aa r X i v : . [ m a t h . L O ] O c t TRANSSERIES AS GERMS OF SURREAL FUNCTIONS
ALESSANDRO BERARDUCCI AND VINCENZO MANTOVA
Abstract.
We show that Écalle’s transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbersto surreal numbers. The same holds for a much larger class of formal series,here called omega-series. Omega-series are the smallest subfield of the surrealnumbers containing the reals, the ordinal omega, and closed under the expand log functions and all possible infinite sums. They form a proper class,can be composed and differentiated, and are surreal analytic. The surrealnumbers themselves can be interpreted as a large field of transseries containingthe omega-series, but, unlike omega-series, they lack a composition operatorcompatible with the derivation introduced by the authors in an earlier paper.
Contents
1. Introduction 22. Preliminaries 52.1. Hahn fields 52.2. Surreal numbers 62.3. Summability 82.4. Hahn fields embedded in No
83. Surreal analytic functions 93.1. Products and powers of summable families 103.2. Sums of power series 113.3. Composition of power series 134. Transseries 144.1. Log-atomic numbers 144.2. Omega-series, LE-series, EL-series 154.3. Isomorphism with classical LE-series 164.4. Adding more log-atomic numbers 184.5. Inductive generation of transseries fields and associated ranks 195. Substitutions 205.1. Pre-substitutions 215.2. Trees 225.3. Extending pre-substitutions to substitutions 23
Date : March 6th, 2017. Revised on September 20th, 2017.2010
Mathematics Subject Classification.
Key words and phrases. surreal numbers, transseries, composition.A.B. was partially supported by PRIN 2012 “Logica, Modelli e Insiemi” and by Progetto diRicerca d’Ateneo 2015 “Connessioni fra dinamica olomorfa, teoria ergodica e logica matematicanei sistemi dinamici”.V.M. was supported by ERC AdG “Diophantine Problems” 267273 and partially supported bythe research group INdAM GNSAGA..
6. Composition 277. Taylor expansions 287.1. Transserial derivations 287.2. A Taylor theorem 297.3. Surreal analyticity 318. A negative result 339. Proof of the summability lemma (Lemma 5.21) 359.1. A property of pre-substitutions 359.2. Further properties of the extensions 369.3. Bad sequences 379.4. Two types of sequences of trees 399.5. No bad sequences of type (A) 399.6. Pruning trees 409.7. No bad sequences 42Acknowledgments 43References 431.
Introduction
Fields of transseries are an important tool in asymptotic analysis and played acrucial role in Écalle’s approach to the problem of Dulac [Dul23, É92]. They ap-pear in various versions, see for instance [DG87, DMM97, Hoe97, Kuh00, DMM01,Sch01, KS05, Hoe06, Hoe09] and the bibliography therein. In [BM] we proved thatConway’s field No of surreal numbers [Con76] admits the structure of a field oftransseries (in the sense of [Sch01]) and a compatible derivation (in fact more thanone). We also proved the existence of “integrals”, in the sense of anti-derivatives,for the “simplest” surreal derivation on No . This makes No into a Liouville closedH-field in the sense of [AD02]. We recall that an H-field is an ordered differentialfield with some compatibility properties between the derivation ∂ and the order; inparticular if f is greater than any constant, then ∂f > . A basic example is thefield of rational functions R ( x ) , ordered by x > R , with constant field R = ker ∂ and ∂x = 1 . The notion of H-field arises as an attempt to axiomatize some of theproperties of Hardy fields, where a Hardy field is a field of germs at + ∞ of eventu-ally C -functions f : R → R closed under derivation. Such fields have been studiedsince the 70’s, see for instance [Bou76, Ros83b, Ros83a, Ros87]. Any o-minimalstructure on the reals gives rise to an H-field, namely the field of germs at + ∞ ofits definable unary functions. In [ADH] van den Dries, Aschenbrenner and van derHoeven proved that, with the “simplest” derivation ∂ introduced in [BM], the sur-reals are a universal H-field; more precisely, every H-field with “small derivations”and constant field R embeds in No as a differential field. Moreover, they provedthat ( No , ∂ ) satisfies the complete first order theory of the logarithmic-exponentialseries of [DMM97, DMM01] and therefore, by the model completeness of the theory[ADH], it admits solutions to all the differential equations that can be solved in abigger model.Another approach to derivation and integration on the surreal numbers was takenby Costin, Ehrlich and Friedman [CEF15] in a more analytic vein, possibly suitable RANSSERIES AS GERMS OF SURREAL FUNCTIONS 3 for asymptotic analysis, namely they consider derivatives and definite integrals offunctions, rather than derivatives of “numbers” (elements of No ).This paper is a first attempt to reconcile the algebraic and the analytic approachto surreal derivation and integration through a notion of composition. The specialsession on surreal numbers at the joint AMS-MAA meeting in Seattle (6-9 Jan.2016) was a timely occasion to discuss these developments and some of the resultsof this paper were presented during that meeting.To discuss our contribution in more detail, we need some definitions. We recallthat in No , as in any Hahn field, there is a formal notion of summability, andone can associate to each summable family ( x i ) i ∈ I its “sum” P i ∈ I x i ∈ No . Wecan thus define the field of omega-series R ⟪ ω ⟫ as the smallest subfield of No containing R ( ω ) and closed under exp , log and sums of summable families. Here ω is the first infinite ordinal and plays the role of a formal variable with deriva-tive . It turns out that R ⟪ ω ⟫ is a very big exponential field (in fact a properclass) properly containing an isomorphic copy of the logarithmic-exponential se-ries of [DMM97, DMM01] (LE-series) and their variants, such as the exponential-logarithmic series of [Kuh00, KT12] (EL-series). More precisely, we can isolate twosubfields R (( ω )) LE ⊂ R (( ω )) EL of R ⟪ ω ⟫ which are isomorphic to the LE and EL-series respectively. The field R (( ω )) LE is a countable union S n ∈ N X n ⊆ No , where X := R ( ω ) and X n +1 is the set of all sums of summable sequences of elementsin X n ∪ exp( X n ) ∪ log( X n ) . In other words, a surreal number is a LE-series if itcan be obtained from R ( ω ) by finitely many applications of P , exp , log (Theorem4.11). This remarkably simple characterization of the LE-series, which should becompared with the original definition, is made possible by working inside the surre-als, with its notion of summability and exponential structure. The EL-series admita similar characterization (Proposition 4.12).We show that each omega-series f ∈ R ⟪ ω ⟫ , hence in particular each LE orEL-series, can be interpreted as a function from positive infinite surreal numbers tosurreal numbers (Corollary 5.23). The idea is simply to substitute ω with a positiveinfinite surreal and evaluate the resulting expression, but the proof of summability(Lemma 5.21) is rather long and technical and it is carried out in Section 9. Similarproblems were tackled in [Sch01] and in some of the cited works by van der Hoeven,although not in the context of surreal numbers. We shall borrow from those papersthe idea of isolating the contributions coming from different “trees”, but with enoughdifferences to warrant an independent treatment. This will give rise to a naturalcomposition operator ◦ : R ⟪ ω ⟫ × No > R → No (Theorem 6.3) which restricts toa composition ◦ : R ⟪ ω ⟫ × R ⟪ ω ⟫ > R → R ⟪ ω ⟫ extending the usual composition ofordinary power series. Formally, we define a composition on R ⟪ ω ⟫ to be a function ◦ : R ⟪ ω ⟫ × No > R → No satisfying the following conditions for all f, g ∈ R ⟪ ω ⟫ and x ∈ No > R :(1) if f = P i<α r i e γ i , then f ◦ x = (cid:0)P i<α r i e γ i (cid:1) ◦ x = P i<α r i e γ i ◦ x ;(2) f ◦ g ∈ R ⟪ ω ⟫ and ( f ◦ g ) ◦ x = f ◦ ( g ◦ x ) ;(3) f ◦ ω = f , ω ◦ x = x .We then prove the following. Theorem 6.3.
There is a (unique) composition ◦ : R ⟪ ω ⟫ × No > R → No . In the last part of the paper we study the interaction between the derivation ∂ : No → No introduced in [BM] and the composition on R ⟪ ω ⟫ . Let us recall that ALESSANDRO BERARDUCCI AND VINCENZO MANTOVA in [BM] we proved the existence of several “surreal derivations” ∂ : No → No andwe studied in detail the “simplest” such derivation [BM, Def. 6.21]. It is easy tosee that all surreal derivations coincide on the subfield R ⟪ ω ⟫ , so the latter admitsa unique surreal derivation ∂ : R ⟪ ω ⟫ → R ⟪ ω ⟫ . The derivation ∂ on R ⟪ ω ⟫ makes itinto a H-field, although not a Liouville closed one because ∂ : R ⟪ ω ⟫ → R ⟪ ω ⟫ is notsurjective. There are, however, many subfields of R ⟪ ω ⟫ which are Liouville closed,among which R (( ω )) LE .We will show that the formal derivative ∂f of an omega-series f ∈ R ⟪ ω ⟫ can beinterpreted as the derivative of the function ˆ f : No > R → No defined by ˆ f ( x ) = f ◦ x ,namely we have ∂f ◦ x = lim ε → f ◦ ( x + ε ) − f ◦ xε , where x and ε range in No (Corollary 7.6). Since ∂f ◦ ω = ∂f , this shows inparticular that the derivative can be defined in terms of the composition: ∂f =lim ε → f ◦ ( ω + ε ) − f ◦ ωε . Other compatibility conditions then follow, such as the chainrule ∂ ( f ◦ g ) = ( ∂f ◦ g ) · ∂g (Corollary 7.7).These results tells us that any omega-series f ∈ R ⟪ ω ⟫ , hence in particular everylogarithmic-exponential series, can be interpreted as a differentiable function ˆ f : No > R → No from positive infinite surreal numbers to surreal numbers. We shallprove that all such functions are surreal analytic in the following sense. Theorem 7.14.
Every f ∈ R ⟪ ω ⟫ is surreal analytic, namely for every x ∈ No > R and every sufficiently small ε ∈ No we have f ◦ ( x + ε ) = X n ∈ N n ! ( ∂ n f ◦ x ) · ε n . It is tempting to raise the conjecture that the exponential field No , enrichedwith all the functions ˆ f : No > R → No for f ∈ R ⟪ ω ⟫ (possibly restricted to someinterval ( a, + ∞ ) ) has a good model theory. For instance, the restricted versioncould yield an o-minimal structure on No . Indeed, note that the family of allfunctions ˆ f : No > R → No (for f ∈ R ⟪ ω ⟫ ) yields a sort of non-standard Hardy fieldon No , namely a field of functions closed under differentiation (it is also closedunder exp , log and composition).We do not know up to what extent the above results can be extended beyond R ⟪ ω ⟫ , namely whether we can introduce a composition operator on the whole of No , thus giving a functional interpretation to all surreal numbers. Concerning thisproblem, we have a negative result. Say that a derivation ∂ and a composition ◦ are compatible if the function x f ◦ x is constant when ∂f = 0 and strictlyincreasing when ∂f > , and if the chain rule ∂ ( f ◦ g ) = ( ∂f ◦ g ) · ∂g holds for all f, g ∈ No (see Definition 8.1). Theorem 8.4.
The simplest derivation ∂ : No → No of [BM] cannot be compatiblewith a composition on No . We conclude with some questions. The first is to study possible notions ofcompositions and compatible derivations on the whole of No (see Question 8.3).This is also connected with the long-standing question of the existence of trans-exponential o-minimal structures; a good composition on No may provide a non-archimedean example. Another related question is to understand whether No hasnon-trivial field automorphisms preserving infinite sums and the function exp . RANSSERIES AS GERMS OF SURREAL FUNCTIONS 5 Preliminaries
In this section, we recall a few well known constructions and facts regardingordered fields and surreal numbers, and above all, we shall establish some of thenotations that will be used throughout the rest of the paper. Since surreal numbersform a proper class, we implicitly work in a set theoretic framework which allowsto talk about classes as first class objects, such as NBG. Therefore, in the followingdefinitions all objects are allowed to be proper classes, unless specified otherwise.Given a class C , we shall say that C is small if it is a set and not a proper class.2.1. Hahn fields.Definition 2.1.
Let K be an ordered field, R ⊆ K a subfield, and f, g ∈ K . Welet: (1) f (cid:22) R g , or f ∈ O R ( g ) , if there is c ∈ R such that | f | ≤ c | g | , and we saythat f is R -dominated by g ;(2) f ≺ R g , or f ∈ o R ( g ) , if c | f | < | g | for every c ∈ R , and we say that f is R -strictly dominated by g ;(3) f is R -finite (or R -bounded ) if f (cid:22) R ;(4) f is R - infinitesimal if f ≺ R ;(5) f ≍ R g if f (cid:22) R g and g (cid:22) R f , namely f /g is R -finite and not R -infinitesimal, and we say that f is R -comparable to g ;(6) f ∼ R g if f − g ≺ R g , and we say that f is R - asymptotic to g .When R ⊆ R we suppress the “ R ”. For instance we write f (cid:22) g if there is c ∈ Q such | f | ≤ c | g | and we say that f is dominated by g , or we write f ∈ O (1) if f is finite , namely f is dominated by . We say that f and g are in the sameArchimedean class if f ≍ g , namely f (cid:22) g and g (cid:22) f .Finally, we say that Γ ⊆ K > is a group of monomials for K if it is amultiplicative subgroup and for every x ∈ K there is a unique m ∈ Γ such that x ≍ m . It can be proved that any real closed field admits a group of monomials. Example 2.2.
The field of Laurent series R (( x Z )) consists of all formal series ofthe form P n ≥ n a n x n , where a n ∈ R and n ∈ Z , ordered according to the sign ofthe leading coefficient a n . The multiplicative subgroup x Z := { x n : n ∈ Z } is agroup of monomials for R (( x Z )) . Remark . Given two monomials m , n , we have m < n if and only if m ≺ n . Definition 2.4.
Let (Γ , · , < ) be an ordered abelian group written in multiplicativenotation. Let R be an ordered field. The Hahn field R ((Γ)) consists of all formalsums x = P m ∈ Γ x m m with coefficients x m ∈ R , whose support Supp( x ) := { m ∈ Γ : x m = 0 } is reverse well-ordered , namely every non-empty subset of thesupport has a maximal element. If x m = 0 we say that x m m is a term of x . Wedenote by R ((Γ)) small ⊆ R ((Γ)) the subclass of all formal sums x = P m ∈ Γ x m m whose support is small (it coincides with R ((Γ)) when Γ is small).The addition in R ((Γ)) is defined component-wise and the multiplication is givenby the usual convolution formula: ( P m x m m ) ( P n y n n ) = ( P o z o o ) where z o = P mn = o x m y n ∈ R . The fact that the supports are reverse well-ordered ensures thatthe latter sum is finite.The leading monomial LM( x ) of x is the maximal monomial in Supp( x ) . The leading term LT( x ) is the leading monomial multiplied by its coefficient, and the ALESSANDRO BERARDUCCI AND VINCENZO MANTOVA leading coefficient is the coefficient of the leading monomial. R ((Γ)) is orderedas follows: x is positive if and only if its leading coefficient is positive. We denoteby Term( x ) := { x m m : m ∈ Supp( x ) } the class of the terms of x . Fact 2.5.
Both R ((Γ)) and R ((Γ)) small are ordered fields. Remark . Note that Γ is a multiplicative subgroup of R ((Γ)) , where we identify m ∈ Γ with m ∈ R ((Γ)) . It follows from the definitions that Γ ⊆ R ((Γ)) containsone and only one representative for each equivalence class modulo ≍ R . In particular,taking R = R , we have that Γ is a group of monomials for R ((Γ)) . The same istrue for R ((Γ)) small .2.2. Surreal numbers.
We denote by No the ordered field of surreal numbers[Con76, Gon86]. A minimal introduction to No , containing all the prerequisitesfor this paper, is contained in [BM]. However, there is no need to assume a priorknowledge of the surreal numbers (the definition itself will not be needed), if one iswilling to take for granted the following fact. Fact 2.7.
We have:(1) No is an ordered real closed field equipped with an exponential function exp : No → No , x e x := exp( x ) , making it into an elementary extensionof ( R , <, + , · , exp) [DE01a]; in particular, exp : No → No is an increasingisomorphism from the additive to the positive multiplicative group.(2) No contains an isomorphic copy of the ordered class On of all ordinalnumbers (hence No is a proper class). The addition and multiplicationrestricted to On coincide with the Hessenberg sum and product.(3) There is a representation of surreal numbers as binary sequences of anyordinal length. The relation of being an initial segment, called simplicity ,is well founded and makes No into a binary tree. This gives us a canonicalchoice for a group M ⊆ No > of monomials: the monomials are thesimplest positive representatives of the Archimedean classes (they form aproper class).(4) The ordinal ω belongs to M (it will later play the role of a formal variablewith derivative ). If ≺ m ∈ M , then e m ∈ M . In particular e ω and e − ω are monomials, but e /ω is not.(5) There is a canonical isomorphism (written as an identification) No = R (( M )) small ⊂ R (( M )) . (6) A surreal number P m ∈ M x m m is purely infinite if all the monomials m in its support are infinite, namely m ≻ . Letting J ⊆ No be the class ofall purely infinite surreal numbers, there is a direct sum decomposition of R -vector spaces No = J ⊕ R ⊕ o (1) . (7) We have M = exp( J ) = { e γ : γ ∈ J } , so we can write No = R (( e J )) small . In other words, every surreal number x ∈ No can be uniquely written inthe form x = X i<α r i e γ i RANSSERIES AS GERMS OF SURREAL FUNCTIONS 7 where α ∈ On , r i ∈ R ∗ , and ( γ i ) i<α is a decreasing sequence in J indexedby an ordinal α ∈ On . We call this the Ressayre normal form of x .(8) The exponential function on o (1) can be calculated using the Taylor seriesof exp , namely exp( ε ) = ∞ X n =0 ε n n ! for all ε ∈ o (1) (see Subsection 2.3 for the meaning of the above infinitesum). Likewise, the inverse log satisfies log(1 + ε ) = ∞ X n =1 ( − n +1 ε n n . Remark . For infinite x , the equality exp( x ) = P ∞ n =0 x n n ! does not hold. In fact,the right-hand side does not even represent a surreal number (see Subsection 2.3).Likewise for log(1 + x ) . Definition 2.9.
