Model completeness for the differential field of transseries with exponentiation
aa r X i v : . [ m a t h . L O ] M a y MODEL COMPLETENESS FOR THE DIFFERENTIAL FIELD OF TRANSSERIESWITH EXPONENTIATION
ELLIOT KAPLAN
Abstract.
Let T be the differential field of logarithmic-exponential transseries. We show that the expansionof T by its natural exponential function is model complete and locally o-minimal. We give an axiomatizationof the theory of this expansion that is effective relative to the theory of the real exponential field. We adaptour results to show that the expansion of T by this exponential function and by its natural restricted sineand restricted cosine functions is also model complete and locally o-minimal. Contents
Introduction 11. Preliminaries 32. Linear independence of ILD-sequences. 63. Logarithmic and exponential H -fields 94. Extensions of logarithmic H -fields 125. Model completeness for T exp and applications 196. Restricted trigonometric functions 237. Final Remarks and Future Directions 25References 27 Introduction
The differential field T of logarithmic-exponential transseries is a real closed ordered field extension of R which was introduced independently by ´Ecalle [15] in his solution to the Dulac conjecture and by Dahn andG¨oring [9] in their work on Tarskis problem on real exponentiation. This structure T serves as a sort ofuniversal domain for studying the asymptotic behavior of non-oscillating solutions to differential equationsover the reals. In [3], Aschenbrenner, van den Dries, and van der Hoeven show that the elementary theoryof T as an ordered valued differential field is model complete.There is a natural exponential exp on the field T that makes T an elementary extension of the real exponentialfield R exp [13]. Let T exp be the expansion of the differential field T by exp. Then T exp is a proper expansionof T in terms of definability, since exp is not definable in T by [3, 16.6.7]. In this paper, we take the firststep towards an analysis of the model theory of T exp . Theorem A (Corollary 5.6) . The elementary theory of T exp is model complete in the language of orderedvalued differential exponential fields. The elementary theory of T is axiomatized by one of two completions of T nl —the theory of ω -free newtonianLiouville closed H -fields [3]. Introduced by van den Dries and Aschenbrenner [1], H -fields form a class ofordered valued differential fields that includes T , all differential subfields of T containing R , and any Hardyfield containing R . The theory T nl is the model companion of the theory of H -fields and the study of H -fieldsand their extensions is a key part of the proof that T is model complete.In studying T exp (or valued exponential fields more generally), it is convenient to work with the logarithmfunction instead of the exponential function. One has a lot of control over the value group of valued field Date : May 7, 2020. xtensions which are only closed under taking logarithms, but not that much control over those extensionswhich are also closed under taking exponentials; see Section 4 of [12] for evidence supporting these heuristics.Accordingly, we study logarithmic H -fields , a class of H -fields which are equipped with a (not necessarilysurjective) logarithm. We call a logarithmic H -field with a surjective logarithm an exponential H -field andwe show that every logarithmic H -field has an exponential H -field extension with the same constant field.We further show that every exponential H -field can be extended to an ω -free newtonian exponential H -fieldwith the same constant field. Every ω -free newtonian exponential H -field is automatically Liouville closed;see Remark 3.7.Let T log extend the theory of logarithmic H -fields by axioms asserting that the constant field C equippedwith the restricted logarithm log | C > is elementarily equivalent to R exp . Let T ntexp extend the theory of ω -freenewtonian exponential H -fields by the same constant field axioms. Using Wilkie’s celebrated theorem that R exp is model complete [30], we show the following: Theorem B (Theorem 5.4) . T ntexp is model complete and it is the model companion of T log . Since T exp | = T ntexp , the first theorem is an immediate corollary of the second theorem. The theory T ntexp hasother models of interest: any maximal Hardy field is a model of T ntexp , as is Conway’s field of surreal numbersequipped with the Kruskal-Gonshor exponential and the Berarducci-Mantova derivation. Our proof closelyfollows the proof that T nl is the model companion of the theory of H -fields. Many of the tools used for thatresult go through in our case with little change, but extending the results in Sections 16.1 and 16.6 of [3]to our setting requires us to prove a nontrivial result about differential field extensions. Using the modelcompleteness result above, we are able to describe the completions of T ntexp . An H -field is said to have smallderivation if the derivative of every infinitesimal element is infinitesimal and large derivation otherwise. Theorem C (Theorem 5.5) . T ntexp has two completions: T ntexp , sm , whose models are the models of T ntexp withsmall derivation, and T ntexp , lg , whose models have large derivation. This mirrors the situation with T nl whose two completions are T nlsm (small derivation) and T nllg (large deriva-tion). Since T exp has small derivation, T ntexp , sm completely axiomatizes the theory of T exp . The only partof this axiomatization that is not known to be effective is the part that states that the constant field iselementarily equivalent to R exp ; see [24]. Thus, the theory of T exp is decidable relative to the theory of R exp .We are also able to use our model completeness result to show that unary definable sets in models of T ntexp are topologically tame. Theorem D (Theorem 5.13) . If K | = T ntexp , then for each y ∈ K and each definable X ⊆ K , there is anopen interval ( a, b ) around y such that X ∩ ( a, b ) is a finite union of points and intervals. In the literature, this property is called local o-minimality ; see [29]. In the future, we hope to betterunderstand the definable sets of arbitrary arity. We also hope to show that T ntexp is combinatorially tame(NIP or even distal). This likely requires us to first prove a quantifier elimination result in some reasonableextended language. At the end of this paper, we give some ideas about what this language might be.Given a real analytic function f : U → R where U ⊆ R n is an open neighborhood of the compact box [ − , n ,we associate to f its restriction f | [ − , n . The expansion of the real field by all of these restricted analyticfunctions and the unrestricted exponential function, denoted R an , exp , is model complete [14]. There is anatural expansion of the exponential field T by restricted analytic functions using Taylor series, and thisexpansion is an elementary extension of R an , exp ; see [13] for details. Ultimately, we would like to prove amodel completeness result for T an , exp : the expansion of the differential field T by these restricted analyticfunctions and the unrestricted exponential. As a step in this direction, we modify our model completenessproof to study the expansion of T exp by restricted sine and cosine functions. We call this expansion T rt , exp ,where the subscript rt stands for “restricted trigonometric functions”. Theorem E (Corollary 6.6) . The elementary theory of T rt , exp is model complete. This theorem uses that the corresponding expansion R rt , exp of the real field by restricted sine and cosinefunctions and the unrestricted exponential function is model complete, a theorem of van den Dries andMiller [14]. We use our result to show that T rt , exp is also locally o-minimal. utline. Section 1 is devoted to preliminaries on valued fields, differential fields, and especially H -fields.In Section 2, we prove a somewhat technical result about differential field extensions. Logarithmic andexponential H -fields are introduced in Section 3 and we study logarithmic H -field extensions and logarithmic H -field embeddings in Section 4. We use these results in Section 5 to prove our main theorems about T ntexp and in Section 6, we study restricted trigonometric functions. In Section 7, we provide an alternativeaxiomatization for T ntexp , indicate a language in which T ntexp might eliminate quantifiers, and briefly discussthe issue of proving model completeness for T an , exp . Acknowledgements.
I would like to thank Lou van den Dries, Allen Gehret, and Nigel Pynn-Coates fortheir many helpful comments on earlier drafts of this paper.1.
Preliminaries
We draw heavily from [3] and for brevity, we cite the results that we use from there only by their number.For example, we say [3, 3.5.19] instead of [3, Corollary 3.5.19]. The book [3] is almost entirely self-contained,so some of the results we cite (like [3, 3.5.19]) were not originally proven there. We use the same notationalconventions as [3], but we will repeat what is needed in this paper.We let m , n and r range over N = { , , , . . . } . All fields are assumed to be of characteristic zero and if K is a field, then we set K × := K \ { } . By “ordered set” we mean “totally ordered set”. If S is an orderedset and a ∈ S , then we set S >a := { s ∈ S : s > a } ; similarly for S > a , S := S > and we define S > , S < , S , and S = analogously. For A ⊆ S , we set A ↓ := { s ∈ S : s < a for some a ∈ A } and we say that A is downward closed in S if A = A ↓ . Ordered fields and ordered exponential fields.
Let K be an ordered field. We let [ − , K denote theclosed interval { x ∈ K : − x } . We let K rc denote the real closure of K . If L is a real closed fieldextension of K , then there is a unique ordered field embedding K rc → L over K .An exponential on K is an ordered group isomorphism exp : K → K > . We define an ordered exponentialfield to be an ordered field K equipped with an exponential exp. In other literature, exponentials aresometimes not required to be surjective onto K > , but it is convenient for us to impose this condition. Welet log : K > → K denote the inverse of exp. If K and L are ordered exponential fields, then an ordered fieldembedding ı : K → L is said to be an ordered exponential field embedding if ı (exp a ) = exp ı ( a ) for all a ∈ K .One important ordered exponential field is the real exponential field R exp . Let Th( R exp ) be the first-ordertheory of R exp in the natural language of ordered exponential fields. This theory is model complete ando-minimal by Wilkie’s theorem [30]. Valued fields.
Let K be a valued field with valuation v : K × → Γ. Then the value group of K is theordered (additively written) abelian group Γ. We let O := (cid:8) a ∈ K : va > (cid:9) denote the valuation ring of K , we let O := (cid:8) a ∈ K : va > (cid:9) denote the unique maximal ideal of O , and we let res( K ) := O / O denotethe residue field of K . Our standing assumption that all fields are of characteristic zero applies to res( K ) aswell, so all valued fields are of equicharacteristic zero. For a, b ∈ K we set a ≍ b : ⇐⇒ va = vb, a b : ⇐⇒ va > vb, a ≺ b : ⇐⇒ va > vb, a ∼ b : ⇐⇒ a − b ≺ a. Note that a ∼ b if and only if a, b = 0 and a/b ∈ O . We let K h denote the henselization of K . Thereis a unique valued field embedding of K h over K into any henselian valued field extending K . If L isalso a valued field, then we denote its value group, valuation ring, and maximal ideal by Γ L , O L , and O L respectively. Let L be a valued field extension of K . We identify Γ with a subgroup of Γ L and we identifyres( K ) with a subfield of res( L ) = O L / O L in the natural way. The Zariski-Abhyankar Inequality states thatthe transcendence degree of L over K is at least the transcendence degree of res( L ) over res( K ) plus the Q -linear dimension of Q Γ L over Q Γ, where Q Γ is the divisible hull of Γ. We say that L is an immediateextension of K if Γ L = Γ and res( L ) = res( K ). ifferential fields. Let K be a differential field with derivation ∂ : K → K . For a ∈ K we often write a ′ instead of ∂ ( a ) and a ′′ instead of ∂ ( a ). More generally, we write a ( n ) for ∂ n ( a ). We set a † := a ′ /a if a = 0and we call a † the logarithmic derivative of a . We define the iterates of the logarithmic derivative asfollows: a h i := a, a h n +1 i := ( (cid:0) a h n i (cid:1) † if a h n i is defined and nonzeroundefined if a h n i is undefined or zero . We let C := ker( ∂ ) denote the constant field of K and if L is also a differential field, then we denote itsconstant field by C L . Given φ ∈ K × , we let K φ be the differential field with underlying field K and derivation φ − ∂ . We call K φ the compositional conjugate of K by φ . Note that C K φ = C .We set K ′ := { a ′ : a ∈ K } and we set ( K × ) † := { a † : a ∈ K × } . Note that K † is a subgroup of K since a † + b † = ( ab ) † for a, b ∈ K × . Note also that K ′ is a C -vector subspace of K . We say that a ∈ K can be integrated if a ∈ K ′ and we call an element b ∈ K with b ′ = a an integral of a . If b and b are integralsof a , then b − b ∈ C . We say that a ∈ K can be exponentially integrated if a ∈ ( K × ) † and we call anelement b ∈ K × with b † = a an exponential integral of a . If b and b are exponential integrals of a , then b /b ∈ C × . In this paper, we will make use of Ostrowski’s Theorem about the algebraic independence ofintegrals [25]. Theorem (Ostrowski) . Let L be a differential field extension of K with C L = C . Let a , . . . , a n ∈ L with a ′ , . . . , a ′ n ∈ K . Then either a , . . . , a n are algebraically independent over K or a , . . . , a n are C -linearlydependent over K . If y is an element of a differential field extension of K , then we let K h y i := K ( y, y ′ , y ′′ , . . . ) denote thedifferential field extension of K generated by y . We say that y is d -transcendental over K if the sequence y, y ′ , y ′′ , . . . is algebraically independent over K (in the field-theoretic sense) and we say that y is d -algebraicover K otherwise. If y is d-algebraic over K , then K h y i has finite transcendence degree over K . Asymptotic fields.
