extension f ◦ l g on Ω.Moreover, for each r ≥
0, the germ g − r = exp − r ◦ (log ◦ g ) has boundedholomorphic extension exp − r ◦ l g on Ω. On the other hand, writing g = am (1 + (cid:15) ) with a > m ∈ L the leading monomial of g and (cid:15) ∈ A small, Iget g − r = a − r m − r ( P − r ◦ (cid:15) ) , where P − r is the Taylor series expansion of x (cid:55)→ (1 + x ) − r at 0. It followsfrom Lemma 4.5(2) that g − r ∈ F with strong asymptotic expansion domainΩ such that T ( g − r ) = a − r m − r ( P − r ◦ T ( (cid:15) )) = T ( g ) − r by (L1). Setting d := ord( g ) <
0, it follows in particular that ord( g − r ) = − rd .Now say that T ( f ) = (cid:80) r ≥ a r exp − r , and let r ≥
0. Since f has strongasymptotic expansion T ( f ) in ∆, we have f − (cid:88) s ≤ r a s exp − s = o ( exp − r ) in H (0) , so that f ◦ l g − (cid:88) s ≤ r a s ( exp − s ◦ l g ) = o (cid:0) exp − r ◦ l g (cid:1) in Ω . By the previous paragraph, we have a s g − s ∈ F with strong asymptoticexpansion domain Ω, for each s ≥
0, and ord( a s g − s ) = − sd → + ∞ as s → + ∞ . Since T ( f ) has L -natural support, it follows from Lemma 3.4 that f ∈ A with T ( f ) = (cid:80) a r T ( g ) − r . On the other hand, since T ( f ) ◦ log = (cid:80) a r x − r , we have T ( f ) ◦ log ◦ T ( g ) = T ( f ), and the lemma is proved. (cid:3) Next, let k, l ∈ N , f ∈ F k and g ∈ F l , and assume that g is infinitelyincreasing. The remaining difficulty in the proof of Theorem 1.3(2) lies inmaking sense of the strong asymptotic expansion of f ◦ log ◦ g . Remarks 6.5.
Set s := ord( g ) ≤
0, and let g s ∈ F (cid:48) l and (cid:15) ∈ A l be suchthat g = g s exp − s (1 + (cid:15) ) and ord( (cid:15) ) >
0. There are two cases to consider: Case 1: s <
0. Say τ k ( f ) = (cid:80) f r exp − r and let r ∈ supp( τ k ( f )). Since f r ∈ F (cid:48) k , there exists m ( r ) ∈ N such that x − m ( r ) ≤ | f r | ≤ x m ( r ) ; andsince g ∈ F l , there exists n ( r ) ∈ N such that x − n ( r ) ≤ log ◦ g ≤ x n ( r ) .Hence there exists N ( r ) ∈ N such that x − N ( r ) ≤ f r ◦ log ◦ g ≤ x N ( r ) . If I already know (by induction on k , say) that each f r ◦ log ◦ g belongsto F j for some j ∈ N independent of r then, by Corollary 4.11(1),there exist h r ∈ F (cid:48) j and d ( r ) ∈ R such that f r ◦ log ◦ g ∼ h r exp d ( r ) .Since (as above for f r ) the germ h r is also polynomially bounded, itfollows that d ( r ) = ord( f r ◦ log ◦ g ) = 0, so thatord (cid:0) τ j ( f r ◦ log ◦ g ) τ l ( g ) − r (cid:1) = − rs . Since exp − r ◦ log ◦ g = g − r for each r , this suggests that the series (cid:88) r ∈ R τ j ( f r ◦ log ◦ g ) τ l ( g ) − r is a candidate for the strong asymptotic expansion of f ◦ log ◦ g inthis case.Case 2: s = 0. The assumption that g is infinitely increasing then impliesthat g ∈ F (cid:48) l is infinitely increasing as well; in particular, we musthave l >
0. By Taylor’s Theorem, since log ◦ g = log ◦ g + F log ◦ (cid:15) and log ◦ g is infinitely increasing while F log ◦ (cid:15) is small, we have f ◦ log ◦ g = ∞ (cid:88) i =0 f ( i ) ◦ log ◦ g i ! ( F log ◦ (cid:15) ) i . This suggests the following: if I already know (by induction on l ,say) that each f ( i ) ◦ log ◦ g belongs to F (cid:48) j for some j ≥ l independentof i , then the series ∞ (cid:88) i =0 f ( i ) ◦ log ◦ g i ! τ l ( F log ◦ (cid:15) ) i is a candidate for the strong asymptotic expansion of f ◦ log ◦ g inthis case.In view of Case 2 above, I need a formal version of the Taylor expansiontheorem. It relies on the observation that the logarithmic generalized powerseries belong to the set T of transseries as defined by van der Hoeven in [13]. Lemma 6.6.