By the decomposition No = J ⊕ R ⊕ o (1) , for every surreal number x ∈ No we can write uniquely x = x ↑ + x = + x ↓ where x ↑ ∈ J , x = ∈ R and x ↓ ≺ . We also write x ↑ = for x ↑ + x = . Definition 2.10.
Thanks to Fact 2.7(5) we can apply to No the definitions alreadyintroduced for Hahn fields (support, leading term, etc.). In particular, if x = P i<α r i e γ i is in normal form, its leading monomial is e γ and its leading term is r e γ ; in this case we define log ↑ ( x ) := γ . Note that log ↑ ( x ) = log( x ) ↑ , as in fact log( x ) = log( r e γ (1 + ε )) = γ + log( r )+ P ∞ n =1 ( − n +1 ε n n where ε ≺ . Moreover, x ≺ y if and only if log ↑ ( x ) < log ↑ ( y ) (so − log ↑ is a Krull valuation). Definition 2.11. If x = P i<α r i e γ i and β ≤ α , the number P i<β r i e γ i is called a truncation of x . A subclass A ⊆ No is truncation closed if for every x in A , alltruncations of x are also in A .Note that x ↑ is a truncation of x and it coincides with the sum of all the terms r i e γ i of x with γ i > (if there are no such terms, then x ↑ = 0 ). Notation . Given
A, B ⊆ No we shall use some self-explanatory notations likethe following: • A > is the set of positive elements of A ; • A ≻ is the set of elements a ∈ A satisfying a ≻ ; • A < B means a < b for all a ∈ A and b ∈ B ; • exp( A ) := { exp( x ) : x ∈ A } and log( A ) := { log( x ) : x ∈ A > } , where log : No > → No is the inverse of exp . Example 2.13.
Since M = exp( J ) , we have M ≻ = exp( J > ) and M ≺ =exp( J < ) . ALESSANDRO BERARDUCCI AND VINCENZO MANTOVA
Summability.
Any Hahn field, and in particular No by Fact 2.7(5), admitsa natural notion of infinite sum, as follows. Definition 2.14.
Let I be a set (not a proper class) and ( x i : i ∈ I ) be an indexedfamily of elements of No .We say that ( x i : i ∈ I ) is summable if S i ∈ I Supp( x i ) is reverse well-orderedand for each m ∈ S i ∈ I Supp( x i ) , there are only finitely many i ∈ I such that m ∈ Supp( x i ) . In this case, the sum P i ∈ I x i is the unique surreal number y = P m y m m such that Supp( y ) ⊆ S i ∈ I Supp( x i ) and, for every m ∈ M , y m = P i ∈ I ( x i ) m (notethat there are finitely many i ∈ I with x i = 0 by the hypothesis of summability).Similar definitions apply replacing No with any field of the form R ((Γ)) small .We shall also say that P i ∈ I x i exists to mean that ( x i ) i ∈ I is summable. Remark . A family ( x i : i ∈ I ) is summable if and only if there are no injectivesequences ( i n ) n ∈ N in I and monomials m n ∈ Supp( x i n ) (not necessarily distinct)such that m n (cid:22) m n +1 for each n ∈ N (where N is the set of non-negative integers).Equivalently, for every injective sequence ( i n ) n ∈ N in I and for any choice of mono-mials m n ∈ Supp( x i n ) , there is a subsequence ( i f ( n ) ) n ∈ N such that m i f ( n ) ≻ m i f ( n +1) for every n ∈ N .2.4. Hahn fields embedded in No . Given a subfield R of No and a multiplica-tive subgroup Γ of the monomials M = e J , we will sometimes be interested inthe class of all surreal numbers that can be written as a sum P r m m for r m ∈ R and m ∈ Γ . Under suitable assumptions on R and Γ , this subclass of No can beidentified with the Hahn field R ((Γ)) . Proposition 2.16.
Let Γ be a small multiplicative subgroup of M = e J and R bea truncation closed subfield of No . If R < Γ > , there is a unique field embedding R ((Γ)) → No sending r m (as an element of R ((Γ)) ) to r m (as an element of No )and preserving infinite sums.Proof. Suppose that
R < Γ > . It suffices to check that the embedding exists.Without loss of generality, we may assume that R ⊆ R , as the compositum R · R isclearly truncation closed and it also satisfies R · R < Γ > .Let P r m m be an element of R ((Γ)) . We wish to prove that ( r m m ∈ No : r m = 0) is summable. Take an injective sequence ( r m n m n ) n ∈ N and a choice of n n ∈ Supp( r m n m n ) . We can write n n = m n o n , where o n ∈ Supp( r m n ) . Note that o n ∈ R , since R contains R and is closed under truncation.After extracting a subsequence, we may assume that ( m n ) n ∈ N is strictly decreas-ing. We can now easily check that ( n n ) n ∈ N is also strictly decreasing: indeed, n n n n +1 = m n m n +1 · o n o n +1 > , as o n +1 o n ∈ R < Γ > . (cid:3) Notation . By Proposition 2.16, given a small multiplicative group Γ of M = e J (the class of monomials of No ) and a truncation closed subfield R ⊆ No such that R < Γ > , we can identify the field R ((Γ)) with the class of surreal numbers thatare of the form P r m m with r m ∈ R and m ∈ Γ . Lemma 2.18.
Let Γ and Γ be subgroups of a given ordered abelian multiplicativegroup. Suppose Γ < Γ > . Then Γ Γ is naturally isomorphic, as an ordered group,to the direct product Γ × Γ with the reverse lexicographic order. RANSSERIES AS GERMS OF SURREAL FUNCTIONS 9
Proof.
Clearly, Γ ∩ Γ = { } , so the map sending ab ∈ Γ Γ to ( a, b ) ∈ Γ × Γ is awell-defined isomorphism of abelian groups. We can easily verify that it preservesthe ordering. Indeed, let a, a ′ ∈ Γ and b, b ′ ∈ Γ be such that b < b ′ . It sufficesto show that ab < a ′ b ′ . This can be rewritten as a/a ′ < b ′ /b . Since b ′ /b > , thedesired result follows by the hypothesis Γ < Γ > . (cid:3) Using the above notation, Proposition 2.16, and Lemma 2.18, we can then deducethe following well-known result (see for instance [DMM01, 1.4]). However, notethat the result contains an equality rather than just an isomorphism, thanks to theidentifications of Notation 2.17.
Corollary 2.19.
Let Γ , Γ be small subgroups of M . If Γ < Γ > , then we have R ((Γ ))((Γ )) = R ((Γ Γ )) ∼ = R ((Γ × Γ )) .Proof. We first note that R ((Γ )) < Γ > , from which it follows at once that R ((Γ ))((Γ )) ⊆ R ((Γ Γ )) by Proposition 2.16. On the other hand, let x = P m ∈ Γ Γ r m m be an element of R ((Γ Γ )) . Since Γ Γ ∼ = Γ × Γ , each m ∈ Γ Γ decomposes uniquely as a product m = no with n ∈ Γ and o ∈ Γ . But then it iseasy to verify that x = X m r m m = X o ∈ Γ X n ∈ Γ r no n ! o ∈ R ((Γ ))((Γ )) . (cid:3) Remark . If one drops the assumption that Γ is small, then the conclusion of2.16 holds with R ((Γ)) small in place of R ((Γ)) . In particular, we may canonicallyidentify R ((Γ)) small with a subfield of No , as in Notation 2.17. As a special case,one recovers the already mentioned identification No = R (( M )) small of Fact 2.7(5).The conclusion of Corollary 2.19 also holds, provided one uses R ((Γ i )) small insteadof R ((Γ i )) for i = 1 , . 3. Surreal analytic functions
A real function is analytic at a point in its domain if there is a neighborhood ofthe point in which it coincides with the limit of a power series. Such notion does notgeneralize directly to surreal numbers, as No does not have a good notion of limitfor series. However, we can replace the limit with the natural notion of infinitesum from Definition 2.14. This leads to a theory of “surreal analytic function”developed in [All87]. In this section we isolate and extend some of those results ina form suitable for our goals.Infinite sum bears some resemblance with the usual notion of absolute conver-gence. On the one hand, like absolute convergence, it enjoys some good algebraicproperties, such as being independent on the “order” in which we sum the elementsof the family. On the other hand, it is not related to the order topology; for in-stance, even if a family ( x i ) i ∈ I is summable, and ( y i ) i ∈ I is such that | y i | ≤ | x i | , itdoes not necessarily follow that ( y i ) i ∈ I is summable. Lemma 3.1.
Let ( a i : i ∈ I ) be a summable family of surreal numbers. Thenfor any partition I = F j ∈ J I j of the set I , each sum P i ∈ I j a i exists, the family ( P i ∈ I j a i : j ∈ J ) is summable, and X j ∈ J X i ∈ I j a i = X i ∈ I a i . Proof.
Clearly, since ( a i : i ∈ I ) is summable, so is each ( a i : i ∈ I j ) for j ∈ J .Moreover, it also follows easily that ( P i ∈ I j a i : j ∈ J ) is summable, as each mono-mial m in Supp( P i ∈ I j a i ) must appear in Supp( a i ) for some i ∈ I j . To check that itssum is indeed equal to P i ∈ I a i , for a given monomial m , let a i, m be the coefficient of m in a i . Then the coefficient of m in P j ∈ J P i ∈ I j a i is P j ∈ J P i ∈ I j a i, m = P i ∈ I a i, m ,which in turn is the coefficient of m in P i ∈ I a i , proving the conclusion. (cid:3) Corollary 3.2.
Let ( a i,j : ( i, j ) ∈ I × J ) be a summable family of surreal numbers.Then both P i ∈ I P j ∈ J a i,j and P j ∈ J P i ∈ I a i,j exist and X i ∈ I X j ∈ J a i,j = X j ∈ J X i ∈ I a i,j = X ( i,j ) ∈ I × J a i,j . Remark . The assumption of summability of ( a i,j : ( i, j ) ∈ I × J ) is necessary,or the equality may not hold. For instance, take a i,i = ω , a i,i +1 = − ω , and a i,j = 0 otherwise for i, j ∈ N , which is clearly not summable. Then P i ∈ N P j ∈ N a i,j = 0 while P j ∈ N P i ∈ N a i,j = ω . Moreover, one of the two sums may not even exists; forinstance, P i ∈ N P j =0 ( − j ω clearly exists and is equal to , while P j =0 P i ∈ N ( − j ω does not exist. It can also happen that the two sums P i P j and P j P i ex-ists and are equal, but the sum P ( i,j ) ∈ I × J does not exists: take a i,i = 2 ω and a i +1 ,i = a i,i +1 = − ω , with all other terms a i,j being zero.3.1. Products and powers of summable families.
The following is well known.
Remark . If ( x i ) i ∈ I and ( y j ) i ∈ J are summable, then so is ( x i y j : ( i, j ) ∈ I × J ) .Its sum P ( i,j ) ∈ I × J x i y j coincides with the product ( P i ∈ I x i )( P j ∈ J y j ) .Using Remark 3.4, one can easily express the n -th power of a sum as follows. Proposition 3.5.
Let ( x i ) i ∈ I be a summable family of surreal numbers and let n ∈ N . Then the family (cid:0)Q m By induction on n ∈ N based on Remark 3.4. (cid:3) Corollary 3.6. If ( a i ε i ) i ∈ N is summable, then for every n ∈ N , X i ∈ N a i ε i ! n = X k ∈ N X i + ... + i n = k a i a i . . . a i n ! ε k . Proof. By Proposition 3.5, (cid:0)P i ∈ N a i ε i (cid:1) n = P τ : n → N Q m Sums of power series. We shall now define how to evaluate a surreal powerseries on a surreal number, and the corresponding notion of surreal analytic func-tion. This is similar to how real analytic functions are extended to No , with thedifference that we now allow power series to have surreal coefficients. Definition 3.7. Given a surreal power series P ( X ) = P ∞ i =0 a i X i ∈ No [[ X ]] , wedefine P ( ε ) := X i ∈ N a i ε i for any ε ∈ No such that the sum on the right hand side exists.Given a function f : U → No from an open subset U of No , we say that f is surreal analytic at x if there are a neighborhood V ⊆ U of x and a power series P ( X ) ∈ No [[ X ]] such that f ( y ) = P ( y − x ) for all y ∈ V .Unlike the case of real analytic functions, in which some power series are notconvergent and thus do not yield analytic functions, we shall now verify that every power series with surreal coefficients induces a surreal analytic function.By Neumann’s lemma [Neu49], if ( a i ) i ∈ N is a sequence of real coefficients and ε ≺ , then ( a i ε i ) i ∈ N is summable. Therefore, for every power series P ( X ) ∈ R [[ X ]] , P ( ε ) is well defined for any ε ≺ . We can easily extend this result to series withsurreal coefficients. We start with the following variant of Neumann’s lemma. Itsproof is an adaptation of a similar argument in [Gon86, p. 52]. Lemma 3.8. Let R be a subfield of No and ε ≺ R . Let ( n i ) i ∈ N , ( m i,j ) i ∈ N ,j ≤ k i besequences of monomials in respectively R and Supp( ε ) , where ( k i ) i ∈ N is a sequenceof natural numbers with lim i →∞ k i = ∞ . Then the sum P i ∈ N n i m i, . . . m i,k i exists.Proof. Suppose by contradiction that there are two family as in the hypothesissuch that P i ∈ N n i m i, . . . m i,k i does not exist. By taking a subsequence, we mayassume that ( n i m i, . . . m i,k i ) i ∈ N is weakly increasing. We may picture m i,j as the ( i, j ) -entry of an infinite table, where i is the row index and j is the column index.Rearranging the terms, we can assume that each row is weakly increasing, namely m i, ≤ m i, ≤ . . . ≤ m i,k i for all i ∈ N .Taking a subsequence we may further assume that ( k i ) i ∈ N is strictly increasing,so in particular k i ≥ i . Choosing a further subsequence we can assume that the firstcolumn ( m i, ) i ≥ is weakly decreasing, since all these monomials are in the supportof ε . Similarly we can assume that ( m i, ) i ≥ is weakly decreasing. Continuing inthis fashion, by a diagonalization argument we can assume that, for any fixed k ,the k -th column ( m i,k ) i ∈ N becomes weakly decreasing after its k -th entry, namely m k,k ≥ m k +1 ,k ≥ m k +2 ,k ≥ . . . . Note that these terms exist since k i ≥ k for all i ≥ k .Now fix i ∈ N and let j > i (so k j > k i ). By construction, n i m i, . . . m i,k i ≤ n j m j, . . . m j,k i m j,k i +1 . . . m j,k j . Since m j,k i +1 . . . m j,k j ≺ R , we must have n i > n j m j,k i +1 . . . m j,k j . It follows that m i, . . . m i,k i < m j, . . . m j,k i , so in particularthere is some k ≤ k i with m i,k < m j,k . Now recall that the k -th column is weaklydecreasing after its k -th entry, hence necessarily i < k . We have thus proved thatfor each i ∈ N and j > i there is some k with i < k ≤ k i such that m i,k < m j,k .Taking j = k i , and recalling that all the rows are weakly increasing, we obtain m i,i ≤ m i,k < m k i ,k ≤ m k i ,k i for all i ∈ N . Iterating we obtain an infinite increasingchain of elements of the form m l,l , contradicting the fact that { m i,j : i ∈ N , j ≤ k i } is in Supp( ε ) . (cid:3) Corollary 3.9. Let R be a truncation closed subfield of No and ε ≺ R . Let ( a i ) i ∈ N be a sequence of coefficients in R . Then ( a i ε i ) i ∈ N is summable.Proof. Without loss of generality, we may assume that R ⊆ R . Indeed, we mayreplace R with the compositum R · R , which is also closed under truncation, as ε ≺ R trivially implies ε ≺ R · R . In particular, we may assume that Supp( a i ) ⊆ R for all a i ∈ R . Note that for all i ∈ N , any monomial in the support of a i ε i has theform n i m i, . . . m i,i − where n i ∈ Supp( a i ) ⊆ R and m i,j ∈ Supp( ε ) for j ≤ i − .The conclusion then follows easily from Lemma 3.8. (cid:3) Corollary 3.10. For every power series P ( X ) ∈ No [[ X ]] , the partial function ε P ( ε ) is surreal analytic at .Proof. Given a power series P ( X ) = P ∞ i =0 a i X i , it suffices to apply Corollary 3.9with the ring R generated by the monomials in the supports Supp( a i ) . The function ε P ( ε ) is then defined at least on o R (1) , which is a nonempty convex subclasscontaining as R is necessarily small. (cid:3) Proposition 3.11. Suppose that f is a surreal analytic function at some x ∈ No .Then f is infinitely differentiable at x and f ( x + ε ) = ∞ X i =0 f ( i ) ( x ) i ! ε i . Proof. Let f be surreal analytic at x , with power series P ( X ) = P ∞ i =0 a i X i . Thenfor every sufficiently small δ we have f ′ ( x + δ ) = lim ε → f ( x + δ + ε ) − f ( x + δ ) ε = lim ε → ∞ X i =0 a i ( δ + ε ) i − δ i ε = lim ε → ∞ X i =0 a i · δ i + iδ i − ε + (cid:0) i (cid:1) δ i − ε + · · · + iδε i − + ε i − δ i ε = ∞ X i =1 ia i δ i − + lim ε → ε · ∞ X i =2 (cid:18)(cid:18) i (cid:19) δ i − + · · · + ε i (cid:19) = ∞ X i =1 ia i δ i − . Therefore, f is differentiable at x and its derivative f ′ is surreal analytic at x .Moreover, the above equation also shows that f ′ ( x ) = a . By induction, it followsthat f is infinitely differentiable, and that a i = f ( i ) ( x ) i ! , as desired. (cid:3) Moreover, we also observe that Neumann’s lemma, already in its original formu-lation, implies the following statement for power series with real coefficients, whichwill prove useful later on. Corollary 3.12. Let ( ε i ) i ∈ I be a summable family such that ε i ≺ for all i ∈ N .Let P i ( X ) = P ∞ n =1 a i,n X n ∈ R [[ X ]] be real power series for i ∈ I . Then the family ( P i ( ε i ) : i ∈ I ) is summable.Proof. Suppose by contradiction that there is a weakly increasing sequence of mono-mials ( m n ) n ∈ N such that m n ∈ Supp( P i n ( ε i n )) . Then for all n ∈ N there is a positiveinteger k n such that m n ∈ Supp( a i n ,k n ε k n i n ) . After extracting a subsequence, we mayeither assume that lim n →∞ k n = ∞ , and we reach a contradiction by Lemma 3.8,or we may assume that the sequence ( k n ) n ∈ N is constant, so that m n ∈ Supp( ε ki n ) for some fixed k ∈ N and all n ∈ N . RANSSERIES AS GERMS OF SURREAL FUNCTIONS 13 In the latter case, write m n = n n, · · · · · n n,k with n n,j ∈ Supp( ε i n ) . Since ( ε i ) i ∈ I is summable, we may extract a subsequence and assume that ( n n,j ) n ∈ N isstrictly decreasing for each j = 1 , . . . , k . But then ( m n ) n ∈ N is strictly decreasing, acontradiction. (cid:3) Remark . Since No is totally disconnected, the present notion of surreal ana-lyticity does not have a good theory of analytic continuation. For instance, one candefine a surreal analytic function on all finite numbers by choosing a power series P r ( X ) ∈ R [[ X ]] for each r ∈ R and defining f ( r + ε ) = P r ( ε ) for each r ∈ R and ε ≺ . Moreover, one can choose the series P r such that the restriction of f to R is itself a real analytic function, but with yet other Taylor expansions. It would beinteresting to develop an analogous of rigid analytic geometry for surreal numbersthat prevents such pathological behavior.3.3. Composition of power series. By Corollary 3.10, there is a morphism from No [[ X ]] to germs at zero of surreal functions defined by evaluating a formal powerseries P ( x ) = P i ∈ N a i X i ∈ No [[ X ]] at X = ε for any sufficiently small ε ∈ No .As for traditional power series, we can show that this morphism behaves well withrespect to composition of power series. Definition 3.14. Let R be a subfield of No . Given two formal power series P ( X ) := P ∞ n =0 a n X n and Q ( X ) := P ∞ m =1 b m X m in R [[ X ]] , where Q ( X ) hasno constant term, their composition ( P ◦ Q )( X ) is defined as the power series P k ∈ N c k X k ∈ R [[ X ]] where c = a and, for k > , c k = k X n =1 a n X m + ... + m n = k b m · · · b m n . Lemma 3.15. Let R be a truncation closed subfield of No and ε ≺ R . Let ( a i,j : ( i, j ) ∈ I × J ) be a family of surreal numbers in R such that, for any fixed j ∈ J , P i ∈ I a i,j exists. Then P ( i,j ) ∈ I × J a i,j ε j exists.Proof. As in the proof of Corollary 3.9, we may assume that Supp( a i,j ) ⊆ R for all ( i, j ) ∈ I × J . For a contradiction, suppose that there is an injective sequence of pairs ( i n , j n ) n ∈ N and a weakly increasing sequence of monomials m n ∈ Supp( a i n ,j n ε j n ) .After extracting a subsequence, we may assume that either lim n ∈ N j n = + ∞ , inwhich case we reach a contradiction by Corollary 3.9, or the sequence ( j n ) n ∈ N isconstant, so that there is some j ∈ J such that m n ∈ a i n ,j ε j for every n ∈ N . Inthis case, it follows that ( a i,j ε j ) i ∈ I is not summable, which is absurd since P i ∈ I a i,j exists, hence so does ε j ( P i ∈ I a i,j ) = P i ∈ I a i,j ε j . (cid:3) Proposition 3.16. Let R be a truncation closed subfield of No and ε ≺ R . Let P ( X ) := P ∞ n =0 a n X n and Q ( X ) := P ∞ m =1 b m X m be two power series in R [[ X ]] (where Q ( X ) has no constant term). Then ( P ◦ Q )( ε ) = P ( Q ( ε )) .Proof. The three sums P ( ε ) , Q ( ε ) and ( P ◦ Q )( ε ) exist by Corollary 3.9. Since Q ( ε ) ≺ R , P ( Q ( ε )) exists as well. Let d n,k = P m + ... + m n = k b m · · · b m n for k ∈ N ∗ .By Corollary 3.6, P ( Q ( ε )) = ∞ X n =0 a n ∞ X m =1 b m ε m ! n = a + ∞ X n =1 a n ∞ X k =1 d n,k ε k . Note that d n,k = 0 for k < n , so the family ( a n d n,k : n ∈ N ) is summable for any k ∈ N ∗ . By Lemma 3.15, the family ( a n d n,k ε k : ( n, k ) ∈ N × N ∗ ) is summable.Therefore, by Corollary 3.2 we have a + ∞ X n =1 a n ∞ X k =1 d n,k ε k = a + ∞ X k =1 ∞ X n =1 a n d n,k ε k = ( P ◦ Q )( ε ) . (cid:3) Transseries With the help of the surreal numbers we shall attempt a general definition of“field of transseries”. Definition 4.1. We say that T is a transserial subfield of No if T is a truncationclosed subfield of No (Definition 2.11) containing R and such that log( T > ) ⊆ T .More generally, let F be an ordered logarithmic field (not necessarily includedin No ) containing R and endowed with a partial operator P from small indexedfamilies of elements of F to F . We say that F is a field of transseries if it isisomorphic to a transserial subfield T of No through a field isomorphism f : F → T preserving R , log and P (the latter condition means that ( x i : i ∈ I ) is the domainof P if and only if ( f ( x i )) i ∈ I is summable in No and P i ∈ I f ( x i ) = f ( P i ∈ I x i ) ).We shall call f an isomorphism of transseries .In [Sch01] an axiomatic definition of transseries field is given. The critical axiom,there called “T4”, is rather technical. One of the main results in [BM] is that No satisfies T4, hence it is a field of transseries in the sense of [Sch01]. More generally,since T4 is inherited by taking subfields, it follows that a field of transseries in thesense of Definition 4.1 is also a field of transseries in the sense of [Sch01] (we alsoexpect the converse to be true, but it is beyond the scope of this paper).4.1. Log-atomic numbers. We write log n ( x ) for the n -fold iterate of log( x ) ,namely log ( x ) = x , log n +1 ( x ) = log(log n ( x )) . Likewise, we write exp ( x ) = x , exp n +1 ( x ) = exp(exp n ( x )) . Definition 4.2. A positive infinite surreal number x ∈ No is log-atomic if forevery n ∈ N , log n ( x ) is an infinite monomial. We call L the class of all log-atomicnumbers. Note that log( L ) = exp( L ) = L .A subclass of the log-atomic numbers, the so called κ -numbers, was isolated by[KM15]. The ordinal ω is a κ -number, hence in particular it is log-atomic. In [BM]we gave a parametrization { λ x : x ∈ No } of L and we proved that there is exactlyone log-atomic numbers in each “level” of No . Definition 4.3. Given x, y > R we write x ≍ L y , and we say that x, y are in thesame level if for some n ∈ N we have log n ( x ) ≍ log n ( y ) . Remark . For all x, y > R , x ≍ y implies log( x ) ∼ log( y ) , so in the abovedefinition we can equivalently require log n ( x ) ∼ log n ( y ) . Fact 4.5 ([BM]) . We have:(1) for each x ∈ No with x > R , there are n ∈ N and λ ∈ L such that log n ( x ) ≍ λ [BM, Prop. 5.8]; in particular, every level contains a log-atomicnumber; RANSSERIES AS GERMS OF SURREAL FUNCTIONS 15 (2) for each λ, µ ∈ L , if λ ≍ L µ , then λ = µ ; in particular, every level containsa unique log-atomic number;(3) for every x > R and every positive n ∈ N , we have x ≍ L x n , but x L e x ;(4) in particular, for λ, µ ∈ L , if λ < µ , then λ n < µ for every n ∈ N ;(5) there are log-atomic numbers strictly between ω and e ω ; there are alsolog-atomic numbers smaller than log n ( ω ) for every n ∈ N or bigger than exp n ( ω ) for every n ∈ N , such as the ordinal ε .4.2. Omega-series, LE-series, EL-series . In this section we shall introducethree subfields R (( ω )) LE ⊂ R (( ω )) EL ⊂ R ⟪ ω ⟫ of No . We shall see that first twoare naturally isomorphic to the exponential fields of respectively the LE-series of[DMM97, DMM01] and the EL-series generated by logarithmic words of [Kuh00,KT12], while the third one is a very big field properly containing both (the ordinal ω plays the role of a formal variable > R ). Definition 4.6. Given a subclass X of No , we write P X for the family of allsurreal numbers x ∈ No which can be written in the form x = P i ∈ I y i for somesummable family ( y i ) i ∈ I of elements of X indexed by a set I . Note that P is aclosure operator, as X ⊆ P X = P P X . Definition 4.7. We define R ⟪ ω ⟫ , the field of omega-series , as the smallest sub-field of No containing R ∪ { ω } and closed under P , exp and log .We shall prove later that R ⟪ ω ⟫ is a proper class. Definition 4.8. Let R (( ω )) LE ⊂ R ⟪ ω ⟫ be the union S n ∈ N X n , where X = R ∪{ ω } and X n +1 = P ( X n ∪ exp( X n ) ∪ log( X n )) . In other words, a surreal number x belongs to R (( ω )) LE if and only if x can be obtained in finitely many steps startingfrom R ∪ { ω } and using the set-operations P , exp , log . Definition 4.9. Let R (( ω )) EL be defined as R (( ω )) LE but starting with X ′ = R ∪ { ω, log( ω ) , log ( ω ) , . . . } instead of X = R ∪ { ω } . In other words, a surrealnumber belongs to R (( ω )) EL if and only if it can be obtained in finitely manysteps from X ′ using P , exp , log (in this case it turns out that log is not actuallynecessary). Remark . Unlike R ⟪ ω ⟫ , the subfields R (( ω )) LE and R (( ω )) EL are not closedunder P ; for instance P n ∈ N log n ( ω ) belongs to R ⟪ ω ⟫ but not to R (( ω )) LE . In-deed, one needs k steps to generate log k ( ω ) starting from R ∪ { ω } , so the wholesum P n ∈ N log n ( ω ) cannot be generated in finitely many steps. The same examplewitnesses that the inclusion R (( ω )) LE ⊂ R (( ω )) EL is proper, as the latter fielddoes contain P n ∈ N log n ( ω ) . Finally note that P n ∈ N / exp n ( ω ) belongs to R ⟪ ω ⟫ but not to R (( ω )) EL .Both R (( ω )) LE and R (( ω )) EL are elementary extensions of the real exponentialfield ( R , + , · , exp) , but they are no longer elementary equivalent if we add the differ-ential operator ∂ of [BM] to the language (see Subsection 7.1): indeed in R (( ω )) LE (and in No itself) the derivation ∂ is surjective, while in R (( ω )) EL it is not. Forinstance one can show that exp( − P n ∈ N log n ( ω )) is an element of R (( ω )) EL with-out anti-derivative in R (( ω )) EL , and in fact not even in R ⟪ ω ⟫ . Indeed, for thesimplest surreal derivation ∂ ([BM, Def. 6.7]), which has anti-derivatives, we have ∂κ − = exp( − P n ∈ N log n ( ω )) , where κ − ∈ No is the simplest log-atomic numbersmaller than log n ( ω ) for each n ∈ N . Such a number cannot belong to R ⟪ ω ⟫ , andsince ker ∂ = R , there cannot be any x ∈ R ⟪ ω ⟫ with ∂x = exp( − P n ∈ N log n ( ω )) . There are many interesting subfields between R (( ω )) LE and R ⟪ ω ⟫ whose domainis a set, for instance the series in R ⟪ ω ⟫ with hereditarily countable support.The definition of R (( ω )) LE as a union S n ∈ N X n suggests the possibility of pro-longing the sequence X n along the transfinite ordinals, setting X = R ∪ { ω } , X α +1 = P ( X α ∪ exp( X α ) ∪ log( X α )) and X λ = S i<λ X i for each limit ordinal α .One can verify that the union S α ∈ On X α along all the ordinals would then coincidewith R ⟪ ω ⟫ .4.3. Isomorphism with classical LE-series. It is well known that there is aunique embedding of the field of LE-series into No sending x to ω , R to R , andpreserving exp and infinite sums (see [ADH]). This subsection will be devoted tothe long, but straightforward proof that R (( ω )) LE is naturally isomorphic to thefield of LE-series, so in particular it is the image of such embedding. This providesa simple characterization of the LE-series, which should be compared with theoriginal definition. Theorem 4.11. R (( ω )) LE is a field of transseries and it is isomorphic to the fieldof logarithmic-exponential series R (( x )) LE of [DMM97, DMM01] ; the isomorphismsending ω to x is unique. Similarly we have: Proposition 4.12. The field R (( ω )) EL is naturally isomorphic to the field of EL-series generated by logarithmic words [KT12, Def. 6.2, Example 4.6] (see also Re-mark 4.33). We leave the verification of Proposition 4.12 to the reader, but we shall givea detailed proof of Theorem 4.11. To this aim we shall first give an equivalentdescription of R (( ω )) LE (recall from Notation 2.17 that we are identifying Hahnfields R ((Γ)) with subfields of No ). Definition 4.13. Let λ ∈ L (a log-atomic number). We define:(1) M ,λ := λ R , K ,λ := R (( M ,λ )) , J ,λ := R (( M ≻ ,λ )) ;(2) M n +1 ,λ := e J n,λ , K n +1 ,λ = K n,λ (( M n +1 ,λ )) , J n +1 ,λ := K n,λ (( M ≻ n +1 ,λ )) ;(3) R (( λ )) E := S n ∈ N K n,λ .The