Let K be a valued differential field (that is, a valued field of equicharacteristic zeroequipped with a derivation). Let Γ, O , and O be the value group, valuation ring, and maximal ideal of K respectively. If a ′ ∈ O for all a ∈ O , then we say that K has small derivation . We say that K has largederivation if K does not have small derivation.We say that K is asymptotic if f ≺ g ⇐⇒ f ′ ≺ g ′ for all nonzero f, g ∈ O . Note that if K is asymptotic,then the constant field C of K is contained in O . Thus, any asymptotic field with nontrivial valuation mustalso have nontrivial derivation. On the other hand, any valued differential field with trivial valuation isautomatically asymptotic. Let K be asymptotic. For f ∈ K × with f
1, the values v ( f ′ ) and v ( f † ) onlydepend on vf , so for γ = vf , we set γ † := v ( f † ) , γ ′ := v ( f ′ ) = γ + γ † . This gives us a map ψ : Γ = → Γ , ψ ( γ ) = γ † and, following Rosenlicht [27], we call the pair (Γ , ψ ) the asymptotic couple of K . We have the followingimportant subsets of Γ:(Γ < ) ′ := { γ ′ : γ ∈ Γ < } , (Γ > ) ′ := { γ ′ : γ ∈ Γ > } , Ψ := { γ † : γ ∈ Γ = } . It is always the case that (Γ < ) ′ < (Γ > ) ′ and that Ψ < (Γ > ) ′ . If there is β ∈ Γ with Ψ < β < (Γ > ) ′ , then wecall β a gap in K . There is at most one such β , and if Ψ has a largest element, then there is no such β .If K has trivial valuation, then the three important subsets above are empty and 0 is a gap in K . We saythat K is grounded if Ψ has a largest element and we say that K is ungrounded otherwise. Finally, wesay that K has asymptotic integration if Γ = (Γ < ) ′ ∪ (Γ > ) ′ .We say that K is H -asymptotic if K is asymptotic and f g = ⇒ f † < g † for all nonzero f, g ∈ O . Wehave an important trichotomy for the structure of H -asymptotic fields: act 1.1 ([3], 9.2.16) . If K be an H -asymptotic field, then exactly one of the following is true:(1) K has asymptotic integration;(2) K has a gap;(3) K is grounded. Let K be an ungrounded H -asymptotic field with asymptotic couple (Γ , ψ ). Then Ψ ⊆ (Γ < ) ′ , so we havea contraction map χ : Γ < → Γ < where χ ( α ) is the unique element in Γ < with χ ( α ) ′ = α † . We say that s ∈ K is a gap creator over K if vf is a gap in K ( f ) for some f in an H -asymptotic field extension of K with f † = s . In the lemma below, we summarize some facts about gap creators from Section 11.5 in [3]. Lemma 1.2.
Let K be an H -asymptotic field with asymptotic integration and divisible value group. Then K has an immediate H -asymptotic field extension with a gap creator. If s ∈ K is a gap creator over K , then vf is a gap in K ( f ) for any nonzero f in an H -field extension of K with f † = s .Proof. By [3, 11.4.10], K has a spherically complete immediate H -field extension. By [3, 11.5.14], thisextension has a gap creator. The second part of the lemma is the remark after [3, 11.5.14]. (cid:3) H -fields. Let K be an ordered differential field with constant field C . We say that K is an H -field if(H1) f ′ > f ∈ K with f > C ;(H2) O = C + O , where O is the convex hull of C in K and O is the unique maximal ideal of O .Let K be an H -field. For φ ∈ K > , the compositional conjugate K φ of K is also an H -field. Any H -field isa valued field with valuation ring O as defined in (H2), and we view H -fields as ordered valued differentialfields. Any ordered field with trivial derivation is an H -field, and every H -field with nontrivial derivationhas a nontrivial valuation ring.For the remainder of this section, let K be an H -field with valuation ring O , value group Γ, and constantfield C . By (H2), the projection map O → res( K ) maps C isomorphically onto res( K ). Consequently, an H -field extension L of K is an immediate extension of K if and only if Γ L = Γ and C L = C . As a valueddifferential field, K is H -asymptotic, so K has an asymptotic couple and the trichotomy in Fact 1.1 appliesto K . Here is a useful lemma about adjoining integrals to henselian H -fields. Lemma 1.3.
Let K be henselian and let s ∈ K \ K ′ . Suppose that either(1) K has asymptotic integration, or(2) vs ∈ (Γ > ) ′ .Then there is an immediate H -field extension K ( a ) of K with a and a ′ = s . If L is also an H -fieldextension of K and b ∈ L satisfies b and b ′ = s , then there is a unique H -field embedding K ( a ) → L over K that sends a to b .Proof. By [3, 10.2.5], the set (cid:8) v ( s − y ′ ) : y ∈ K (cid:9) has no largest element. Thus K ( a ) is an immediate H -fieldextension of K by either [3, 10.2.4] or [3, 10.2.6]. The universal property of K ( a ) follows from the universalproperties in [3, 10.2.4] and [3, 10.2.6]. (cid:3) A major result in [3] is that the theory of H -fields has a model companion, namely the theory T nl of ω -free newtonian Liouville closed H -fields. We say that K is Liouville closed if K is real closed and every a ∈ K can be integrated and exponentially integrated in K . The axioms “ ω -free” and “newtonian” are moretechnical and we will not define these axioms precisely, but we will list some facts about these axioms thatwill be useful later in this paper.The axiom of ω -freeness is a rather subtle axiom that, among other things, rules out the existence of gapcreators. If K is ω -free, then K is ungrounded by definition and K has no gap creator. In particular, K hasno gap, so by Fact 1.1 we see that every ω -free H -field has asymptotic integration. The property of ω -freenessis quite robust; it passes to d-algebraic H -field extensions and it is inherited by certain H -subfields: Fact 1.4 ([3], Section 11.7 and 13.6.1) . If K is ω -free and L is a d -algebraic H -field extension of K , then L is ω -free. If E is an ungrounded H -subfield of K and Γ Let K be ω -free and newtonian and let K h y i be an immediate H -field extension of K . Let L bealso an H -field extension of K and let z ∈ L realize the same cut as y over K . Then there is an H -fieldembedding K h y i → L over K that sends y to z . If K is ω -free, then a newtonization of K is by definition an immediate newtonian H -field extension of K that embeds over K into any ω -free newtonian H -field extension of K . Fact 1.8. If K is ω -free, then K has a newtonization K nt which is d -algebraic over K . Any two newtoniza-tions of K are isomorphic over K . Fact 1.8 was shown under the assumption that K also have divisible value group [3, 14.3.12 and 14.5.4] butagain, this divisibility assumption can be removed by [26]. Both ω -freeness and newtonianity are preservedunder compositional conjugation by elements of K > .If K is ω -free, then a Newton-Liouville closure of K is by definition a newtonian liouville closed H -fieldextension of K that embeds over K into any newtonian Liouville closed H -field extension of K . Fact 1.9 ([3], 14.5.10) . If K is ω -free, then K has a Newton-Liouville closure K nl . Any such K nl is d -algebraic over K and the constant field of K nl is a real closure of K . We have a sort of converse to Fact 1.9: Lemma 1.10. Let K be ω -free and let L is a newtonian Liouville closed H -field extension of K . If L is d -algebraic over K and if C L is a real closure of C , then L is a Newton-Liouville closure of K .Proof. Let K nl be a Newton-Liouville closure of K . Then there is an embedding K nl → L over K and C L is contained in the image of this embedding. By [3, 16.2.1], this embedding is surjective. (cid:3) Finally, we note that Newton-Liouville closures are unique and minimal. Fact 1.11 ([3], 16.2.2) . Let K be ω -free. Any two Newton-Liouville closures of K are isomorphic over K .If K nl is a Newton-Liouville closure of K , then the only newtonian Liouville subfield of K nl containing K is K nl itself. Linear independence of ILD-sequences. Let K be a differential field with constant field C and let y be an element in a differential field extension of K that is d-transcendental over K . Definition 2.1. An ILD-sequence for y over K is a sequence ( y n ) in K h y i where y = y and where y n +1 /y † n ∈ K × for each n . In the above definition, ILD stands for “iterated logarithmic derivative.” The simplest ILD-sequence is thesequence (cid:0) y h n i (cid:1) . In this section, we will prove the following proposition. roposition 2.2. If ( y n ) is an ILD-sequence for y over K , then ( y † n ) is C -linearly independent over K h y i ′ . The proof of this proposition is inspired by ideas of Srinivasan [28]. This proposition will come in handy ina couple of places: in Subsections 4.4 and 4.5, we will encounter situations where we need to show that thelogarithms of certain ILD-sequences for y over K are algebraically independent over K . Using Ostrowski’sTheorem, we can reduce this problem to showing that these logarithms are C -linearly independent over K , and then by taking derivatives, we can solve this problem by applying the above proposition. Towardsproving Proposition 2.2, we begin with a weaker version: Lemma 2.3. y † K h y i ′ .Proof. Set L := K h y ′ i , so K h y i = L ( y ) and L ( y ) is isomorphic to the field of rational functions L ( Y ).Suppose towards a contradiction that y † ∈ L ( y ) ′ , so there are coprime polynomials P, Q ∈ L [ Y ] = with y ′ y = y † = (cid:18) P ( y ) Q ( y ) (cid:19) ′ = P ( y ) ′ Q ( y ) − P ( y ) Q ( y ) ′ Q ( y ) . Multiplying by y and Q ( y ) gives Q ( y ) y ′ = y (cid:0) P ( y ) ′ Q ( y ) − P ( y ) Q ( y ) ′ (cid:1) and so y divides Q ( y ) in the ring L [ y ]. Let k > y k divides Q ( y ) in L [ y ]. Then y k divides Q ( y ) and so y k − divides P ( y ) ′ Q ( y ) − P ( y ) Q ( y ) ′ in L [ y ]. In particular, y k divides P ( y ) ′ Q ( y ) − P ( y ) Q ( y ) ′ since 2 k − > k . Since y k divides Q ( y ) but not P ( y ) we see that y k must divide Q ( y ) ′ in L [ y ].Take R ∈ L [ Y ] with Q ( y ) = y k R ( y ). We have Q ( y ) ′ = ky k − y ′ R ( y ) + y k R ( y ) ′ so y must divide R ( y ) in L [ y ].Then y k +1 divides Q ( y ) in L [ y ], contradicting that we chose k to be maximal. (cid:3) Remark 2.4. In Lemma 2.3, we can relax the assumption that y is d -transcendental over K ; in the proofwe only use that y is transcendental over K h y ′ i . However, in this paper we only apply this lemma to d -transcendental elements. For the remainder of this section, let ( y n ) be an ILD-sequence for y over K and for each n , take d n ∈ K × with y † n = d n y n +1 , so y ′ n = d n y n y n +1 . For each n we set K n := K ( y n , y n +1 , . . . ) = K h y n i . By [3, 4.1.5],each element in K h y i \ K is d-transcendental over K . In particular, each y n is d-transcendental over K and C K h y i = K . Lemma 2.5. The sequence ( y n ) is algebraically independent over K .Proof. An easy induction on n gives that y ( n ) ∈ K [ y , . . . , y n ] for each n . In particular, K ( y, . . . , y ( n ) ) ⊆ K ( y , . . . , y n ) . Thus, K ( y , . . . , y n ) has transcendence degree n + 1 over K for each n , since y is assumed to be d-transcendental over K . This shows that the sequence ( y n ) is algebraically independent over K . (cid:3) It follows from Lemma 2.5 that the ring K n +1 [ y n ] is isomorphic to the polynomial ring K n +1 [ Y ] for each n . Lemma 2.6. Let n, k > and a , . . . , a k ∈ K n +1 . If a + · · · + a k y kn ∈ K n +1 [ y n ] ′ , then a i y in ∈ ( K n +1 y in ) ′ for each i ∈ { , . . . , k } .Proof. Take m > k and P = P + · · · + P m Y m ∈ K n +1 [ Y ] with k X i =0 a i y in = P ( y n ) ′ = m X j =0 P ′ j y jn + P j ( y jn ) ′ . Since ( y jn ) ′ = jd n y n +1 y jn for each j , we have k X i =0 a i y in = m X j =0 ( P ′ j + P j jd n y n +1 ) y jn , so a i y in = ( P ′ i + P i id n y n +1 ) y in = ( P i y in ) ′ for each i ∈ { , . . . , k } . (cid:3) emma 2.7. For each n we have K ′ ∩ K n +1 [ y n ] = K n +1 [ y n ] ′ .Proof. We need the following two claims: Claim 1. For each n we have K ′ n ∩ K n +1 [ y n ] ⊆ K n +1 [ y n ] ′ .Proof of Claim 1. Since y n is transcendental over K n +1 , the ring K n +1 [ y n ] is isomorphic to the polynomialring K n +1 [ Y ]. Let P, Q ∈ K n +1 [ Y ] = be coprime polynomials with Q monic such that (cid:18) P ( y n ) Q ( y n ) (cid:19) ′ = P ( y n ) ′ Q ( y n ) − P ( y n ) Q ( y n ) ′ Q ( y n ) ∈ K n +1 [ y n ] . We will show that Q = 1. Note that Q ( y n ) must divide P ( y n ) ′ Q ( y n ) − P ( y n ) Q ( y n ) ′ in the ring K n +1 [ y n ].Since Q ( y n ) divides P ( y n ) ′ Q ( y n ) in K n +1 [ y n ] and since no factor of Q ( y n ) divides P ( y n ), we see that Q ( y n ) must divide Q ( y n ) ′ in K n +1 [ y n ]. Let k = deg Q and take Q , . . . , Q k − , ∈ K n +1 with Q ( Y ) = Y k + Q k − Y k − + · · · + Q . Then Q ( y n ) ′ = kd n y n +1 y kn + k − X i =0 (cid:0) Q ′ i + Q i id n y n +1 (cid:1) y in , so kd n y n +1 Q ( y n ) = Q ( y n ) ′ . For i < k , we have kd n y