Let F ∈ R (( L )) , let k > , and let G ∈ R (( L (cid:48) )) and H ∈ R (( L )) be such that G is infinitely increasing and H is small. Then, as elements of T , we have F ◦ ( G + H ) = ∞ (cid:88) i =0 F ( i ) ◦ Gi ! H i . Proof.
By [13, Theorem 5.12], there exists a transseries G − ∈ T such that G ◦ G − = x . Since H is small, so is the transseries δ := H ◦ G − ; that is, wehave δ ≺ m ∈ L we have m † := (log m ) (cid:48) is bounded, so that m † δ is small as well. It follows from [13,Proposition 5.11(c)] that F ◦ ( x + δ ) = ∞ (cid:88) i =0 F ( i ) i ! δ i . Composing on the right with G then proves the lemma. (cid:3) Theorem 6.7.
Let k, l ∈ N , f ∈ F k and g ∈ F l , and assume that g isinfinitely increasing. Then f ◦ log ◦ g ∈ F k + l and T ( f ◦ log ◦ g ) = ( T f ) ◦ log ◦ ( T g ) . Moreover, writing g = g s exp − s (1 + (cid:15) ) with s = ord( g ) and ord( (cid:15) ) > ,and writing τ k ( f ) = (cid:80) f r exp − r , we have τ k + l ( f ◦ log ◦ g ) = (cid:40)(cid:80) r ∈ R τ k − l ( f r ◦ log ◦ g ) τ l ( g ) − r if s < , (cid:80) i ∈ N f ( i ) ◦ log ◦ g i ! τ l ( F log ◦ (cid:15) ) i if s = 0 , where F log is the Taylor series at 0 of the function x (cid:55)→ log(1 + x ) . Proof.
Since F k is the fraction field of A k , I may assume that f ∈ A k . ByLemma 6.1 there is a strong F -asymptotic expansion domain Ω of g suchthat l g (Ω) ⊆ ∆, where ∆ is a strong F -asymptotic expansion domain of f .In particular, the germ h := f ◦ log ◦ g has a holomorphic extension h := f ◦ l g on Ω.We proceed by induction on the pair ( k, l ) ∈ N with respect to thelexicographic ordering of N . The case k = l = 0 corresponds to Lemma 6.4,so I assume ( k, l ) > (0 ,
0) and the theorem holds for lower values of ( k, l ).Let f r ∈ F (cid:48) k be such that τ k ( f ) = (cid:80) r ≥ f r exp − r , and let g r ∈ F (cid:48) l be suchthat τ l ( g ) = (cid:80) r ∈ R g r exp − r . Set s := ord( g ) ≤
0; we distinguish two cases:
Case 1: s < . By the inductive hypothesis, each f r ◦ log ◦ g belongs to F k − l . Since f r ∈ R if k = 0 and F (cid:48) l ⊆ F (cid:48) k − l if k >
0, it follows fromRemark 6.5(1) that the series H := (cid:88) r ≥ τ k − l ( f r ◦ log ◦ g ) τ l ( g ) − r belongs to F (cid:48) k − l (( E )) ⊆ F (cid:48) k + l (( E )), and I claim that τ k + l ( h ) = H .To prove the claim, let r ∈ supp( τ k ( f )); it suffices, by Lemma 3.4, toshow that h − (cid:88) s ≤ r ( f s ◦ l g ) g − s = o (cid:0) ( f r ◦ l g ) g − r (cid:1) in Ω . However, by assumption I have f − (cid:80) s ≤ r f s exp − s = o (cid:16) exp − r (cid:48) (cid:17) in ∆, forany r (cid:48) > r such that r (cid:48) < ord (cid:16) f − (cid:80) s ≤ r f s exp − s (cid:17) ; in particular, h − (cid:88) s ≤ r ( f s ◦ l g ) g − s = o (cid:16) g − r (cid:48) (cid:17) in Ω . On the other hand, by Case 1 of Remark 6.5, the germ f r ◦ log ◦ g is poly-nomially bounded, so that g − r (cid:48) = o (( f r ◦ l g ) g − r ) in Ω, which proves theclaim.Finally, by the inductive hypothesis I have, for r ≥
0, that T (cid:88) s ≤ r f s ◦ log ◦ gg s = (cid:88) s ≤ r T ( f s ) ◦ log ◦ T ( g ) T ( g ) s = ( T ( f )) r ◦ log ◦ T ( g ) . Since ord (( f s ◦ log ◦ g ) g − s ) → + ∞ as s → + ∞ , we get T ( h ) = T ( f ) ◦ log ◦ T ( g ), and the theorem is proved in this case. Case 2: s = 0 . Then l > h ∈ F l − such that g = h ◦ log. By the inductive hypothesis and Proposition 5.2, each f ( i ) ◦ log ◦ h belongs to F k + l − , so that f ( i ) ◦ log ◦ g belongs to F (cid:48) k + l ; in particular, theseries H := (cid:88) i ∈ N f ( i ) ◦ log ◦ g i ! τ l ( F log ◦ (cid:15) ) i belongs to F (cid:48) k + l (( E )), where (cid:15) := ( g − g ) /g . Based on Case 2 of Remark6.5, I now claim that τ k + l ( h ) = H. To prove the claim, note first that it is clear from Case 2 of Remark 6.5if f ( n ) = 0 for some n ∈ N , since the series H is given by a finite sum in thiscase. So assume from now on f ( n ) (cid:54) = 0 for all n ; since ord( F log ◦ (cid:15) ) >
