aa r X i v : . [ m a t h . L O ] J a n ON THE VALUE GROUP OF THE TRANSSERIES
ALESSANDRO BERARDUCCI AND PIETRO FRENI
Abstract.
We prove that the value group of the field of transseriesis isomorphic to the additive reduct of the field.
Contents
1. Introduction 12. Hahn fields 33. Transseries 64. Stratification 75. Levels 96. Summability 117. The crucial isomorphism 128. The group of monomials is order isomorphic to its positivecone 149. Generalizing a result on omega-maps 18References 201.
Introduction
Given a real closed field ( K, + , · , < ), the possibility of defining anordered exponential on it, that is, a ordered group isomorphism exp :( K, + , < ) → ( K > , · < ), is strictly connected to the properties of itsnatural valuation (i.e. the valutation whose valuation ring is given bythe elements bounded in absolute value by some natural number). Forexample if an exponential exists, then the valuation group v ( K ) isisomorphic to an additive complement of the valuation ring. In [9] itis shown that for ordered fields that are maximal with respect to theirnatural valuation this condition fails unless the value group is trivial (in Date : December 11, 2020. ArXiv version January 3, 2021.2010
Mathematics Subject Classification.
Primary 16W60; Secondary 03C64.The first author was partially supported by the Italian research projectPRIN 2017, “Mathematical logic: models, sets, computability”, Prot.2017NWTM8RPRIN.. which case K ⊆ R ), so maximal non-archimedean ordered fields do notadmit an exponential. In [10] the same property is used, the other wayaround, to show that, given a regular uncountable cardinal κ , there isa non-trivial group M such that the field K = R (( M )) κ ⊆ R (( M )) of κ -bounded generalized series admits an exponential map.In [3] the authors study the related property of admitting an isomor-phism Ω : ( K, + , < ) → ( M , + , < ), where M ⊆ K > is an embeddedmultiplicative copy of the value group. They call such an isomorphism omega-map , in analogy to Conway’s omega map on Surreal numbers[4], and prove that, for fields of the form R (( M )) κ , its existence impliesthe existence of an exponential map. Moreover, for fields of the sameform, the converse holds under the additional hypothesis that the valuegroup M is order isomorphic to its positive cone M > .The above results leave open the question whether the field T ofLE-transseries [5] admits an omega-map, as it is not a field of the form R (( M )) κ . Besides R , transseries are arguably the most important exam-ple of an exponential field. They are an important tool in asymptoticanalysis and have been used by Ecalle [6] to give a positive solution toDulac’s conjecture (the finiteness of limit cycles in planar polynomialvector fields).The main result of the paper is that the value group of T is iso-morphic to T itself as an ordered additive group. This is proved byexplicitly constructing an omega-map Ω : T → M LE where M LE is thegroup of transmonomials.Finally, abstracting some tools which have been useful in the con-struction we generalize results in [3] to a much wider class of fields.We describe below the main ideas of the paper. In the case ofexponential fields of the form K = R (( M )) κ treated in [3], any or-der isomorphism η : M → M > naturally induces an isomorphism H : K → R (( M > )) of ordered R -vector spaces from the field to thecanonical complement of its valuation ring. This in turn induces anomega-map.When considering the case of T , we follow a similar approach butthere are many additional complications. We recall that T is a subfieldof R (( M LE )). Any order isomorphism M LE → M LE,> induces anembedding of ordered R -vector spaces T → R (( M LE,> )), however thereis no guarantee that its image is T ↑ := T ∩ R (( M LE,> )). In this paperwe show that M LE is order isomorphic to M LE,> and we produce aparticular order isomorphism η : M LE → M LE,> which induces an N THE VALUE GROUP OF THE TRANSSERIES 3 isomorphism of ordered R -vector spaces T → T ↑ . This in turn willinduce an omega-map (Theorem 8.7).In the above process we are lead to consider an ideal of subsets of M LE (generated by subgroups) and characterize T as the field of seriesin R (( M LE )) with support in the ideal (Section 6). A subset of M LE will be called bounded if it belongs to the ideal and a map η : M LE → M LE,> will be called bounded if it maps bounded sets to boundedsets. The proof of the main result is then reduced to the problem ofconstructing a bounded order isomorphism η : M LE → M LE,> withbounded inverse. This is achieved in Lemma 8.6.Endowing groups of monomials with suitable ideals of subsets yieldsflexible constructions of many intereresting fields, encompassing T , the κ -bounded series and the field of Puiseux series (Definition 9.1). Someresults of [3] generalize easily to this new setting (see Theorem 9.5).For simplicity of notation throughout the paper we work with fieldswhose residue field is R , but with minor modifications we could havetaken any other model of the first order theory of ( R , exp).2. Hahn fields
