Valuation theory of exponential Hardy fields^{∗}

Franz-Viktor and Salma Kuhlmann

Dedicated to Murray Marshall on the occasion of his 60th birthday

29. 6. 2000

1 Introduction

In this paper, we analyze the structure of the Hardy fields associated with o-minimal expansions of the reals with exponential function. In fact, we work in the following more general setting. We take T to be the theory of a polynomially bounded o-minimal expansion P of the ordered field of real numbers by a set F_{T} of real-valued functions. We assume that F_{T} contains a symbol for every 0-definable function. Further, we assume that T defines the restricted exponential and logarithmic functions (cf. [D–M–M1]). Then also T (exp) is o-minimal (cf. [D–S2]). Here, T (exp) denotes the theory of the expansion (P, exp) where exp is the un-restricted real exponential function. Finally, we take any model R of T (exp) which contains (ℝ, +, ·, <, F_{T} , exp) as a substructure. Then we consider the Hardy field H(R) (see Section 2.2 for the definition) as a field equipped with convex valuations. Under the given assumptions, it is equal to the closure LE_{R,F_{T}} (x) of its subfield R(x) under F_{T} , exp and its inverse log; here, x denotes the germ of the identity function. We analyze its valuation theoretical structure by explicitly showing how LE_{R,F_{T}} (x) can be built up (cf. Section 3.3). Our construction method yields the following result (see Section 3.4 for definitions):

Theorem 1.1 Every model R as chosen above is levelled.

This implies that T (exp) has levels with parameters, in the sense of [M–M], and is exponentially bounded (cf. Theorem 3.11). We can determine the level of a function explicitly: it is the difference of two numbers which come up naturally in our construction method.

In Section 3.5 we use our main structure theorem (Theorem 3.11) to deduce:

Theorem 1.2 Suppose that for all r ∈ ℝ, F_{T} contains the power function

P_{r} : (0, ∞) ⟶ ℝ
x ↦ x^{r} .

∗This paper was written while both authors were partially supported by a Canadian NSERC research grant.

Let R_{T} denote the reduct of R to the language of T. Then the Hardy field H(R_{T}) is maximal among the Hardy subfields of H(R) associated with polynomially bounded reducts of R.

L. v. d. Dries conjectured that

ℝ_{an,powers} = (ℝ, +, ·, 0, 1, <, F_{an}, {P_{r} | r ∈ ℝ}) ,

the expansion of the ordered field of real numbers by the set F_{an} of restricted analytic functions and the power functions P_{r}, is a maximal polynomially bounded reduct of

ℝ_{an,exp} = (ℝ, +, ·, 0, 1, <, F_{an}, exp}) ,

At least on the level of Hardy fields, this is true: since the elementary theory of ℝ_{an,powers} is polynomially bounded and o-minimal and the power functions are definable in ℝ_{an,exp} , the foregoing theorem shows (cf. Theorem 3.16 for a more general result): H(ℝ_{an,powers}) is maximal among the Hardy subfields of H(ℝ_{an,exp}) associated with polynomially bounded reducts of ℝ_{an,exp} .

In Sections 4 and 5 we answer a question raised by A. Macintyre. In the paper [D–M– M2], the authors give an explicit construction of a nonarchimedean model of the theory of the reals with restricted analytic functions and exponentiation, called the logarithmic exponential power series field. They use the results of [R–M] about truncation-closed embeddings in generalized power series fields to answer a problem raised by Hardy, and to show that certain functions, including the Gamma-function and the Riemann Zetafunction, cannot be defined using exponential function, logarithm and restricted analytic functions. Macintyre asked whether the results of [D–M–M2] can be deduced by a “more invariant” version of truncation. Indeed, we establish the results of [D–M–M2] without using embeddings in the logarithmic exponential power series field. We replace truncation results by an intrinsic property of the Hardy field of the expansion ℝ_{an,exp} of the reals by restricted analytic functions and the exponential function. This property is expressed by structure theorems for the residue fields of arbitrary convex valuations. It is invariant because it does not depend on an embedding in logarithmic exponential power series fields. Note that there is an abundance of convex valuation rings that are not T (exp)-convex (cf. Theorem 3.11). For these, the methods of [D–L] are not applicable.

It is well known that the residue field of a real closed ordered field K is (up to order preserving isomorphism) a real closed ordered subfield of K. Now the question arises: if K has more structure, how much of it can be preserved on the embedded residue field? Take K = H(R) and w a convex valuation which is trivial on R. If O_{w} is not T (exp)-convex, then the residue field will not be closed under exp. This problem can be approached as follows. Almost like building up H(R) = LE_{R,F_{T}} (x) with our construction method, we can build up a subfield LE_{R,F_{T}}^{w} (x) of O_{w} by starting with any subfield of O_{w} which properly contains R, and closing under the same functions as before, except for exp. For the function exp we apply the method only as long as it does not produce elements outside of O_{w}. See Theorem 4.7.

Hardy conjectured that the compositional inverse of (log x)(log log x) is not asymptotic to an element of the Hardy field LE. This is defined to be the smallest subfield of H(ℝ_{an,exp}) which is real closed, exp- and log-closed and contains ℝ(x). It coincides with the field of the germs of all compositions of semialgebraic functions, exp and log. For our intrinsic solution of the Hardy problem, we also need to know the residue fields of convex valuations w on the Hardy field LE. But this field is not definably closed in H(ℝ_{an,exp}). In fact, the compositional inverse of (log x)(log log x) is 0-definable over LE, but not an element of it. Hence, LE is not of the form LE_{R,F_{T}} (x). But from its definition we see that it is the closure of ℝ(x) under a subset of F_{T} (for instance, the set of semi-algebraic functions), exp and log. Therefore, for F ⊆ F_{T} we explicitly construct the smallest field which is real closed and closed under F, exp and log; we denote it by LE_{R,F}(x). Similarly, we construct corresponding subfields LE_{R,F}^{w}(x) of O_{w}. The only necessary condition on F for our construction is that it contains the restricted exp and log.

Under certain additional conditions on F (see Section 4), Theorem 4.7 then tells us that LE_{R,F}^{w}(x) coincides with the residue field LE_{R,F}(x)w. The conditions are fulfilled by any set F of restricted analytic functions which is closed under partial derivations and contains the restricted exp and log. Furthermore, H(ℝ_{an,exp}) is equal to LE_{ℝ,F_{an}}(x) (cf. Section 5 of [D–M–M1]). If F_{LE} is the smallest subset of F_{an} which contains the restricted exp and log and is closed under partial derivations, then LE is equal to LE_{ℝ,F_{LE}}(x) (cf. Section 3 of [D–M–M2]). Thus, Theorem 4.7 gives us information about the residue fields of H(ℝ_{an,exp}) and of LE. An important point for our solution of the Hardy problem is that by our construction we obtain the residue field LEw inside of H(ℝ_{an,exp})w (cf. Corollary 4.8). This is clear since if F_{1} ⊆ F_{2} ⊆ F_{T} , then LE_{R,F_{1}}^{w}(x) ⊆ LE_{R,F_{2}}^{w}(x).

It would be interesting to verify that our condition (COMP) given in Section 4 is satisfied by sets of Gevrey functions (as it is the case for sets of restricted analytic functions), or by sets of convergent generalized power series for which the exponents of each variable form a sequence cofinal in ℝ (indexed by the natural numbers), cf. [D–S]. Although the condition on the exponents is quite restrictive, it holds for the presently known applications of interest. In particular, the function ζ(− log x) = ∑_{n=1}^{∞} x^{log n} on [0, e^{−2}] (with ζ the Riemann zeta function) satisfies the condition. It is not known whether the results on residue fields can be established without the restriction on the exponents.