0, wehave ord (cid:0) ( F log ◦ (cid:15) ) i (cid:1) → ∞ as i → ∞ . Shrinking Ω if necessary, we may assume that Ω is also a strong F -asymptoticexpansion domain of (cid:15) and of log ◦ g , with corresponding holomorphic ex-tensions e and l g , respectively. By Lemma 3.4, it therefore suffices to showthat h − n (cid:88) i =0 f ( i ) ◦ l g i ! ( F log ◦ e ) i = o (cid:32) f ( n ) ◦ l g n ! ( F log ◦ e ) n (cid:33) in Ω, for n ∈ N . However, it follows from Corollary 4.11(1) that (cid:12)(cid:12) f ( n +1) ( z ) (cid:12)(cid:12) ≤ e p | z | for some p ∈ N and sufficiently large z ∈ Ω. Also, since T ( g ) ∈ F (cid:48) l and g is infinitely increasing, the leading monomial of g belongs to L (cid:48) , so theleading monomial of log ◦ g is log i for some i ≥
1; hence | l g ( z ) | ≤ q log | z | for some q ∈ N and sufficiently large z ∈ Ω. Finally, since ord( (cid:15) ) >
0, itfollows that | ( F log ◦ e )( z ) | ≤ | z | r | e − sz | for sufficiently large z ∈ Ω, where s = ord( F log ◦ (cid:15) ) > r ∈ N . Combining these three estimates withTaylor’s formula, one obtaines (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h − n (cid:88) i =0 f ( i ) ◦ l g i ! ( F log ◦ e ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ K (cid:12)(cid:12)(cid:12) x t exp − ( n +1) s (cid:12)(cid:12)(cid:12) in Ω, for some t ∈ N and K >
0. On the other hand, since (cid:12)(cid:12) f ( n ) ( z ) (cid:12)(cid:12) ≥ e − p | z | for some p ∈ N and sufficiently large z ∈ Ω, since | l g ( z ) | ≤ q log | z | for some q ∈ N and sufficiently large z ∈ Ω, and since | ( F log ◦ e )( z ) | ≥ | z | − r | e − sz | forsufficiently large z ∈ Ω for some r ∈ N , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( n ) ◦ l g n ! ( F log ◦ e ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ K (cid:48) (cid:12)(cid:12) x − u exp − ns (cid:12)(cid:12) in Ω, for some u ∈ N and K (cid:48) >
0. By Lemma 2.4, we have x t exp − ( n +1) s = o ( x − u exp − ns ) in Ω , so the claim follows.Finally, since ord( F log ◦ (cid:15) ) i → ∞ as i → ∞ , it follows from the inductivehypothesis, Proposition 5.2 and Lemma 6.6 that T ( h ) = σ ( τ k ( h ))= (cid:88) i ∈ N T (cid:0) f ( i ) ◦ log ◦ g (cid:1) i ! T ( F log ◦ (cid:15) ) i = (cid:88) i ∈ N T ( f ) ( i ) ◦ log ◦ T ( g ) i ! F log ◦ T ( (cid:15) ) i = T ( f ) ◦ (log ◦ T ( g ) + F log ◦ T ( (cid:15) ))= T ( f ) ◦ log ◦ T ( g ) , so the theorem follows in this case as well. (cid:3) Concluding remarks
As mentioned in the introduction, the purpose of this paper is to extendIlyashenko’s construction in [3] of the class of almost regular maps to ob-tain a qaa field containing them. My reason for doing so is the conjecturethat this class generates an o-minimal structure over the field of real num-bers. This conjecture, in turn, might lead to locally uniform bounds on thenumber of limit cycles in subanalytic families of real analytic planar vectorfields all of whose singularities are hyperbolic; see [6] for explanations anda positive answer in the special case where all singularities are, in addition,non-resonant. (For a different treatment of the general hyperbolic case, seeMourtada [7].)My hope is to settle the general hyperbolic case by adapting the procedurein [6], which requires three main steps:(1) extend