Given a multiplicatively written ordered abelian group M denote by R (( M )) the field of Hahn’s generalized series (cfr. [7]) with monomialsfrom M , real coefficients and reverse-well-ordered support R (( M )) = { f ∈ R M : supp( f ) is reverse-well-ordered } where supp( f ) = { m ∈ M : f ( m ) = 0 } denotes the support of the series. The value of f at m is referred to asthe coefficient of the monomial m in the series f or the coefficient of f at m and written as f m := f ( m ).The set R (( M )) is naturally an ordered field extension of R . Sumsand multiplication by scalars in R are defined termwise and order islexicographic, or equivalently f > f max supp( f ) > f g ) m = X no = m f n g o and the fact that the supports of f and g are reverse well orderedensures that the sum on the right hand side has only finitely manynon-zero terms and that the support supp( f g ) ⊆ supp( f ) supp( g ) isstill reverse-well-ordered.Given a set I , an I -indexed family of series ( f i ∈ R (( M )) : i ∈ I )is said to be summable if the union of the supports of its elements ALESSANDRO BERARDUCCI AND PIETRO FRENI S i ∈ I supp( f i ) is reverse well ordered and for every m ∈ M the set { i : f i, m = 0 } is finite: in such a case the formal sum is defined as X i ∈ I f i := g where g m := X m ∈ supp( f i ) f i, m . A monomial m ∈ M is usually identified with the series having coef-ficient 1 at m and 0 at other monomials. A term is then a series ofthe form t = k m with k ∈ R . With such a convention, for every re-verse well ordered set S of monomials, a family of terms of the form( k m m : m ∈ S ) is summable: elements of R (( M )) are thus usually writ-ten as P m ∈ S f m m for some reverse well ordered S ⊆ M or as P i<α k i m i where α is an ordinal number and ( m i : i < α ) is a strictly decreasing α -sequence in M .For every element f of R (( M )) \ { } it makes sense to talk about the leading monomial, coefficient and term of f , which we denote respec-tively aslm( f ) := max supp( f ) , lc( f ) := f lm( f ) , lt( f ) := lm( f )lc( f ) . Since R (( M )) is an ordered field it makes sense to define dominance andarchimedean equivalence on non-zero elements. These notions have aneasy characterization in terms of the leading term of a series. Definition 2.1.
For x, y ∈ R (( M )) \{ } , we have the following relations(1) dominance: x (cid:22) y ⇐⇒ ∃ n ∈ N , | x | < n | y | ⇐⇒ lm( x ) ≤ lm( y );(2) comparability: x ≍ y ⇐⇒ (cid:0) x (cid:22) y & y (cid:22) x (cid:1) ⇐⇒ lm( x ) =lm( y );(3) strict dominance: x ≺ y ⇐⇒ (cid:0) x (cid:22) y & x y (cid:1) ⇐⇒ lm( x ) < lm( y );(4) asymptotic equivalence: x ∼ y ⇐⇒ x − y ≺ x ⇐⇒ lt( x ) =lt( y ).Series that are ≻ infinite whereas elements ≺ infinitesimal . Remark . The function lm : R (( M )) \ { } → M , where we endow M with the opposite order, is a field valuation whose residue field is R and whose valuation ring is the set of series f such that supp( f ) ≤ archimedean valuation . Remark . We recall that an extension of valued fields is immediateif it preserves the value group and residue field. It is worth mentioningthat R (( M )) is maximal in the sense that it has no proper immediate N THE VALUE GROUP OF THE TRANSSERIES 5 extension (see [8, p.193, Satz 26]). Thus every proper ordered fieldextension F ⊇ R (( M )) has non-zero elements that are not comparableto any element of R (( M )). Remark . Every element of f = P i<α k i m i of R (( M )) decomposesuniquely as f = f ↑ + f ◦ + f ↓ where supp( f ↑ ) > f ◦ ∈ R and f ↓ ≺
1, whereas every non-zeroelement f decomposes multiplicatively as f = lm( f )lc( f ) (cid:18) f − lt( f )lt( f ) (cid:19) where lm( f ) ∈ M , lc( f ) ∈ R \ { } and f − lt( f )lt( f ) ≺ Remark . It may worth remarking that in the definitions above onecould have considered any ordered field k instead of R : in this caseinstead of (cid:22) one must consider the k -dominance relation defined as x (cid:22) k y ⇔ ∃ k ∈ k , | x | < k | y | .If ( M , · , , ≺ ) is an orderd group we will denote by M ≻ the set ofelements of M greater than 1. Fact 2.6. If N is a multiplicatively written ordered abelian group, k is a subfield of R (( N )) and M ⊆ N is a subgroup such that M ≻ > k ,then for every ordinal α , every strictly decreasing sequence ( m i ) i<α ofelements of M , and every sequence ( k i ) i<α of elements of k , the family k i m i is summable.Proof. Just note that supp( k i m i ) < supp( k j m j ) for i > j . (cid:3) Definition 2.7.
In the hypothesis of Fact 2.6 we denote the set ofelements of the form P i<α k i m i with k i ∈ k and m i ∈ M as k (( M )):this is a subfield of R (( N )) isomorphic to the field of generalized serieswith coefficients from k and monomials from M .More generally if Γ is a subset of N , k is a subfield of R (( N )), and k < h Γ i ≻ , where h Γ i is the subgroup generated by Γ, then k ((Γ)) willdenote the k -vector subspace of R (( N )) consisting of the series of theform P i<α k i m i with all m i laying in Γ and k i laying in k .For example if k = R then every Γ ⊆ N satisfies the hypothesis, sogiven any Γ ⊆ N , R ((Γ)) is the set of series with support included in Γ. Remark § . Let k ⊆ R (( N )) be a subfieldand let M , M ⊆ N be subgroups such that M ≻ > M ≻ > k . Then k (( M ))(( M )) = k (( M M )) . ALESSANDRO BERARDUCCI AND PIETRO FRENI
In particular every series f ∈ k (( M M )) has a unique representationas f = X i<α k i m i with k i ∈ k (( M )) and m i ∈ M for every i < α. Transseries
The field T of transseries is a subfield of a field of the form R (( M LE ))where M LE is a suitable ordered multiplicative group called the groupof transsmonomials. We shall not define T , but in this and the followingsection we list all the properties needed in this paper. In particular weshall need the fact that T is the union of the subfields T n,λ in Definition4.2. This representation of T will be used to introduce the ideal ofsubsets of M LE mentioned in the introduction. Definition 3.1.