In Section 6, we introduce an intrinsic form of power series expansions for the elements of LE_{R,F}(x). For this, we use monomials (which are obtained from elements in the image of an arbitrary cross-section by multiplication with reals) together with coefficients from significant residue fields LE_{R,F}(x)w. From such an expansion of a function h ∈ H(ℝ_{an,exp}), one can define the principal part of h, which turns out to carry information about the asymptotic behaviour of the function exp h(x) (Theorem 6.4). This puts the particular solution of the Hardy problem in a more general framework (Corollary 6.5).

2 Some preliminaries

If (K, w) is a valued field, then we write wa for the value of a ∈ K and wK for its value group {wa | 0 ≠ a ∈ K}. Further, we write aw for the residue of a, and Kw for the residue field. The valuation ring is denoted by O_{w}. For generalities on valuation theory, see [R].

2.1 Convex valuations

A valuation w on an ordered field K is called convex if O_{w} is convex. The convex valuation rings of an ordered field are linearly ordered by inclusion. If O_{w} ⊂_{≠} O_{w′} then w is said to be finer than w′. There is always a finest convex valuation, called the natural valuation. It is characterized by the fact that its residue field is archimedean. A valuation w on an ordered field is convex if and only if the natural valuation is finer or equal to w. Throughout this paper, v will always denote the natural valuation, unless stated otherwise.

If a, b are elements of an ordered group or an ordered field, then we write a ≪ b < 0 if a < b < 0 and ∀n ∈ ℕ : a < nb. Similarly, a ≫ b > 0 if a > b > 0 and ∀n ∈ ℕ : a > nb. We set |a| := max{a, −a}. Then the natural valuation is characterized by:

va < vb ⇔ |a| ≫ |b| .  (1)

Note that if ℝ ⊂ K and a ∈ K with va = 0, then there is some r ∈ ℝ such that v(a − r) > 0. Further, wr = 0 for every r ∈ ℝ and every convex valuation w.

Lemma 2.1 Let v, w be arbitrary valuations on some field K. Suppose that v is finer than w. Then for all a, b ∈ K,

va ≤ vb ⇒ wa ≤ wb .  (2)

In particular, wa > 0 ⇒ va > 0. Further, H_{w} := {vz | z ∈ K ∧ wz = 0} is a convex subgroup of the value group vK of v. We have that vz ∈ H_{w} ⇔ z ∈ O_{w}^{×}. There is a canonical isomorphism wK ≃ vK/H_{w}. Conversely, every convex subgroup of vK is of the form H_{w} for some valuation w such that v finer or equal to w.

The valuation v of K induces a valuation v/w on Kw. There are canonical isomorphisms v/w(Kw) ≃ H_{w} and (Kw)v/w ≃ Kv. If Kw is embedded in O_{w} such that the restriction of the residue map is the identity on Kw, then v/w = v|_{Kw} (up to equivalence). Writing v instead of v|_{Kw}, we then have that v(Kw) = H_{w} and (Kw)v = Kv.

We will call H_{w} the convex subgroup associated with w and w the valuation associated with H_{w}. Since the isomorphism is canonical, we will write wK = vK/H_{w}.

The order type of the chain of nontrivial convex subgroups of an ordered abelian group G is called the rank of G. If finite, then the rank is not bigger than the maximal number of rationally independent elements in G. In particular, G has finite rank if it is finitely generated or equivalently, if its divisible hull is a ℚ-vector space of finite dimension.

From (1) and (2) it follows that for every convex valuation w,

|a| ≤ |b| ⇒ wa ≥ wb .  (3)

Lemma 2.2 Let w be any valuation on K(x_{i} | i ∈ I_{1} ∪ I_{2}) such that the values wx_{i}, i ∈ I_{1} , are rationally independent over wK, and the residues x_{i}w, i ∈ I_{2} , are algebraically independent over Kw. Then the elements x_{i}, i ∈ I_{1} ∪ I_{2} are algebraically independent over K. Moreover,

wK(x_{i} | i ∈ I_{1} ∪ I_{2}) = wK ⊕ ⨁_{i∈I_{1}} ℤwx_{i} and K(x_{i} | i ∈ I_{1} ∪ I_{2})w = Kw(x_{i}w | i ∈ I_{2}) .

(4)

For the proof, see [B], chapter VI, §10.3, Theorem 1.

Corollary 2.3 Suppose that ℝ(x_{i} | i ∈ I) is an ordered field such that the values vx_{i}, i ∈ I are rationally independent. Let w be a convex valuation on ℝ(x_{i} | i ∈ I). Assume that there is a subset I_{w} ⊂ I such that wx_{i} = 0 for all i ∈ I_{w} and that the values wx_{i}, i ∈ I \ I_{w} are rationally independent. Then

wℝ(x_{i} | i ∈ I) = ⨁_{i∈I\I_{w}} ℤwx_{i} and ℝ(x_{i} | i ∈ I)w = ℝ(x_{i} | i ∈ I_{w}) .

Proof: For i ∈ I_{w}, wx_{i} = 0 implies that vx_{i} ∈ H_{w}. By the foregoing lemma, vℝ(x_{i} | i ∈ I_{w}) = ⨁_{i∈I_{w}} ℤvx_{i} ⊂ H_{w}. This proves that w is trivial on ℝ(x_{i} | i ∈ I_{w}). So we can assume that the residue map is the identity on ℝ(x_{i} | i ∈ I_{w}). Now apply the foregoing lemma with K = ℝ(x_{i} | i ∈ I_{w}) (then Kw = K), I_{1} = I \ I_{w} and I_{2} = ∅. □

A sequence of elements a_{ν} ∈ K, ν < λ (λ some limit ordinal), is called a pseudo Cauchy sequence in (K, w) if w(a_{ρ} −a_{σ}) < w(a_{σ} −a_{τ}) for all ρ, σ, τ with ρ < σ < τ < λ. It follows from the ultrametric triangle law that w(a_{ν} − a_{τ}) = w(a_{ν} − a_{ν+1}) whenever ν < τ < λ. The element a is called a (pseudo) limit of this pseudo Cauchy sequence if w(a_{ν} − a) = w(a_{ν} − a_{ν+1}) for all ν < λ. In general, there may be several distinct limits:

Lemma 2.4 Let a be a limit of (a_{ν})_{ν<λ}. Then b is also a limit of (a_{ν})_{ν<λ} if and only if w(a − b) > w(a_{ν} − a_{ν+1}) for all ν < λ.

An extension (K, w) ⊂ (L, w) of valued fields is called immediate if the canonical embedding of wK in wL and the canonical embedding of Kw in Lw are surjective (we then write wK = wL and Kw = Lw). The henselization of a valued field is an immediate extension.

Lemma 2.5 Assume that (K, w) ⊂ (L, w) is immediate and that a ∈ L \ K. Then there is a pseudo Cauchy sequence in (K, w) with limit a, but not having a limit in K.

The next lemma follows from the Lemma of Ostrowski (cf. [R]) and the results of Kaplansky’s important paper [KA]:

Lemma 2.6 Let K be any real closed field and w a convex valuation on K. Assume that (a_{ν})_{ν<λ} is a pseudo Cauchy sequence in (K, w), not having a limit in K. Assume further that in some extension of (K, w), there exists a limit a. Then the extension of w to K(a) is uniquely determined and immediate.

If (K_{1}, w) ⊂ (K_{2}, w) is an immediate algebraic extension of ordered fields with convex valuation w, then their henselizations (in a fixed henselian extension field) are equal.

If the values w(a_{ν}−a_{ν+1}) are cofinal in wK, then (a_{ν})_{ν<λ} is called a Cauchy sequence in (K, w). Lemma 2.4 shows that if this sequence has a limit in K, then this limit is uniquely determined. Indeed, if a, b ∈ K are limits, then w(a − b) > wK, that is, w(a − b) = ∞, or in other words, a = b. All elements in the completion of a valued field are limits of Cauchy sequences (and in particular, the completion is an immediate extension). Conversely:

Lemma 2.7 Let the situation be as in Lemma 2.6, with (a_{ν})_{ν<λ} a Cauchy sequence. Then there is a unique embedding of (K(a), w) over K in the completion of (K, w).