n +1 Q i = Q ′ i + Q i id n y n +1 . We claim that Q i = 0 for each i < k . Suppose towards a contradiction that there is i < k with Q i = 0.Then Q † i = ( k − i ) d n y n +1 = ( k − i ) y † n and so y k − in = cQ i for some c ∈ C × K n = C × . Since cQ i ∈ K n +1 andsince y n is transcendental over K n +1 by Lemma 2.5, this is a contradiction. Thus Q , . . . , Q k − are all 0 and Q ( y n ) = y kn .We now claim that k must be 0, so Q = 1. We have (cid:18) P ( y n ) Q ( y n ) (cid:19) ′ = (cid:18) P ( y n ) y kn (cid:19) ′ = P ( y n ) ′ y kn − P ( y n ) ky k − n y ′ n y kn = P ( y n ) ′ − P ( y n ) kd n y n +1 y kn . Let m = deg P and take P , . . . , P m ∈ K n +1 , P m = 0 with P ( Y ) = P m Y m + P m − Y m − + · · · + P . Then P ( y n ) ′ − P ( y n ) kd n y n +1 = m X i =0 (cid:0) P ′ i + P i id n y n +1 − P i kd n y n +1 (cid:1) y in . Suppose towards a contradiction that k > 0. Then y kn must divide P ( y n ) ′ − P ( y n ) kd n y n +1 in K n +1 [ y n ], sothe i = 0 term P ′ − P kd n y n +1 in the above sum must be 0. If P = 0, then y n divides P ( y n ) in K n +1 [ y n ],contradicting the assumption that P and Q are coprime. However if P = 0, then P † = kd n y n +1 = ky † n , so y kn = cP ∈ K n +1 for some c ∈ C × contradicting that y n is transcendental over K n +1 . Thus k = 0, so Q = 1 and (cid:18) P ( y n ) Q ( y n ) (cid:19) ′ = P ( y n ) ′ ∈ K n +1 [ y n ] ′ . (cid:3) Claim 2. For each n we have K ′ ∩ K n ⊆ K ′ n .Proof of Claim 2. We will show by induction on m n that K ′ ∩ K n ⊆ K ′ m . This is clear for m = 0.Suppose this holds for a given m < n . Then K ′ ∩ K n ⊆ K ′ m ∩ K n ⊆ (cid:0) K ′ m ∩ K m +1 [ y m ] (cid:1) ∩ K m +1 ⊆ K m +1 [ y m ] ′ ∩ K m +1 , where the last containment follows from Claim 1. By Lemma 2.6 with k = 0 we have K m +1 [ y m ] ′ ∩ K m +1 ⊆ K ′ m +1 . Therefore, K ′ ∩ K n ⊆ K ′ m +1 as required. (cid:3) ow we turn to the statement of the lemma. We have K ′ ∩ K n +1 [ y n ] = ( K ′ ∩ K n ) ∩ K n +1 [ y n ] ⊆ K ′ n ∩ K n +1 [ y n ] ⊆ K n +1 [ y n ] ′ by Claims 2 and 1. The other containment K n +1 [ y n ] ′ ⊆ K ′ ∩ K n +1 [ y n ] is clear. (cid:3) We are now ready to prove the proposition. Proof of Proposition 2.2. We need to show that the sequence ( y † n ) is C -linearly independent over K h y i ′ = K ′ .Suppose this is not the case and take r > 0, indices n < · · · < n r , and nonzero constants c , . . . , c r ∈ C × such that c y † n + · · · + c r y † n r ∈ K ′ . Since c i y † n i = c i d n i y n i +1 for each i ∈ { , . . . , r } , we also have c y † n + · · · + c r y † n r = c d n y n +1 + · · · + c r d n r y n r +1 ∈ K n +2 [ y n +1 ] . By Lemma 2.7 with n + 1 in place of n , we have c d n y n +1 + · · · + c r d n r y n r +1 ∈ K n +2 [ y n +1 ] ′ . By Lemma 2.6 with k = 1, n = n + 1, a = P ri =2 c i d n i y n i +1 , and a = c d n , we have c y † n = c d n y n +1 ∈ ( K n +2 y n +1 ) ′ ⊆ K ′ n +1 . In particular, y † n ∈ K ′ n = K h y n i ′ . However, Lemma 2.3 with y n in place of y gives us that y † n K h y n i ′ ,a contradiction. (cid:3) Logarithmic and exponential H -fields We will be working only with H -fields (possibly with additional structure) for the remainder of this paper,so we make the following convention. Convention. For the remainder of this paper, K is an H -field with valuation ring O , maximal ideal O ,derivation ∂ , constant field C , and asymptotic couple (Γ , ψ ) . In this section, we look at H -fields that are equipped with a logarithm. Logarithms on H -fields werepreviously studied in [2]. Definition 3.1. A logarithm on K is a map log : K > → K such that(L1) log embeds the multiplicative group K > into the additive group of K ;(L2) log(1 + O ) ⊆ O ;(L3) (log f ) ′ = f † for all f ∈ K > . We use exp to denote the inverse of log where it is defined. Let log be a logarithm on K . Then( K × ) † = ( K > ) † = (log K > ) ′ ⊆ K ′ , so K is ungrounded. Condition (L3) tells us that the trace of the logarithm on Γ < is given by the contractionmap: for a ∈ K > with a ≻ v (log a ) = χ ( va ). By [3, 9.2.18] we have α < nχ ( α ) < α ∈ Γ < and each n > 0. This shows that any logarithm on K is sufficiently “slow-growing”: Lemma 3.2. If log is a logarithm on K , then a ≻ (log a ) n ≻ for each a ∈ K > with a ≻ and each n > . Here is another consequence of (L3). Lemma 3.3. If log is a logarithm on K , then log( K > ) ∩ C = log( C > ) .Proof. If c ∈ C > , then (log c ) ′ = c † = 0 so log( C > ) ⊆ log( K > ) ∩ C . For the other containment, let a ∈ K > and suppose log a ∈ C . Then a † = (log a ) ′ = 0 so a ∈ C > . (cid:3) Though we only require that a logarithm on K is a group embedding, any logarithm on K is actually an ordered group embedding provided that its restriction to C > is. emma 3.4. Let log be a logarithm on K and suppose log c > for each c ∈ C > . Then log f > for each f ∈ K > .Proof. Let f > 1. First, consider the case that f ≻ 1. Then log f ≻ f ) ′ = f † > 0, so log f ispositive by the H -field axiom (H1). Next consider the case that f ≍ f 1. Then f = c (1 + ε ) forsome c ∈ C > and some ε ∈ O so log f = log c + log(1 + ε ) . We have log c > | C > and condition (L2) in Definition 3.1 gives us that log(1+ ε ) ≺ 1, so log f ∼ log c > 0. Finally, consider the case that f ∼ 1. Then f = 1 + ε for some ε ∈ O > , so(log f ) ′ = f † ∼ ε ′ < . Since log f ∈ O axiom (H1) again gives log f > (cid:3) Definition 3.5. A logarithmic H -field is a henselian H -field K equipped with a logarithm log on K suchthat the inverse exp of log is defined on all of C and such that C equipped with exp | C is a real closed orderedexponential field. Let K be a logarithmic H -field with logarithm log. Then log | C > is order-preserving by our definition ofordered exponential fields, so log is order-preserving by Lemma 3.4. For all φ ∈ K > , the compositionalconjugate K φ of K is also a logarithmic H -field with the same logarithm as K . Definition 3.6. An exponential H -field is a logarithmic H -field K with log( K > ) = K . Let K be an exponential H -field. Then Γ is divisible since v exp (cid:16) log an (cid:17) = van for each n > 0. Then K is real closed, since C is assumed to be real closed and K is assumed to behenselian; see [3, 3.5.19]. If an element a ∈ K can be integrated, then it can be exponentially integratedsince b ′ = (exp b ) † for all b ∈ K . To summarize: Remark 3.7. If K is an exponential H -field, then K is real closed. If in addition K ′ = K , then K isLiouville closed. In particular, every ω -free newtonian exponential H -field is Liouville closed. Logarithmic H -field embeddings are defined in the obvious way: Definition 3.8. Let K and L be logarithmic H -fields and let ı : K → L be an H -field embedding. We saythat ı preserves logarithms if ı (log f ) = log ı ( f ) for all f ∈ K > . A logarithmic H -field embedding is an H -field embedding that preserves logarithms. The notions of a logarithmic H -field extension and alogarithmic H -subfield are defined analogously. The extension K ℓ . Recall our convention that K is an H -field. In this subsection, we show that K has an H -field extension K ℓ that admits a definable logarithm on 1 + O , and we use this extension to developsome tools that will be used in the next section. Lemma 3.9. K has an immediate H -field extension K ℓ such that(1) K ℓ is henselian;(2) (1 + O K ℓ ) † ⊆ O ′ K ℓ ;(3) K ℓ embeds uniquely over K into any henselian H -field extension L of K with (1 + O L ) † ⊆ O ′ L .Proof. We define an ℓ -tower on K to be an increasing chain of henselian H -fields ( K µ ) µ ν such thati. K = K h ;ii. K µ = S λ<µ K λ for each limit ordinal µ ;iii. K µ +1 = K µ ( y µ ) h where y ′ µ = (1 + ε µ ) † for some ε µ ∈ O K µ with (1 + ε µ ) † / ∈ O ′ K µ .We claim that if ( K µ ) µ ν is an ℓ -tower on K , then K µ is an immediate extension of K for each µ ν . Ofcourse, K is an immediate extension of K and if K λ is an immediate extension of K for each λ below somelimit µ ν , then K µ is also an immediate extension of K . Suppose K µ is an immediate extension of K forsome given µ < ν and let ε µ ∈ O K µ be as in the definition of K µ +1 . Then v (1 + ε µ ) † = v ( ε µ ) ′ ∈ (Γ >K µ ) ′ , sothe hypothesis of Lemma 1.3 is satisfied and K µ +1 is an immediate extension of K µ . n easy induction on µ shows that each K µ is also a Liouville extension of K . That is, C K µ is algebraicover C and each a ∈ K µ is contained in a field extension K ( t , . . . , t n ) ⊆ K µ where for i ∈ { , . . . , n } , either t i ∈ K ( t , . . . , t i − ) rc , or t ′ i ∈ K ( t , . . . , t i − ) , or t i = 0 and t † i ∈ K ( t , . . . , t i − ) . In particular, | K µ | = | K | for each µ by [3, 10.6.8], so maximal ℓ -towers exist. Let ( K µ ) µ ν be a maximal ℓ -tower on K . Then K ν is a henselian immediate extension of K and (1 + O K ν ) † ⊆ O ′ K ν .Now let L be a henselian H -field extension of K with (1 + O L ) † ⊆ O ′ L . We claim that there is a unique H -field embedding ı µ : K µ → L over K for each µ ν . This holds for K by the universal property of thehenselization and if it holds for K λ for each λ below some limit µ ν , then it holds for K µ . Suppose thisholds for K µ for some given µ < ν and let ε µ ∈ O K µ be as in the definition of K µ +1 . Let a ∈ L with a ≺ a ′ = ı µ (1 + ε µ ) † . The universal property in Lemma 1.3 and universal property of the henselization givea unique H -field embedding ı µ +1 : K µ +1 → L over K µ that sends y µ to a . Since a is the unique integralof ı µ (1 + ε µ ) † in L with nonzero valuation, the uniqueness of ı µ +1 does not depend on the condition that ı µ +1 ( y µ ) = a . Then K ν has all of the desired properties and we may take K ℓ := K ν . (cid:3) If K = K ℓ , then we define a function ln : 1 + O → O by letting ln a be the unique element of O satisfying(ln a ) ′ = a † for each a ∈ O . Given ε ∈ O = , we have ln(1 + ε ) ′ ∼ ε ′ , so ε > ε ) > 0. From thisit is straightforward to check that ln is an ordered group embedding. If K = K ℓ and M is an H -subfield of K , then we identify M ℓ with its unique image in K . The extension K ℓ is related to logarithms on K in thefollowing way: Lemma 3.10. Let K be a henselian H -field. If log is a logarithm on K , then K = K ℓ and log a = ln a forall a ∈ O .Proof. If log is a logarithm on K , then log(1 + O ) ⊆ O by (L2), so(1 + O ) † = log(1 + O ) ′ ⊆ O ′ and K = K ℓ . For a ∈ O , we have (log a ) ′ = a † = (ln a ) ′ , so log a − ln a ∈ C . Since log a and ln a are bothin O , they must be equal. (cid:3) In particular, if K is a logarithmic H -field, then K = K ℓ , since K is henselian. The map ln gives us acriterion for checking whether a given H -field embedding preserves logarithms. Lemma 3.11. Let K and L be logarithmic H -fields and let ı : K → L be an H -field embedding. Let f, g ∈ K > with f ∼ g . If ı (log g ) = log ı ( g ) , then ı (log f ) = log ı ( f ) .Proof. Take ε ∈ O such that f = g (1 + ε ). Thenlog f = log g + log(1 + ε ) = log g + ln(1 + ε )by Lemma 3.10. We have ı (log g ) = log ı ( g ) by assumption, so it remains to show that ı (cid:0) ln(1+ ε ) (cid:1) = ln ı (1+ ε ).To see this, note that ı (cid:0) ln(1 + ε ) (cid:1) ∈ O L and that ı (cid:0) ln(1 + ε ) (cid:1) ′ = ı (cid:0) (1 + ε ) † (cid:1) = ı (1 + ε ) † . Since ln ı (1 + ε ) is the unique element of O L with derivative ı (1 + ε ) † , we have ı (cid:0) ln(1 + ε ) (cid:1) = ln ı (1 + ε ). (cid:3) The extension K ℓ is useful both for constructing logarithmic H -field extensions and for checking whether H -subfields are actually logarithmic H -subfields. We detail these methods below. Corollary 3.12. Let K be a logarithmic H -field and let E be an H -subfield of K with C E = C . Assumethat for each f ∈ E > there is g ∈ E > with f ≍ g and log g ∈ E . Then E ℓ is a logarithmic H -subfield of K .Proof. Since E ℓ is an immediate extension of E , the conditions on E also hold for E ℓ so we assume withoutloss of generality that E = E ℓ . We need to show that log( E > ) ⊆ E . Let f ∈ E > and take g ∈ E > with f ≍ g and log g ∈ E . Then f = cg (1 + ε ) for some c ∈ C >E = C > and some ε ∈ O E solog f = log c + log g + log(1 + ε ) = log c + log g + ln(1 + ε ) . Our assumption gives that log c and log g are in E and, since E = E ℓ , we have ln(1 + ε ) ∈ E as well. (cid:3) emma 3.13. Let K be a logarithmic H -field and let M be an H -field extension of K with C M = C . Let ( a i ) i ∈ I be a family of elements in M > with a i for each i such that Γ M = Γ ⊕ M i ∈ I Z va i and let ( b i ) i ∈ I be a family of elements in M such that b ′ i = a † i for each i ∈ I . Then there is a unique logarithm log on M ℓ extending the logarithm on K such that log a i = b i for each i ∈ I . With this logarithm, M ℓ is alogarithmic H -field extension of K . If L is also a logarithmic H -field extension of K and ı : M → L is an H -field embedding over K , then the unique H -field embedding M ℓ → L extending ı preserves logarithms ifand only if ı ( b i ) = log ı ( a i ) for each i ∈ I .Proof. Since M ℓ is an immediate extension of M , the conditions on M also hold for M ℓ , so we assumewithout loss of generality that M = M ℓ . Let f ∈ M > . Our assumption on Γ M and C M gives f = g (1 + ε ) Y i ∈ I a k i i for some g ∈ K > , some ε ∈ O M , and some family ( k i ) i ∈ I of integers where only finitely many k i are nonzero.We set log f := log g + ln(1 + ε ) + X i ∈ I k i b i . It is routine to show that this does not depend on the choice of g . Note that(log f ) ′ = (log g ) ′ + ln(1 + ε ) ′ + X i ∈ I k i b ′ i = g † + (1 + ε ) † + X i ∈ I k i a † i = f † . Using that log | K > and ln are group embeddings, it is straightforward to show that log is a group embedding.Then log is indeed a logarithm on M since log(1 + O ) = ln(1 + O ) ⊆ O , and M equipped with log is alogarithmic H -field since M = M ℓ and C M = C . For uniqueness, let log ∗ is an arbitrary logarithm on M . Then log ∗ (1 + ε ) = ln(1 + ε ) = log(1 + ε ) by Lemma 3.10, so if log ∗ extends the logarithm on K andlog ∗ a i = b i , then log ∗ f = log f .Now let L be a logarithmic H -field extension of K and let ı : M → L be an H -field embedding over K .We continue to assume that M = M ℓ and we assume that ı ( b i ) = log ı ( a i ) for each i ∈ I . We need to showthat ı (log f ) = log ı ( f ) where f is as above. Using the fact that f ∼ g Q i ∈ I a k i i and Lemma 3.11, we mayassume that f = g Q i ∈ I a k i i . Since log (cid:16) g Y i ∈ I a k i i (cid:17) = log g + X i ∈ I k i log( a i )and since g ∈ K , this further reduces to showing that ı (log a i ) = log ı ( a i ) for each i ∈ I . This holds by ourassumption, since log a i = b i for each i . The other implication, that ı ( b i ) = log ı ( a i ) if ı preserves logarithms,is clear. (cid:3) The conditions on C M and Γ M in the above lemma are always satisfied when M is an immediate H -fieldextension of K : Corollary 3.14. Let K be a logarithmic H -field and let M be an immediate H -field extension of K . Thenthere is a unique logarithm log on M ℓ extending the logarithm on K . With this logarithm, M ℓ is a logarithmic H -field extension of K and if L is also a logarithmic H -field extension of K and ı : M → L is an H -fieldembedding over K , then the unique embedding M ℓ → L extending ı preserves logarithms. Extensions of logarithmic H -fields In this section, we prove a variety of extension and embedding results about logarithmic H -fields for use inSection 5. For the remainder of this section, K is a logarithmic H -field with logarithm log. .1. d -algebraic extensions. In this subsection we deal with various d-algebraic logarithmic H -field ex-tensions of K . We will show that any ω -free logarithmic H -field has a minimal ω -free newtonian exponential H -field extension. We begin with the newtonization. Corollary 4.1. Let K be ω -free. Then there is a unique logarithm on K nt extending the logarithm on K . If L is an ω -free newtonian logarithmic H -field extension of K , then there is a logarithmic H -field embedding K nt → L over K .Proof. We have ( K nt ) ℓ = K nt since K nt is asymptotically d-algebraically maximal by Fact 1.6. Since K nt isan immediate extension of K , the logarithm on K extends uniquely to a logarithm on K nt by Corollary 3.14.For L as in the statement of the Corollary, there is an H -field embedding K nt → L over K by Fact 1.8, andthis embedding preserves logarithms by Corollary 3.14. (cid:3) Next, we deal with constant field extensions. Lemma 4.2. Let M be a logarithmic H -field extension of K . Then K ( C M ) ℓ is a logarithmic H -subfieldof M with value group Γ . Let L be also a logarithmic H -field extension of K and let ı : C M → C L be anordered exponential field embedding over C . Then ı extends uniquely to a logarithmic H -field embedding K ( C M ) ℓ → L over K .Proof. The value group of K ( C M ) ℓ is the same as the value group of K ( C M ), which is the same as thevalue group of K by [3, 10.5.15]. Thus, for each f ∈ K ( C M ) ℓ with f > g ∈ K > with f ≍ g .Since K is a logarithmic H -field, Corollary 3.12 with M and K ( C M ) in place of K and E gives us that K ( C M ) ℓ is a logarithmic H -subfield of M . Now let L and ı be as in the statement of the lemma. By [3,10.5.15 and 10.5.16] there is a unique H -field embedding K ( C M ) → L over K that extends ı . This in turnextends uniquely to an embedding : K ( C M ) ℓ → L . It remains to show that preserves logarithms. Let f ∈ K ( C M ) ℓ with f > 0. Then there is c ∈ C >M and g ∈ K > with f ∼ cg . By Lemma 3.11 it suffices to showthat (cid:0) log( cg ) (cid:1) = log ( cg ). This follows from our assumption on ı : (cid:0) log( cg ) (cid:1) = (log c + log g ) = ı (log c ) + log g = log ı ( c ) + log g = log ( cg ) . (cid:3) Now we move on to real closures. Lemma 4.3. Set K rc ,ℓ := ( K rc ) ℓ . Then K rc ,ℓ is a real closed H -field with constant field C and there isa unique logarithm on K rc ,ℓ extending the logarithm on K . If L is also a real closed logarithmic H -fieldextension of K , then there is a unique logarithmic H -field embedding ı : K rc ,ℓ → L over K .Proof. Set M := K rc ,ℓ . Then M is an immediate extension of K rc , so Γ M = Q Γ and C M = C rc = C since C is assumed to be real closed. Thus, M is real closed since M is henselian; see [3, 3.5.19]. For each f ∈ M > there is g ∈ K > and q ∈ Q with f ∼ g q . Take ε ∈ O M with f = g q (1 + ε ). We setlog f := q log g + ln(1 + ε ) . It is routine to check that log is indeed a logarithm on M extending the logarithm on K . Any logarithm log ∗ on M which extends the logarithm on K must satisfy log ∗ ( g q ) = q log g ∈ K and log ∗ (1 + ε ) = ln(1 + ε ), sothis extension is unique. Now let L be a real closed logarithmic H -field extending K . Then by the universalproperty of the real closure there is a unique H -field embedding K rc → L over K and this in turn extendsuniquely to an embedding ı : M → L . To see that ı preserves logarithms, let g ∈ K > and q ∈ Q and notethat ı (log g q ) = q log g = log ı ( g q ) . We are done in light of Lemma 3.11. (cid:3) Finally, we deal with adding exponentials. Lemma 4.4. Let a ∈ K \ log( K > ) with a ≺ . Then K has an immediate H -field extension K ( f ) with f ∼ and f † = a ′ . There is a unique logarithm on K ( f ) ℓ extending the logarithm on K , and this logarithmsatisfies log f = a . If L is also a logarithmic H -field extension of K and a ∈ log( L > ) , then there is a uniquelogarithmic H -field embedding ı : K ( f ) ℓ → L over K . roof. We claim that a ′ ( K × ) † . Suppose not and let b ∈ K × with a ′ = b † . Then b ≍ va ′ ∈ (Γ > ) ′ , soby replacing b with cb for some c ∈ C × , we may assume that b ∼ 1. Lemma 3.10 gives us that a = ln b = log b ,contradicting our assumption that a log( K > ).With this claim out of the way, we may apply [3, 10.5.18] with a ′ in place of s to get an immediate H -field extension K ( f ) of K where f ∼ f † = a ′ . By Corollary 3.14 there is a unique logarithm logon K ( f ) ℓ . We have log f = ln f = a , since a ′ = f † . Now let L be a logarithmic H -field extension of K with a ∈ log( L > ). Since (exp a ) † = a ′ , [3, 10.5.18] gives us a unique H -field embedding K ( f ) → L that sends f to exp a . This in turn extends uniquely to an H -field embedding ı : K ( f ) ℓ → L , and ı preserves logarithmsby Corollary 3.14. Since any logarithmic H -field embedding K ( f ) ℓ → L must send f to exp a , the map ı isunique as claimed. (cid:3) Lemma 4.5. Let K be real closed with asymptotic integration. Let a ∈ K and suppose that a − log b ≻ for all b ∈ K > . Then K has an H -field extension K ( f ) with constant field C where f † = a ′ . There is alogarithm on K ( f ) ℓ that extends the logarithm on K and is uniquely determined by the condition log f = a .If L is also a logarithmic H -field extension of K and a ∈ log( L > ) , then there is a unique logarithmic H -fieldembedding ı : K ( f ) ℓ → L over K .Proof. We may assume that a < 0. Then a ′ < a ≻ 1. We claim that v ( a ′ − b † ) ∈ Ψ ↓ forall b ∈ K × . If not, then there is b ∈ K × with either a ′ = b † or v ( a ′ − b † ) ∈ (Γ > ) ′ since K has asymptoticintegration. By replacing b with − b if necessary, we may assume that b > 0, so either a ′ = (log b ) ′ or v ( a ′ − log b ) ′ ∈ (Γ > ) ′ . In either case we have a − log b 1, a contradiction.With this claim out of the way, we may apply [3, 10.5.20] with a ′ in place of s to get an H -field K ( f )extending K where f † = a ′ , f > C, Γ K ( f ) = Γ ⊕ Z vf, C K ( f ) = C. By Lemma 3.13 there is a unique logarithm on K ( f ) ℓ with log f = a . Now let L be a logarithmic H -fieldextension of K with a ∈ log( L > ). Then (exp a ) † = a ′ so by [3, 10.5.20], there is a unique H -field embedding K ( f ) → L that sends f to exp a . This in turn extends uniquely to an H -field embedding ı : K ( f ) ℓ → L .Since a = log ı ( f ), Lemma 3.13 gives that ı preserves logarithms. Since any logarithmic H -field embedding K ( f ) ℓ → L must send f to exp a , the map ı is unique as claimed. (cid:3) We now show that each ω -free logarithmic H -field has a minimal exponential H -field extension. Corollary 4.6. If K is ω -free, then K has an ω -free exponential H -field extension K e with constant field C and with the following property: for any exponential H -field L extending K , there is a unique logarithmic H -field embedding K e → L over K . Moreover, K e is d -algebraic over K and the only exponential H -subfieldof K e containing K is K e itself.Proof. We define an e -tower on K to be increasing chain of ω -free logarithmic H -fields ( K µ ) µ ν such thati. K = K ;ii. K µ = S λ<µ K λ for each limit ordinal µ ;iii. If K µ is not real closed, then K µ +1 = K rc ,ℓµ equipped with the logarithm in Lemma 4.3;iv. If K µ is real closed, then K µ +1 = K µ ( f µ ) ℓ where log f µ K µ and either log f µ ≺ K µ +1 is as in Lemma 4.4) or log f µ − log b ≻ b ∈ K >µ (in which case K µ +1 is as in Lemma 4.5).Since the extensions in Lemmas 4.3, 4.4, and 4.5 are all d-algebraic, they all preserve ω -freeness by Fact 1.4.This ensures that the assumptions in Lemma 4.5 are met whenever this lemma is applied, since ω -free H -fields have asymptotic integration. Induction on µ gives that each K µ is a Liouville extension of K , so | K µ | = | K | and C K µ = C for each µ ; see the proof of Lemma 3.9 for the definition of a Liouville extension.Thus, maximal e -towers exist, and we let ( K µ ) µ ν be a maximal e -tower on K .To see that K ν is an exponential H -field, let a ∈ K ν . If a − log b ≻ b ∈ K >ν , then ( K µ ) µ ν canbe extended by Lemma 4.5, so we take b ∈ K > with a − log b 1. If a − log b ≍ 1, then there is c ∈ C > with a − log b ∼ log c ∈ C × since log( C > ) = C . We have a − log( cb ) = a − log b − log c ≺ , so we may arrange that a − log b ≺ b with cb . We may take f ∈ K >µ with a − log b = log f ,since otherwise ( K µ ) µ ν can be extended using Lemma 4.4. Then log( f b ) = a , so a ∈ log( K >µ ). et K e := K ν . The proof that K e has the desired universal property goes the same way as the proof ofLemma 3.9, where we now appeal to the universal properties in Lemmas 4.3, 4.4, and 4.5. Minimality of K e follows from the universal property: if E is an exponential H -subfield of K e containing K , then there is aunique logarithmic H -field embedding K e → E ⊆ K e over K . Since the unique embedding K e → K e over K is the identity map, E must be all of K e . (cid:3) Remark 4.7. The assumptions in Corollary 4.6 can be weakened a bit: we can instead assume that K is λ -free, instead of ω -free. See [18] for the definition of λ -freeness and an indication of how the proof ofCorollary 4.6 should be changed to prove this stronger result. If K is ω -free, then a Newton-exponential closure of K is by definition a newtonian exponential H -field extension of K which embeds over K into any other newtonian exponential H -field extension of K .Alternating Corollaries 4.6 and 4.1, we see that Newton-exponential closures exist. Proposition 4.8. If K is ω -free, then K has a Newton-exponential closure K nt ,e which is d -algebraic over K and which has constant field C . Since any other Newton-exponential closure of K embeds into K nt ,e over K , we see that any Newton-exponential closure of K is d-algebraic over K and has constant field C . By Remark 3.7, any Newton-exponential closure is Liouville closed, so by Lemma 1.10, the underlying H -field of any Newton-exponentialclosure of K is a Newton-Liouville closure of the H -field K . Since any two Newton-Liouville closures of K are