Denote by T the field of LE-transseries in a formalvariable x as described in [5] and let M LE be the group of LE-transserialmonomials. Note that T ⊆ R (( M LE )) . Let T ↑ = R (( M LE, ≻ )) ∩ T be the R -vector space of the transseries whose support only containsinfinite monomials and observe that T = T ↑ ⊕ R ⊕ o (1) where o (1) isthe set of infinitesimal transseries. The elements of T ↑ are called purelyinfinite . Fact 3.2.
Recall that T admits an exponential function exp : T → T > making it into an elementary extension of the ordered field of real num-bers with the natural exponential function. The function exp restrictsto an ordered group isomorphisms exp | : (cid:0) T ↑ , + , , < (cid:1) ≃ (cid:0) M LE , · , , < (cid:1) and this suffices to determine exp on the whole T via the formula (3.1) exp( f ) = exp( f ↑ ) exp( f ◦ ) X n ∈ N ( f ↓ ) n n ! The compositional inverse of exp is called logarithm, log : T > → T , and has an analogous piecewise characterization in terms of themultiplicative decomposition: for f > one has (3.2)log( f ) = log(lm( f )) + log(lc( f )) + X n> ( − n +1 εn ε = f − lt( f )lt( f ) For g > we define f g := exp( g log( f )) . N THE VALUE GROUP OF THE TRANSSERIES 7
Definition 3.3 (Normal form) . Since M LE = exp( T ↑ ), every element f ∈ T has a unique representation as f = X i<α r i e γ i where α is an ordinal, r i ∈ R \{ } for every i < α , ( γ i ) i<α is a strictly de-creasing sequence of elements of T ↑ and e γ i = exp( γ i ); we call P i<α r i e γ i the normal form of f . 4. Stratification
Below we work in the field T of LE-transseries in the formal variable x . Definition 4.1.
For n ∈ N , let log n be the n -fold composition oflog and let exp n be the n -fold composition of exp. We extend thisnotation to the case n ∈ Z with the convention that exp n = log − n . Forexample log ( x ) = log( x ), log ( x ) = x and log − ( x ) = exp( x ). Now letexp Z ( x ) = { exp n ( x ) | n ∈ Z } . For λ ∈ exp Z ( x ) and n ∈ Z , we define λ n = exp n ( λ ) so that λ − n = log n ( λ ) . Definition 4.2.
For λ ∈ exp Z ( x ) and n ∈ N let us consider the follow-ing inductively defined subsets of T :(1) M ,λ := λ R , T ,λ = R (( M ,λ )), J ,λ = R (( M ≻ ,λ )).(2) M n +1 ,λ := e J n,λ , T n +1 ,λ = T n,λ (( M n +1 ,λ )), J n +1 ,λ := T n,λ (( M ≻ n +1 ,λ )).For this to be well defined one needs to observe that for each n ∈ N we have T n,λ < M ≻ n +1 ,λ (see Definition 2.7 and Remark 2.8) and that T n,λ (( M n +1 ,λ )) ⊆ T . For the verification of these facts the reader mustrefer to the original definition of the LE-transseries in [5] or to theequivalent definition in [2, Prop. 4.12] (see in particular [2, Lemma4.14]). From [5] or [2, Prop. 4.18] it also follows that T = [ n,λ T n,λ where n ∈ N and λ ∈ exp Z ( x ). In fact in the union it suffices to take λ of the form x − k with k ∈ N (rather than k ∈ Z ). This depends on thefact that T n, exp( λ ) ⊆ T n +1 ,λ . Notice that in [5] T and exp are definedby a simultanous induction, while in [2] the transseries are defined asa subfield of Conway’s surreal numbers No with the exponentiationcoming from No . For a short account of the latter approach and allthe relevant definitions see also [1]. ALESSANDRO BERARDUCCI AND PIETRO FRENI
Definition 4.3.
Although T is not a maximal valued field, its subfields T n,λ are maximal, indeed T n,λ = R (( N n,λ ))where N n,λ = T n,λ ∩ M LE . The subgroups N n,λ ⊆ M LE can be induc-tively generated as follows:(1) N ,λ = M ,λ .(2) N n +1 ,λ = N n,λ M n +1 ,λ . Remark . For n ∈ N we have M ≻ n +1 ,λ > N n,λ and a direct lexico-graphic product N n +1 ,λ = M ,λ M ,λ · . . . · M n +1 ,λ , so R (( N n,λ ))(( M n +1 ,λ )) = R (( N n +1 ,λ )). Definition 4.5.