Note that if wK is archimedean, then it follows from Newton’s method together with this lemma that the henselization of (K, w) is embeddable in the completion of (K, w). If w and v are arbitrary valuations such that v is finer than w and Kw ⊂ K, then (K, v) is henselian if and only if (K, w) and (Kw, v) are henselian (cf. [R]). From these facts, one obtains:

Lemma 2.8 Let K be an ordered field with convex valuation w. Suppose that Kw ⊂ K and that (Kw, v) is henselian. Then the henselization of K with respect to v is equal to the henselization of K with respect to w. If in addition wK is archimedean, this henselization is embeddable in the completion of (K, w).

If K is a formally real field, then K^{r} will denote its real closure. For the proof of the next lemma, see [P].

Lemma 2.9 Let K be an ordered field with convex valuation w. Then K is real closed if and only if (K, w) is henselian, wK is divisible and Kw is real closed. Further, wK^{r} = ℚ ⊗_{ℤ} wK (the divisible hull of wK), and K^{r}w = (Kw)^{r}. If wK is divisible and Kw is real closed, then the real closure of K is equal to the henselization of K with respect to w (and embeddable in the completion of (K, w) if wK is archimedean).

If x is a positive element in the real closed field K, then it has a unique positive k-th root, for every k ∈ ℕ. So if K contains the real closure of a field ℝ(x_{i} | i ∈ I), with all x_{i} positive, then x_{i}^{q} ∈ K for all i ∈ I and all q ∈ ℚ. This can be used to show that every real closed field K, with its natural (or any convex) valuation v, admits a cross-section, i.e., an embedding π of the group vK in the multiplicative group K^{×} such that vπα = α for all α ∈ vK. Indeed, take any maximal set X = {x_{i} | i ∈ I} ⊂ K such that the values vx_{i} are rationally independent. By the maximality of the set, together with Lemma 2.9, it follows that vK is the divisible hull of vℝ(x_{i} | i ∈ I) = ⨁_{i∈I} ℤvx_{i}. For every α ∈ vℝ(x_{i} | i ∈ I) there is a unique element x of the multiplicative group ⟨X ⟩ generated by the x_{i}, such that vx = α. Consequently, there is a unique cross-section π of (K, v) whose image contains X, and this image πvK is the divisible hull ˜⟨X ⟩ = {∏_{i∈I_{0}} x_{i}^{q_{i}} | I_{0} ⊂ I finite, q_{i} ∈ ℚ} of ⟨X ⟩. If we have fixed a cross-section π, or a set X and take π to be the associated cross-section, then we call ℝ^{×} · πvK the set of monomials of K. Hence the monomials are the elements of the form

d = r ∏_{i∈I_{0}} x_{i}^{q_{i}} with 0 ≠ r ∈ ℝ, I_{0} ⊂ I finite, and q_{i} ∈ ℚ for every i ∈ I_{0} .

For the rest of this section, we will assume that (M, exp) is a model of the elementary theory of (ℝ, +, ·, 0, 1, <, exp) such that ℝ ⊂ M and the restriction of exp to ℝ is the natural exponential exp on ℝ. Further, we take w to be any convex valuation on M. Then the exponential exp of M is an order preserving isomorphism from the additive group of M onto its multiplicative group of positive elements. Its inverse is the logarithm log; it is order preserving and defined for all positive elements. Consequently, if z ∈ M is positive infinite, that is, z > ℝ, then log z > log({r ∈ ℝ | r > 0}) = ℝ. In other words,

vz < 0 ∧ z > 0 ⇒ v log z < 0 ∧ log z > 0 .  (5)

Further, exp satisfies the Taylor axiom scheme:

(TA)  |z| ≤ 1 ⇒ | exp z − ∑_{n=0}^{m} \frac{z^{n}}{n!} | < |z^{m}|  (m ∈ ℕ) .

In order to derive a valuation theoretical property from this axiom, we need the following simple lemma:

Lemma 2.10 Let K be an ordered field and w a convex valuation on K. Suppose that h ∈ K satisfies

|h − ∑_{k=0}^{m} s_{k}z_{k}| < |s′_{m}z_{m}| for all m ∈ ℕ,  (6)

where s_{k}, s′_{k} ∈ ℝ \ {0}, and z_{k} ∈ K are such that wz_{k+1} > wz_{k}. Write

S_{m} := ∑_{k=0}^{m} s_{k}z_{k} .

Then (S_{m})_{m∈ℕ} is a pseudo Cauchy sequence in (K, w). Further,

w(h − S_{m}) = wz_{m+1} = w(S_{m+1} − S_{m}) ,  (7)

which shows that h is a limit of this sequence.

Proof: Recall that ws = 0 for 0 ≠ s ∈ ℝ, and that w|a| = wa for every a in K. By (6) and (3), we have that

w(h − S_{m} − s_{m+1}z_{m+1} − s_{m+2}z_{m+2}) = w(h − S_{m+2}) ≥ ws′_{m+2}z_{m+2} = wz_{m+2}
> wz_{m+1} = ws_{m+1}z_{m+1} .

By the ultrametric triangle law,

w(s_{m+1}z_{m+1} + s_{m+2}z_{m+2}) = min{ws_{m+1}z_{m+1}, ws_{m+2}z_{m+2}} = ws_{m+1}z_{m+1} .

Hence, again by the ultrametric triangle law,

w(h − S_{m}) = min{w(h − S_{m} − s_{m+1}z_{m+1} − s_{m+2}z_{m+2}), w(s_{m+1}z_{m+1} + s_{m+2}z_{m+2})}
= ws_{m+1}z_{m+1} = w(S_{m+1} − S_{m}) .
□

Lemma 2.11 For every z ∈ M,

wz > 0 ⇒ w exp z = 0 ∧ w(exp z − 1) = wz  (8)
vz = 0 ⇒ v exp z = 0 .  (9)

Proof: By Lemma 2.1, wz > 0 implies vz > 0, that is, z is infinitesimal. In particular, |z| < 1, and (TA) holds. Applying (7) of Lemma 2.10 with m = 1 and z_{m} = z^{m}, we find that w(exp z − 1 − z) = wz^{2} = 2wz > wz. By the ultrametric triangle law, this implies that w exp z = w(1 + z) = w1 = 0 and w(exp z − 1) = wz. This proves (8).

Now assume that vz = 0. Then there is some r ∈ ℝ ⊂ M such that v(z − r) > 0. We have that exp r ∈ ℝ, hence v exp r = 0. By (8) with w = v, v exp(z − r) = 0. Thus, v exp z = v exp r exp(z − r) = v exp r + v exp(z − r) = 0. This proves (9). □

With M as before, exp also satisfies the following growth axiom scheme:

(GA)  z > m^{2} ⟹ exp z > z^{m}  (m ∈ ℕ) .

From this, we derive:

Lemma 2.12 For every z ∈ M,

wz < 0 ∧ z > 0 ⇒ w exp z ≪ wz ≪ w log z < 0  (10)
wz = 0 ∧ z > 0 ⇒ w log z ≥ 0  (11)
vz ≥ 0 ⇔ v exp z = 0 .  (12)

Proof: If wz < 0 and z > 0, then z > ℝ and thus, z > m^{2} for every m ∈ ℕ. So by (GA), exp z > z^{m} > 0 for all m. Hence by (3), w exp z ≤ mwz for all m, i.e., w exp z ≪ wz < 0. In view of (5), we can replace z by log z to get that wz ≪ w log z < 0. This proves (10).