isomorphic over K by Fact 1.11, we see that any Newton-Liouville closure L of K admits a logarithmthat makes L a Newton-exponential closure of K . Fact 1.11 also gives us uniqueness and minimality ofNewton-exponential closures. Corollary 4.9. If K is ω -free, then any two Newton-exponential closures of K are isomorphic over K .If K nt ,e is a Newton-exponential closure of K , then the only newtonian exponential H -subfield of K nt ,e containing K is K nt ,e itself.Proof. Let K nt ,e and L be Newton-exponential closures of K . Then there is a logarithmic H -field embedding K nt ,e → L over K and the image of K nt ,e is in particular a newtonian Liouville closed H -subfield of L . Since L is a Newton-Liouville closure of K , the image of K nt ,e must equal L by Fact 1.11. Minimality also followsfrom Fact 1.11: any newtonian exponential H -subfield E of K nt ,e is in particular a newtonian Liouvilleclosed H -subfield of K nt ,e , so E must equal K nt ,e . (cid:3) Constructing ω -free logarithmic H -field extensions. In this subsection we will show that anylogarithmic H -field can be extended to an ω -free logarithmic H -field. Lemma 4.10. K has a logarithmic H -field extension with a gap and with constant field C .Proof. If K has a gap, then we are done, so we may assume that K has asymptotic integration. By Lemma 4.3there is a unique logarithm on K rc ,ℓ extending the logarithm on K . If K rc ,ℓ has a gap, then we are done. Ifnot, then we replace K by K rc ,ℓ and we assume that K is real closed. Then in particular, Γ is divisible so byLemma 1.2, K has an immediate H -field extension L with a gap creator s . By Corollary 3.14, the logarithmon K extends uniquely to a logarithm on L ℓ . Then L ℓ is real closed and has asymptotic integration since itis an immediate henselian extension of K . Thus we may replace K with L ℓ and assume that s ∈ K is a gapcreator over K . If s K ′ , then by Lemma 1.3, K has an immediate H -field extension K ( a ) where a ′ = s .Again by Corollary 3.14, the logarithm on K extends uniquely to a logarithm on K ( a ) ℓ so we may assumethat there is a ∈ K with a ′ = s . If there is b ∈ K > with a − log b 1, then v ( s − b † ) ∈ (Γ > ) ′ , contradicting [3,11.5.10]. Thus, a − log b ≻ b ∈ K > and we may apply Lemma 4.5 to get a logarithmic H -field K ( f ) ℓ extending K with log f = a . Then f † = a ′ = s so vf is a gap in K ( f ) ℓ by Lemma 1.2. (cid:3) Lemma 4.11. Let s ∈ K > and suppose that vs is a gap in K . Then K has an ω -free logarithmic H -fieldextension K y that contains an element y ≻ with y ′ = s , has constant field C , and has the followingproperty: If L is also a logarithmic H -field extension of K containing z ≻ with z ′ = s , then there is aunique logarithmic H -field embedding K y → L over K that sends y to z . roof. By [3, 10.5.11], K has an H -field extension K ( y ) with y ≻ y > 0, and y ′ = s . This extension hasthe following properties: C K ( y ) = C, Γ K ( y ) = Γ ⊕ Z vy, Ψ K ( y ) = Ψ ∪ (cid:8) ψ ( vy ) (cid:9) , Ψ < ψ ( vy ) . Set K := K ( y ). Since K is grounded there is no way to define a logarithm on K (or even on K ℓ ). Insteadwe use [3, 10.5.12] to build an H -field extension K ( y ) with y ≻ y > 0, and y ′ = y † . We have C K = C, Γ K = Γ K ⊕ Z vy , Ψ K = Ψ K ∪ (cid:8) ψ ( vy ) (cid:9) , Ψ K < ψ ( vy ) . Continuing in this manner, we build an H -field K y = K ( y , y , . . . ) ℓ where y = y and where y n ≻ y n > y ′ n +1 = y † n for each n . Then C K y = C, Γ K y = Γ ⊕ M n Z vy n , Ψ K y = Ψ ∪ (cid:8) ψ ( vy ) , ψ ( vy ) , . . . (cid:9) .K y is ω -free since it is the increasing union of its grounded subfields K ( y , . . . , y n ) ℓ ; see [3, 11.7.15]. ByLemma 3.13 there is a unique logarithm on K y with log y n = y n +1 for each n . Let L and z be as in thestatement of the lemma and for each n , let log n z denote the n th iterated logarithm of z . Then the universalproperties in [3, 10.5.11 and 10.5.12] give a unique H -field embedding K ( y , y , . . . ) → L over K that sends y n to the iterated logarithm log n z for each n . This extends uniquely to an H -field embedding ı : K y → L .Since log y n = y n +1 for each n , it is straightforward using Lemma 3.13 to check that ı preserves logarithms.Note ı as a logarithmic H -field embedding is uniquely determined by its restriction to K and by the conditionthat ı ( y ) = z . (cid:3) We made a choice in Lemma 4.11 to give s an infinite integral. Below, we see that we can choose to insteadgive s an infinitesimal integral (after replacing s with − s for convenience). Lemma 4.12. Let s ∈ K < and suppose that vs is a gap in K . Then K has an ω -free logarithmic H -fieldextension K y that contains an element y ≺ with y ′ = s , has constant field C , and has the followingproperty: If L is also a logarithmic H -field extension of K containing z ≺ with z ′ = s , then there is aunique logarithmic H -field embedding K y → L over K that sends y to z .Proof. By [3, 10.5.10], K has an H -field extension K ( y ) with y ≺ y > 0, and y ′ = s . This extension hasthe following properties: C K ( y ) = C, Γ K ( y ) = Γ ⊕ Z vy, Ψ K ( y ) = Ψ ∪ (cid:8) ψ ( vy ) (cid:9) , Ψ < ψ ( vy ) . Just as in the proof of Lemma 4.11, we build an H -field K y = K ( y , y , . . . ) ℓ where y n ≻ y n > 0, and y ′ n +1 = y † n for each n but where this time, y = y − . Since ψ ( vy − ) = ψ ( vy ), we have C K y = C, Γ K y = Γ ⊕ M n Z vy n , Ψ K y = Ψ ∪ (cid:8) ψ ( vy ) , ψ ( vy ) , . . . (cid:9) . Again, K y is ω -free since it is the increasing union of its grounded subfields K ( y , . . . , y n ) ℓ . The proofthat K y has the desired universal property is the same as the proof of Lemma 4.11 except for the followingchanges: we use [3, 10.5.10] in place of [3, 10.5.11] and we send each y n to log n ( z − ) instead of log n z . (cid:3) The following is immediate from Lemmas 4.10 and 4.11. Corollary 4.13. K has an ω -free logarithmic H -field extension with constant field C . Corollary 4.13 and Proposition 4.8 give us the following: Corollary 4.14. K has an ω -free newtonian exponential H -field extension with constant field C . The cuts Ψ ↓ and Γ < . In this subsection, we assume that K is ω -free and that Γ is divisible. We let z > H -field extension of K with Ψ ↓ < vz < (Γ > ) ′ . By [3, 13.4.10 and 13.4.12], K h z i is an H -field extension of K with C K h z i = C, Γ K h z i = Γ ⊕ Z vz, Ψ K h z i = Ψ . By [3, 9.8.6], vz is a gap in K h z i and so K h z † i is an immediate extension of K by Lemma 1.5. Moreover, z is d-transcendental over K by [3, 13.6.1]. emma 4.15. K has a logarithmic H -field extension K z that contains z and has the following property: if L is also a logarithmic H -field extension of K containing an element z ∗ > with Ψ ↓ < vz ∗ < (Γ > ) ′ , thenthere is a unique logarithmic H -field embedding K z → L over K that sends z to z ∗ .Proof. Since z is d-transcendental over K , Lemma 2.3 gives that z † K h z i ′ . In particular, z † K h z † i ′ .We build K z in three steps. First, set M := K h z † i h . We claim that z † M ′ . Suppose otherwise, so z † = y ′ for some y ∈ M . Then y is algebraic over K h z † i and so it is in K h z † i by Ostrowski’s Theorem, acontradiction. Since M is an immediate extension of K , it has asymptotic integration, so by Lemma 1.3, M has an immediate H -field extension M ( a ) where a ′ = z † . By Corollary 3.14, the logarithm on K extendsuniquely to a logarithm on M ( a ) ℓ . Set N := M ( a ) ℓ and note that N , being an immediate extension of K ,has asymptotic integration. If there is b ∈ N > with a − log b 1, then v ( s − b † ) ∈ (Γ > ) ′ , contradicting [3,11.5.10]. Thus, a − log b ≻ b ∈ N > and we may apply Lemma 4.5 to uniquely extend the logarithm on N to a logarithm on N ( z ) ℓ where log z = a . We let K z be the logarithmic H -field N ( z ) ℓ with this logarithm.Below, we include a diagram of the H -fields in this proof. All arrows are inclusions and all starred arrowsmarked are immediate extensions. The two unmarked arrows are not immediate extensions. K K h z † i K h z i K z M N ∗ ∗ ∗∗ Now let L and z ∗ be as in the statement of the lemma. By [3, 13.4.11] there is a unique H -field embedding K h z i → L over K that sends z to z ∗ . This restricts to an embedding K h z † i → L which then extends uniquelyto an embedding M → L . By Lemma 1.3 this in turn extends uniquely to an embedding M ( a ) → L thatsends a to log z ∗ and this further extends uniquely to an embedding N → L . This last embedding preserveslogarithms by Corollary 3.14 and, using Lemma 4.5, we extend this embedding a unique logarithmic H -fieldembedding K z → L that sends z to z ∗ . Since any logarithmic H -field embedding that sends z to z ∗ mustsend a = log z to log z ∗ , we see that this embedding is uniquely determined by the condition that it sends z to z ∗ . (cid:3) Note that K z above has a gap, namely vz . We now let y > H -field extension of K with Γ < < vy < Corollary 4.16. K has an ω -free logarithmic H -field extension K y that contains y and has the followingproperty: if L is also a logarithmic H -field extension of K containing an element y ∗ > with Γ < < vy ∗ < ,then there is a unique logarithmic H -field embedding K y → L over K that sends y to y ∗ .Proof. First, note y ′ > ↓ < vy ′ < (Γ > ) ′ . We let K y ′ be the logarithmic H -field extension of K containing y ′ as constructed in Lemma 4.15. We now use Lemma 4.11 with y ′ in place of s to construct an ω -free logarithmic H -field K y that extends K y ′ and contains y .Let L be also a logarithmic H -field extension of K containing an element y ∗ > < < vy ∗ < ↓ < v ( y ∗ ) ′ < (Γ > ) ′ so, by Lemma 4.15, there is a unique logarithmic H -field embedding K y ′ → L over K that sends y ′ to ( y ∗ ) ′ . By Lemma 4.11 this extends uniquely to a logarithmic H -field embedding K y → L that sends y to y ∗ . Of course, any H -field embedding K y → L over K that sends y to y ∗ must send y ′ to ( y ∗ ) ′ , so the embedding K y → L is uniquely determined by the condition that y be sent to y ∗ . (cid:3) Extensions controlled by asymptotic couples. In this subsection we assume that K is an ω -freenewtonian exponential H -field and we let L be a logarithmic H -field extension of K with C L = C . Weassume that K is maximal in L in the sense that there is no y ∈ L \ K for which K h y i is an immediateextension of K . By Section 11.4 in [3], the set (cid:8) v ( f − a ) : a ∈ K (cid:9) ⊆ Γ L has a largest element for each f ∈ L \ K . We define a best approximation to f ∈ L \ K to be an element b ∈ K such that v ( f − b ) = max (cid:8) v ( f − a ) : a ∈ K (cid:9) . Note that b is a best approximation to f if and only if v ( f − b ) Γ (this uses the fact that C L = C ).Let f ∈ L \ K . We set f := f , we let b be a best approximation to f , and we set f := ( f − b ) † . Then f K since K is Liouville closed and C L = C , so we may repeat this process: let b be a best approximation o f and set f := ( f − b ) † . Continuing in this way we construct a sequence ( f n ) of elements in L \ K anda sequence ( b n ) of elements in K such that b n is a best approximation of f n and such that f n +1 = ( f n − b n ) † .Now for each n > 0, choose an element a n ∈ K × such that a † n = b n and such that a n and f n − − b n − havethe same sign. For each n , we set m n := a − n +1 ( f n − b n ), so m n > 0. Then f n +1 − b n +1 = (cid:18) f n − b n a n +1 (cid:19) † , so m † n = a n +2 m n +1 for each n . Note that K h f i = K h m i = K ( m , m , . . . ). We will be using the followingfacts about K h f i , established in [3, 16.1.2 and 16.1.3]. Fact 4.17. K h f i is ω -free and Γ K h f i = Γ ⊕ M n Z v ( m n ) . By Fact 4.17 and the Zariski-Abhyankar Inequality, we see that m is d-transcendental over K . Thus,( m n ) is an ILD-sequence for m over K , as defined in Section 2. We have the following consequence ofProposition 2.2. Corollary 4.18. The sequence (log m n ) is algebraically independent over K h m i .Proof. Suppose towards a contradiction that (log m n ) is algebraically dependent over K h m i . Since (log m n ) ′ = m † n ∈ K h m i for each n and since L has the same field of constants C as K h m i , we may use Ostrowski’sTheorem to deduce that the sequence (log m n ) is C -linearly dependent over K h m i . Then the sequence (cid:0) (log m n ) ′ (cid:1) = ( m † n ) is C -linearly dependent over K h m i ′ , contradicting Proposition 2.2. (cid:3) We now build a sequence ( K n ) of henselian H -subfields of L as follows: K := K h m i h = K h f i h , K n +1 := K n (log m n ) h . Corollary 4.18 gives that log m n K n for each n . We set K ∞ := S n K n and we set K f := K ℓ ∞ . ByLemma 1.3, each K n is an immediate extension of K h f i , so K f is an immediate extension