We define T ↑ n,λ = R (( N ≻ n,λ )) and observe that T ↑ = S n,λ T ↑ n,λ . Remark . Since M ≻ n +1 ,λ > N n,λ we have N ≻ n +1 ,λ = (cid:0) N n,λ M n +1 ,λ ) ≻ = N ≻ n,λ ∪ ( N n,λ M ≻ n +1 ,λ ) and N ≻ n,λ < N n,λ M ≻ n +1 ,λ ,so applying R (( − )) we get T ↑ n +1 ,λ = T ↑ n,λ + J n +1 ,λ and by induction weeasily obtain(4.1) T ↑ n,λ = J ,λ + . . . + J n,λ . It follows that N n +1 ,λ = M ,λ M ,λ · . . . · M n +1 ,λ = λ R exp( T ↑ n,λ ) , thus T n +1 ,λ = R (( λ R exp( T ↑ n,λ ))). In other words every f ∈ T n +1 ,λ canbe written as f = X i<α r i λ s i e α i where α is an ordinal, r i ∈ R \ { } , s i ∈ R , α i ∈ T ↑ n,λ and ( λ s i e α i ) i<α isstrictly decreasing. We can also write it in the form f = X i<α r i e β i where β i = s i log( λ ) + α i (this is the normal form of Definition 3.3).Recalling that T n +1 ,λ = T n,λ (( M n +1 ,λ )) = T n,λ (( e J n,λ )) we also have arepresentation of the form f = X i<α k i e γ i where k i ∈ T n,λ and γ i ∈ J n,λ . N THE VALUE GROUP OF THE TRANSSERIES 9
Definition 4.7.
It is convenient to extend Definition 4.2 to the casewhen n ∈ Z , so we put(1) M − ,λ = 1.(2) T − ,λ = R .(3) J − ,λ = R log( λ ).and for n < − M n,λ = 1 , T n,λ = R and J n,λ = { } . Proposition 4.8.
For all n ∈ Z and λ ∈ exp Z ( x ) , (1) M n +1 ,λ − ⊇ M n,λ . (2) T n +1 ,λ − ⊇ T n,λ . (3) J n +1 ,λ − ⊇ J n,λ .Proof. The case n = − • M ,λ − = λ R − ⊇ M − ,λ ; • T ,λ − = R (( λ R − )) ⊇ R = T − ,λ ; • J ,λ − = R (( λ R > − )) ⊇ J − ,λ .We can then conclude by an easy induction argument. (cid:3) Levels
Our next goal is to represent the ordered vector space T ↑ as a lexico-graphic direct sum L n ∈ Z J n of suitable subspaces which can be char-acterized in terms of “levels”. Since exp( T ↑ ) = M LE , this will alsoinduce a decomposition of M LE as a direct sum of multiplicative sub-roups M n = exp( J n − ).Recall that x is the formal variable of T and x − k = log k ( x ). Mapping x to x − k will induce an automorphism of T sending J n to J n − k and M n in M n − k . Definition 5.1.
For n ∈ Z we define:(1) T n = S k ∈ Z T n + k, x − k .(2) M n = S k ∈ Z M n + k, x − k .(3) N n = S k ∈ Z N n + k, x − k .(4) J n = S k ∈ Z J n + k, x − k .By Proposition 4.8 all the unions are increasing. Remark . We have M LE = S n ∈ N N n and exp( J n ) = M n +1 by Defi-nitions 5.1 and 4.2The following definion is needed to prove that the vector spaces J n are in direct sum. Definition 5.3 (Levels) . Let f, g ≻
1. We say that f and g have thesame level if there is n ∈ N such that log n ( | f | ) ≍ log n ( | g | ). We saythat f has level n ∈ Z if it has the same level of exp n ( x ) and we writein this case lv( f ) = n . For f ≺
1, we define lv( f ) = lv(1 /f ). Byconvention we also stipulate that the level of an element f ≍ −∞ and the level of 0 is undefined. We have:(1) If 1 P i<α r i e γ i ∈ T is in normal form, then lv( P i<α r i e γ i ) =lv( γ ) + 1.(2) lv( x s ) = 1 for all s ∈ R ∗ . Remark . The function lv : T ∗ → Z ∪ {−∞} satisfies:(1) lv( f g ) = max { lv( f ) , lv( g ) } (2) if 1 (cid:22) f (cid:22) g then lv( f ) ≤ lv( g )(3) if 1 ≺ f, g and lv( f ) < lv( g ), then | f | < | g | .It follows that { m ∈ M LE | lv( m ) ≤ n } is a convex subgroup of M LE for all n ∈ Z . Proposition 5.5.
For n ∈ Z we have: (1) J n is the field of transseries whose support only consists of infi-nite monomials of level exactly n . (2) N n = { m ∈ M LE | lv( m ) ≤ n } ; (3) T n is the field of transseries whose support only contains mono-mials of level less or equal than n .Proof. One easily sees by induction that if n + k ≥ = m ∈ M n + k, x − k then lv( m ) = n . It follows in particular that if n ∈ N n + k, x − k then lv( n ) ≤ n and that equality holds if and only if n / ∈ N n + k − , x − k .Hence the monomials of level n are exactly those contained in M n N n − = N n \ N n − . All of (1)-(2)-(3) easily follow from this. (cid:3) Proposition 5.6.
We have J > n +1 > J n and T ↑ is the direct sum T ↑ = M n ∈ Z J n as an ordered R -vector space .Proof. The inequality J > n +1 > J n follows from Remark 5.4(3). The sumis direct by Proposition 5.5(1) and it is equal to T ↑ by Remark 4.6,Equation 4.1. (cid:3) Corollary 5.7.