next Lemma shows that the above definition is well posed, namely at eachstep M n +1 ,λ is a subgroup of M and K n,λ < M > n +1 ,λ , so that under the conventionsof Notation 2.17, each K n +1 ,λ is again in No ; in particular, in clause (2) we areallowed to use the exponential function of No to define e J n +1 ,λ . Note moreover that J n,λ ⊆ J , as we shall verify in a moment. Lemma 4.14. For each n ∈ N , M n,λ is a well defined divisible subgroup of M andmoreover M ≻ n +1 ,λ > K n,λ .Proof. We proceed by induction on n . Trivially, M ,λ is a well defined divisiblesubgroup of M . Now fix n and assume that M n,λ is well defined and that M ≻ n,λ >K n − ,λ (an empty condition if n = 0 ). Then J n,λ is a well defined subset of No byProposition 2.16, and in particular it is a divisible additive subgroup of K n,λ . In [DMM01], the field of logarithmic-exponential series is denoted either by R (( x − )) LE or by R (( t )) LE , where x > R and t = x − is infinitesimal. We prefer here to use the notation R (( x )) LE for the LE-series, with x > R , as in [ADH17], to better match the notation R (( ω )) LE . RANSSERIES AS GERMS OF SURREAL FUNCTIONS 17 We claim that J n,λ is consists only of purely infinite numbers. Indeed, let m be amonomial in the support of J n,λ . Then m = no for some n ∈ M ≻ n,λ and o ∈ K n − ,λ (with o = 1 if n = 0 ). By inductive hypothesis, o − ∈ K n − ,λ < n , so m > ,proving the claim. It follows that M n +1 ,λ is a divisible multiplicative subgroup of M .Finally, let e γ ∈ M ≻ n +1 ,λ . We wish to prove that e γ > M n,λ . Let m be theleading monomial of γ . As before, we can write m = no for some n ∈ M ≻ n,λ and o ∈ K n − ,λ (with o = 1 if n = 0 ). By inductive hypothesis, we also know that n > K n − ,λ . Since γ > n , it follows that m = e γ > e K n − ,λ , so in particular, m > e J n − ,λ = M n,λ , as desired. (cid:3) Remark . By Corollary 2.19 we have K n,λ = R (( M ,λ M ,λ . . . M n,λ )) . Lemma 4.16. For all n ∈ N we have:(1) exp( K n,λ ) ⊆ K n +1 ,λ ;(2) K n,λ ⊆ K n +1 , log( λ ) ;(3) log( K > n,λ ) ⊆ K n +1 , log( λ ) .In particular, R (( λ )) E is closed under exp and log( R (( λ )) E ) ⊆ R ((log( λ ))) E .Proof. We work by induction on n .For (1), let x ∈ K n,λ . We can write uniquely x = γ + r + ε where γ ∈ J n,λ , and if n > , r ∈ K n − ,λ and ε ≺ K n − ,λ , otherwise simply r ∈ R and ε ≺ . In any case, e x = e γ · e r · P ∞ i =0 ε i i ! . But then it suffices to note that e γ ∈ M n +1 ,λ ⊆ K n +1 ,λ bydefinition, while e r is either already in R or in K n,λ by inductive hypothesis, andthe remaining sum is in K n,λ because K n,λ is a Hahn field. Therefore, e x ∈ K n +1 ,λ ,as desired.Concerning (2), note that M ,λ = λ R = e R log( λ ) ⊆ e J , log( λ ) = M , log( λ ) . Itfollows that J ,λ ⊆ J , log( λ ) and K ,λ ⊆ K , log( λ ) . By a straightforward induction,it follows that M n,λ ⊆ M n +1 , log( λ ) , J n,λ ⊆ J n +1 , log( λ ) and K n,λ ⊆ K n +1 , log( λ ) ,proving the desired conclusion.Finally, for (3), let x ∈ K > n,λ . We can write uniquely x = m · r · (1 + ε ) where m ∈ M n,λ , and if n > , r ∈ K > n − ,λ and ε ≺ K n − ,λ , otherwise simply r ∈ R and ε ≺ . We have log( x ) = log( m ) + log( r ) + P ∞ i =1 ( − i +1 ε i i . Since K n,λ isa Hahn field, the rightmost sum is in K n,λ , which is contained in K n +1 , log( λ ) by(2), while log( r ) is either already in R or in K n, log( λ ) by inductive hypothesis. For log( m ) , we simply note that if n = 0 , then log( m ) = s · log( λ ) ∈ K , log( λ ) for some s ∈ R , otherwise log( m ) ∈ K n − ,λ , which is contained in K n, log( λ ) by (2). Therefore, log( x ) ∈ K n +1 , log( λ ) , as desired. (cid:3) Proposition 4.17. For each λ ∈ L , R (( λ )) E is (uniquely) isomorphic to the expo-nential field R (( x )) E defined in [DMM97, DMM01] through an isomorphism sending λ to x and preserving exp , P and R .Proof. It suffices to note that Definition 4.13 is formally identical to the definitionof R (( x )) E , except that in our case the various Hahn fields are identified withsubfields of No (Notation 2.17) and the role of the formal variable is taken by λ .The uniqueness follows trivially. (cid:3) Proposition 4.18. For each λ ∈ L , S k ∈ N R ((log k ( λ ))) E is (uniquely) isomorphicto the exponential field R (( x )) LE defined in [DMM97, DMM01] through an isomor-phism sending λ to x and preserving exp , P and R .Proof. In [DMM01], R (( x )) LE is defined as a direct limit of a suitable system ofself-embeddings Φ k : R (( x )) E → R (( x )) E . The embedding Φ k sends x to exp k ( x ) .In turn, when composed with the isomorphism R (( x )) E ∼ = R ((log k ( λ ))) E of Propo-sition 4.17, it gives the embedding of R (( x )) E into R ((log k ( λ ))) E sending x to λ .Therefore, the image of such direct limit is the directed union S k ∈ N R ((log k ( λ ))) E ,as desired. The uniqueness follows trivially. (cid:3) Proposition 4.19. S k ∈ N R ((log k ( ω ))) E is equal to R (( ω )) LE . In particular, thereis a unique isomorphism of transseries from S k ∈ N R ((log k ( ω ))) E to R (( x )) LE send-ing ω to x .Proof. Note that each K n,λ is closed under infinite sums, while by Lemma 4.16, exp( K n,λ ) ⊆ K n +1 ,λ and log( K > n,λ ) ⊆ K n +1 , log( λ ) . Since X ⊆ K ,ω , it fol-lows at once that X n ⊆ K n, log n ( ω ) for all n ∈ N , so in particular R (( ω )) LE ⊆ S k ∈ N R ((log k ( ω ))) E .Conversely, it is clear that each element of K n, log n ( ω ) can be obtained from X = R ∪ { ω } by finitely many applications of exp , log and infinite sums. It followsat once that S k ∈ N R ((log k ( ω ))) E ⊆ R (( ω )) LE . (cid:3) Theorem 4.11 then follows at once by Propositions 4.19 and 4.18. Remark . If we modify Definition 4.13 putting M ,λ := λ Z instead of λ R , theunion S k ∈ N R ((log k ( ω ))) E will be the same, since λ R = exp( R log λ ) . So in thedefinition of the LE-series in [DMM01] one may start with x Z instead of x R .4.4. Adding more log-atomic numbers.Definition 4.21. Consider the class L ⊆ No of log-atomic numbers and let R ⟪ L ⟫ be the smallest subfield of No containing R ∪ L and closed under exp , log and P (in the sense of Definition 4.6).In [BM, Thm. 8.6] we showed that R ⟪ L ⟫ is the largest subfield of transseriessatisfying axiom ELT4 of [KM15, Def. 5.1]. We also showed that No itself does notsatisfy ELT4, hence R ⟪ L ⟫ = No [BM, Thm. 8.7]. The derivative ∂ : No → No introduced in [BM, Def. 6.21] can be restricted to R ⟪ L ⟫ and remains surjective onthis subfield. We thus have the inclusions R (( ω )) LE ⊂ R (( ω )) EL ⊂ R ⟪ ω ⟫ ⊂ R ⟪ L ⟫ ⊂ No with R (( ω )) LE , R ⟪ L ⟫ and No having a surjective derivation, while the derivationon R (( ω )) EL and R ⟪ ω ⟫ is not surjective. It would be interesting to study thecomplete first order theories of these structures, both as differential fields, and asdifferential fields with an exponentiation. The only known result so far is that No and R (( ω )) LE are elementary equivalent as differential fields [ADH], and probablythe same proof can be used to deduce that R ⟪ L ⟫ has the same first order theoryas well. RANSSERIES AS GERMS OF SURREAL FUNCTIONS 19 Inductive generation of transseries fields and associated ranks. Forthe purposes of Section 5, it is useful to inductively construct R ⟪ ω ⟫ and othersubfields of R ⟪ L ⟫ with a limited use of the log function, and to introduce a rankfunction reflecting the stages of the inductive construction. We need the followingdefinition. Definition 4.22. Let ∆ ⊆ L be a subclass with log(∆) ⊆ ∆ and let R ⟪ ∆ ⟫ be thesmallest subfield of No containing R ∪ ∆ and closed under P , exp and log .As we shall see Corollary 4.30, R ⟪ ∆ ⟫ coincides with the smallest subclass of No containing R ∪ ∆ and closed under P and exp (or even just exp ↾ J ); the closureunder log can be automatically deduced. Taking ∆ = L , we obtain the field R ⟪ L ⟫ seen in Subsection 4.4. On the other hand, when ∆ = { log n ( ω ) : n ∈ N } , weobtain R ⟪ ω ⟫ (Definition 4.7). Notation . Given a subclass A ⊆ M , we denote by R (( A )) small (or just R (( A )) if A is a set) the class of all surreal numbers with support contained in A . Noticethat if A is a group, R (( A )) small is a field, but we occasionally use the notationwithout assuming that A is a group. Definition 4.24. Let log(∆) ⊆ ∆ ⊆ L . We define by induction on the ordinal α ∈ On a subclass ∆ α ⊆ No as follows: ∆ = ∅ , ∆ = ∆ ∪{ } ; ∆ α +1 = R (( e ∆ α ∩ J )) small for α ≥ ; ∆ λ = S α<λ ∆ α for λ a limit ordinal. Given x ∈ S α ∈ On ∆ α , we definethe (exponential) rank ER ∆ ( x ) as the least ordinal β such that x ∈ ∆ β +1 . Remark . Note that ∆ is not an additive group. For α ≥ , ∆ α is an R -linearsubspace of No (and it is closed under P ); for α ≥ , ∆ α is a field, and a Hahnfield when α is a successor ordinal. Moreover, all the classes ∆ α are truncationclosed. Proposition 4.26. For all α < β we have ∆ α ⊆ ∆ β .Proof. It suffices to prove that ∆ α ⊆ ∆ α +1 for all α ∈ On . This is clear for α = 0 .Since log(∆) ⊆ ∆ , we have ∆ ⊆ e ∆ ⊆ R (( e ∆ )) small , thus ∆ ⊆ ∆ , proving thecase α = 1 . We then proceed by induction. If α = β + 1 , then ∆ β ⊆ ∆ β +1 holdsby inductive hypothesis, so ∆ α = R (( e ∆ β ∩ J )) small ⊆ R (( e ∆ β +1 ∩ J )) small = ∆ α +1 . If α is a limit ordinal, take some x ∈ ∆ α . By definition of ∆ α , there is some β < α such that x ∈ ∆ β , so by inductive hypothesis, x ∈ ∆ β +1 = R (( e ∆ β ∩ J )) small ⊆ R (( e ∆ α ∩ J )) small = ∆ α +1 . Since x is arbitrary, we obtain ∆ α ⊆ ∆ α +1 , as desired. (cid:3) The following corollary provides an equivalent definition of the rank. Its proofis easy and left to the reader. Corollary 4.27. For x ∈ S α ∈ On ∆ α we have(1) if x ∈ ∆ ∪ { } , then ER ∆ ( x ) = 0 ;(2) otherwise, ER ∆ ( x ) = sup { ER ∆ ( γ ) + 1 : e γ ∈ Supp( x ) } .Moreover, x ∈ ∆ β if and only if ER ∆ ( x ) < β . Proposition 4.28. We have:(1) for all α ≥ , P ∆ α +1 ⊆ ∆ α +1 (in particular, P ∆ α ⊆ ∆ α +2 for all α );(2) for all α ≥ , log(∆ > α ) ⊆ ∆ α ;(3) for all α ∈ On , e ∆ α ⊆ ∆ α +1 (in particular, e ∆ α ⊆ ∆ α for all limit α ). In particular, ∆ α is a transserial subfield of No for all α ≥ , and S α ∈ On ∆ α isclosed under exp , log and infinite sums.Proof. (1) Trivial, since by definition ∆ α +1 = R (( e ∆ α ∩ J )) small for α ≥ .(2) Without loss of generality, we may assume that α is of the form β + 1 with β ≥ , so that ∆ α is a Hahn field (see Remark 4.25). Take any x ∈ ∆ > α . We canwrite uniquely x = re γ (1 + ε ) , where r ∈ R > , γ ∈ ∆ β ∩ J and ε ∈ ∆ α ∩ o (1) .Then log( x ) = γ + log( r ) + P ∞ n =1 ( − n ε n n ∈ ∆ β + ∆ α = ∆ α by Proposition 4.26.It follows that log(∆ > α ) ⊆ ∆ α , as desired.(3) Note that the conclusion is trivially true for α = 0 , , so we may assumethat α ≥ . Take any x ∈ ∆ α . Since ∆ α is closed under truncation (see againRemark 4.25), we can write uniquely x = γ + r + ε , with γ ∈ ∆ α ∩ J , r ∈ R and ε ∈ ∆ α ∩ o (1) . Since ∆ α +1 is a Hahn field, we have e x = e γ · e r · P ∞ n =0 ε n n ! ∈ R (( e ∆ α ∩ J )) small = ∆ α +1 , as desired. (cid:3) Corollary 4.29. R ⟪ ∆ ⟫ = S α ∈ On ∆ α .Proof. By Proposition 4.28, S α ∈ On ∆ α contains R and ∆ (as both are containedin ∆ ) and it is closed under exp , log and infinite sums. It follows that R ⟪ ∆ ⟫ ⊆ S α ∈ On ∆ α . On the other hand, one can easily verify by induction that ∆ α ⊆ R ⟪ ∆ ⟫ for all α ∈ On , and the conclusion follows. (cid:3) Corollary 4.30. S α ∈ On = R ⟪ ∆ ⟫ is the smallest class containing ∆ ∪ { } andsuch that whenever the exponents γ i ∈ J of x = P i<α r i e γ i are in the class, thenalso x is in the class. The ordinal ER ∆ ( x ) measures the number of steps needed toobtain x with this inductive construction. Corollary 4.31. R ⟪ ∆ ⟫ is truncation closed, so it is a field of transseries in thesense of Definition 4.1.Proof. Immediate from the equality R ⟪ ∆ ⟫ = S α ∈ On ∆ α . (cid:3) Corollary 4.32. R ⟪ ∆ ⟫ is a proper class. In particular, R ⟪ ω ⟫ is a proper class.Proof. Let Γ = M ∩ R ⟪ ∆ ⟫ be the class of monomials of R ⟪ ∆ ⟫ . Since R ⟪ ∆ ⟫ isclosed under P and truncations, we have R ⟪ ∆ ⟫ = R ((Γ)) small . If for a contradic-tion R ⟪ ∆ ⟫ were a set, then R ⟪ ∆ ⟫ = R ((Γ)) . Since on the other hand R ⟪ ∆ ⟫ isan exponential subfield of No , R ((Γ)) would then carry a compatible exponentialfunction, contradicting [KKS97]. (cid:3) The following remark is implicit in our previous observations, but it is worth torecord it: Remark . Let ∆ = { log n ( ω ) : n ∈ N } . Then R (( ω )) EL = S n ∈ N ∆ n = ∆ ω .5. Substitutions Before defining the full notion of composition, we first define substitutions (alsocalled right-compositions in [Sch01]). Definition 5.1. Let T be a field of transseries. We say that f : T → No is strongly additive if for every summable sequence ( x i : i ∈ I ) in T , the sequence ( f ( x i ) : i ∈ I ) in No is summable and f ( P i ∈ I x i ) = P i ∈ I f ( x i ) . RANSSERIES AS GERMS OF SURREAL FUNCTIONS 21 Definition 5.2. Let T a field of transseries. A substitution c : T → No isa strongly additive map which is the identity on R and preserves log , namely c (log( x )) = log( c ( x )) for all x ∈ T .It is fairly easy to check that the substitutions are well behaved functions. Proposition 5.3. Let c : T → No be a substitution. Then c is an ordered fieldisomorphism fixing R . In particular, for all x, y ∈ T we have x < y → c ( x ) < c ( y ) and therefore x ≺ y → c ( x ) ≺ c ( y ) .Proof. Fix some x, y ∈ T . Clearly, c is additive. Moreover, if x > , then log( x ) ∈ T , so c (log( x )) = log( c ( x )) , so c ( x ) > , and in particular, c preserves the ordering.If x, y > , then c ( xy ) = c ( e log( xy ) ) = e c (log( x )+log( y )) = e c (log( x )) · e c (log( y )) = c ( x ) c ( y ) , and it follows easily that c is multiplicative. Therefore, c is an ordered fieldisomorphism which by definition fixes R . In particular, if x < y , then c ( x ) < c ( y ) .Moreover, if x ≺ y , then r | x | < | y | for all r ∈ R , so r | c ( x ) | < | c ( y ) | for all r ∈ R , so c ( x ) ≺ c ( y ) . (cid:3) In this section, we show how to construct inductively substitutions on fields ofthe form R ⟪ ∆ ⟫ starting from their values on some subclass ∆ ⊆ L . The proof thatthe construction is well defined is fairly complicated and technical; for the sakeof readability, the proof of one of the intermediate statement, the “summabilitylemma” 5.21, will be postponed to Section 9.5.1. Pre-substitutions. To build a substitution on R ⟪ ∆ ⟫ , we start with a certainassignment of values to each element of ∆ satisfying some suitable compatibilityconditions. We call such assignment a pre-substitution. Definition 5.4. A map c : ∆ → No is a pre-substitution if(1) the domain ∆ is a subclass of L closed under log ;(2) c ( λ ) > and c (log( λ )) = log( c ( λ )) for all λ ∈ ∆ ;(3) for any decreasing sequence ( λ i ∈ ∆) i ∈ N , the family ( c ( λ i )) i ∈ N is summa-ble;(4) for any increasing sequence ( λ i ∈ ∆) i ∈ N , the family (cid:0) c ( λ i ) − (cid:1) i ∈ N is sum-mable;(5) for all λ, µ ∈ ∆ , if λ < µ , then c ( λ ) ≺ c ( µ ) . Remark . By (1) and (2) it follows by induction on n ∈ N that c ( λ ) > exp n (0) for every λ ∈ ∆ , and therefore for all λ ∈ ∆ we have ≺ c ( λ ) . Moreover, if λ < µ , then c ( λ ) < c ( µ ) and c ( λ ) n ≺ c ( µ ) for all n ∈ N (since log( c ( λ )) = c (log( λ )) ≺ c (log( µ )) = log( c ( µ )) ).Clearly, if ∆ ⊆ L is a class closed under log and c : R ⟪ ∆ ⟫ → No is a substitution,then c ↾ ∆ is a pre-substitution. We shall prove that the converse holds, namelythat every pre-substitution with domain ∆ extends to a (unique) substitution withdomain R ⟪ ∆ ⟫ (Theorem 5.22), and as a corollary we shall deduce the existenceof substitutions on R ⟪ ω ⟫ (Corollary 5.23). We first give an explicit example ofpre-substitution on ∆ = { log i ( ω ) : i ∈ N } . Proposition 5.6. Let x ∈ No > N . Then the sequence (log i ( x )) i ∈ N is summable.Proof. By [BM, Prop. 5.8], there is an integer k ∈ N and some log-atomic number µ ∈ L such that log k ( x ) = µ + ε for some ε ≺ . Thus, it suffices to show that the sequence (log i ( µ + ε )) i ∈ N is summable. Let P ( y ) be the Taylor series of log(1 + y ) ,namely P ( y ) := P ∞ n =1 ( − n n y n . Then log( µ + ε ) = log( µ ) + log (cid:18) εµ (cid:19) = log( µ ) + P (cid:18) εµ (cid:19) = µ + ε where µ := log( µ ) ∈ L and ε = P (cid:16) εµ (cid:17) ≺ . We define inductively µ := µ , µ i +1 := log( µ i ) , ε := ε and ε i +1 := P (cid:16) ε i µ i (cid:17) . By construction, ε i ≺ and log i ( µ + ε ) = µ i + ε i for all i ∈ N . Since ( µ i ) i ∈ N is a decreasing sequence ofmonomials, P i µ i exists. To finish the proof it suffices to show that P i ε i exists.Let m be a monomial in the support of ε i +1 = P (cid:16) ε i µ i (cid:17) . Then there is an integer m ≥ such that m ∈ Supp (cid:16)(cid:16) ε i µ i (cid:17) m (cid:17) ⊆ µ mi Supp( ε i ) m . By an easy induction itfollows that m = 1 µ n · · · · · µ n i i · o where n ≥ . . . ≥ n i ≥ and o is a product of finitely many elements of Supp( ε ) .Note that o varies in the set S ∞ m =1 Supp( ε ) m , which is reverse well-ordered byLemma 3.8. Therefore, it suffices to prove that the family (cid:16) µ n ····· µ nii : i ∈ N (cid:17) issummable.Letting δ = P i ∈ N µ µ ··· µ i , we have that µ n ····· µ nii is in the support of δ n .Since δ ≺ , by Corollary 3.9, ( δ n : n ∈ N ) is summable, so (cid:16) µ n ····· µ nii : i ∈ N (cid:17) issummable, hence ( ε i ) i ∈ N is summable, as desired. (cid:3) Corollary 5.7. Let x ∈ No > R , let ∆ = { log i ( ω ) : i ∈ N } and let c x : ∆ → No bethe map that sends log i ( ω ) to log i ( x ) . Then c x is a pre-substitution. Trees. We now aim at extending each pre-substitution c : ∆ → No to asubstitution c : R ⟪ ∆ ⟫ → No . For this, we introduce the notion of tree , whoseaim is to keep track of the monomials that may appear in the support of c ( x ) byexpressing c ( x ) in terms of the values of c . To justify the definition of tree, considerthe following heuristic argument.Suppose we wish to calculate c ( x ) for some x ∈ R ⟪ ∆ ⟫ . If ER ∆ ( x ) = 0 , then wesimply use the equations c ( λ ) = c ( λ ) = P t ∈ Term( c ( λ )) t and c (0) = 0 . Now assume ER ∆ ( x ) > and write x = P i<α r i e γ i in normal form. First, we observe that wemust have c ( x ) = P i<α c ( r i e γ i ) , so our problem reduces to calculating c ( r i e γ i ) foreach i < α . Fix one γ = γ i and consider the following equation: c ( re γ ) = re c ( γ ) = re c ( γ ) ↑ = · exp( c ( γ ) ↓ ) = re c ( γ ) ↑ = · ∞ X n =0 (cid:0) c ( γ ) ↓ (cid:1) n n ! . Note that ER ∆ ( γ ) < ER ∆ ( x ) , so we may assume to already have obtained c ( γ ) ,and that c ( γ ) ↓ is presented as a sum c ( γ ) ↓ = P j ∈ J t j for some family ( t j ) j ∈ J ofterms (i.e. elements of R ∗ M ), where J = J i is some index set. Using Proposi-tion 3.5, we get c ( re γ ) = re c ( γ ) ↑ = · ∞ X n =0 X τ : n → J Y m Note that the right-hand side can be seen as a sum of terms . We use the aboveequation to present c ( re γ ) as a sum of terms indexed by the set { ( n, τ ) : n ∈ N , τ : n → J } . By taking the sum over all terms r i e γ i , we obtain a presentation of c ( x ) as a sum of terms indexed by the set { ( r i e γ i , n, τ ) : i < α, n ∈ N , τ : n → J i } .We then proceed inductively and assume that the index sets J i are themselvesconstructed in the same way (unless ER ∆ ( γ i ) = 0 , in which case we use the equa-tions c ( λ ) = c ( λ ) = P t ∈ Term( c ( λ )) t and c (0) = 0 ). One can then picture theindex ( r i e γ i , n, τ ) as a tree with root r i e γ i and children τ (0) , . . . , τ ( n − , as in thefollowing definition. Definition 5.8. Fix a pre-substitution c : ∆ → No . We define inductivelythe class of trees as follows. A tree is an ordered triple T = h R( T ) , n, τ i where R( T ) ∈ R ⟪ ∆ ⟫ ∩ R ∗ M is a term, called the root of T , and n, τ are defined as follows:(1) if R( T ) = λ ∈ ∆ , then n = 0 and τ is a term of c ( λ ) , so in this case T = h λ, , t i with t ∈ Term( c ( λ )) ;(2) if R( T ) = re γ / ∈ ∆ , then n ∈ N and τ is a function with domain n = { , , . . . , n − } such that τ (0) , . . . , τ ( n − are trees, called the children of T ( n can be zero, in which case T has no children); we also require that,for each i < n , the root R( τ ( i )) of τ ( i ) is a term of γ = log ↑ (R( T )) (where log ↑ is as in Definition 2.10).The descendants of T are T itself, its children, and the descendants of itschildren. The proper descendants are the descendants different from T itself.The leaves of T are the descendants U of T without children (for instance thedescendants with root in ∆ ).Note that by induction on ER ∆ , the above definition of tree is well founded. Definition 5.9. Let T = h R( T ) , n, τ i be a tree. We define size( T ) ∈ N as thenumber of descendants of T , namely:(1) size( T ) := 1 if T has no children, namely n = 0 ;(2) size( T ) := 1 + P i Fix a pre-substitution c :∆ → No . We shall now define a substitution c : R ⟪ ∆ ⟫ → No extending the given σ (0) = ( λ, , t ) . . . σ ( m − 1) = ( λ, , t m − ) τ (0) = ( se λ , m, σ ) T = ( re γ , n, τ ) τ (1) = . . . . . .. . . . . . Figure 5.1. An example of tree with root R ( T ) = re γ , where se λ is a term of γ , λ ∈ ∆ , t , . . . , t m − are terms of c ( λ ) , and thecontribution c ( T ) of T is c ( T ) = re c ( γ ) ↑ = n ! c ( τ (0)) c ( τ (1)) . . . = re c ( γ ) ↑ = n ! se c ( λ ) ↑ = m ! c ( σ (0)) . . . c ( σ ( m − c ( τ (1)) . . . = re c ( γ ) ↑ = n ! se c ( λ ) ↑ = m ! t . . . t m − c ( τ (1)) . . . pre-substitution c . To this aim, we shall define simultaneously by induction on α ∈ On the following objects: • the set of admissible trees A( x ) of each x ∈ ∆ α (which are trees inthe sense of Definition 5.8 with root R( T ) ∈ Term( x ) and some furtherrequirements); • the contribution c ( T ) ∈ R ∗ M of each T ∈ A( x ) ; • the extension c : ∆ α → No (which is obtained by summing the contribu-tions of the admissible trees in A( x ) , that is c ( x ) = P T ∈ A( x ) c ( T ) ).The main difficulty will be in proving that each family ( c ( T ) : T ∈ A( x )) is summa-ble, which is needed to show that c ( x ) = P T ∈ A( x ) c ( T ) is well defined (Lemma 5.21). Definition 5.10. Let α ∈ On be given. Let I ( α ) be the hypothesis For all x ∈ ∆ α , ( c ( T ) : T ∈ A( x )) is summable where A( x ) and c ( T ) for T ∈ A( x ) are inductively defined as in Definition 5.11(assuming I ( β ) for β < α ). Definition 5.11. First, we let A(0) := ∅ , and for λ ∈ ∆ , we define:(1) A( λ ) := {h λ, , t i : t ∈ Term( c ( λ )) } (namely every tree with root in ∆ isadmissible);(2) c ( h λ, , t i ) := t (the value of a tree with root in ∆ is its third component);(3) c ( λ ) := P T ∈ A( λ ) c ( T ) .This defines A( x ) , c ( T ) and c ( x ) for all x ∈ ∆ and T ∈ A( x ) .Now let α > and assume I ( β ) for all β < α . For general x ∈ ∆ α we define:(4) A( x ) := S t ∈ Term( x ) A( t ) ;(5) A ◦ ( x ) := { T ∈ A( x ) : c ( T ) ≺ } .When x = t = re γ is a term in ∆ α \ ∆ , let β < α be such that re γ ∈ ∆ β +1 \ ∆ β .We observe that γ ∈ ∆ β , and we define:(6) A( re γ ) := {h re γ , n, τ i : n ∈ N , τ : n → A ◦ ( γ ) } ;(7) for T = h re γ , n, τ i ∈ A( re γ ) , c ( T ) := re c ( γ ) ↑ = · n ! Y i Assuming that I ( α ) holds for every α ∈ On , we define c : R ⟪ ∆ ⟫ → No as the union of the functions c : ∆ α → No for α ∈ On . Remark . The present notion of tree should be compared with the similarnotion of labeled trees in [Sch01]. In this comparison, the admissible trees play thesame role as the well-labeled trees.We shall now prove that c : R ⟪ ∆ ⟫ → No is well defined and that it is theunique substitution on R ⟪ ∆ ⟫ extending c . The most technical and difficult partwill be proving that if I ( α ) holds, then ( c ( T ) : T ∈ A( x )) is summable for all RANSSERIES AS GERMS OF SURREAL FUNCTIONS 25 x ∈ ∆ α +1 \ ∆ α (Lemma 5.21). As anticipated, the proof of this fact is postponedto Section 9.First, we check that c is indeed an extension of c , and that it fixes R . Proposition 5.15. For all λ ∈ ∆ , ( c ( T ) : T ∈ A( λ )) is summable and c ( λ ) = X T ∈ A( λ ) c ( T ) = c ( λ ) . In particular, I (0) and I (1) hold, and c extends c .Proof. For any λ ∈ ∆ and T ∈ A( λ ) , we have T = h λ, , t i for some t ∈ Term( λ ) and c ( T ) = t . Moreover, ( c ( T ) : T ∈ A( λ )) coincides with ( t : t ∈ Term( c ( λ ))) ,hence it is summable and by definition c ( λ ) = X T ∈ A( λ ) ¯ c ( T ) = X t ∈ Term( c ( λ )) t = c ( λ ) . (cid:3) Proposition 5.16. If r ∈ R , then ( c ( T ) : T ∈ A( r )) is summable and c ( r ) = r .Proof. Note first that I (1) holds by 5.15, and that R ⊆ ∆ , so A( r ) is well definedfor each r ∈ R . Now observe that A(0) = ∅ , so c (0) = P T ∈ A(0) c ( T ) is an emptysum (equal to zero) and we get c (0) = 0 . For r = 0 the only admissible tree T ∈ A( r ) is given by T = h re , , ∅i . By definition, c ( T ) = re c (0) = re = r , hence c ( r ) = r . (cid:3) We now prove that assuming I ( α ) , the extension c : ∆ α → No preserves log and infinite sums. For α ≥ , since ∆ α is a field of transseries, this says that c : ∆ α → No is a substitution. Note that in the following statement the hypothesisis I ( α ) , but the conclusion is about terms in ∆ α +1 . Proposition 5.17. Assume I ( α ) . Let re γ ∈ ∆ α +1 be a term. Then ( c ( T ) : T ∈ A( re γ )) is summable, so c ( re γ ) = P T ∈ A( re γ ) c ( T ) is well defined and c ( re γ ) = re c ( γ ) . Proof. The result is clear if re γ ∈ ∆ = ∆ ∪ { } , for in that case c coincides with c by Proposition 5.15, so we can assume re γ / ∈ ∆ . Then ER ∆ ( γ ) < ER ∆ ( re γ ) , andby the inductive hypothesis, c ( γ ) = P T ′ ∈ A( γ ) c ( T ′ ) . By definition of A ◦ ( γ ) we have c ( γ ) ↓ = X T ′ ∈ A ◦ ( γ ) c ( T ′ ) . Unraveling the definitions we have: re c ( γ ) = re c ( γ ) ↑ = e c ( γ ) ↓ = X n ∈ N re c ( γ ) ↑ = n ! (cid:0) c ( γ ) ↓ (cid:1) n = X n ∈ N re c ( γ ) ↑ = n ! X T ′ ∈ A ◦ ( γ ) ¯ c ( T ′ ) n = X n ∈ N re c ( γ ) ↑ = n ! X τ : n → A ◦ ( γ ) Y i Assume I ( α ) . Let P i<β r i e γ i ∈ ∆ α . Then c X i<β r i e γ i = X i<β r i e c ( γ i ) . Proof. It follows at once from Proposition 5.17 and the equality A( P i<β r i e γ i ) = S i<β A( r i e γ i ) . (cid:3) Corollary 5.19. Assume I ( α ) , with α ≥ . Then c : ∆ α → No is a substitution.Proof. Since α ≥ , ∆ α is a transserial subfield of No by Proposition 4.28. ByProposition 5.16, c fixes R , and by Proposition 5.18, it is strongly additive. More-over, c preserves log . Indeed, let x = re γ (1 + ε ) ∈ ∆ α , where r ∈ R , γ ∈ J and ε ≺ . We have c (log( x )) = c γ + log( r ) + ∞ X i =1 ( − n ε n n ! = c ( γ ) + log( r ) + ∞ X i =1 ( − n c ( ε ) n n . By Proposition 5.17, c ( γ ) = log( c ( e γ )) , so the right hand side is log( c ( x )) , asdesired. (cid:3) Corollary 5.20. Assume I ( α ) for all α ∈ On . Then c : R ⟪ ∆ ⟫ → No is asubstitution extending c . Finally, we need to prove inductively that I ( α ) holds for all α ∈ On . The maindifficulty is in proving the successor stage, namely that I ( α ) implies I ( α + 1) . Thisis contained in the following lemma, the proof of which is postponed to Section 9. Lemma 5.21 (Summability) . Assume I ( α ) . Then ( c ( T ) : T ∈ A( x )) is summablefor all x ∈ ∆ α +1 \ ∆ α . In particular, I ( α ) implies I ( α + 1) .Proof. Postponed to Section 9. (cid:3) Theorem 5.22. Any pre-substitution c : ∆ → No extends uniquely to a substitu-tion c : R ⟪ ∆ ⟫ → No . RANSSERIES AS GERMS OF SURREAL FUNCTIONS 27 Proof. Fix a pre-substitution c : ∆ → No . By Proposition 5.15, I (0) and I (1) hold. It is also clear by the definition of I ( α ) that whenever α is a limit ordinal, I ( α ) is implied by, and in fact equivalent to, V β<α I ( β ) . Moreover, by Lemma 5.21, I ( α ) implies I ( α + 1) . Therefore, I ( α ) holds for all α ∈ On . By Corollary 5.20, c is a substitution extending c . The uniqueness follows by an easy induction on ER ∆ . (cid:3) Corollary 5.23. Given x ∈ No > , there is a unique substitution c x : R ⟪ ω ⟫ → No sending ω to x .Proof. Let ∆ = { log i ( ω ) : i ∈ N } and let c x : ∆ → No be the map that sends log i ( ω ) to log i ( x ) . Then c x is a pre-substitution by Corollary 5.7. By Theorem5.22, there is a unique substitution c x : R ⟪ ∆ ⟫ = R ⟪ ω ⟫ → No extending c x . (cid:3) Composition We prove that omega-series can be composed in a meaningful way. Intuitively,for f, g ∈ R ⟪ ω ⟫ , with g > R , f ◦ g is the result of substituting g for ω in f . Forinstance, we will have X i ∈ N log i ( ω ) ! ◦ X i ∈ N log i ( ω ) ! = X i ∈ N log i X j ∈ N log j ( ω ) . Note that the right-hand side exists in No by the results in Section 5 and it is infact an element of R ⟪ ω ⟫ . Definition 6.1. Let T ⊆ No be a transserial subfield containing ω . A composi-tion on T is a function ◦ : T × No > R → No which satisfies the following axioms:(1) for all x ∈ No > R , the map f f ◦ x is a substitution, namely:(a) for any summable ( f i ) i ∈ I in T , the family ( f i ◦ x ) i ∈ I is summable and X i ∈ I f i ! ◦ x = X i ∈ I ( f i ◦ x ); (b) r ◦ x = r for all r ∈ R ;(c) log( f ) ◦ x = log( f ◦ x ) for all f ∈ T ;(2) T is closed under composition: for all f ∈ T , g ∈ T > R we have f ◦ g ∈ T ;(3) associativity: ( f ◦ g ) ◦ x = f ◦ ( g ◦ x ) for all f ∈ T , g ∈ T > R , x ∈ No > R ;(4) ω is the identity: for all x ∈ No > R and f ∈ T we have ω ◦ x = x , f ◦ ω = f .The axioms are modeled on the usual composition of real valued functions, wherewe interpret ω as the identity function. The restriction on the second argument tobe positive infinite is necessary for a composition to exist; for instance we cannothope to define P n ∈ N ω − n ◦ (1 / in any reasonable way, as the axioms imply thatthe result should be P n ∈ N n . Recall that by Proposition 5.3, for all x ∈ No > N ,the map f ◦ x is increasing and it preserves the dominance relation (cid:22) .When T ⊆ R ⟪ ω ⟫ , the list of axioms can be shortened. More precisely, we have: Proposition 6.2. If T is a transserial field included in R ⟪ ω ⟫ , there is at most onefunction ◦ : T × No > R → No satisfying the following conditions:(1) for all x ∈ No > R , the map f f ◦ x is a substitution;(2) for all x ∈ No > R , ω ◦ x = x . If any such function ◦ exists, it satisfies f ◦ ω = f for any f ∈ T . If moreover T isclosed under ◦ , then ◦ is associative, so it is a composition.Proof. Suppose that ◦ is a function satisfying the above properties. Let ∆ = { log i ( ω ) : i ∈ N } ⊆ T , and fix some x ∈ No > R . We claim that the values of thesubstitution f f ◦ x for f ∈ ∆ are uniquely determined by the requirement ω ◦ x = x . We shall prove this by induction on ER ∆ ( f ) ; at the same time, we willalso verify associativity when T is closed under ◦ .Note first that log i ( ω ) ◦ x = log i ( x ) by definition of substitution. Moreover, log i ( ω ) ◦ ( g ◦ x ) = log i ( g ◦ x ) = log i ( g ) ◦ x = (log i ( ω ) ◦ g ) ◦ x for any g ∈ T > N , and also log i ( ω ) ◦ ω = log i ( ω ) . It now follows by induction on ER ∆ ( f ) that the value of f ◦ x is also uniquely determined, f ◦ ω = f , and if T is closed under ◦ , then f ◦ ( g ◦ x ) = ( f ◦ g ) ◦ x for any g ∈ T > R . Indeed, if f = P i<α r i e γ i , where ER ∆ ( f ) > , then we must have f ◦ x = X i<α r i e γ i ◦ x where ER ∆ ( γ i ) < ER ∆ ( f ) . The value of f ◦ x is then uniquely determined by thevalues γ i ◦ x , which are themselves uniquely determined by inductive hypothesis,and clearly f ◦ ω = f as again by induction γ i ◦ ω = γ i . Moreover, if T is closedunder ◦ , then f ◦ ( g ◦ x ) = X i<α r i e γ i ◦ ( g ◦ x ) = X i<α r i e ( γ i ◦ g ) ◦ x = X i<α r i e γ i ◦ g ! ◦ x = ( f ◦ g ) ◦ x. Therefore, ◦ is unique, f ◦ ω = f for any f ∈ T , and if T is closed under ◦ , then itis associative, so it is a composition. (cid:3) Theorem 6.3. There is a unique composition ◦ : R ⟪ ω ⟫ × No > R → No .Proof. Let ∆ = { log i ( ω ) : i ∈ N } . Fix x ∈ No > R and f ∈ R ⟪ ω ⟫ . By Corol-lary 5.23, there exists a unique substitution c x on R ⟪ ∆ ⟫ = R ⟪ ω ⟫ such that c x (log i ( ω )) =log i ( x ) for all i ∈ N . We then define f ◦ x := c x ( f ) . Clearly, this function is theunique one satisfying the hypothesis of Proposition 6.2. One can easily verify byinduction on ER ∆ that R ⟪ ω ⟫ is closed under ◦ , so it is a composition. (cid:3) Taylor expansions In this section, let ◦ be the unique composition on R ⟪ ω ⟫ . We shall now provethat for every f ∈ R ⟪ ω ⟫ , the function x f ◦ x is surreal analytic in the sense ofDefinition 3.7. Moreover, the coefficients will coincide with the iterated derivativesof f divided by n ! , when using the unique surreal derivation on R ⟪ ω ⟫ .7.1. Transserial derivations. Recall the notion of derivation from [Sch01, BM]. Definition 7.1. Given a field T , we recall that a map ∂ : T → T is a derivation ifit is additive ( ∂ ( x + y ) = ∂x + ∂y ) and satisfies the Leibniz rule ( ∂ ( xy ) = x · ∂y + ∂x · y ) .If T is a field of transseries we say that ∂ : T → T is a transserial derivation ifit is a derivation satisfying the following additional properties:(1) ∂ is strongly additive;(2) ∂e x = e x · ∂x ;(3) ∂ω = 1 ; RANSSERIES AS GERMS OF SURREAL FUNCTIONS 29 (4) ∂r = 0 if r ∈ R .As in [BM], we call surreal derivation a transserial derivation with ker ∂ = R .In [BM], the authors proved that there exist surreal derivations on No , and infact several of them. However, just like we proved that there is a unique compositionon R ⟪ ω ⟫ , we can easily verify that there exists a unique transserial derivation on R ⟪ ω ⟫ . Proposition 7.2. The field of omega-series admits a unique transserial derivation ∂ : R ⟪ ω ⟫ → R ⟪ ω ⟫ , which is in fact a surreal derivation.Proof. Suppose first that there exists a transserial derivation ∂ : R ⟪ ω ⟫ → R ⟪ ω ⟫ .Since ∂ω = 1 , an easy induction on ER ∆ shows that in fact the values of ∂ areuniquely determined, and that ker( ∂ ) = R . Therefore, if there is one such deriva-tion, it is unique, and it is a surreal derivation.For the existence, let ∂ be any surreal derivation, which exists by the results of[BM]. By the same argument as above, since ∂ω = 1 ∈ R ⟪ ω ⟫ , an easy inductionon ER ∆ shows that ∂ ( R ⟪ ω ⟫ ) ⊆ R ⟪ ω ⟫ . Therefore, the restriction of ∂ to R ⟪ ω ⟫ isthe unique transserial derivation on R ⟪ ω ⟫ . (cid:3) Remark . Unlike the subfield R (( ω )) LE , but like R (( ω )) EL , the field of omega-series R ⟪ ω ⟫ is not closed under anti-derivatives. For instance, it contains no integralfor the monomial exp( − P n ∈ N log n ( ω )) .7.2. A Taylor theorem. From now on, let ∂ : R ⟪ ω ⟫ → R ⟪ ω ⟫ be the uniquetransserial derivation on R ⟪ ω ⟫ . Recall that for any x ≺ we have exp( x ) = P n ∈ N x n n ! . When x ≻ , the equality does not hold, as the right hand side clearlydoes not exist. However, we can still approximate exp( x ) with Taylor polynomials.In particular we have the following: Proposition 7.4. Given x ∈ No , there are A ∈ No and ε ∈ No > (dependingon x ) such that, for every ε ∈ No smaller in modulus than ε , we have exp( x + ε ) = exp( x ) + exp ′ ( x ) ε + O ( Aε ) where exp ′ ( x ) := exp( x ) and O ( Aε ) is a surreal number (cid:22) Aε . Similarly, we canwrite log( x + ε ) = log( x ) + log ′ ( x ) ε + O ( Aε ) where log ′ ( x ) := x .Proof. Immediate from the fact that No is an elementary extension of R exp . (cid:3) The next theorem extends the above remark to a much larger class of functions. Theorem 7.5. Given f ∈ R ⟪ ω ⟫ and x ∈ No > R , there are A ∈ No and ε ∈ No > (both depending on f and x ) such that, for every ε ∈ No smaller in modulus than ε , we have f ◦ ( x + ε ) = f ◦ x + ( ∂f ◦ x ) · ε + O ( Aε ) , where O ( Aε ) is a surreal number (cid:22) Aε .Proof. We reason by induction on the ordinal ER ∆ ( f ) , where ∆ = { log i ( ω ) : i ∈ N } .Case 1. The theorem is clear if f ∈ R or f = ω , as in this case f ◦ ( x + ε ) = f ◦ x + ( ∂ ( f ) ◦ x ) ε for every ε and we can take A = 0 . Case 2. Now consider the case when f = log( g ) where g > , and assume thatconclusion holds for g . Then there are B ∈ No and ε ∈ No > (depending on g, x )such that g ◦ ( x + ε ) = g ◦ x + ( ∂ ( g ) ◦ x ) ε + O ( Bε ) whenever | ε | ≤ | ε | . Taking the log of both sides, and recalling that log( g ◦ ( x + ε )) =log( g ) ◦ ( x + ε ) = f ◦ ( x + ε ) , we obtain f ◦ ( x + ε ) = log( g ◦ x + ( ∂ ( g ) ◦ x ) ε + O ( Bε )) . Using the second order Taylor expansion of log at g ◦ x , we can find A ∈ No ,depending on g and x , such that, for all sufficiently small ε , log( g ◦ x + ( ∂ ( g ) ◦ x ) ε + O ( Bε )) = log( g ◦ x ) + 1 g ◦ x ( ∂ ( g ) ◦ x ) ε + O ( Aε )= log( g ) ◦ x + (cid:18) ∂ ( g ) g ◦ x (cid:19) ε + O ( Aε )= f ◦ x + ( ∂ ( f ) ◦ x ) ε + O ( Aε ) . Combining the equations we obtain f ◦ ( x + ε ) = f ◦ x + ( ∂ ( f ) ◦ x ) ε + O ( Aε ) , asdesired.Case 3. When f = log n ( ω ) for some n ∈ N , the desired result follows from theprevious cases by induction on n . We have thus established the conclusion when ER ∆ ( f ) = 0 , namely f ∈ ∆ = ∆ ∪ { } .Case 4. Consider now the case when f = exp( g ) and assume that the conclusionholds for g . We can then proceed as in case 2 using the second order Taylorexpansion of exp at g ◦ x .Case 5. Consider the case when f = P i ∈ I f i and assume by induction that theresult holds for each f i . By definition f ◦ ( x + ε ) = P i ∈ I ( f i ◦ ( x + ε )) . By inductionthere are ε i,x ∈ No > and A i,x ∈ No such that f i ◦ ( x + ε ) = f i ◦ x + ( ∂ ( f i ) ◦ x ) ε + O ( A i,x ε ) for all ε < ε i,x . Now let ε ∈ No > be smaller than ε i,x for every i ∈ I and let A (cid:23) A i,x for every i ∈ I . Then for every ε smaller in modulus than ε we have f ◦ ( x + ε ) = f ◦ x + ( ∂ ( f ) ◦ x ) · ε + O ( Aε ) , as desired.Finally, observe that the above cases suffices to establish inductively the theoremfor every f ∈ R ⟪ ω ⟫ . (cid:3) Corollary 7.6. For every f ∈ R ⟪ ω ⟫ and every x ∈ No > R we have ∂f ◦ x = lim ε → f ◦ ( x + ε ) − f ◦ xε In particular, taking x = ω , we obtain ∂f = lim ε → f ◦ ( ω + ε ) − f ◦ ωε , so the derivativeis definable in terms of the composition. Corollary 7.7. The unique composition on R ⟪ ω ⟫ satisfies ∂ ( f ◦ g ) = ( ∂f ◦ g ) · ∂g .Proof. Thanks to Corollary 7.6, it suffices to show that that for all sufficiently small ε we have ( f ◦ g ) ◦ ( x + ε ) = ( f ◦ g ) ◦ x + (( ∂f ◦ g ) · ∂g ) ε + O ( Aε ) RANSSERIES AS GERMS OF SURREAL FUNCTIONS 31 where A ∈ No depends on f , g , x but not on ε . Applying Theorem 7.5 first to g and then to f , there are C, D ∈ No , not depending on ε , such that ( f ◦ g ) ◦ ( x + ε ) = f ◦ ( g ◦ ( x + ε ))= f ◦ ( g ◦ x + ( ∂g ◦ x ) ε + O ( C · ε ))= f ◦ ( g ◦ x ) + ( ∂f ◦ ( g ◦ x )) · ( ∂g ◦ x ) ε + O ( D · ε ) , and we conclude by noting that ( ∂f ◦ ( g ◦ x )) · ( ∂g ◦ x ) = (( ∂f ◦ g ) · ∂g ) ◦ x . (cid:3) Surreal analyticity. We now extend in the obvious way the notion of surrealanalyticity of Definition 3.7 to the numbers in R ⟪ ω ⟫ . Definition 7.8. Let f ∈ R ⟪ ω ⟫ . We say that f is surreal analytic at x ∈ No > R if the function y f ◦ y is surreal analytic in a neighborhood of x is the sense ofDefinition 3.7. We say that f is surreal analytic if y f ◦ y is surreal analyticat every x ∈ No > R .For instance, exp( ω ) and log( ω ) are surreal analytic. Proposition 7.9. Let x ∈ No > R . Then for every ε ≺ we have exp( x + ε ) = P ∞ i =0 e x i ! ε i . In particular, exp( ω ) is surreal analytic.Proof. Indeed, exp( x + ε ) = exp( x ) · exp( ε ) = exp( x ) · P ∞ i =0 ε i i ! . (cid:3) Proposition 7.10. Let x ∈ No > R . Then for every ε ≺ x we have log( x + ε ) =log( x ) + P ∞ i =1 ( − i +1 ix i ε i . In particular, log( ω ) is surreal analytic.Proof. It suffices to write x + ε = x (cid:0) εx (cid:1) , so that δ := εx ≺ , and recall that log( x + ε ) = log (cid:16) x (cid:16) εx (cid:17)(cid:17) = log( x ) + log(1 + δ ) = log( x ) + ∞ X i =1 ( − i +1 i δ i . (cid:3) Moreover, surreal analyticity is preserved under compositions. Lemma 7.11. If g ∈ R ⟪ ω ⟫ is surreal analytic at x ∈ No > R and f ∈ R ⟪ ω ⟫ issurreal analytic at y := g ◦ x , then f ◦ g is surreal analytic at x .Proof. Fix f, g, x, y as in the hypothesis. By assumption there are two sequences ( a i ) i ∈ N and ( b j ) j ∈ N in No such that, for every sufficiently small ε, δ we have g ◦ ( x + ε ) = g ◦ x + ∞ X j =1 b j ε j and f ◦ ( y + δ ) = X i ∈ N a i δ i . Note that ( f ◦ g ) ◦ ( x + ε ) = f ◦ ( y + P ∞ j =1 b j ε j ) = P i ∈ N a i ( P ∞ j =1 b j ε j ) i for every suf-ficiently small ε . To finish the proof it suffices to observe that, by Proposition 3.16,there is a sequence ( c m ) m ∈ N in No such that, for every sufficiently small ε , we have X k ∈ N a k ( ∞ X n =1 b n ε n ) k = X m ∈ N c m ε m . (cid:3) Corollary 7.12. For all i ∈ N , log i ( ω ) is surreal analytic. We can also verify that if f ∈ R ⟪ ω ⟫ is surreal analytic, the coefficients of itsTaylor expansions can be calculated using the derivation ∂ just like with classicalanalytic functions. Proposition 7.13. If f ∈ R ⟪ ω ⟫ is surreal analytic at x ∈ No > R , then for everysufficiently small ε ∈ No we have f ◦ ( x + ε ) = X n ∈ N n ! ( ∂ n f ◦ x ) · ε n where ∂ f = f and ∂ n +1 f = ∂ ( ∂ n f ) .Proof. Let f ∈ R ⟪ ω ⟫ be analytic at x ∈ No > R . Let ˆ f be associated function x + ε f ◦ ( x + ε ) , which by assumption is also surreal analytic (in the sense ofDefinition 3.7). By Proposition 3.11, we know that f ◦ ( x + ε ) = ˆ f ( x + ε ) = ∞ X i =0 ˆ f ( i ) ( x ) i ! ε i . By Corollary 7.6, it follows by induction on i that in fact ˆ f ( i ) ( x ) = ∂ i f ◦ x , provingthe desired conclusion. (cid:3) We can then conclude that every omega-series is surreal analytic. Theorem 7.14. Every f ∈ R ⟪ ω ⟫ is surreal analytic, and for every x ∈ No > R andevery sufficiently small ε ∈ No we have f ◦ ( x + ε ) = X i ∈ N i ! ( ∂ i f ◦ x ) · ε i . Proof. Let f ∈ R ⟪ ω ⟫ . We reason by induction on ER ∆ ( f ) , where ∆ = { log i ( ω ) : i ∈ N } .The case f = 0 is trivial, while the case f = log n ( ω ) follows from Corollary 7.12and Proposition 7.13. This shows the conclusion for ER ∆ ( f ) = 0 , namely for f ∈ ∆ = ∆ ∪ { } .Now suppose ER ∆ ( f ) > . Write f = P