Now assume that wz = 0 and z > 0. If vz < 0, then by (10), vz < v log z < 0. If vz > 0, then vz^{−1} < 0 and by (10), vz^{−1} < v log z^{−1} = v(− log z) = v log z < 0. In both cases, it follows from Lemma 2.1 that 0 = wz = wz^{−1} ≤ w log z ≤ 0, i.e., w log z = 0. Now let vz = 0. If v log z < 0, then by (10), vz = v exp log z < 0 if log z > 0, and vz = −vz^{−1} = −v exp(− log z) > 0 if log z < 0. Hence, v log z ≥ 0, and again by Lemma 2.1, w log z ≥ 0. This proves (11).

Implication “⇒” of (12) follows from (8) with w = v, together with (9). The converse implication follows from (11), where we take w = v and replace z by exp z. □

For positive infinite elements z ∈ M and m ∈ ℤ, we set log_{0} z = z, log_{m+1} z = log(log_{m} z) if m ≥ 0, and and log_{m−1} z = exp(log_{m} z) if m ≤ 0; note that every log_{m} z is again positive infinite. Similarly, we define exp_{m} z for every z ∈ M.

Corollary 2.13 Assume that R is an exp-closed subfield of M. If x ∈ M such that wx < wR and x > 0, then for m > 1,

wx ≪ w log x ≪ … ≪ w log_{m} x ≪ … < wR .  (13)

Proof: The part “wx ≪ w log x ≪ … ≪ w log_{m} x” follows from (10) by induction on m. Now suppose that there is a positive integer m and some α ∈ wR such that α ≤ w log_{m} x. Replacing α by 2α ∈ wR if necessary, we may assume that α < w log_{m} x. Take a positive element a ∈ R such that wa = α. Then by virtue of (3), 0 < log_{m} x < a. It follows that x < exp_{m} a, which implies that wx ≥ w exp_{m} a ∈ wR. This proves that if wx < wR then w log_{m} x < wR for all m. □

The valuation v is a homomorphism from the multiplicative group M^{>0} of positive elements onto the value group vM. Its kernel is U^{>0} = {z ∈ M | vz = 0 ∧ z > 0}, the subgroup of positive units. So v induces an isomorphism M^{>0}/U^{>0} ≃ vM. (3) shows that it is order reversing. The exponential exp is an order preserving isomorphism from the additive group of M onto the multiplicative group M^{>0}. By (12), the preimage of U^{>0} under exp is precisely O_{v}. Hence,

Lemma 2.14 The map z ↦ v exp(−z) induces an order preserving isomorphism M/O_{v} ≃ vM of ordered abelian groups. In particular, if va < 0, then the map ℝ ∋ r ↦ v exp(−ra) ∈ vM is order preserving.

If the elements z_{j}, j ∈ J, are rationally independent over O_{v} in the additive group of M, then the values v exp z_{j}, j ∈ J, are rationally independent in vM.

For further details on the valuation theory of exponential fields, see [KS] and [K–K1].

2.2 Hardy fields

Let us recall some basic facts about Hardy fields. Initially, they were only defined as fields consisting of germs at ∞ of real-valued functions. But we will work with a more general definition that has also been used by other authors lately. Assume that T is the theory of any o-minimal expansion of the ordered field of real numbers by real-valued functions, and that R is a model of T. The Hardy field of R, denoted by H(R), is the set of germs at ∞ of unary R-definable functions f : R → R. Then H(R) is an ordered differential field which contains R. Let x ∈ H(R) be the germ of the identity function. Then H(R) is the closure of R(x) under all 0-definable function of R.

By v_{R} we will denote the finest convex valuation on H(R) which is trivial on R. Then v_{R}a < 0 if and only if a > R. If f, g are non-zero unary R-definable functions on R, then we will denote their germs in H(R) by the same letters. With this convention, the following holds:

v_{R}f = v_{R}g ⟺ lim_{x→∞} \frac{f(x)}{g(x)} is a non-zero constant in R .  (14)

(Note that “x → ∞” means letting x outgrows every element of R.) The functions f and g are asymptotic on R if and only if this constant is 1, and we have:

v_{R}(f − g) > v_{R}g ⟺ f and g are asymptotic on R,  (15)

or in other words,

v (\frac{f}{g} − 1) > vR ⟺ f and g are asymptotic on R,  (16)

3 Closures of R(x) under F, log and exp

General assumptions: Throughout this section, we will assume that T is the theory of a polynomially bounded o-minimal expansion P of the ordered field of real numbers by real-valued functions. Further, we assume that T defines the restricted exp and log. Then also T (exp) is o-minimal (cf. [D–S2]). Here, T (exp) denotes the theory of the expansion (P, exp) where exp is the un-restricted real exponential function.

The archimedean field

Q := {r ∈ ℝ | the function x ↦ x^{r} : (0, ∞) ⟶ ℝ is 0-definable in P}

is called the field of exponents of T.

We let F_{T} denote the set of function symbols in the language of T and assume that there is a function symbol in F_{T} for each 0-definable function of P. This implies that T admits quantifier elimination and a universal axiomatization (cf. [D–L], §2). We let F denote any subset of F_{T} .

Further, we assume that M is a model of T. Suppose that K is a subfield of M and a model of T. Take x_{i} ∈ M, i ∈ I. By K⟨x_{i} | i ∈ I⟩ we denote the definable closure of K ∪ {x_{i} | i ∈ I} in M. By our choice of F_{T} , it is the closure of K ∪ {x_{i} | i ∈ I} under F_{T} , that is, the smallest subfield of M containing K ∪ {x_{i} | i ∈ I} and closed under all functions which interpret the function symbols of F_{T} in M. Since T admits quantifier elimination, K⟨x_{i} | i ∈ I⟩ is an elementary substructure of M and hence a model of T.

For an arbitrary subfield F ⊂ M, the real closure F^{r} of F can be taken to lie in M since M is real closed. We denote by F^{h} the henselization of (F, v). It can be taken to lie in M since by Lemma 2.9, (M, v) is henselian.

We let F^{F} denote the smallest subfield of M which is F-closed, that is, closed under all functions on M which are interpretations of function symbols in F. Similarly, F^{Q} will denote the smallest subfield of M which is closed under the exponents from Q. Further, we let F^{rQF} denote the smallest real closed subfield of M which is F-closed, closed under the exponents from Q and contains F; we will say that F is rQF-closed if F = F^{rQF}. Analogously, we define F^{hF} to be the smallest subfield of M which is F-closed, henselian w.r.t. v and contains F. Note that F^{F} ⊂ F^{hF} ⊂ F^{rQF}.

If F is Q-closed, then for every convex valuation w, the value group wF is a Q-vector space with scalar multiplication defined by qw(a) = w(a^{q}) for q ∈ Q. If α ∈ wF, then Qα shall denote the Q-subvector space generated by α. As Q always contains ℚ, we see that wF^{Q} is always divisible.

3.1 Value groups

The following property (Lemma 3.1) of polynomially bounded o-minimal expansions of the reals was proved in full generality in [D] (Lemma 5.4); see also Corollary 3.7 of [D–M– M1]. Note that in the case of a polynomially bounded expansion, every convex valuation of a model is T -convex (cf. [D–L], §4).

Lemma 3.1 Assume that R is a subfield of M and a model of T. If x ∈ M such that vx ∉ vR, then vR⟨x⟩ = vR ⊕ Qvx.

Lemma 3.2 Assume that R is a subfield of M and a model of T. Take any convex valuation w on M and elements x_{i} ∈ M, i ∈ I, such that the values wx_{i}, i ∈ I, are Q-linearly independent over wR. Then

wR(x_{i} | i ∈ I)^{rQF} = wR(x_{i} | i ∈ I)^{Q} = wR ⊕ ⨁_{i∈I} Qwx_{i} .  (17)

Proof: If our assertion holds for w = v, then it holds for every convex valuation w on M. This is seen as follows. Let H_{w} be the convex subgroup of vM associated with w. The hypothesis that the values wx_{i}, i ∈ I, are Q-linearly independent over wR yields in particular that vx_{i}, i ∈ I, are Q-linearly independent over vR + H_{w}. If we have proved our assertion for w = v, we thus obtain that (17) holds for v in the place of w. Passing to cosets modulo H_{w}, we obtain (17). Hence, it now suffices to prove our assertion for w = v.