The group M LE is the multiplicative direct sum of thesubgroups M n . N THE VALUE GROUP OF THE TRANSSERIES 11 Summability
In this section we introduce the ideal of subsets of M LE mentionedin the introduction. Definition 6.1.
Given a family ( m i ) i<α of monomials in M LE , wesay that ( m i ) i<α is T -summable if ( m i ) i<α is summable and the Hahnseries P i<α m i belongs to T ⊆ R (( M LE )). Note that if ( m i ) i<α is T -summable, then for every sequence of non-zero real numbers ( r i ) i<α wehave P i<α r i m i ∈ T .We give below a reformulation of T -summability which is more con-venient for our treatment. Definition 6.2.
Given a set X , we say that X is a bornology on X ifit is an ideal in the posets of subsets of X whose union is X , that is:(1) whenever Z ⊆ Y ∈ X one has Z ∈ X (2) whenever Z, Y ∈ X one has Z ∪ Y ∈ X (3) S X = X .Given X X = ( X, X ) we say that a subset S ⊆ X is X -bounded if itis a subset of some element of X .A map between sets endowed with a bornology f : X X → Y Y is saidto be bounded if the image of any bounded subset is a bounded subset.A bijection is said to be bi-bounded if it is bounded with boundedinverse.Let Y ⊆ X and X a bornology on X , then Y naturally carries abornology X | Y = { Z ∩ Y : Z ∈ X } consisting of those subsets that are X -bounded when regarded as sub-sets of X . It is the largest bornology making the inclusion a boundedmap.Given a set X and a family of subsets F whose union is X , thesmallest bornology containing F is said to be the bornology generated by F . If F is upward directed, the bornology generated by F is thefamily of subsets of X that are contained in some F ∈ F . Definition 6.3.
We introduce the following bornologies on M LE , T , T [ t ± ].(1) M is the bornology on M LE generated by the subgroups N n, x − k for n, k ∈ N (2) T is the bornology on T generated by the subfields T n, x − k for n, k ∈ N .(3) We also consider on T [ t ± ] the bornology generated by thesubgroups E [ t [ m,n ] ] = t n E + · · · + t m E where E is a T -boundedsubfield of T and m ≤ n are in Z . Remark . With the above definition it follows that a set of mono-mials S ⊆ M LE is T -summable if and only if it is reverse well orderedand M -bounded. Remark . Note that M = T | M LE and that that exp and log arebi-bounded maps with respect to the bornologies we just introduced asexp( T n, x − k ) ⊆ T n +1 , x − k and log( T > n, x − k ) ⊆ T n, x − k − .7. The crucial isomorphism
We recall that T n is the field of transseries whose support only con-tains monomials of level less or equal than n and J n is the field oftransseries whose support only consists of infinite monomials of levelexactly n . We shall prove that there is an isomorphism T n ∼ = J n ofordered vector spaces. Proposition 7.1.
For each n ∈ Z , there is an isomorphism of or-dered R -vector spaces f n : T n ≃ J n . Moreover the isomorphism maps T n + k, x − k ⊆ T n onto J n + k, x − k ⊆ J n for n + k ≥ − so it is bi-boundedwith respect to the bornologies T | T n and T | J n .Proof. It suffices to show that for each k, n ∈ Z such that n + k ≥ − f n + k, x − k : T n + k, x − k ≃ J n + k, x − k and that theseisomorphisms can be glued together so to define the f n s as f n := [ k ≥− n − f n + k, x − k : T n −→ J n . Easing the notation, we shall consider isomorphisms f n,λ : T n,λ ∼ −→ J n,λ for any λ ∈ log Z ( x ) and n ≥ −
1, and we define them by induction on n starting at n = − T n +1 ,λ = T n,λ (( e J n,λ )) holds for every n ≥ − h : T → T be defined as(7.1) h ( x ) = ( x + 1 if x ≥ − x if x ≤ . We shall use only the fact that h is an order isomorphism T ≃ T > mapping 0 to 1 and restricting to h | : T n,λ ∼ −→ T > n,λ for every n ∈ Z and λ ∈ exp Z ( x ).We build inductively f n,λ as follows: for n = − f − ,λ : T − ,λ = R ∼ −→ J − ,λ = log( λ ) R f − ,λ ( r ) = log( λ ) r N THE VALUE GROUP OF THE TRANSSERIES 13