j<α r j e γ j , and recall that by definition ER ∆ ( γ j ) < ER ∆ ( f ) for all j < α . Therefore, by inductive hypothesis, we canassume that γ j is surreal analytic for every j < α . Since exp( ω ) is surreal analyticby Proposition 7.9, it follows that exp( ω ) ◦ γ j = exp( γ j ) is surreal analytic byLemma 7.11, hence so is f j := r j e γ j . This means that for each x , there is some ε j > such that for all ε smaller than ε j in absolute value, we have f j ◦ ( x + ε ) = P i ∈ N i ! ( ∂ i f j ◦ x ) · ε i .Since ∂ is strongly additive, and ( f j : j < α ) is summable, the family ( ∂f j : j < α ) is also summable and P j ∂f j = ∂ (cid:16)P j f j (cid:17) . In turn, ( ∂f j ◦ x : j < α ) mustbe summable, and P j ( ∂f j ◦ x ) = ( P j ∂f j ) ◦ x = ∂ (cid:16)P j f j (cid:17) ◦ x = ∂f ◦ x . Similarly,by induction on i ∈ N , ( ∂ i f j ◦ x : j < α ) is summable and P j ( ∂ i f j ◦ x ) = ∂ i f ◦ x .By Lemma 3.15, for every sufficiently small ε , (cid:0) ( ∂ i f j ◦ x ) · ε i : ( i, j ) ∈ N × α (cid:1) issummable and therefore, by Corollary 3.2, we have X j X i i ! ( ∂ i f j ◦ x ) · ε i = X i X j i ! ( ∂ i f j ◦ x ) · ε i = X i i ! (cid:0) ∂ i f ◦ x (cid:1) · ε i . RANSSERIES AS GERMS OF SURREAL FUNCTIONS 33 Recalling that f j ◦ ( x + ε ) = P i ∈ N i ! ( ∂ i f j ◦ x ) · ε i , it follows that f ◦ ( x + ε ) = X j ( f j ◦ ( x + ε )) = X j X i i ! ( ∂ i f j ◦ x ) · ε i = X n i ! ( ∂ i f ◦ x ) · ε n thus proving that f is surreal analytic. (cid:3) Remark . When f ∈ R (( ω )) LE and x ∈ R (( ω )) LE , one can verify that thereexists an n ∈ N such that the equation of Theorem 7.14 holds for any ε (cid:22) e − exp n ( ω ) .Indeed, note that the subfields K m, log i ( ω ) (see Definition 4.13) are closed under thederivation ∂ , and that there is some k ∈ N such that g ◦ x ∈ K m + k, log i + k ( ω ) for any g ∈ K m, log i ( ω ) . Then all the coefficients ∂ i f ◦ x/i ! live in some fixed K n, log n ( ω ) , andit suffices to apply Corollary 3.9 to get the desired conclusion. In particular, onecan give a meaningful definition of analyticity for LE-series by staying inside thefield of LE-series, without resorting to No .In full generality, Corollary 3.9 guarantees that the equation of Theorem 7.14holds for any ε that is infinitesimal with respect to any non-zero omega-series g ∈ R ⟪ ω ⟫ . In some cases, this is the best we can do. Take for instance f = P ∞ n =0 e − exp n ( ω ) . Then one can easily verify that ( ∂ i f ◦ ω ) i ∈ N = ( ∂ i f ) i ∈ N is notsummable, and in fact that ( ∂ i f · ε ) i ∈ N is not summable for any ε such that ε (cid:23) e − exp n ( ω ) for some n ∈ N , and in particular for any ε ∈ R ⟪ ω ⟫ ∗ . Therefore, theexpansion of f ◦ ( ω + ε ) given by Theorem 7.14 only exists for the numbers ε withabsolute value smaller than any omega-series. Corollary 7.16. Given f ∈ R ⟪ ω ⟫ and x ∈ No > R , we have f ◦ ( x + ε ) = f ◦ x + ( ∂f ◦ x ) · ε + O (( ∂ f ◦ x ) · ε ) whenever ε ∈ No satisfies ( ∂ i +2 f ◦ x ) · ε i (cid:22) ∂ f ◦ x for all i ∈ N . A negative result The interaction between the unique composition ◦ on R ⟪ ω ⟫ and the uniquetransserial derivation on R ⟪ ω ⟫ suggests looking for compositions that are compat-ible with a transserial derivation. Definition 8.1. Given a transserial subfield T ⊆ No , a transserial derivation ∂ : T → T , and a composition ◦ : T × No > R → No , we say that ∂ and ◦ are compatible if the following holds:(1) if ∂f = 0 , then f ◦ x = f for every x ;(2) ∂f > ⇒ f ◦ x < f ◦ y whenever x < y ;(3) ∂ ( f ◦ g ) = ( ∂f ◦ g ) · ∂g . Theorem 8.2. The unique surreal derivation ∂ on R ⟪ ω ⟫ is compatible with theunique composition on R ⟪ ω ⟫ .Proof. Condition (1) follow at once from ker( ∂ ) = R .For condition (2), let f ∈ R ⟪ ω ⟫ . We reason by induction on ER ∆ ( f ) , where ∆ = { log i ( ω ) : i ∈ N } . If ER ∆ ( f ) = 0 , then the conclusion is easy: for instance if f = log i ( ω ) , then f ◦ g = log i ( g ) and the chain rule in (3) can be verified as in theclassical case, recalling also Corollary 7.6. Now suppose that ER ∆ ( f ) > . Write f = P i<α r i e γ i , where ER ∆ ( γ i ) < ER ∆ ( f ) for all i < α . Suppose that f ◦ x ≥ f ◦ y for some x < y . Since the maps g g ◦ x , g g ◦ y are substitutions, they preserve the relation (cid:22) (Proposition 5.3), so we must have ( r e γ ) ◦ x ≥ ( r e γ ) ◦ y ,so r e γ ◦ x ≥ r e γ ◦ y .Without loss of generality, we may assume that γ = 0 (by replacing f with f − r ) and that r > (by replacing f with − f ). Under these assumptions, wemust have γ ◦ x ≥ γ ◦ y , so by inductive hypothesis ∂γ ≤ . Note moreover thatsince γ ∈ J =0 , we must have ∂γ = 0 . In turn, since ∂f ∼ r e γ ∂γ , it follows that ∂f ≥ , as desired.Point (3) is Corollary 7.7. (cid:3) Question 8.3. We do not know whether there is a composition and a compatibletransserial derivation (possibly with ker( ∂ ) bigger than R ) on the whole of No .Note that the present notion of compatibility is rather weak, and for instanceit does not require the conclusion of Theorem 7.14 to hold, or even just Theorem7.5. However, even such a weak notion does not allow the “simplest” derivation ∂ : No → No of [BM] to be compatible with a composition. Theorem 8.4. The “simplest” surreal derivation ∂ : No → No in [BM] cannot becompatible with a composition ◦ : No × No > R → No .Proof. Let y ∈ No , and observe that the rules of transserial derivations yield ∂ (log n ( y )) = Q i We will now give a proof of Lemma 5.21. We work under the notations of Section5. Suppose that c : ∆ → No is a given pre-substitution. Then we wish to provethe following: Lemma 5.21. Assume I ( α ) . Then ( c ( T ) : T ∈ A( x )) is summable for all x ∈ ∆ α +1 \ ∆ α . In particular, I ( α ) implies I ( α + 1) . For the rest of this section, let c : ∆ → No be a pre-substitution, and assumethat the inductive hypothesis I ( α ) holds. Then c : ∆ α → No is well defined, andthe objects A( x ) , c ( T ) and A ◦ ( x ) are clearly well defined for all x ∈ ∆ α +1 and all T ∈ A( x ) . Moreover, recall that by Proposition 5.17, c ( t ) is also well defined for allterms t ∈ ∆ α +1 ∩ R ∗ M .9.1. A property of pre-substitutions. We start by observing a rather technical,but crucial fact on pre-substitutions. Lemma 9.1. Let x ∈ No and m be the leading monomial of x . Then Supp( x ) ⊆ ∞ [ n =0 m n +1 · Supp( x − ) n . Proof. Let t = r m be the leading term of x . Write x − = t − (1 + ε ) , where ε ≺ .Then x = t (1 + ε ) = t · ∞ X n =0 ( − n ε n , hence every element in the support of x has the form m · n · . . . · n n with n ≥ and n i ∈ Supp( ε ) . On the other hand, since ε = tx − − r m x − − and ε ≺ , wehave Supp( ε ) ⊆ m · Supp( x − ) , and the conclusion follows. (cid:3) Lemma 9.2. Let c : ∆ → No be a pre-substitution. Let ( λ i ) i ∈ N , ( m i ) i ∈ N be twosequences such that λ i ∈ ∆ and m i ∈ Supp( c ( λ i )) for all i ∈ N . Then there is anincreasing sequence of indexes ( i j ) j ∈ N such that one of the following holds:(1) the subsequence ( λ i j ) j ∈ N is decreasing and for all j ∈ N m i j +1 m i j ≺ c ( λ i j +1 ) c ( λ i j ) ≺ (2) the subsequence ( λ i j ) j ∈ N is increasing and for all j ∈ N m i j +1 m i j ≺ c ( λ i j +1 ) ; (3) the subsequence ( λ i j ) j ∈ N is constant and for all j ∈ N m i j +1 m i j (cid:22) . Note that in all three cases we have m ij +1 m ij ≺ c ( λ i j +1 ) .Proof. Let λ i =: e µ i . Note that µ i ∈ ∆ . We have c ( λ i ) = e c ( µ i ) = e c ( µ i ) ↑ = e c ( µ i ) ↓ . Thus n i := m i e − c ( µ i ) ↑ ∈ Supp (cid:0) exp( c ( µ i ) ↓ = ) (cid:1) = Supp(exp( c ( µ i ) ↓ ) , and thereforethere is some n i ∈ N such that n i ∈ Supp(( c ( µ i ) ↓ ) n i ) . After extracting a subse-quence we may assume that ( λ i ) i ∈ N is monotone, so either increasing, decreasing,or constant.(1) Suppose that ( λ i ) i ∈ N is decreasing. Then ( µ i ) i ∈ N is also decreasing, hence thefamily ( c ( µ i ) : i ∈ N ) is summable. In particular, ( c ( µ i ) ↓ : i ∈ N ) is summable,and by Corollary 3.12, (exp( c ( µ i ) ↓ ) : i ∈ N ) is summable. We may thereforeextract a subsequence and assume that ( n i ) i ∈ N is decreasing, so that m i +1 e − c ( µ i +1 ) ↑ ≺ m i e − c ( µ i ) ↑ . Since c ( λ i ) = e c ( µ i ) , it follows that m i +1 m i ≺ e c ( µ i +1 ) ↑ e c ( µ i ) ↑ ≍ c ( λ i +1 ) c ( λ i ) ≺ . (2) Consider now the case when ( λ i ) i ∈ N is increasing. Let o i := LM( c ( µ i )) . ByLemma 9.1, applied with x = c ( µ i ) , we deduce that Supp( c ( µ i )) ⊆ ∞ [ m =0 o m +1 i · Supp( c ( µ i ) − ) m . Since n i = m i e c µi ) ↑ ∈ Supp(( c ( µ i ) ↓ ) n i ) , it follows that there is an m i ∈ N such that m i e c ( µ i ) ↑ ∈ o n i ( m i +1) i · Supp( c ( µ i ) − ) n i m i and therefore m i · e − c ( µ i ) ↑ · o − n i ( m i +1) i ∈ Supp( c ( µ i ) − ) n i m i .Now observe that c ( µ i ) − ≺ and that the family ( c ( µ i ) − : i ∈ N ) issummable because ( µ − i ) i ∈ N is decreasing. By Corollary 3.12, applied with ε i = c ( µ i ) − , the family (cid:16) m i · e − c ( µ i ) ↑ · o − n i ( m i +1) i : i ∈ N (cid:17) is summable. We maytherefore extract a subsequence and assume that m i e c ( µ i ) ↑ · o n i ( m i +1) i ≻ m i +1 e c ( µ i +1 ) ↑ · o n i +1 ( m i +1 +1) i +1 . Since c ( µ i ) is positive infinite, e c ( µ i ) ↑ ≻ c ( µ i ) n ≍ o ni for any n ∈ N , so m i +1 e c ( µ i +1 ) ↑ ≺ m i +1 e c ( µ i +1 ) ↑ · o n i +1 ( m i +1 +1) i +1 ≺ m i e c ( µ i ) ↑ . Therefore, m i +1 m i ≺ e c ( µ i +1 ) ↑ e c ( µ i ) ↑ ≍ c ( λ i +1 ) c ( λ i ) (cid:22) c ( λ i +1 ) . (3) Finally, suppose that there is a λ ∈ ∆ such that λ i = λ for all i ∈ N . In thiscase all the monomials m i are in the support of c ( λ ) ∈ No , hence obviously wemay extract a subsequence and assume that m i +1 (cid:22) m i for all i ∈ N . (cid:3) Further properties of the extensions. Recall that I ( α ) implies that c :∆ α → No is a substitution when α ≥ (Corollary 5.20). In particular, c preservesthe ordering and the dominance relation ≺ by Proposition 5.3. We observe that I ( α ) implies similar monotonicity properties for α < , and also for terms in ∆ α +1 . Proposition 9.3. For all x, y ∈ ∆ α , and for all x, y ∈ ∆ α +1 ∩ R ∗ M , we have x < y → c ( x ) < c ( y ) and x ≺ y → c ( x ) ≺ c ( y ) . RANSSERIES AS GERMS OF SURREAL FUNCTIONS 37 Proof. If α is or , then for all x, y ∈ ∆ α we have x < y → c ( x ) < c ( y ) and x ≺ y → c ( x ) ≺ c ( y ) by definition of pre-substitution. The same conclusionholds for α ≥ by Corollary 5.19 and Proposition 5.3. For α = 2 , note that byProposition 5.18, if we expand some x ∈ ∆ \ R as x = r e λ + P ≤ i<β r i e λ i + s (where r i , s ∈ R , λ i ∈ ∆ , and λ i > λ j for all i ≤ j < β ), we have c ( x ) = r e c ( λ ) + X ≤ i<β r i e c ( λ i ) + s, while c ( r ) = r for all r ∈ R by Proposition 5.16. By definition of pre-substitution,it follows at once that c ( x ) ∼ r e c ( λ ) , and in turn, that c ( x ) > if and only if x > (and obviously c ( r ) > if and only if r > ). Since ∆ is an additive group,we have x < y → c ( x ) < c ( y ) for all x, y ∈ ∆ . By the same argument, it alsofollows that x ≺ y → c ( x ) ≺ c ( y ) for all x, y ∈ ∆ .Now take some x, y ∈ ∆ α +1 ∩ R ∗ M . Write x = re γ , y = se δ , with r, s ∈ R ∗ and γ, δ ∈ J . By Proposition 5.17, c ( re γ ) and c ( se δ ) are well defined and equalto respectively re c ( γ ) , se c ( δ ) . We observe that if γ < δ , then c ( γ ) < c ( δ ) , and ifmoreover < γ , then < c ( γ ) and γ ≺ δ , so c ( γ ) ≺ c ( δ ) . This easily implies that x < y → c ( x ) < c ( y ) and x ≺ y → c ( x ) ≺ c ( y ) . (cid:3) We also need the following properties of admissible trees. Lemma 9.4. Let x ∈ ∆ α +1 and T = h re γ , n, τ i ∈ A( x ) . We have:(1) re c ( γ ) ↑ = ≍ re c ( γ ) = c ( re γ ) = c (R( T )) ;(2) if re γ = R( T ) / ∈ ∆ , then c ( T ) ≍ c (R( T )) · Q i In order to prove that the family ( c ( T ) : T ∈ A( x )) issummable for any x ∈ ∆ α +1 , by Remark 2.15, one could try to verify that thereis no injective sequence ( T i ) i ∈ N of trees in A( x ) such that c ( T i ) (cid:22) c ( T i +1 ) for all i ∈ N . However, we will actually prove the stronger statement that there are nobad sequences , which are defined as follows: Definition 9.5. Let x ∈ ∆ α +1 and let ( T i ) i ∈ N be a sequence of trees in A( x ) . Wesay that the sequence is bad if it is injective, R( T i ) (cid:23) R( T i +1 ) for each i ∈ N , and (cid:18) c ( T i ) c ( T i +1 ) (cid:19) n (cid:22) c (R( T i )) c (R( T i +1 )) for all i, n ∈ N .For instance, Lemma 9.2(1) and (3) immediately imply that there are no badsequences in A( x ) for any x ∈ ∆ . The non-existence of bad sequences in a given A( x ) quickly implies the desired summability. Proposition 9.6. Let x ∈ ∆ α +1 . If there are no bad sequences in A( x ) , then ( c ( T ) : T ∈ A( x )) is summable.Proof. Suppose that ( c ( T ) : T ∈ A( x )) is not summable. Then there is an injectivesequence of trees ( T i ) i ∈ N in A( x ) such that c ( T i ) (cid:22) c ( T i +1 ) for all i ∈ N . After extracting a subsequence, we may assume that R( T i ) (cid:23) R( T i +1 ) for every i ∈ N , as all these roots are terms of x . Therefore, c (R( T i )) (cid:23) c (R( T i +1 )) for all i ∈ N by Proposition 9.3. It follows that for all i, n ∈ N we have (cid:18) c ( T i ) c ( T i +1 ) (cid:19) n (cid:22) (cid:22) c (R( T i )) c (R( T i +1 )) , so the sequence ( T i ) i ∈ N is bad. (cid:3) Remark . If ( T i ) i ∈ N is a bad sequence, then all its subsequences are bad. Thisfollows from the fact that for all i, k, n ∈ N we have (cid:18) c ( T i ) c ( T i + k +1 ) (cid:19) n = k Y j =0 c ( T i + j ) c ( T i + j +1 ) n (cid:22) k Y j =0 c (R( T i + j )) c (R( T i + j +1 )) = c (R( T i )) c (R( T i + k +1 )) . We start with a few special cases in which it is easy to prove that sequences oftrees are not bad. Proposition 9.8. Let x ∈ ∆ α +1 . Let ( T i ) i ∈ N be a sequence of distinct trees in A( x ) . If R( T i ) ∈ ∆ for all i ∈ N , then ( T i ) i ∈ N is not bad.Proof. Write T i = h λ i , , t i i , where t i = c ( T i ) is a term of c ( λ i ) . Since λ i ∈ Term( x ) for each i ∈ N , after extracting