Since every element of R(x_{i} | i ∈ I)^{rQF} already lies in R(x_{i} | i ∈ I_{0})^{rQF} for a finite subset I_{0} ⊆ I and a similar assertion is true for the fields R(x_{i} | i ∈ I)^{Q} and R⟨x_{i} | i ∈ I⟩, it suffices to prove our assertion for the case of I finite. We may write I = {1, …, n}. By induction on n one shows that

vR⟨x_{1}, …, x_{n}⟩ = vR ⊕ ⨁_{i=1}^{n} Qvx_{i} .  (18)

Indeed, for n = 1, this is the assertion of Lemma 3.1. Suppose we have already shown that the assertion holds for some j < n in the place of n. Then by our assumption on the values of the x_{i}, we know that vx_{j+1} ∉ vR ⊕ ⨁_{i=1}^{j} Qvx_{i} = vR⟨x_{1}, …, x_{j}⟩. Hence, Lemma 3.1 implies that

vR⟨x_{1}, …, x_{j+1}⟩ = v (R⟨x_{1}, …, x_{j}⟩⟨x_{j+1}⟩)
= vR⟨x_{1}, …, x_{j}⟩ ⊕ Qvx_{j+1} = vR ⊕ ⨁_{i=1}^{j+1} Qvx_{i} ,

i.e., (18) also holds for j + 1 in the place of n.

Since R⟨x_{1}, …, x_{n}⟩ is rQF-closed, we have that

R(x_{1}, …, x_{n})^{Q} ⊆ R(x_{1}, …, x_{n})^{rQF} ⊆ R⟨x_{1}, …, x_{n}⟩ .

As vR(x_{1}, …, x_{n})^{Q} is a Q-vector space and contains vx_{1}, …, vx_{n},

vR ⊕ ⨁_{i=1}^{n} Qvx_{i} ⊆ vR(x_{1}, …, x_{n})^{Q} .

We obtain that

vR ⊕ ⨁_{i=1}^{n} Qvx_{i} ⊆ vR(x_{1}, …, x_{n})^{Q} ⊆ vR(x_{1}, …, x_{n})^{rQF}
⊆ vR⟨x_{1}, …, x_{n}⟩ = vR ⊕ ⨁_{i=1}^{n} Qvx_{i} ,

which shows that equality must hold everywhere. □

3.2 Linear independence of generating values

From now on, let M always be a model of T (exp), and R a submodel of M containing (ℝ, +, ·, <, F_{T} , exp) as a substructure. By v_{R} we will denote the finest convex valuation on M which is trivial on R. We take F as before, but always assume in addition that F contains function symbols for the restricted exp and log. Hence, if a subfield F of M is F-closed, then exp ε ∈ F and log(1 + ε) ∈ F for every infinitesimal ε in F. Since ℝ ⊆ R, we have that Rv = ℝ.

For every subfield K of O_{w}, its multiplicative group K^{×} is contained in the multiplicative group O_{w}^{×} of all units of O_{w}. We will say that K is relatively exp-closed in O_{w}^{×} if a ∈ K and exp(a) ∈ O_{w}^{×} implies that exp(a) ∈ K. For example, ℝ is relatively exp-closed in O_{w}^{×} for every convex valuation w of M.

Lemma 3.3 Let K be a log- and rQF-closed subfield of M. Let w be a convex valuation of M. Assume that the residue field Kw is a subfield of O_{w} ∩ K, relatively exp-closed in O_{w}^{×}. Take any a ∈ K such that exp a ∉ K. Then w exp a is Q-linearly independent over wK.

Proof: Suppose that w exp a is not Q-linearly independent over wK. Since K is Qclosed, wK is a Q-vector space, and it follows that w exp a = wb ∈ wK for some positive b ∈ K. Then w\frac{exp a}{b} = 0 and by Lemma 2.12, w(a − log b) = w log(\frac{exp a}{b}) ≥ 0. Since K is log-closed, log b ∈ K. Hence, there is c ∈ Kw such that w(a − log b − c) > 0. By Lemma 2.11, this shows that w \frac{exp a}{b exp c} = w exp(a − log b − c) = 0. In particular, we find that w exp c = w\frac{exp a}{b} = 0, that is, exp c ∈ O_{w}^{×}. By assumption on Kw, exp c ∈ Kw ⊂ K.

By Lemma 2.1, w(a − log b − c) > 0 yields that v(a − log b − c) > 0. Therefore, exp(a − log b − c) ∈ K^{F} = K, showing that exp a = exp(a − log b − c) · b · exp c ∈ K. We conclude: if exp a ∉ K, then w exp a is Q-linearly independent over wK. □

Lemma 3.4 Assume that K = R(x_{i} | i ∈ I)^{rQF} ⊂ M such that

1) the values vx_{i}, i ∈ I, are Q-linearly independent over vR,

2) x_{i} > 0 and log x_{i} ∈ K for all i ∈ I. Then K is log-closed.

Proof: Take a positive b ∈ K. By virtue of Lemma 3.2, there is a finite subset I_{0} ⊂ I and q_{i} ∈ Q such that vb = vr′ + ∑_{i∈I_{0}} q_{i}vx_{i} for some positive r′ ∈ R. So we can write b = r′ ∏_{i∈I_{0}} x_{i}^{q_{i}} · r · (1 + ε) with positive r ∈ ℝ and some ε ∈ K such that vε > 0. We have that log(1 + ε) ∈ K since K is F-closed. Moreover, log r′ ∈ R ⊂ K and log r ∈ ℝ ⊂ K. Therefore,

log b = log r′ + ∑_{i∈I_{0}} q_{i} log x_{i} + log r + log(1 + ε) ∈ K .
□

Lemma 3.5 Assume that K is of the form

\begin{matrix}R(x_{i} | i ∈ I)^{rQF} log-closed, with x_{i} > 0 and \\ vx_{i}, i ∈ I, Q-linearly independent over vR. \end{matrix} \}  (19)

Take any a ∈ K such that exp a ∉ K. Then v exp a is Q-linearly independent over vK,

vK(exp a)^{rQF} = vK ⊕ Q v exp a .  (20)

Moreover, K(exp a)^{rQF} is again log-closed, and therefore of the form (19). It contains exp b whenever b ∈ K(exp a)^{rQF} and v exp b is Q-linearly dependent over vK(exp a)^{rQF}.

Proof: Applying Lemma 3.3 with w = v and Kw = ℝ, we obtain that v exp a is Q-linearly independent over vK and that exp b ∈ K(exp a)^{rQF} whenever b ∈ K(exp a)^{rQF} and v exp b is Q-linearly dependent over vK(exp a)^{rQF}. Equation (20) follows by an application of Lemma 3.2 to K and to K(exp a)^{rQF}. Finally, we infer from Lemma 3.4 that K(exp a)^{rQF} is log-closed. □

Lemma 3.6 Assume that K is of the form (19) and that w is a convex valuation such that Kw = R. Then the values wx_{i}, i ∈ I, are Q-linearly independent.

Proof: From Kw = R it follows that the natural valuation v of K is the composition of w with the natural valuation of Kw = R (which is simply the restriction of v to R ⊂ K). Thus, vR is a convex subgroup of vK and there is a canonical isomorphism wK ≃ vK/vR. Hence ∑_{i∈I} q_{i}wx_{i} = 0 (where q_{i} ∈ Q, almost all of them zero) implies ∑_{i∈I} q_{i}vx_{i} ∈ vR. By (19), this implies that q_{i} = 0 for all i ∈ I. □

3.3 A basic construction

First, we show how to construct log-closed fields K as in (19). From now on, we always assume that x ∈ M such that x > R, that is, vx < vR and x > 0.