of K h f i as well.Corollary 3.12 and Fact 4.17 gives that K h f i is a logarithmic H -subfield of L . Since K f is a d-algebraicextension of K h f i , it is ω -free by Fact 1.4. Proposition 4.19. Let M be a logarithmic H -field extension of K and let g ∈ M realize the same cut as f over K . Then there is a unique logarithmic H -field embedding K f → M over K that sends f to g .Proof. By [3, 16.1.5] and the universal property of the henselization, there is a unique H -field embedding ı : K → M over K that sends f to g . For each n , set n n := ı ( m n ). Let n > H -fieldembeddings ı m : K m → M for each m n such that ı n | K m = ı m and ı n (log m m ) = log n m for each m < n .Since log m n K n , we use Lemma 1.3 and the universal property of the henselization to get a unique H -fieldembedding ı n +1 : K n +1 → M that extends ı n and sends log m n to log n n . The union of these embeddings isan embedding ı ∞ : K ∞ → M that sends log m n to log n n for each n . This extends to an H -field embedding K f → M that preserves logarithms in light of Lemma 3.13 and Fact 4.17. Since ı is the unique H -fieldembedding K → M which sends f to g and since any logarithmic H -field embedding K f → M must sendlog m n to log ı ( m n ) = log n n for each n , our embedding is unique as claimed. (cid:3) Adding elements at infinity. In this subsection we assume that K is ω -free and that Ψ is downwardclosed in Γ. This subsection will not be used in the proof of model completeness, but it will be used in theproof of local o-minimality. Let L be a logarithmic H -field extension of K with C L = C and let a ∈ L with a > K . Lemma 4.20. K h a i is ω -free and Γ K h a i = Γ ⊕ M n Z v (cid:0) a h n i (cid:1) . Proof. If we assume that K is Liouville closed, then this is just [3, 16.6.9 and 16.6.10]. However, the onlyconsequence of being Liouville closed that is used in the proof of [3, 16.6.9 and 16.6.10] is that Ψ is downwardclosed. (cid:3) t follows from the above lemma and the Zariski-Abhyankar Inequality that a is d-transcendental over K .Thus, (cid:0) a h n i (cid:1) is an ILD-sequence for a over K and we have the following: Corollary 4.21. The sequence (cid:0) log a h n i (cid:1) is algebraically independent over K h a i .Proof. Suppose not. Since (cid:0) log a h n i (cid:1) ′ = a h n +1 i ∈ K h a i for each n and since L has the same field ofconstants C as K h a i , we may use Ostrowski’s Theorem to deduce that the sequence (cid:0) log a h n i (cid:1) is C -linearlydependent over K h a i . Then the sequence (cid:0) (log a h n i ) ′ (cid:1) = (cid:0) ( a h n i ) † (cid:1) must be C -linearly dependent over K h a i ′ ,contradicting Proposition 2.2. (cid:3) As in the previous subsection, we build a sequence ( K n ) of henselian H -subfields of L as follows: K := K h a i h , K n +1 := K n (cid:0) log a h n i (cid:1) h . Corollary 4.21 gives that log a h n i K n for each n . We set K ∞ := S n K n and we set K a := K ℓ ∞ . Then K a is an immediate d-algebraic extension of K h a i , so K a is a logarithmic H -subfield of L by Corollary 3.12 andLemma 4.20 and K a is ω -free by Fact 1.4. Proposition 4.22. Let M be a logarithmic H -field extension of K and let b ∈ M with b > K . Then thereis a unique logarithmic H -field embedding K a → M over K that sends a to b .Proof. This is proven in much the same way as Proposition 4.19. By the proof of [3, 16.6.10] and theuniversal property of the henselization, there is a unique H -field embedding ı : K → M over K that sends a to b . Repeated applications of Lemma 1.3 and the universal property of the henselization gives us aunique H -field embedding K ∞ → M that extends ı and sends log a h n i to log b h n i for each n . This extendsuniquely to an H -field embedding : K a → M which preserves logarithms in light of Lemmas 3.13 and 4.20.As with Proposition 4.19, is unique as a logarithmic H -field embedding since ı is unique as an H -fieldembedding. (cid:3) Model completeness for T exp and applications In this section, we axiomatize a model complete theory of logarithmic H -fields. We show that T exp is amodel of this theory and we list some other models of this theory. Finally, we examine some consequencesof our model completeness result. The main ingredient is the following proposition. Proposition 5.1. Let K and L be ω -free newtonian exponential H -fields and assume the underlying orderedset of L is | K | + -saturated and the cofinality of Γ 0. By our cofinality assumptionon Γ 0. Using Corollary 4.16, we extend ı to a logarithmic H -fieldembedding of some ω -free logarithmic H -subfield E y ⊆ K containing y . We assume for the remainder of theproof that Γ Corollary 5.2. Let K and L be as in the statement of Proposition 5.1. If both K and L have small derivation,then any ordered exponential field embedding ı : C → C L extends to a logarithmic H -field embedding K → L .The same is true if both K and L have large derivation.Proof. We first consider the case that K and L have small derivation. Since K is ω -free and newtonian,there is x ∈ K with x ′ = 1. Additionally, x ≻ K has small derivation. We view C as a logarithmic H -subfield of K with trivial derivation and gap 0 and we let C x be the ω -free logarithmic H -field extensionof C containing x constructed in Lemma 4.11. We may identify C x with a logarithmic H -subfield of K . Since L is also ω -free and newtonian with small derivation, there is f ∈ L with f ≻ f ′ = 1, so ı extendsto a logarithmic H -field embedding C x → L which sends x to f by Lemma 4.11. This further extends to alogarithmic H -field embedding K → L by Proposition 5.1.Now suppose K and L have large derivation. Again, since K is ω -free and newtonian, there is y ∈ K with y ′ = − 1. Then y K has large derivation, so by subtracting a constant from y , we may assume that y ≺ 1. Let C y be the ω -free logarithmic H -field extension of C containing y constructed in Lemma 4.12,and identify C y with a logarithmic H -subfield of K . Take g ∈ L with g ≺ g ′ = − ı toa logarithmic H -field embedding C y → L which sends y to g . This further extends to a logarithmic H -fieldembedding K → L , again by Proposition 5.1. (cid:3) Model completeness and completeness. To get a model completeness result, we need to removethe assumption that C E = C in Proposition 5.1. In order to do this, we impose some additional requirementson the constant fields of K , L , and E . A logarithmic H -field K is said to have real exponential constantfield if its constant field C equipped with exp | C models Th( R exp ). Corollary 5.3. Let E , K , L and ı be as in the statement of Proposition 5.1, except we drop the assumptionthat C E = C . Assume in addition that the underlying ordered set of C L is | C | + -saturated and that E , K ,and L all have real exponential constant fields. Then ı extends to a logarithmic H -field embedding K → L .Proof. The theory of R exp is model complete and o-minimal by Wilkie’s theorem [30]. The saturationassumption on the underlying ordered set of C L gives us that C L is saturated as an ordered exponential fieldby o-minimality. By model completeness, the ordered exponential field embedding ı | C E : C E → C L extendsto an ordered exponential field embedding : C → C L . By Lemma 4.2 there is a unique logarithmic H -fieldembedding E ( C ) ℓ → L that extends both ı and . Since E ( C ) ℓ is d-algebraic over E , it is ω -free by Fact 1.4.Now apply Proposition 5.1 with E ( C ) ℓ in place of E . (cid:3) Let L log := { + , × , , , , , ∂ , log } , where and are binary relation symbols and where ∂ and log areunary function symbols. We view each logarithmic H -field K as an L log -structure in the obvious way, wherelog is defined to be identically zero on K . Let T log be the L log -theory of logarithmic H -fields with realexponential constant field and let T ntexp be the L log -theory of ω -free newtonian exponential H -fields with realexponential constant field. The theory T ntexp is consistent since it has a model; see Corollary 5.6. Theorem 5.4. The L log -theory T ntexp is model complete and it is the model companion of T log .Proof. The saturation assumptions for L and C L and the cofinality assumption for Γ L in Corollary 5.3 allhold when L is | K | + -saturated as an L log -structure. Model completeness for T ntexp follows from Corollary 5.3and a standard model completeness test; see [3, B.10.4]. By Corollary 4.14, every logarithmic H -field withreal exponential constant field can be extended to an ω -free newtonian exponential H -field with the samereal exponential constant field. Thus, T ntexp is the model companion of T log . (cid:3) We can use Theorem 5.4 and Corollary 5.2 to characterize the completions of T ntexp . Let T ntexp , sm be the L log -theory extending T ntexp whose models have small derivation and let T ntexp , lg be the L log -theory extending T ntexp whose models have large derivation. Let K | = T ntexp . Then K φ | = T ntexp , sm for any φ ∈ K > with vφ ∈ (Γ < ) ′ and K ψ | = T ntexp , lg for any ψ ∈ K > with vψ ∈ (Γ > ) ′ . Thus, T ntexp , sm and T ntexp , lg are both consistent since T ntexp is consistent. heorem 5.5. T ntexp , sm and T ntexp , lg are both the two completions of T ntexp .Proof. Let K, L | = T ntexp , sm and assume L is | K | + -saturated. Then C L is | C | + -saturated, so there is anordered exponential field embedding ı : C → C L since C and C L are elementarily equivalent; see [3, B.9.5].This in turn extends to an embedding : K → L by Corollary 5.2. Then is elementary since T ntexp is modelcomplete, so K and L are elementarily equivalent. This shows that T ntexp , sm is complete, and the same proofshows that T ntexp , lg is complete. (cid:3) Transseries. Let T exp be the natural expansion of the field of logarithmic exponential transseries to an L log -structure. In the introduction, T exp was the expansion of T by the exponential function and here, it isthe expansion of T by the logarithm function, but there is really no difference in terms of model completenessor definability. Corollary 5.6. The L log -theory of T exp is model complete and completely axiomatized by T ntexp , sm .Proof. The logarithm log on T exp extends the usual logarithm on its constant field R and satisfies (L1)–(L3)of Definition 3.1; see [13]. Moreover, log is surjective and by [3, 15.0.2], T is an ω -free newtonian H -fieldwith small derivation, so T exp | = T ntexp , sm . (cid:3) The only part of this axiomatization of T exp that is not known to be effective is the condition that modelsof T ntexp , sm have real exponential constant field. Of course, any effective axiomatization of Th( R exp ) can beused to give an effective axiomatization of T exp , but for the time being, such an axiomatization seems to beout of reach; see [24]. Many important H -subfields of T are logarithmic H -subfields of T exp . To help us findexamples, we use the following criterion: Lemma 5.7. Let log denote the logarithm on T exp and let K ⊇ R be an H -subfield of T exp . If K is henselianand log( K > ) ⊆ K , then K expanded by log | K > is a model of T log . If K is ω -free, newtonian, and Liouvilleclosed, then log( K > ) ⊆ K and K expanded by log | K > is a model of T ntexp , sm .Proof. Since the axioms in Definition 3.1 are universal, if K is henselian and closed under log, then K | = T log .Now assume that K is ω -free, newtonian, and Liouville closed. We need to show that K is closed under expand log. To see that K is closed under log, let a ∈ K > . Since (log a ) ′ = a † and since T nl is model complete,there is b ∈ K with b ′ = a † . Then log a = b + r for some r ∈ R , so log a ∈ K . Showing that K is closedunder exp is similar. (cid:3) By Lemma 5.7, the subfield T da of transseries that are d-algebraic over Q is an ω -free exponential H -subfield of T exp , as is the subfield T g of grid-based transseries. Thus, T da and T g are both elementary L log -substructures of T exp . The field T log of logarithmic transseries is an ω -free newtonian logarithmic H -subfield of T exp , but T log is not an exponential H -subfield of T exp . See [22] for more information about T g and see [19] for more information about T log .Using Theorem 5.5 and Corollary 5.6, we can transfer some known facts about T exp into formal consequencesof T ntexp . Corollary 5.8. Let K | = T ntexp . Then the underlying ordered exponential field of K models Th( R exp ) .Proof. If this corollary holds for K φ where φ ∈ K > , then it holds for K , so by compositionally conjugatingby a suitable element of K > , we may assume that K | = T ntexp , sm . Then K is elementarily equivalent to T exp ,and the underlying ordered exponential field of T exp is a model of Th( R exp ) by [13]. (cid:3) Let