a subsequence, we may assume that ( λ i : i ∈ N ) is either constant or decreasing. In the former case, all the contributions c ( T i ) are distinct elements of Term( c ( λ )) for some fixed λ ∈ ∆ , so after extracting asubsequence we may assume c ( T i ) ≻ c ( T i +1 ) for all i ∈ N , so the sequence is notbad. In the latter case, by Lemma 9.2, we may extract a further subsequence andassume that c ( T i ) c ( T i +1 ) = t i t i +1 ≻ c ( λ i ) c ( λ i +1 ) = c (R( T i )) c (R( T i +1 )) . Therefore, ( T i ) i ∈ N is not bad. (cid:3) Proposition 9.9. Let t be a term in ∆ α +1 . Then there are no bad sequences in A( t ) .Proof. Let ( T i ) i ∈ N be a sequence of distinct trees in A( t ) . We want to prove that ( T i ) i ∈ N is not bad. Since t is a term, by Proposition 5.17 ( c ( T ) : T ∈ A( t )) issummable. Thus, extracting a subsequence, we can assume that c ( T i ) ≻ c ( T i +1 ) for every i ∈ N . Observing that R( T i ) = t for every i ∈ N , it follows that c ( T i ) c ( T i +1 ) ≻ c (R( T i )) c (R( T i +1 )) , and therefore ( T i ) i ∈ N is not bad. (cid:3) RANSSERIES AS GERMS OF SURREAL FUNCTIONS 39 Two types of sequences of trees. We now distinguish two special types ofsequences of trees, and verify that every injective sequences of trees in some given A( x ) has at least one subsequence of one of the two types. Definition 9.10. Let x ∈ ∆ α +1 and let T i = h r i e γ i , n i , τ i i ∈ A( x ) be distinct treesfor i ∈ N such that ( γ i ) i ∈ N is weakly decreasing.We say that the sequence ( T i ) i ∈ N has type: (A) if R( τ i ( j )) ≻ γ − γ i for all i ∈ N , j < n i ; (B) if n ≥ and for all i ∈ N > there is k < n i such that R( τ i ( k )) (cid:22) γ i − − γ i .Note that a sequence ( T i ) i ∈ N may be of neither type. A sequence with n i = 0 for all i ∈ N , or with ( γ i ) i ∈ N constant, is vacuously of type (A). Moreover, for asequence of type (B), ( γ i ) i ∈ N is necessarily strictly decreasing and n i ≥ for all i ∈ N . Lemma 9.11. If ( T i ) i ∈ N is a sequence of type (A) or (B), then all its subsequenceshave type (A) or (B) respectively.Proof. Suppose ( T i ) i ∈ N is of type (A) and let ( T i j ) j ∈ N be a subsequence. Since ( γ i ) i ∈ N is weakly decreasing, for all k < n i j we have R( τ i j ( k )) ≻ γ − γ i j (cid:23) γ i − γ i j , so the subsequence is of type (A).Now let ( T i ) i ∈ N be a sequence of type (B). Write T i = h r i e γ i , n i , τ i i . Using againthe fact that ( γ i ) i ∈ N is weakly decreasing, if k is such that R( τ i ( k )) (cid:22) γ i − − γ i ,then R( τ i ( k )) (cid:22) γ j − γ i for all j < i , so any any subsequence of ( T i ) i ∈ N is of type(B). (cid:3) Proposition 9.12. Let x ∈ ∆ α +1 and let T i = h r i e γ i , n i , τ i i ∈ A( x ) be distincttrees for i ∈ N . Then ( T i ) i ∈ N has a subsequence of type (A) or (B).Proof. After extracting a subsequence, we may assume that ( γ i ) i ∈ N is weakly de-creasing. If n i = 0 for every i ∈ N , then ( T i ) i ∈ N is of type (A) and we are done.We can therefore suppose without loss of generality that n ≥ .We proceed by trying to construct a subsequence ( T i j ) j ∈ N of type (B), and checkthat when the construction fail we find a subsequence of type (A). We define T i j by induction on j ∈ N . For j = 0 , we let T i j = T i := T .Assuming that T i j has been defined, we have two cases. If R( τ i ( k )) ≻ γ i j − γ i for all i > i j , k < n i , then the sequence T i j , T i j +1 , T i j +2 , . . . , has type (A), and weare done. Otherwise, we let i j +1 be the minimum i for which there exists k suchthat R( τ i ( k )) (cid:22) γ i j − γ i .Clearly, either the procedure fails after a finite number of steps, and we finda subsequence of type (A), or it defines a subsequence ( T i j ) j ∈ N of type (B), asdesired. (cid:3) No bad sequences of type (A). As a start, it is fairly easy to see that badsequences of type (A) do not exist. Proposition 9.13. Let x ∈ ∆ α +1 . Then A( x ) contains no bad sequences of type(A).Proof. For a contradiction let ( T i ) i ∈ N be a bad sequence in A( x ) of type (A). ByProposition 9.9 the sequence of terms (R( T i ) : i ∈ N ) cannot be constant, so by taking a subsequence we can assume that the terms R( T i ) are distinct, and sincethey are all terms of x , we may also assume (taking another subsequence) that R( T ) ≻ R( T ) ≻ R( T ) ≻ . . . . By Proposition 9.3 it then follows that c (R( T n )) ≻ c (R( T n +1 )) for every n ∈ N .Let i ∈ N and write T i = h r i e γ i , n i , τ i i . By assumption, for any child U = τ i ( j ) of T i we have R( U ) ≻ γ − γ i (this holds vacuously if T i has no children). Weclaim that for any such U we must have R( U ) ∈ Term( γ ) . Indeed by construction R( U ) ∈ Term( γ i ) ; therefore, if R( U ) / ∈ Term( γ ) , then R( U ) would be a term ofthe difference γ − γ i , contradicting the assumption R( U ) ≻ γ − γ i .We have thus proved that all the roots of the children of the trees T i are termsof γ = log ↑ (R( T )) ; hence, we can replace the root of each T i with e γ obtaining anew sequence T ′ i := h e γ , n i , τ i i in A( e γ ) . Since T i and T ′ i have the same children,by Lemma 9.4(2) we have: c ( T i ) c ( T ′ i ) ≍ c (R( T i )) c (R( T ′ i )) ≍ c ( e γ i ) c ( e γ ) . By Proposition 5.17, the family ( c ( T ′ ) : T ′ ∈ A( e γ )) is summable. Therefore, afterextracting a subsequence we may assume that c ( T ′ i ) c ( T ′ i +1 ) (cid:23) (note that the inequalityis not necessarily strict, because the trees T ′ i might not be distinct). It follows that c ( T i ) c ( T i +1 ) ≍ c (R( T i )) c (R( T i +1 )) · c ( T ′ i ) c ( T ′ i +1 ) (cid:23) c (R( T i )) c (R( T i +1 )) ≻ . Therefore, ( T i ) i ∈ N is not bad. (cid:3) Pruning trees. In the sequel we consider trees in A( x ) for some x ∈ ∆ α +1 .We establish a procedure to “prune” a tree T , that is, to remove some descendants,in such a way that its contribution c ( T ) changes only by a small amount. Definition 9.14. Let T = h re γ , n, τ i be an admissible tree (i.e. T ∈ A( re γ ) ), U bea child of T (necessarily admissible), and U ′ be an admissible tree with the sameroot as U . Let j be the minimum integer such that τ ( j ) = U .(1) We define T [ U ′ /U ] as T with U replaced by U ′ . More precisely, T [ U ′ /U ] := h re γ , n, τ ∗ i where τ ∗ ( i ) := τ ( i ) for i = j and τ ∗ ( j ) := U ′ . Note that if c ( U ′ ) ≺ , then T [ U ′ /U ] is again an admissible tree.(2) We define T \ U as the admissible tree obtained from T by removing thechild U . More precisely, T \ U := h re γ , n − , τ ∗ i where τ ∗ ( i ) := τ ( i ) for i < j and τ ∗ ( i ) := τ ( i + 1) for i ≥ j . Definition 9.15. Let T = h re γ , n, τ i ∈ A( x ) with size( T ) > . If L is a leaf of T ,we define the minimal child of T with leaf L to be the child U = τ ( j ) of T suchthat:(1) L is a leaf of U (possibly L = U );(2) among such children, R( U ) is minimal with respect to (cid:22) ;(3) among such children, j is minimal. RANSSERIES AS GERMS OF SURREAL FUNCTIONS 41 Definition 9.16. Let T = h re γ , n, τ i ∈ A( x ) with size( T ) > and let L be a leafof T . We define T L by induction on size( T ) as follows. Let U be the minimal childof T with leaf L . We define:(1) if size( U ) = 1 (namely L = U ), let T L := T \ L ;(2) if size( U ) > and c ( U L ) ≺ , let T L := T [ U L /U ] ;(3) if size( U ) > and c ( U L ) (cid:23) , let T L := T \ U . Remark . Note that in all three cases, T L is still an admissible tree; in partic-ular, in (2) this is guaranteed by the condition c ( U L ) ≺ , as for all children S ofan admissible tree the contribution c ( S ) must be infinitesimal. Lemma 9.18. Let L be a leaf in T ∈ A( x ) , with size( T ) > , and let U be theminimal child of T with leaf L . We have:(1) size( T L ) < size( T ) and R( T L ) = R( T ) ;(2) T L ∈ A( x ) ;(3) if T L = T \ U , then c ( T ) ≍ c ( T L ) · c ( U ) ;(4) if T L := T [ U L /U ] , then c ( T ) = c ( T L ) · c ( U ) c ( U L ) ;(5) c ( T L ) ≻ c ( T ) ;Proof. We work by induction on size( T ) . Point (1) is straightforward and point (2)is Remark 9.17.For (3), let T =: h re γ , n, τ i and let j < n be minimal such that U = τ ( j ) . Bydefinition we have c ( T ) = re c ( γ ) ↑ = · c ( U ) · n ! Y i Let T be an admissible tree and U be a proper descendant of T .Then R( U ) ≻ , and if U ′ is a proper descendant of U we have ≺ R( U ′ ) n ≺ R( U ) for every n ∈ N .Proof. Suppose first that U is a child of T . Write R( T ) = re γ , so that R( U ) is a termof γ = log ↑ (R( T )) . Since γ ∈ J , R( U ) is of the form se δ with < δ ∈ J , so R( U ) ≻ ,proving the first conclusion. Moreover, it follows that δ n ≺ e δ ≍ R( U ) for all n ∈ N .If now U ′ is a child of U , then R( U ′ ) is a term of δ , so R( U ′ ) n (cid:22) δ n ≺ R( U ) , whileby the previous argument R( U ′ ) ≻ . The general conclusion with U a descendantof T and U ′ a descendant of U now follows by transitivity of (cid:22) . (cid:3) Proposition 9.20. Let L be a leaf in a tree T of size > and let U be the minimalchild of T with leaf L . Then c ( T ) ≍ c ( T L ) · c ( L ) · t where (cid:22) t (cid:22) c (R( U )) . Proof. We work by induction on size( T ) .Case 1. If size( U ) = 1 (namely U = L ), then T L = T \ L and c ( T ) ≍ c ( T L ) · c ( L ) ,so it suffices to take t = 1 .Case 2. Assume size( U ) > and c ( U L ) (cid:23) . Then T L = T \ U , and therefore c ( T ) ≍ c ( T L ) · c ( U ) . We may assume by induction that c ( U ) ≍ c ( U L ) · c ( L ) · u ,where (cid:22) u (cid:22) c (R( U ′ )) and U ′ is the minimal child of U with leaf L . Substitutingwe obtain c ( T ) ≍ c ( T L ) · c ( L ) · c ( U L ) · u. By Lemma 9.4 we have c ( U L ) (cid:22) c (R( U L )) = c (R( U )) , and by Lemma 9.19 u (cid:22) c (R( U ′ )) ≺ c (R( U )) , hence we can take t := c ( U L ) · u .Case 3. Finally, assume size( U ) > and c ( U L ) ≺ . Then T L = T [ U L /U ] ,and by Lemma 9.18 we have c ( T ) = c ( T L ) · c ( U ) c ( U L ) . By inductive hypothesis wehave c ( U ) ≍ c ( U L ) · c ( L ) · u , where reasoning as above we have (cid:22) u ≺ c (R( U )) .Substituting we get c ( T ) ≍ c ( T L ) · c ( L ) · u, hence we can take t = u . (cid:3) No bad sequences. We can finally prove that there are no bad sequences atall in any A( x ) . Proposition 9.21. Let x ∈ ∆ α +1 . If ( T i ) i ∈ N is a bad sequence in A( x ) , then thereare a bad sequence ( S j ) j ∈ N in A( x ) and some k ∈ N such that size( S ) < size( T k ) , R( S ) = R( T k ) and c ( S ) ≻ c ( T k ) .Proof. By Proposition 9.12 and Proposition 9.13, there is a subsequence ( P j ) j ∈ N of ( T i ) i ∈ N of type (B). Recall that by definition of type (B), size( P j ) > for all j ∈ N .Write P j = h r j e γ j , n j , τ j i . Let L be a leaf of P . For j ≥ , let U j be a childof P j with R( U j ) (cid:22) γ j − − γ j , which exists by definition of type (B), and let L j be a leaf of U j . We may then assume that U j is the minimal child with leaf L j (ifnot, just replace U j with the minimal child U with leaf L j , and observe that thecondition R( U ) (cid:22) γ j − − γ j is still satisfied because R( U ) (cid:22) R( U j ) ).We can write L j = h λ j , , s j i , where λ j ∈ ∆ and s j = c ( L j ) ∈ Term( c ( λ j )) . ByLemma 9.19 we have λ j (cid:22) R( U j ) ; therefore, since c preserves (cid:22) by Proposition 9.3, c ( λ j ) (cid:22) c (R( U j )) (cid:22) c ( γ j − − γ j ) for all j ≥ .By Lemma 9.2, we may extract a further subsequence of ( P j ) j ∈ N and assumethat for all j ∈ N we have (cid:16) s j +1 s j (cid:17) ≺ c ( λ j +1 ) , so (cid:18) s j +1 s j (cid:19) (cid:22) c ( γ j − γ j +1 ) . Now let S j := P L j j , which is well defined since size( P j ) > for all j ∈ N . We shallprove that ( S j ) j ∈ N has the desired properties.By Proposition 9.20, for all j ∈ N we have c ( P j ) = c ( P Ljj ) · c ( L j ) · t j = c ( P L j j ) · s j · t j where (cid:22) t j (cid:22) c (R( U j )) for all j ∈ N . In particular, t j +1 t j (cid:22) t j +1 (cid:22) c (R( U j +1 )) ,so t j +1 t j (cid:22) c ( γ j − γ j +1 ) . RANSSERIES AS GERMS OF SURREAL FUNCTIONS 43 It follows that c ( P L j j ) c ( P L j +1 j +1 ) = c ( P j ) c ( P j +1 ) · s j +1 s j · t j +1 t j (cid:22) c ( γ j − γ j +1 ) · c ( P j ) c ( P j +1 ) . Since ( P j ) j ∈ N is bad, for all j, n ∈ N we have (cid:18) c ( P j ) c ( P j +1 ) (cid:19) n (cid:22) c (R( P j )) c (R( P j +1 )) . Likewise, for all j, n ∈ N we also have ( c ( γ j − γ j +1 )) n (cid:22) e c ( γ j − γ j +1 ) ≍ c (R( P j )) c ( R ( P j +1 )) using Lemma 9.4, Proposition 9.3 and the fact that γ j − γ j +1 ≻ . It follows thatfor all j, n ∈ N we have c ( P L j j ) c ( P L j +1 j +1 ) ! n (cid:22) c (R( P j )) c (R( P j +1 )) . Recalling that R( P L j j ) = R( P j ) for all j ∈ N , it follows that ( S j ) j ∈ N = ( P L j j ) j ∈ N isanother bad sequence in A( x ) .To conclude, let k ∈ N be such that T k = P . By construction, size( S ) =size( P L ) < size( P ) = size( T k ) , and by Lemma 9.18, c ( S ) = c ( P L ) ≻ c ( P ) = c ( T k ) , as desired. (cid:3) Proposition 9.22. Let x ∈ ∆ α +1 . Then A( x ) contains no bad sequences.Proof. Suppose by contradiction that there is a bad sequence of trees in A( x ) .Among all such bad sequences, let ( T i ) i ∈ N be the one such that size( T ) is minimal,and fixed T , size( T ) is minimal, and so on. By Proposition 9.21, there is anotherbad sequence ( S j ) j ∈ N in A( x ) and some k ∈ N such that size( S ) < size( T k ) , R( S ) = R( T k ) and c ( S ) ≻ c ( T k ) .We observe that T , T , . . . , T k − , S , S , . . . is again a bad sequence in A( x ) . Indeed, it suffices to note that for all n ∈ N wehave (cid:18) c ( T k − ) c ( S ) (cid:19) n ≺ (cid:18) c ( T k − ) c ( T k ) (cid:19) n (cid:22) c (R( T k − )) c (R( T k )) = c (R( T k − )) c (R( S )) . However, since size( S ) < size( T k ) , this contradicts our minimality assumption.Therefore, there are no bad sequences in A( x ) , as desired. (cid:3) By Proposition 9.6, this completes the proof of Lemma 5.21, as desired. 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