Then to define f n +1 ,λ from f n,λ we use T n +1 ,λ = T n,λ ((exp( J n,λ ))) and J n +1 ,λ = T n,λ ((exp( J > n,λ ))) . Assuming inductively that we have an isomorphism f n,λ : T n,λ ∼ −→ J n,λ composing with the order isomorphism h | T n,λ : T n,λ ≃ T > n,λ we obtainan induced order isomorphism f n,λ ◦ h ◦ f − n,λ : J n,λ ∼ −→ J > n,λ , which in turn induces an isomorphism f n +1 ,λ : T n +1 ,λ ∼ −→ J n +1 ,λ as follows: every element x of T n +1 ,λ may be written uniquely as x = X i<α k i exp( γ i ) k i ∈ T n,λ \ { } γ i ∈ J n,λ then one sends x to f n +1 ,λ ( x ) = X i<α k i exp (cid:0) f n,λ ◦ h ◦ f − n,λ ( γ i ) (cid:1) This way we end up with a family of isomorphisms f n, x − k : T n, x − k ∼ −→ J n, x − k .The glueing needed then follows from the following claim: for every λ = x − k and every n ≥ − , f n +1 , log( λ ) extends f n,λ .To prove this claim we proceed by induction on n . For n = − r ∈ R we have f , log( λ ) ( r ) = r exp( f − , log( λ ) ◦ h ◦ f − − , log( λ ) (0)) == r exp( f − , log( λ ) ◦ h (0)) == r exp( f − , log( λ ) (1)) = r log( λ ) = f − ,λ ( r ) . As for the inductive case, assume f n +1 , log( λ ) extends f n,λ , and let usprove that f n +2 , log( λ ) extends f n +1 ,λ : let x ∈ T n +1 ,λ ⊆ T n +2 , log( λ ) , thecrucial observation is that if we write x as x = X i<α k i exp( γ i ) k i ∈ T n +1 , log( λ ) \ { } γ i ∈ J n +1 , log( λ ) since x ∈ T n +1 ,λ one has that actually k i ∈ T n,λ and γ i ∈ J n,λ . Hencerecalling the inductive hypothesis we have f n +2 , log( λ ) ( x ) = X i<α k i exp (cid:0) f n +1 , log( λ ) ◦ h ◦ f − n +1 , log( λ ) ( γ i ) (cid:1) == X i<α k i exp (cid:0) f n,λ ◦ h ◦ f − n,λ ( γ i ) (cid:1) = f n +1 ,λ ( x ) . This completes the proof. (cid:3)
Remark . Note that by construction f n is strognly T n − -linear. Morepreciesly, given a transseries of the form P k i e γ i where k i ∈ T n − and γ i ∈ J n − , we have that f n (cid:16)X k i e γ i (cid:17) = X k i e f n − ◦ h ◦ f − n − ( γ i ) . This can be used to compute f n using as a base case f n ↾ R which isgiven by f n ( r ) = r x n . In particular f ( r ) = r x . Let us also note that f n ↾ T , x n is given by f n ( P i<α r i x s i n ) = P i<α r i x h ( s i ) n . Example 7.3.
Consider the transseries exp( x e − x ). Its normal form isexp( x e − x ) = X n ∈ N x n e − n x n ! ∈ T , x ⊆ T , x We compute f and f on exp( x e − x ). • f (exp( x e − x )) = X n ≥ x n e f ◦ h ◦ f − ( − n x ) n ! = X n ≥ x n e n +1 x n ! ∈ J , x , be-cause f ◦ h ◦ f − ( − n x ) = f ◦ h ( − n ) = n +1 x . • f (exp( x e − x )) = exp( x e − x ) e f ◦ h ◦ f − (0) = exp( x e − x ) e e x ∈ J , x The group of monomials is order isomorphic to itspositive cone
In this section we show that there is a bi-bounded order isomorphism M LE ∼ = M LE, ≺ . Since exp( T ↑ ) = M LE and exp is bi-bounded (Remark6.5) this reduces to show that there is a bi-bounded order isomorphism T ↑ ∼ = T ↑ ,> . In turn this depends on the fact that T ↑ is isomorphicto the ordered vector space T [ t ± ] of Laurent polynomials over thefield T . In fact we will show that, for any ordered field K , the vectorspace K [ t ± ] is order isomophic to its positive cone. Since all therelevant isomorphisms are bi-bounded with respect to the appropriatebornologies, combining the isomorphisms we obtain our main result. N THE VALUE GROUP OF THE TRANSSERIES 15
Remark . Notice that for all n ∈ Z there is an automorphism S n of T preserving exp and infinite sums and sending x to x n = exp n ( x ) (see[5]). The restriction of S n is an isomorphism S n | : ( T n , + , · , , , < ) ≃ ( T , + , · , , , < ). Proposition 8.2.
There is a bi-bounded isomorphisms of ordered R -vector spaces F : T [ t ± ] → T ↑ . where T [ t ± ] is the ordered ring of Laurent polynomials with coeffi-cients from T ordered with the condition t > T and is endowed withthe bornology defined in Proposition 6.3.Proof. It suffices to compose the isomorphisms T [ t ± ] ≃ M n ∈ Z T ≃ M n ∈ Z T n ≃ M n ∈ Z J n ≃ T ↑ More precisely, let f n : T n → J n be as in Proposition 7.1, then onedefines F : T [ t ± ] → T ↑ , F ( X k i t i ) = X f n ◦ S n ( k n ) . This is an ordered isomorphism by virtue of Proposition 5.6.Notice that F ( t i T n, x − n − i ) = J n, x − n + i , hence by linearity F ( T n, x − n [ t [ − m,m ]) = X i ∈ [ − m,m ] J n, x − n + i . To prove that F is bi-bounded it suffices to show that, as m ≤ n range in Z , the sets T n, x − n [ t [ − m,m ] and P i ∈ [ − m,m ] J n, x − n + i generate thebornologies of T [ t ± ] and T ↑ respectively. This is clear for T [ t ± ].Now recall that the bornology of T ↑ is generated by the sets T ↑ n, x − k and, by Remark 4.6, Equation 4.1, for n ∈ N and k ∈ Z , setting m = max {| n − k | , | k |} , we get T ↑ n, x − k ⊆ J n, x − n − m + · · · + J n, x − n + m ⊆ T ↑ n +2 m, x − m − n . (cid:3) Proposition 8.3.