T be an o-minimal theory extending the theory of real closed ordered fields in a suitable language L . Let M | = T and let δ be a derivation on M . Following the terminology of [17], we say that δ is a T -derivationon M if for any open U ⊆ M n , any continuously differentiable function f : U → M that is L -definablewithout parameters, and any u = ( u , . . . , u n ) ∈ U , we have δ f ( u ) = ∇ f ( u ) · ( δ u , . . . , δ u n )where ∇ f is the gradient of f . Note that the statement “ δ is a T -derivation” can be expressed by sentencesin the language L ∪ { δ } . orollary 5.9. Let K | = T ntexp . Then ∂ is a Th( R exp ) -derivation on K .Proof. If ∂ is a Th( R exp )-derivation on K , then so is φ − ∂ for any φ ∈ K > . Thus, by compositionallyconjugating by a suitable element of K > , we may assume that K | = T ntexp , sm . Then K is elementarilyequivalent to T exp and the derivation on T exp is a Th( R exp )-derivation; see Example 2.15 in [17]. (cid:3) Surreal numbers. The field No of surreal numbers is a real closed field extension of R introduced byConway [8]. The surreals may be defined in a number of equivalent ways, but for our purposes, we define asurreal number to be a map a : γ → {− , + } , where γ is an ordinal. For such a , the ordinal γ is called the length of a (sometimes called the tree-rank or birthday of a , depending on which definition of the surrealsis being used). The collection of all surreal numbers is a proper class, and each ordinal γ is identified withthe surreal number of length γ which takes constant value +. For each γ , we let No ( γ ) be the set of surrealnumbers of length < γ .The surreals admit a (surjective) logarithm, defined by Kruskal and Gonshor [20] and with this logarithm, No is an elementary extension of R exp [11]. More recently, Berarducci and Mantova equipped the surrealswith a derivation that makes No an exponential H -field with small derivation and real exponential constantfield R [6]. With this derivation, the H -field No is newtonian and ω -free [5], so No is a model of T ntexp , sm .Let κ be a regular uncountable cardinal. Then the set No ( κ ) is an ω -free newtonian Liouville closed H -subfield of No containing R [5, Corollary 4.6]. By adapting the proof of Lemma 5.7, we see that No ( κ ) isclosed under exp and log, so it is an elementary L log -substructure of No . The next proposition, an analogueof [5, Theorem 3], shows that the surreal numbers are universal among models of T log with small derivationand real exponential constant field R . Proposition 5.10. Every set-sized logarithmic H -field with small derivation and real exponential constantfield R admits a logarithmic H -field embedding into No over R .Proof. Let K be a logarithmic H -field with real exponential constant field R and small derivation. It sufficesto show that some logarithmic H -field extension of K with the same constant field R admits a logarithmic H -field embedding into No over R . Since K has small derivation, either 0 ∈ (Γ < ) ′ or K has gap 0. In thesecond case, we can use Lemma 4.11 to extend K to an ω -free logarithmic H -field with 0 ∈ (Γ < ) ′ , so bypassing to an extension, we may assume that 0 ∈ (Γ < ) ′ . Then any logarithmic H -field extension of K hassmall derivation. By Corollary 4.14, K has an ω -free exponential H -field extension with the same constantfield R , so we may assume that K | = T ntexp , sm . Let κ := | K | + . Then No ( κ ) | = T ntexp , sm and by [5, Lemma 5.3],the underlying ordered sets of No ( κ ) and Γ No ( κ ) are κ -saturated. By Corollary 5.2 with No ( κ ) in place of L , the identity map on R extends to a logarithmic H -field embedding K → No ( κ ). (cid:3) There is a natural field embedding ı : T → No which was shown to be an elementary H -field embeddingin [5]. Since ı preserves logarithms, it is even an elementary logarithmic H -field embedding. In [16], ı wasshown to also be initial : if a is in ı ( T ), then so is a | λ for all λ less than the length of a . The followingquestion is similar to a question asked in [5]. Question 5.11. Does every model of T ntexp , sm with real exponential constant field R admit an initial logarith-mic H -field embedding into No over R ? Hardy fields. Recall that a Hardy field is a set of germs of real-valued functions at + ∞ that is closedunder differentiation and that forms a field under addition and multiplication. Every Hardy field containing R (that is, containing the germs of all constant functions) is an H -field with constant field R . Given a Hardyfield H and a germ f ∗ ∈ H > , we define the germ log f ∗ as follows: take any representative function f ( x ) for f ∗ that is strictly positive on an interval ( a, + ∞ ). Then log f ( x ) is also defined on the interval ( a, + ∞ ), andwe let log f ∗ be the germ of the function log f ( x ). Note that log f ∗ is not necessarily in H . Lemma 5.12. Let H be a henselian Hardy field containing R . If log f ∗ ∈ H for each f ∗ ∈ H > , then H equipped with the logarithm f ∗ log f ∗ is a model of T log .Proof. The logarithm in the statement of the lemma is a logarithm on H as defined in Definition 3.1:properties (L1) and (L2) are basic properties of the real logarithm function and property (L3) is just thechain rule. The restriction of this logarithm to R is just the normal logarithm on R , so H has real exponentialconstant field R . Since H is assumed to be henselian, we are done. (cid:3) very Hardy field has small derivation so by Proposition 5.10, every Hardy field satisfying the conditions inLemma 5.12 admits a logarithmic H -field embedding into the surreal numbers over R .Recently, Aschenbrenner, van den Dries and van der Hoeven finished the proof of their conjecture in [4] thatall maximal Hardy fields are ω -free and newtonian. They are currently preparing a paper on this result.By [7], any Hardy field has a Hardy field extension that contains R and is closed under log and exp. Thus,any maximal Hardy field is a model of T nlexp , sm . In particular, all maximal Hardy fields are elementarilyequivalent as differential exponential fields.5.5. Local o-minimality. Let K | = T ntexp . Our model completeness result can be used to show that K is locally o-minimal in the sense of [29]. Theorem 5.13. For each y ∈ K and each X ⊆ K that is L log -definable with parameters from K , there isan interval I around y such that X ∩ I is a finite union of points and intervals.Proof. We begin with the following claim. Claim. Let M be an elementary extension of K and let a ∈ M with a > K . Let N be an | M | + -saturatedelementary extension of K and let b ∈ N with b > K . Then there is a logarithmic H -field embedding ı : M → N over K that sends a to b .Proof of Claim. As in Corollary 5.3, we extend the identity map K → N to a logarithmic H -field embedding K ( C M ) ℓ → N . Set L := K ( C M ) ℓ . Then L is ω -free by Fact 1.4. Our assumption on K gives K = ( K × ) † ,so Ψ is downward closed in Γ. Then Ψ L is downward closed in Γ L as well, since Γ L = Γ. The fact thatΓ L = Γ also gives that a is greater than L and that b is greater than the image of L in N , so the assumptionsin Subsection 4.5 are met with L and M in place of K and L . Let L a be the ω -free logarithmic H -fieldextension of L containing a constructed in that subsection. By Proposition 4.22 with L and N in placeof K and M , there is a logarithmic H -field embedding L a → N that sends a to b . Using the saturationassumption on N and Proposition 5.1, we extend this to a logarithmic H -field embedding M → N . (cid:3) This claim and the fact that T ntexp is model complete gives that K is o-minimal at infinity , that is, for eachdefinable X ⊆ K there is f ∈ K with ( f, + ∞ ) ⊆ X or ( f, + ∞ ) ∩ X = ∅ . Deducing this from the claimis a standard model theoretic argument: suppose towards a contradiction that both X and K \ X containarbitrarily large elements of K . Then we can arrange that a ∈ X M and b ∈ N \ X N , where a , b , M , and N are as in the claim and where X M and X N are the natural extensions of X to M and N . Since T ntexp ismodel complete, the map ı : M → N constructed in the claim is elementary, so b = ı ( a ) ∈ ı ( X M ) ⊆ X N , a contradiction. The statement of the theorem follows from o-minimality at infinity by taking fractionallinear transformations. (cid:3) Restricted trigonometric functions In this section, we examine H -fields with restricted trigonometric functions and we prove that T rt , exp ismodel complete. Definition 6.1. Restricted trigonometric functions on K are functions sin : [ − , K → K and cos: [ − , K → K such that(RT1) sin( a + b ) = sin a cos b + cos a sin b and cos( a + b ) = cos a cos b − sin a sin b for all a, b ∈ [ − , K with a + b ∈ [ − , K ;(RT2) (sin a ) ′ = (cos a ) a ′ and (cos a ) ′ = − (sin a ) a ′ for all a ∈ [ − , K ;(RT3) sin( O ) ⊆ O and cos( O ) ⊆ O . Let sin and cos be restricted trigonometric functions on K . Then sin and cos remain restricted trigonometricfunctions in any compositional conjugate of K . For each c ∈ [ − , C we have (sin c ) ′ = (cos c ) c ′ = 0 sosin c ∈ C . Likewise, cos c ∈ C for all c ∈ [ − , C . With (RT3), this gives us sin(0) = 0 and cos(0) = 1. Thenext lemma shows that the restrictions of sin and cos to O are definable in the underlying H -field of K . emma 6.2. Let sin and cos be restricted trigonometric functions on K , let a ∈ O = , and let A be thehomogeneous linear differential polynomial A ( Y ) = − ( a ′ ) Y + ( a ′ ) † Y ′ − Y ′′ . Then sin a is the unique zero of A in a (1 + O ) and cos a is the unique zero of A in a O .Proof. We will first show that sin a and cos a are both zeros of A . We have A (sin a ) = − ( a ′ ) (sin a ) + ( a ′ ) † (sin a ) ′ − (sin a ) ′′ = − ( a ′ ) (sin a ) + ( a ′ ) † (cos a ) a ′ − (cid:0) (cos a ) a ′ (cid:1) ′ = − ( a ′ ) (sin a ) + ( a ′ ) † (cos a ) a ′ + (sin a )( a ′ ) − (cos a ) a ′′ = 0 . Likewise, A (cos a ) = 0. Now we will show that sin a ∈ a (1 + O ) and that cos a ∈ a O . Since cos a ∼ 1, wehave (sin a ) ′ = (cos a ) a ′ ∼ a ′ . Then sin a ∼ a since K is asymptotic and a, sin a ≺ a ∼ a ≺ 1, we have(cos a − ′ = − (sin a ) a ′ ≺ a ′ . Again, this gives cos a − ≺ a since K is asymptotic.It remains to show uniqueness. Since A is an order two homogeneous linear differential polynomial, theset of zeros of A in K is a C -linear subspace of K of dimension at most 2; see [3, 4.1.14]. Moreover, sin a and cos a are C -linearly independent since c sin a ∈ O for all c ∈ C , so the set { sin a, cos a } forms a basis forthis subspace. Let b = c sin a + c cos a be an arbitrary zero of A in K , where c , c ∈ C . If b ∼ a , then c must be 0, since otherwise b ∼ c ≍ b = c sin a ∼ c a , so c must be 1 and b = sin a . If b − ≺ a , then c must be 1, since otherwise b − ∼ c − ≍ 1. This gives b − c sin a + cos a − ∈ c sin a + a O and so c must be 0 and b = cos a , since otherwise b ∼ c sin a ≍ a . (cid:3) Let L rt , log := L log ∪ { sin , cos } , where sin and cos are unary function symbols. Letˆ L := L rt , log \ { , ∂ } = { + , × , , , , log , sin , cos } . Let R rt , exp be the natural expansion of R exp by restricted trigonometric functions and let Th( R rt , exp ) be theˆ L -theory of R rt , exp . Then Th( R rt , exp ) is model complete and o-minimal by van den Dries and Miller [14].Let T ntrt , exp be the L rt , log -theory asserting that for K | = T ntrt , exp :(1) K is an ω -free newtonian exponential H -field;(2) sin and cos are identically zero off of [ − , K and the restrictions of sin and cos to [ − , K arerestricted trigonometric functions on K ;(3) The expansion of C by exp | C , sin | [ − , C and cos | [ − , C models Th( R rt , exp ).We will see that T ntrt , exp is has a model in Corollary 6.6. If K | = T ntrt , exp , then we view C as an ˆ L -structurein the natural way. Lemma 6.2 gives us a criterion for when a logarithmic H -field embedding is actually an L rt , log -embedding. Corollary 6.3. Let K, L | = T ntrt , exp , let ı : K → L be a logarithmic H -field embedding, and suppose ı | C : C → C L is an ˆ L -embedding. Then ı is an L rt , log -embedding.Proof. We need to show that ı (sin f ) = sin ı ( f ) and ı (cos f ) = cos ı ( f ) for all f ∈ [ − , K . This holds if f ∈ C , so let f ∈ [ − , K and suppose f C . Then there is a unique c ∈ [ − , C and a unique a ∈ O = with f = c + a . We have ı (sin f ) = ı (sin c cos a + cos c sin a ) = ı (sin c ) ı (cos a ) + ı (cos c ) ı (sin a )by (RT1). Likewise, ı (cos f ) = ı (cos c ) ı (cos a ) − ı (sin c ) ı (sin a ). By our assumption on ı , it is enough to showthat ı (sin a ) = sin ı ( a ) and that ı (cos a ) = cos ı ( a ). Let A be the homogeneous linear differential polynomialover K from Lemma 6.2 and let ıA be the image