Ilyashenko’s class A into a qaa algebra; (2) construct such algebras in several variables, such that the corre-sponding system of algebras is stable under various operations (suchas blowings-up, say);(3) obtain o-minimality using a normalization procedure.While this paper contains a first successful attempt at Step (1), Step (2)poses some challenges. For instance, it is not immediately obvious what thenature of logarithmic generalized power series in several variables should be;they should at least be stable under all the operations required for Step (3).In collaboration with Tobias Kaiser, I am taking the approach of enlarg-ing the set of monomials itself, in such a way that this set is already stableunder the required operations; a natural candidate for such a set of mono-mials is the set of all functions definable in the o-minimal structure R an , exp (see van den Dries and Miller [11] and van den Dries et al. [9]). However,working with this large set of monomials requires us to revisit Step (1) andfurther adapt the construction discussed here to the corresponding general-ized power series. A joint paper (in collaboration with Tobias Kaiser andmy student Zeinab Galal) addressing this generalization of Step (1) is inpreparation. References [1] A. A. Borichev and A. L. Volberg,
The finiteness of limit cycles, and uniquenesstheorems for asymptotically holomorphic functions , Algebra i Analiz, (1995), 43–75.[2] Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations ,vol. 86 of Graduate Studies in Mathematics, American Mathematical Society, Provi-dence, RI, 2008.[3] Yu. S. Ilyashenko,
Finiteness theorems for limit cycles , vol. 94 of Translations ofMathematical Monographs, American Mathematical Society, Providence, RI, 1991.Translated from the Russian by H. H. McFaden.[4] Tobias Kaiser,
The Dirichlet problem in the plane with semianalytic raw data, quasianalyticity, and o-minimal structure , Duke Math. J., (2009), 285–314.[5] T. Kaiser,
The Riemann mapping theorem for semianalytic domains and o -minimality , Proc. Lond. Math. Soc. (3), (2009), 427–444.[6] T. Kaiser, J.-P. Rolin, and P. Speissegger, Transition maps at non-resonant hyperbolicsingularities are o-minimal , J. Reine Angew. Math., (2009), 1–45.[7] Abderaouf Mourtada,
Action de derivations irreductibles sur les algebres quasi-regulieres d’hilbert . arXiv:0912.1560 [math.DS], December 2009.[8] Walter Rudin,
Real and complex analysis , McGraw-Hill Book Co., New York,third ed., 1987.[9] Lou van den Dries, Angus Macintyre, and David Marker,
The elementary theory ofrestricted analytic fields with exponentiation , Ann. of Math. (2), (1994), 183–205.[10] ,
Logarithmic-exponential series , in Proceedings of the International Confer-ence “Analyse & Logique” (Mons, 1997), vol. 111, 2001, 61–113.[11] Lou van den Dries and Chris Miller,
On the real exponential field with restrictedanalytic functions , Israel J. Math., (1994), 19–56.[12] Lou van den Dries and Patrick Speissegger, The real field with convergent generalizedpower series , Trans. Amer. Math. Soc., (1998), 4377–4421.[13] J. van der Hoeven,
Transseries and real differential algebra , vol. 1888 of Lecture Notesin Mathematics, Springer-Verlag, Berlin, 2006. [14] A. J. Wilkie, Model completeness results for expansions of the ordered field of realnumbers by restricted Pfaffian functions and the exponential function , J. Amer. Math.Soc., (1996), 1051–1094. Department of Mathematics and Statistics, McMaster University, 1280Main Street West, Hamilton, Ontario L8S 4K1, Canada
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