Given an ordered field K and a bornology on K on K generated by subfields, consider the bornology K [ t ] generated bysubgroups of the form E [ t [ m,n ] ] = Et m + · · · + Et n as E ranges in the K -bounded subfields of K and m ≤ n range in Z . There are: (1) a bi-bounded order isomorphism h : K ≃ K > (2) a bi-bounded order isomorphism H : K [ t ± ] ≃ K [ t ± ] > extend-ing h where K [ t ± ] is the additive group of all Laurent polyno-mials, with the ring order induced by t > K . Proof. (1) An order isomorphism can be defined piecewise, e.g. setting h ( x ) = ( x + 1 if x ≥ − x if x ≤ . One easily sees that since h is defined only in terms of the order andof field operations and constants, for every subfield L ⊆ K it restrictsto h | : L ≃ L > , hence it is bi-bounded.(2) This is a bit more involved. We will define order isomorphisms A and B as in the diagram below K [ t ± ] Z > × K [ t − ] K [ t ± ] > A H B and define H as the composition H := B − ◦ A . Here Z > × K [ t − ]denotes the product of Z and K [ t − ] endowed with the lexicographictotal order, that is ( x, y ) < ( x ′ , y ′ ) if and only if x < x ′ or x = x ′ and y < y ′ .In order to define the isomorphisms A let us observe that the set t n K [ t − ] of Laurent polynomials of degree ≤ n can be partitioned intothree order-convex subsets t n K [ t − ] = L n ∪ t n − K [ t − ] ∪ U n with L n < t n − K [ t − ] < U n where L n is the set of negative Laurent polynomials of degree n and U n is the set of positive Laurent polynomials of degree n , that is: L n := t n − K [ t − ] + K < t n U n := t n − K [ t − ] + K > t n . It follows that we can write K [ t ± ] as the disjoint union K [ t ± ] = [ n> L n ∪ K [ t − ] ∪ [ n> U n . By point (1) we have order isomorphisms K ≃ K > ≃ K < . It followsthat we can write down induced order isomorphisms u n : U n → { n } × K [ t − ] , y + xt n (cid:0) n, t − n y + h − ( x ) (cid:1) l n : L n → {− n } × K [ t − ] , y − xt n (cid:0) − n, t − n y − h − ( x ) (cid:1) where y ranges in t n − K [ t − ] and x in K > . N THE VALUE GROUP OF THE TRANSSERIES 17
Thus we can define an order isomorphism A : K [ t ± ] → Z > × K [ t − ]as the union A = [ n> l n ∪ A ∪ [ n> u n : K [ t ± ] → Z > × K [ t − ]where A : K [ t − ] ≃ { } × K [ t − ] is the obvious isomorphism y (0 , y ).Similarly K [ t ± ] > decomposes as a disjoint union K [ t ± ] > = [ n ∈ Z U n with U n < U n +1 , and again we can define the order isomorphism B : K [ t ± ] > → Z > × K [ t − ] as the union for n ∈ Z of u n : U n ≃ { n } × K [ t − ]: B = [ n ∈ Z u n : K [ t ± ] > → Z > × K [ t − ] . In order to prove that H : B − ◦ A : K [ t ± ] → K [ t ± ] > is bi-boundedit suffices to prove that for any subfield E of K we have E [ t [ − n,n ] ] > ⊆ H ( E [ t [ − n,n ] ]) ⊆ E [ t [ − n, n ] ] > . Computing A and B on E [ t [ − n,n ] ], where E is a subfield of K , wehave A ( E [ t [ − n,n ] ]) = [ | k |≤ n { k } × E [ t [ −| k |− n, ] ,B ( E [ t [ − n,n ] ]) = [ | k |≤ n { k } × E [ t [ − k − n, ] . From this we see that clearly B ( E [ t [ − n,n ] ]) ⊆ A ( E [ t [ − n,n ] ]) ⊆ B ( E [ t [ − n, n ] ]).The claim follows applying B − . (cid:3) Definition 8.4.
By Proposition 8.3 point (2), setting K = T , K = T | T , we get an order isomorphism H : T [ t ± ] → T [ t ± ] > which is bi-bounded with respect to the bornology of Definition 6.3. Definition 8.5.
Let F : T [ t ± ] → T ↑ be the ordered R -vector spaceisomorphism described in Proposition 8.2. We define H and η as fol-lows: H = F ◦ H ◦ F − : T ↑ → T ↑ ,> η = exp ◦H ◦ log : M LE → M LE, ≻ . Lemma 8.6.
The map η : ( M LE , < ) ≃ ( M LE, ≻ , < ) is a bi-bounded order isomorphism.Proof. We know that all the maps exp , H , F are bi-bounded order iso-morphisms hence η , being a composition of them and their inverses hasto be a bi-bounded isomorphism (see Remark 6.5, Proposition 8.2 andProposition 8.3). (cid:3) Theorem 8.7.
The ordered group of transserial monomials M LE isisomorphic to the ordered additive reduct of T .Proof. Consider the isomorphism of ordered R -vectors spaces H : T → T ↑ defined by H ( P r i m i ) = P r i η ( m i ). Note that H is well defined be-cause η is bounded and it is an isomorphism because its inverse isbounded. Now defineΩ : ( T , , + , < ) → ( M LE , , · , < )as the composition Ω = exp ◦ H . Then Ω is the desired isomorphism. (cid:3) Generalizing a result on omega-maps
In [3] it is shown that a field of the form R (( M )) <κ admits an omega-map if and only if admits an exponential and M is order isomorphic to M > . We generalize this result to the case when instead of R (( M )) <κ we have a subfield of R (( M )) induced by a bornology on M . Definition 9.1.