of A under ı , that is, ıA ( Y ) = − ı ( a ′ ) Y + ı ( a ′ ) † Y ′ − Y ′′ . y Lemma 6.2 we know that sin a is a zero of A and that sin a ∼ a , so ı (sin a ) is a zero of ıA and ı (sin a ) ∼ ı ( a ).Then ı (sin a ) = sin ı ( a ), since sin ı ( a ) is the unique zero of ıA in ı ( a )(1+ O L ). Likewise, ı (cos a ) = cos ı ( a ). (cid:3) Theorem 6.4. T ntrt , exp is model complete.Proof. Let K , L , and E be models of T ntrt , exp where E is an L rt , log -substructure of K and where L is | K | + -saturated as an L rt , log -structure. Let ı : E → L be an L rt , log -embedding. To show that T ntrt , exp is modelcomplete, it is enough to show that ı extends to an L rt , log -embedding K → L ; see [3, B.10.4]. We may view ı | C E : C E → C L as an ˆ L -embedding and, using that Th( R rt , exp ) is model-complete, we may extend ı | C E toan ˆ L -embedding : C → C L . By Lemma 4.2 there is a unique logarithmic H -field embedding E ( C ) ℓ → L that extends both ı and . Since E ( C ) ℓ is d-algebraic over E , it is ω -free by Fact 1.4, so by Proposition 5.1with E ( C ) ℓ in place of E , we have a logarithmic H -field embedding K → L which extends both ı and .This is even an L rt , log -embedding by Corollary 6.3. (cid:3) Again, we can characterize the completions of T ntrt , exp . Theorem 6.5. T ntrt , exp has two completions: T ntrt , exp , sm , whose models are the models of T ntrt , exp with smallderivation, and T ntrt , exp , lg , whose models have large derivation.Proof. Consistency of T ntrt , exp , sm and T ntrt , exp , lg follows from consistency of T ntrt , exp and the remarks beforeTheorem 5.5. For completeness, let K, L | = T ntrt , exp , sm and assume L is | K | + -saturated. Then C and C L areelementarily equivalent as ˆ L -structures and C L is | C | + -saturated, so there is an ˆ L -embedding ı : C → C L . Inparticular, ı is a logarithmic H -field embedding, so it extends to a logarithmic H -field embedding : K → L by Corollary 5.2. It follows from Corollary 6.3 and Theorem 6.4 that is an elementary L rt , log -embedding,so K and L are elementarily equivalent. This shows that T ntrt , exp , sm is complete, and the same proof showsthat T ntrt , exp , lg is complete. (cid:3) Let T rt , exp be the natural expansion of T exp to an L rt , log -structure; see [13] for details. One can easily checkthat the restricted trigonometric functions in this expansion satisfy (RT1)–(RT3), so we have the following: Corollary 6.6. The L rt , log -theory of T rt , exp model complete and completely axiomatized by T ntrt , exp , sm . Again, T ntrt , exp , sm is effective relative to Th( R rt , exp ). We can use Theorem 6.5 and Corollary 6.6 to deducethe following analogues of Corollaries 5.8 and 5.9: Corollary 6.7. If K | = T ntrt , exp , then the ˆ L -reduct of K models Th( R rt , exp ) and ∂ is a Th( R rt , exp ) -derivationon K .Proof. Both of these properties are invariant under compositional composition and hold for T rt , exp ; see [13]and [17]. (cid:3) We can also amend our proof of Theorem 5.13 to show that any model of T ntrt , exp is locally o-minimal. Corollary 6.8. Let K | = T ntrt , exp . For each y ∈ K and each X ⊆ K that is L rt , log -definable with parametersfrom K , there is an interval I around y such that X ∩ I is a finite union of points and intervals.Proof. Let M be an elementary extension of K and let a ∈ M with a > K . Let N be an | M | + -saturatedelementary extension of K and let b ∈ N with b > K . As in the proof of Theorem 5.13, it suffices to showthat there is an L rt , log -embedding ı : M → N over K that sends a to b . Following the proof of Theorem 6.4,we extend the identity map K → N to an L rt , log -embedding ı : K ( C M ) ℓ → N . By Theorem 5.13, ı extendsto a logarithmic H -field embedding : M → N that sends a to b . Then is even an L rt , log -embedding byCorollary 6.3. (cid:3) Final Remarks and Future Directions An alternative axiomatization. Let K be an H -field. We say that K has the Intermediate ValueProperty (IVP) if for all r > 0, all P ∈ K [ Y , . . . , Y r ] and all f < g ∈ K with P ( f, f ′ , . . . , f ( r ) ) < < P ( g, g ′ , . . . , g ( r ) ) , there is y ∈ K with f < y < g and P ( y, y ′ , . . . , y ( r ) ) = 0. In [4], it was announced that the theory of ω -freenewtonian Liouville closed H -fields has an alternative axiomatization: act 7.1. A Liouville closed H -field is ω -free and newtonian if and only if it has IVP. This alternative axiomatization relies heavily on the fact that the field T g of grid-based transseries hasIVP [21]. Now let K be an exponential H -field. If K is ω -free and newtonian, then K is Liouville closedby Remark 3.7, so K has IVP by Fact 7.1. Now assume K has nontrivial derivation and IVP. Let a ∈ K and take f ∈ K > with f ≻ f ′ ≻ a (finding such an f uses that K have nontrivial derivation). Then f ′ − a ∼ f ′ > − f ) ′ − a < 0, so by IVP, there is y ∈ K with | y | < f and y ′ = a . This shows that K ′ = K , so K is Liouville closed by Remark 3.7 and K is ω -free and newtonian by Fact 7.1. We summarizebelow: Remark 7.2. An exponential H -field with nontrivial derivation is ω -free and newtonian if and only if it hasIVP. In particular, the models of T ntexp are exactly the exponential H -fields with nontrivial derivation, realexponential constant field, and IVP. One can make an analogous definition for differential exponential polynomials : say that an exponential H -field K has exp-IVP if for all r > 0, all P ∈ K [ Y , . . . , Y r ] and all f < g ∈ K with P (cid:0) f, . . . , f ( r ) , exp f, . . . , exp f ( r ) (cid:1) < < P (cid:0) g, . . . , g ( r ) , exp g, . . . , exp g ( r ) (cid:1) , there is y ∈ K with f < y < g and P (cid:0) y, . . . , y ( r ) , exp y, . . . , exp y ( r ) (cid:1) = 0. Question 7.3. Does every model of T ntexp have exp -IVP? Quantifier elimination. It seems reasonable to believe that the theory T ntexp is combinatorially tame andthat the definable sets in any model of this theory are geometrically tame. For instance, T ntexp is likely NIP oreven distal and the constant field of any model of T ntexp is probably stably embedded as a model of Th( R exp ).The appropriate analogues of all of these properties hold for T nl , but in order to prove that T ntexp enjoys theseproperties, it is invaluable to have a quantifier elimination result at hand. As in the case of T nl , quantifierelimination will almost surely have to involve some additions to our language. Let K | = T ntexp and let L Λ , Ω , dflog be the extension of L log by the following symbols:(1) A unary predicate Λ , to be interpreted in K as the set {− y †† : y ∈ K, y ≻ } ;(2) A unary predicate Ω , to be interpreted in K as the set { a ∈ K : 4 y ′′ + ay = 0 for some y ∈ K × } ;(3) A function symbol e f for each function f : R n → R that is definable without parameters in R exp .Since K | = Th( R exp ) by Corollary 5.8, each of these function symbols has a natural interpretationas a function K n → K . Question 7.4. Does T ntexp eliminate quantifiers in the language L Λ , Ω , dflog ? The additions of Ω and Λ are necessary to prove quantifier elimination for the theory T nl , so they will almostcertainly be required to prove quantifier elimination for T ntexp . It is not currently known whether Th( R exp )admits quantifier elimination in a “nice” language, but it has been known for some time that Th( R exp )does not eliminate quantifiers in the natural language of ordered exponential fields [10]. Since Th( R exp ) iso-minimal, this theory has definable Skolem functions, so we do know that it admits quantifier eliminationafter adding a function symbol for each function that is definable without parameters. A similar approachshould work with T ntrt , exp . We note that it may be possible to get quantifier elimination by only interpretingthe function symbols in (3) as functions from C n → C (and as identically zero off of C n ). Model completeness for T an , exp . We do not expect that our proof that T rt , exp is model complete canbe generalized to show that T an , exp —the expansion of T exp by all restricted analytic functions—is modelcomplete. Indeed, our model completeness result for T rt , exp relies heavily on the fact that the restrictions ofsin and cos to O are definable in the underlying H -field of any model of T ntrt , exp . Our proof that T exp is modelcomplete suggests a model completeness result for T an can be “upgraded” to a model completeness result for T an , exp by “adding” a logarithm function to each step. The proof that R an , exp eliminates quantifiers in [12]further substantiates the philosophy that one should start with restricted analytic functions and add thelogarithms later. First steps towards a model completeness result for T an are considered under the umbrellaof H T -fields in [23], but a full proof of model completeness for T an will likely take quite a bit of work. eferences [1] M. Aschenbrenner and L. van den Dries. H -fields and their Liouville extensions. Math. Z. , 242(3):543–588, 2002.[2] Liouville closed H -fields. J. Pure Appl. Algebra , 197(1):83–139, 2005.[3] M. Aschenbrenner, L. van den Dries, and J. van der Hoeven. Asymptotic Differential Algebra and Model Theory ofTransseries . Number 195 in Annals of Mathematics Studies. Princeton University Press, 2017.[4] On numbers, germs, and transseries. In Proceedings of the International Congress of Mathematicians—Rio deJaneiro 2018. Vol. II. Invited lectures , pages 1–23. World Sci. Publ., Hackensack, NJ, 2018.[5] The surreal numbers as a universal H -field. J. Eur. Math. Soc. (JEMS) , 21(4):1179–1199, 2019.[6] A. Berarducci and V. Mantova. Surreal numbers, derivations and transseries. J. Eur. Math. Soc. (JEMS) , 20:339–390,2018.[7] N. Bourbaki. Fonctions d’une Variable R´eelle, Chapitre V, Appendice Corps de Hardy. Fonctions (H) . Hermann, Paris,1976.[8] J. H. Conway. On Numbers and Games . L.M.S. monographs. Academic Press, 1976.[9] B. Dahn and P. G¨oring. Notes on exponential-logarithmic terms. Fund. Math. , 127(1):45–50, 1987.[10] L. van den Dries. Remarks on Tarski’s problem concerning ( R , + , · , exp). In Logic colloquium ’82 (Florence, 1982) , volume112 of Stud. Logic Found. Math. , pages 97–121. North-Holland, Amsterdam, 1984.[11] L. van den Dries and P. Ehrlich. Fields of surreal numbers and exponentiation. Fund. Math. , 167(2):173–188, 2001.[12] L. van den Dries, A. Macintyre, and D. Marker. The elementary theory of restricted analytic fields with exponentiation. Ann. Math. , 140:183–205, 1994.[13] Logarithmic-exponential power series. J. Lond. Math. Soc. (2) , 56(3):417–434, 1997.[14] L. van den Dries and C. Miller. On the real exponential field with restricted analytic functions. Israel J. Math. , 85(1-3):19–56, 1994.[15] J. Ecalle. Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac . Actualit´esMath´ematiques. Hermann, Paris, 1992.[16] P. Ehrlich and E. Kaplan. Surreal ordered exponential fields. ArXiv Mathematics e-prints , 2020. http://arxiv.org/abs/2002.07739/ .[17] A. Fornasiero and E. Kaplan. Generic derivations on o-minimal structures. ArXiv Mathematics e-prints , 2019. http://arxiv.org/abs/1905.07298/ .[18] A. Gehret. A tale of two Liouville closures. Pacific J. Math. , 290(1):41–76, 2017.[19] Towards a model theory of logarithmic transseries . PhD thesis, University of Illinois at Urbana-Champaign, 2017.[20] H. Gonshor. An Introduction to the Theory of Surreal Numbers . Cambridge University Press, 1986.[21] J. van der Hoeven. A differential intermediate value theorem. In Differential equations and the Stokes phenomenon , pages147–170. World Sci. Publ., River Edge, NJ, 2002.[22] Transseries and real differential algebra , volume 1888 of Lecture Notes in Mathematics . Springer-Verlag, Berlin,2006.[23] E. Kaplan. H T -fields, T -Liouville closures, and step-completions. In preparation .[24] A. Macintyre and A. J. Wilkie. On the decidability of the real exponential field. In Kreiseliana , pages 441–467. A K Peters,Wellesley, MA, 1996.[25] A. Ostrowski. Sur les relations alg´ebriques entre les int´egrales ind´efinies. Acta Math. , 78:315–318, 1946.[26] N. Pynn-Coates. Newtonian valued differential fields with arbitrary value group. Comm. Algebra , 47(7):2766–2776, 2019.[27] M. Rosenlicht. On the value group of a differential valuation. II. Amer. J. Math. , 103(5):977–996, 1981.[28] V. R. Srinivasan. On certain towers of extensions by antiderivatives . PhD thesis, University of Oklahoma, 2009.[29] C. Toffalori and K. Vozoris. Notes on local o-minimality. MLQ Math. Log. Q. , 55(6):617–632, 2009.[30] A. J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functionsand the exponential function. J. Amer. Math. Soc. , 9:1051–1094, 1996.