Lemma 3.7 The field

R(log_{m} x | m ≥ 0)^{rQF}

is log-closed. The convex hull of its value group in vM is equal to the smallest convex subgroup containing vx and vR. If w is a convex valuation on M, trivial on R and such that wx = 0, then the field R(log_{m} x | m ≥ 0)^{rQF} lies in O_{w}.

Proof: From Corollary 2.13 we know that

vx ≪ v log x ≪ … ≪ v log_{m} x ≪ … < vR .  (21)

In particular, the values v log_{m} x lie in distinct archimedean classes. As Q is archimedean, it follows that the values v log_{m} x are Q-linearly independent over vR. So it follows from Lemma 3.4 that R(log_{m} x | m ≥ 0)^{rQF} is log-closed.

From Lemma 3.2 we infer that vR(log_{m} x | m ≥ 0)^{rQF} = vR ⊕ ⨁_{m≥0} Q v log_{m} x. Now (21) yields that this group is contained in the smallest convex subgroup H of vM which contains vx and vR. If w is as in our assumption, then H is contained in the convex subgroup H_{w} of vM associated with w. Thus, w is trivial on R(log_{m} x | m ≥ 0)^{rQF}, that is, this field lies in O_{w}. □

Next, we build up LE_{R,F}(x). As a preparation for what we will need in the next section, we will keep our construction more general. We will construct a variety of fields (described in Lemma 3.8 below) of which LE_{R,F}(x) is just a special case. Let w be a convex valuation on M, trivial on R, and H_{w} its associated convex subgroup of vM. Further, let K_{0}^{w} ⊂ O_{w} be any field of the form (19). For example, if wx = 0, then we can take K_{0}^{w} = R(log_{m} x | m ≥ 0)^{rQF}. We will see later that if w ≠ v_{R} then there always exists such a field K_{0}^{w} which properly contains R.

Now we construct K_{1}^{w} as follows. Assume that a ∈ K_{0}^{w} such that exp a ∉ K_{0}^{w}, but v exp a ∈ H_{w}. Then by Lemma 3.5, K_{0}^{w}(exp a)^{rQF} is again of the form (19), with vK_{0}^{w}(exp a)^{rQF} = vK_{0}^{w} ⊕ Q v exp a ⊂ H_{w}. The latter shows that it is again a subfield of O_{w}. We repeat this procedure until we arrive at a field K_{1}^{w} ⊂ O_{w} of the form (19), which contains exp a for every a ∈ K_{0}^{w} such that exp a ∈ O_{w}^{×}. Then we construct K_{2}^{w} from K_{1}^{w} in the same way as we constructed K_{1}^{w} from K_{0}^{w}. We iterate to obtain fields K_{n}^{w} ⊂ O_{w}, of the form (19). Their union

K_{∞}^{w} := ⋃_{n∈ℕ} K_{n}^{w} ⊂ O_{w}

is rQF-closed and of the form (19). It is also relatively exp-closed in O_{w}^{×}. To see this, let a ∈ K_{∞}^{w}. Then a ∈ K_{n}^{w} for some n. If exp a ∈ O_{w}^{×}, then by construction, exp a ∈ K_{n+1}^{w}. On the other hand, every other log- and rQF-closed subfield of O_{w}, relatively exp-closed in O_{w}^{×} and containing K_{0}^{w}, must also contain K_{∞}^{w}. This proves:

Lemma 3.8 K_{∞}^{w} is the uniquely determined smallest log- and rQF-closed subfield of O_{w}, relatively exp-closed in O_{w}^{×} and containing K_{0}^{w}. It is of the form (19).

We derive some further information from our construction.

Lemma 3.9 Take n ∈ ℕ. If a ∈ K_{n}^{w} with va < 0, a > 0, then

v log a ∈ vK_{n−1}^{w} , and v log_{n} a ∈ vK_{0}^{w} .

Proof: By the construction of K_{n}^{w} from K_{n−1}^{w}, there are elements a_{j} ∈ K_{n−1}^{w}, j ∈ J, such that vK_{n}^{w} = vK_{n−1}^{w} ⊕ ⨁_{j∈J} Qv exp a_{j}. Hence, a ∈ K_{n}^{w} can be written as

a = ∏_{j∈J_{0}}(exp a_{j})^{q_{j}} · c · r · (1 + ε)

with J_{0} a finite subset of J, q_{j} ∈ Q, c ∈ K_{n−1}^{w}, r ∈ ℝ and ε ∈ K_{n}^{w} with vε > 0. Then log a = ∑_{j∈J_{0}} q_{j}a_{j} + log c + log r + log(1 + ε). Since v log a < 0 by Lemma 2.12, but v log(1 + ε) > 0, we find that v log a = v(∑_{j∈J_{0}} q_{j}a_{j} + log c + log r) ∈ vK_{n−1}^{w}. By induction it follows that v log_{n} a ∈ vK_{0}^{w}. □

If w is trivial on R and wx = 0 and we start our construction from K_{0}^{w} = R(log_{m} x | m ≥ 0)^{rQF}, then K_{∞}^{w} will be the uniquely determined smallest log- and rQF-closed subfield of O_{w}, relatively exp-closed in O_{w}^{×} and containing R(x). We denote it by

LE_{R,F}^{w}(x) .

Let u denote the trivial valuation on M. Then O_{u} = M and H_{u} = vM. In this case, LE_{R,F}^{u}(x) is exp-closed and contains x. Therefore,

LE_{R,F}^{u}(x) = LE_{R,F}(x) .

Lemma 3.10 Suppose that x > R. Then for every y ∈ LE_{R,F}(x), y > R, the sequence exp_{m} y, m ≥ 0, is cofinal in LE_{R,F}(x) , and the sequence log_{m} y, m ≥ 0, is coinitial in {z ∈ LE_{R,F}(x) | z > R}.

Proof: It suffices to show the result for y = x. Indeed, if it holds in this case, then there is ν ∈ ℕ such that exp_{ν} x > y > log_{ν} x. It follows that exp_{n} y > exp_{ν+n} x, showing that also the sequence exp_{m} y, m ≥ 0, is cofinal. It also follows that log_{n} x > log_{ν+n} y, showing that also the sequence log_{m} y, m ≥ 0, is coinitial.

Take any a ∈ LE_{R,F}(x), x > R. From Lemma 3.9 with w = u and K_{0}^{w} = R(log_{m} x | m ≥ 0)^{rQF} we infer that v log_{n} a ∈ vR(log_{m} x | m ≥ 0)^{rQF} for some n ∈ ℕ. By Lemma 3.7, every element α < 0 in this value group is either archimedean equivalent to vx, or satisfies vx ≪ α < 0. Since v log_{n} a ≪ v log_{n+1} a < 0 by Lemma 2.12, it follows that vx ≪ v log_{n+1} a < 0. Hence by (1), x > log_{n+1} a and therefore, exp_{n+1} x > a.

Now let a ∈ LE_{R,F}(x), a > R. As before, v log_{n} a ∈ vR(log_{m} x | m ≥ 0)^{rQF} for some n ∈ ℕ. As the sequence v log_{m} x, m ≥ 0, is cofinal in the negative part of this value group, there is some m_{0} such that v log_{n} a < v log_{m_{0}} x. Hence by (1), a ≥ log_{n} a > log_{m_{0}} x. □

Now we deduce our main theorem on the valuation theoretical structure of LE_{R,F}(x). If we take F = F_{T} and M = H(R), then the theorem describes the structure of the Hardy field H(R).