Given an ordered group ( M , · , , < ), if Γ is a subsetof M and G is a bornology on Γ we define R ((Γ G )) := [ S ∈G R (( S )) , that is, the subspace of R ((Γ)) consisting of well founded sums with G -bounded support. Remark . Let ( M , · , , < ) be an ordered abelian group and let M be a bornology generated by subgroups. Then: • R (( M M )) ⊆ R (( M )) is a subfield (as it is a directed union offields). • if ε ∈ R (( M M )) is infinitesimal and ( k n ) n ∈ N is a N -sequence in R , then X n ∈ N k n ε n ∈ R (( M M )) . N THE VALUE GROUP OF THE TRANSSERIES 19
It follows in particular that in order for fields of the form R (( M M )) tohave an exponential it suffices that they have one restricted to purelyinfinite elements, the extension being constructed as in Equation 3.1. Example 9.3.
Let ( M , · , , < ) be a multiplicatively written orderedabelian group.(1) If κ is an uncountable regular cardinal, the family M κ of subsetsof M having cardinality strictly less than κ is a bornology whichcan be generated by subgroups. The field R (( M M κ )) is the fieldof κ -bounded Hahn-series and is also denoted by R (( M )) κ .(2) The family g of subsets contained in some finitely generatedsubgroups of M is the smallest bornology on M generated bysubgroup. The field R (( M g )) is called field of grid based series(cfr [11]).(3) Taking M = x Q we obtain the field of Puiseux series R (( x Q g )).(4) The field T of LE-transseries coincides with R (( M LE M )) where M is the bornology in Definition 6.3. Remark . If f : Γ G → ∆ D is a bounded increasing map between totalorders, then the natural induced map F : R ((Γ)) → R ((∆)) defined as F X i<α k i γ i = X i<α k i f ( γ i )maps R ((Γ G )) into R ((∆ D )).The following results generalizes Theorem 4.1 of [3]. Proposition 9.5.
Let N be a multiplicatively written ordered abeliangroup and let N be a bornology generated by subgroups of N . For thefield K = R (( N N )) , denote by K the bornology on K generated by thesubfields of the form R (( N ′ )) as N ′ ranges in the N -bounded subgroupsof N , then the following are equivalent: (1) there is an ordered bi-bounded isomorphism ( K , + , , <, K ) ≃ ( N , · , , <, N ) between the field and its group of values v ( K ) = N ; (2) K admits a bi-bounded isomorphism exp | : ( K ↑ , + , , <, K| ) ≃ ( N , · , , <, N ) and an ordered bi-bounded map ( N , N , < ) ≃ ( N ≻ , N | , < ) where N | and K| denote the restrictions of the bornologies N and K to N ≻ and K ↑ respectively.Proof. Assume (1) holds, that is we have an isomorphism Ω : ( K , + , , <, K ) ≃ ( N , · , , <, N ) and let h : K → K > be defined by the formula 7.1 (Proof of Proposition 7.1): it is easy to check that thenΩ ◦ h ◦ Ω − : ( N , · , , <, N ) ≃ ( N ≻ , · , , <, N | )is a bi-bounded chain isomorphism, hence it allows us to define anisomorphism G : ( K ↑ , + , , < ) → ( K , + , , < ), as the only strongly R -linear map that restricts to Ω ◦ h − ◦ Ω − : N ≻ → N (see Re-mark 9.4). G is then bi-bounded w.r.t. to K and K| . The compositeΩ ◦ G is the sought exponential restricted to purely infinite elmentsΩ ◦ G = exp | : ( K ↑ , + , , < ) ≃ ( N , · , , < ): it is bi-bounded because itis a composition of bi-bounded isomorphisms.On the other hand assuming (2) if we have an isomorphism η : ( N , N , < ) ≃ ( N ≻ , N | , < ) we can immediately define an isomorphism H :( K , + , , <, K ) → ( K ↑ , + , , <, K| ) as the only strongly R -linear maprestricting to η and set Ω = exp ◦ H : it is again bi-bounded because itis a composition of bi-bounded maps. (cid:3) Remark . By Remark 9.2, the condition (2) of Proposition 9.5 isequivalent to K admitting a surjective exponential which restricts to abi-bounded isomorphism exp | : ( K ↑ , + , , <, K| ) ≃ ( N , · , , <, N ) anda bi-bounded order isomorphism ( N , N , < ) ≃ ( N ≻ , N | , < ). Remark . If we consider the case of a group N endowed with the idealof subgroups N κ consisting of the subgroups with cardinality strictlyless than κ for some fixed regular uncountable cardinal κ , then Propo-sition 9.5 and Remark 9.6 tell us that K = R (( N )) κ admits an isomor-phism Ω : ( K , + , , < ) ≃ ( N , · , , < ) with its archimedean value group v ( K ) ≃ N if and only if N ≃ N ≻ and K admits a surjective exponen-tial such that exp( K ↑ ) = N . This implies Theorem 4.1 of [3] (see alsoTheorem 3.4 therein). References [1] Alessandro Berarducci. Surreal numbers, exponentiation andderivations. arXiv:3325368 , (16 August):1–27, 2020.[2] Alessandro Berarducci and Vincenzo Mantova. Transseries asgerms of surreal functions.
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