Theorem 3.11 LE_{R,F}(x) is of the form

R(x_{i} | i ∈ I)^{rQF} with x_{i} > 0 and v_{R}x_{i}, i ∈ I, Q-linearly independent.  (22)

Moreover,

LE_{R,F}(x)v_{R} = R .  (23)

The elements x_{i} can be chosen so as to include x and log_{m} x for all m ∈ ℕ.

If R = ℝ, then LE_{R,F}(x) has exponential rank 1, in the sense of [K–K2]. In general, exprk LE_{R,F}(x) = exprk R + 1.

Proof: By our construction, we get that LE_{R,F}(x) is of the form (19). Since F ⊆ F_{T}, we have that LE_{R,F}(x) ⊆ LE_{R,F_{T}} (x). By definition of the valuation v_{R}, its valuation ring is the convex hull of R in M. As R is an elementary submodel of LE_{R,F_{T}} (x), we can deduce from [D–L], p. 75, (1), that this valuation ring is T (exp)-convex in LE_{R,F_{T}} (x). Since LE_{R,F_{T}} (x) is the T (exp)-definable closure of R(x) in itself, we can apply Corollary 5.4 of [D–L] to obtain that LE_{R,F_{T}} (x)v_{R} = R. Since R ⊂ LE_{R,F}(x) ⊆ LE_{R,F_{T}} (x), this proves (23). By Lemma 3.6, this also implies that v_{R}x_{i}, i ∈ I, are Q-linearly independent.

The exponential rank is the order type of the set of proper T (exp)-convex valuation rings, ordered by inclusion. Lemma 3.10 shows that LE_{R,F}(x) has exactly one more than R, namely R itself. This proves our assertions about the exponential rank. □

3.4 Levels

An infinitely increasing unary function f on R has level s if s ∈ ℤ and there is N ∈ ℕ such that log_{N+s} ∘f is asymptotic to log_{N} on R. If a denotes the germ of f in H(R), then by (16) the latter is equivalent to

v (\frac{log_{N+s} a}{log_{N} x} − 1) > vR .

We say that R is levelled if every R-definable ultimately strictly increasing and unbounded unary function on R has a level. In this section, we will prove that every definable function on R has a level, and we will determine this level explicitly.

Take any a ∈ LE_{R,F}(x) such that a > R. According to our construction, we write LE_{R,F}(x) = K_{∞} with K_{0} = R(log_{m} x | m ≥ 0)^{rQF}. By Lemma 3.9 there is some n ∈ ℕ such that v log_{n} a ∈ vK_{0} . Similarly as in the proof of Lemma 3.4, we write log_{n} a = r′ ∏_{i≥0}(log_{i} x)^{q_{i}} · r · (1 + ε) with q_{i} ∈ Q, only finitely many of them nonzero, r′ ∈ R, r ∈ ℝ and ε ∈ K such that vε > 0. It follows that

log_{n+1} a = log r′ + ∑_{i≥0} q_{i} log_{i+1} x + log r + log(1 + ε) .

As a > R by assumption, there must be at least on nonzero q_{i}. Let i_{0} be the smallest of all i ≥ 0 for which q_{i} ≠ 0. We have that v log r = 0, v log(1+ε) > 0 and v log_{i_{0}+1} x < v log_{i+1} x for i > i_{0} . Also, v log_{i_{0}+1} x < vr′. Thus, we can write log_{n+1} a = q_{i_{0}} log_{i_{0}+1} x · (1 + ε′) with vε′ > 0. Then

log_{n+2} a = log q_{i_{0}} + log_{i_{0}+2} x + log(1 + ε′) .

Again, v log_{i_{0}+2} x < 0 = v log q_{i_{0}} < vε′ = v log(1 + ε′). Hence,

v (log_{n+2} a − log_{i_{0}+2} x) = v (log q_{i_{0}} + log(1 + ε′)) = v log q_{i_{0}} = 0 .

Thus,

v (\frac{log_{n+2} a}{log_{i_{0}+2} x} − 1) = −v log_{i_{0}+2} x > vR .  (24)

We have now proved a result which in fact constitutes an abstract notion of levels, without referring to Hardy fields:

Proposition 3.12 Take any element a ∈ LE_{R,F}(x) such that a > R. Then a “has level over R” in the following sense: there is some s ∈ ℤ and N ∈ ℕ such that

v_{R}(log_{N+s} a − log_{N} x) > v_{R} log_{N} x .

Now take any R-definable, ultimately strictly increasing and unbounded function f on R. Let a be the germ of f at infinity. Then a > R. Hence, a is an element of the Hardy field H(R) = LE_{R,F_{T}} (x) of R (where x > R). Then (24) shows that log_{n+2} f(x) and log_{i_{0}+2} x are asymptotic as functions on R. That is, the function f has level n − i_{0} . This proves Theorem 1.1.

3.5 A maximality property of the T -definable closure in the T (exp)-definable closure

Lemma 3.13 Assume that T has field of exponents ℝ and that ℝ ⊂ R ⊂ M are models of T (exp). Let x ∈ M, x > R. Then R(x)^{F_{T}} (the T -definable closure of R ∪ {x} in M) has the following maximality property:

1) v_{R}R(x)^{F_{T}} ≃ ℝ,

2) R(x)^{F_{T}} is maximal among all subfields of LE_{R,F_{T}} (x) whose value group w.r.t. v_{R} is archimedean.

Proof: Assertion 1) follows from Lemma 3.2. In order to prove assertion 2), we show the following: Take any a ∈ LE_{R,F_{T}} (x) \ R(x)^{F_{T}}. Then v_{R}R(x)^{F_{T}} (a) is not archimedean.

By Theorem 3.11 we can write LE_{R,F_{T}} (x) = R(x_{i} | i ∈ I)^{F_{T}} with x_{i} > 0 and v_{R}x_{i}, i ∈ I, ℝ-linearly independent, and x among the x_{i}. As a ∈ R(x_{i} | i ∈ I)^{F_{T}}, there are x_{i_{1}}, …, x_{i_{n}} (n ≥ 1) such that a ∈ R(x, x_{i_{1}}, …, x_{i_{n}})^{F_{T}}, and we choose n minimal with this property. By the Exchange Lemma for o-minimal theories ([P–S]) applied to T, we then obtain that

x_{i_{1}} ∈ R(x, a, x_{i_{2}}, …, x_{i_{n}})^{F_{T}} .  (25)

Suppose that v_{R}R(x, a)^{F_{T}} = v_{R}R(x)^{F_{T}}. Then by Lemma 3.2,

v_{R}R(x, a, x_{i_{2}}, …, x_{i_{n}})^{F_{T}} = v_{R}R(x, a)^{F_{T}} (x_{i_{2}}, …, x_{i_{n}})^{F_{T}} = v_{R}R(x, a)^{F_{T}} ⊕ ⨁_{j=2}^{n} ℝv_{R}x_{i_{j}}
= v_{R}R(x)^{F_{T}} ⊕ ⨁_{j=2}^{n} ℝv_{R}x_{i_{j}} = ℝv_{R}x ⊕ ⨁_{j=2}^{n} ℝv_{R}x_{i_{j}} .

But this does not contain v_{R}x_{i_{1}} . This contradiction to (25) shows that

v_{R}R(x, a)^{F_{T}} ≠ v_{R}R(x)^{F_{T}} .

By the Valuation Property ([D–S2], Proposition 9.2) it follows that

v_{R}R(x)^{F_{T}} ⊂_{≠} v_{R}R(x)^{F_{T}} (a) .

Since v_{R}R(x)^{F_{T}} ≃ ℝ it follows that v_{R}R(x)^{F_{T}} (a) is not archimedean. □

Lemma 3.14 Let H ⊂ H(R) be a subfield containing R(x) and closed under compositions and compositional inverses for v_{R}-positive infinite germs (i.e., germs a ∈ H such that a > R). If H is polynomially bounded (i.e., every germ in H is bounded by a power x^{n} for some n ∈ ℕ), then v_{R}(H) is archimedean.

Proof: Assume for a contradiction that there is g ∈ H(R) such that g > R and v_{R}g ≪ v_{R}x or v_{R}x ≪ v_{R}g. The former implies that g > x^{n} for all n ∈ ℕ, a contradiction to the fact that H is polynomially bounded. So assume that v_{R}x ≪ v_{R}g. But this implies that for all n ∈ ℕ,

x^{n} < g^{−1} ,

where g^{−1} denotes the compositional inverse of g. This again contradicts the assumption that H is polynomially bounded. Indeed, let n ∈ ℕ. Since g^{n} < x, there exists r ∈ R (and we may assume r > 1) such that for a ∈ R with a > r we have g(a)^{n} < a. On the other hand, g is invertible, ultimately. So for b large enough, g^{−1}(b) = a exists with a > r. Thus, g(g^{−1}(b))^{n} < g^{−1}(b). □

Corollary 3.15 The field R(x)^{F_{T}} (i.e., the Hardy field associated with the reduct of R to the language of T) is maximal among the polynomially bounded subfields of H(R) which are closed under compositions and compositional inverses for v_{R}-positive infinite germs.

Proof: Let H be a polynomially bounded subfield of H(R) closed under compositions and compositional inverses for v_{R}-positive infinite germs, and containing R(x)^{F_{T}}. Then by Lemma 3.14, v_{R}H is archimedean. Hence by Lemma 3.13, H cannot be a proper extension of R(x)^{F_{T}}. □

Let us note that there exist polynomially bounded subfields of H(R) which properly contain R(x)^{F_{T}}. For instance, R(x, log x)^{F_{T}} and R(log_{m} x | m ≥ 0)^{F_{T}} are such fields. But they are not closed under compositions and compositional inverses for v_{R}-positive infinite germs.

3.6 A maximality property of the Hardy field H(R_{an,powers})

Now we consider the special case where F_{T} is the set of 0-definable functions in ℝ_{an,powers} . We let R_{an,powers} denote the reduct of R to the language of ℝ_{an,powers} , and R_{an,exp} the reduct of R to the language of ℝ_{an,exp} . Since

x^{r} = exp(r log x)

for all r ∈ ℝ, the power functions are ℝ-definable (actually, already 0-definable) in R_{an,exp}. Therefore,

H(R_{an,exp}) = H(R) .

On the other hand, H(R_{an,powers}) is a proper subfield of H(R). It has the following maximality property:

Theorem 3.16 Let H ⊆ H(R) be a polynomially bounded field containing H(R_{an,powers}) and closed under compositions and compositional inverses for v_{R}-positive infinite germs. Then H = H(R_{an,powers}).

In particular, H(R_{an,powers}) is maximal among the Hardy subfields of H(R) associated with polynomially bounded reducts of R.

Proof: We take T to be the elementary theory of R_{an,powers}. We know that H(R_{an,powers}) = R(x)^{F_{T}} with x ∈ H(R), x > R the germ of the identity function. Now our first assertion follows from Corollary 3.15.

If H is the Hardy field of a polynomially bounded reducts of R, then H is closed under compositions and compositional inverses for v_{R}-positive infinite germs. Hence our second assertion follows from the first. □

4 Residue fields of F-closures

In this section we wish to determine the residue fields of LE_{R,F}(x) with respect to any convex valuation which is trivial on R; such a valuation is not necessarily T (exp)-convex. In addition to our earlier assumptions (see Section 3.2), we consider the following conditions: (PADE) F is closed under partial derivations; (COMP) if w is a convex valuation on a model N of T (exp) and F is a subfield of N such that F w ⊂ F is F-closed and wF is archimedean, then either F^{F} is embeddable in the completion of (F, w), or there is some y ∈ F^{F}, y ≠ 0, such that wy > wF. Note that if F^{F} is embeddable in the completion of (F, w), then wF^{F} = wF and F^{F}w = F w. If on the other hand, 0 ≠ y ∈ F^{F} such that wy > wF, then wF^{F} is not archimedean.

We denote by T_{an} the theory of the expansion ℝ_{an} = (ℝ, +, ·, 0, 1, <, F_{an}}.

Lemma 4.1 If F ⊆ F_{an} satisfies condition (PADE), then it satisfies condition (COMP) in each model of T_{an}.

Proof: Assume the hypothesis as given in condition (COMP). By Zorn’s Lemma, we find a maximal subfield F_{0} of F^{F} containing F and embeddable in the completion of (F, w). Suppose that F^{F} is not embeddable in the completion of (F, w). Then F_{0} ≠ F^{F}, that is, F_{0} is not F-closed. So let f(X_{1}, …, X_{k}) ∈ F and a = (a_{1}, …, a_{k}) ∈ F_{0}^{k} with va_{i} > 0 such that f(a) ∈ F^{F} \ F_{0} . We write a_{i} = c_{i} + ε_{i} with c_{i} ∈ F_{0}w = F w and wε_{i} > 0; let c = (c_{1}, …, c_{k}). By the Taylor expansion, the following assertions hold (they are elementary sentences in the language of T_{an} and thus hold in the T_{an}-model N): for all m ∈ ℕ,

|f(a_{1}, …, a_{k}) − ∑_{ν=(0,...,0)}^{(m,...,m)} \frac{∂^{ν}f}{∂X^{ν}} (c_{1}, …, c_{k})\frac{ε^{ν}}{ν!} | ≤ |ε_{1} · . . . · ε_{k}|^{m}

(for ν = (ν_{1}, …, ν_{k}) ∈ ℕ^{k}, \frac{∂^{ν}f}{∂X^{ν}} stands for \frac{∂^{ν_{1}}...∂^{ν_{k}} f}{∂X_{1}^{ν_{1}}...∂X_{k}^{ν_{k}}}, and ν! stands for ν_{1}! · . . . · ν_{k}!). By (3) it follows that for all m ∈ ℕ,

w (f(a_{1}, …, a_{k}) − ∑_{ν=(0,...,0)}^{(m,...,m)} \frac{∂^{ν}f}{∂X^{ν}} (c_{1}, …, c_{k})\frac{ε^{ν}}{ν!} ) ≥ m(wε_{1} + … + wε_{k}) .

Since wF_{0} is archimedean and wε_{i} > 0, the sequence m(wε_{1} + … + wε_{k}), m ∈ ℕ, is cofinal in wF_{0} . This shows that the partial sums form a Cauchy sequence in (F_{0}, w), with limit f(a). Note that since F is closed under partial derivatives and F w is F-closed, the coefficients \frac{∂^{ν}f}{∂X^{ν}} (c_{1}, …, c_{k}) lie in F w ⊂ F_{0} . So the partial sums are indeed elements of F_{0} .

Suppose that the sequence has no limit in F_{0} . Then we can apply Lemma 2.7 to obtain that F_{0}(f(a)) is embeddable in the completion of (F_{0}, w) and hence also in the completion of (F, w). But this contradicts the maximality of F_{0} . Hence, there is some b ∈ F_{0} which is also a limit of this sequence (observe that it is not necessarily a Cauchy sequence in (N, w)). Then by Lemma 2.4, w(f(a) − b) > wF_{0} . With y := f(a) − b ≠ 0, we have found the desired element y which satisfies wy > wF. □

At this point, it may be helpful to give an example which shows that an element y as in the assertion of the above lemma can indeed exist. Take L to be any T_{an}-model with non-archimedean value group. Choose y, t ∈ L such that vy ≫ vt > 0. Then vy > ℚvt = vℝ(t)^{r}. It is well known that in general, (ℝ(t)^{r})^{F} ≠ ℝ(t)^{r}. Take any a ∈ (ℝ(t)^{r})^{F} \ ℝ(t)^{r}. As ℝ(t)^{r} ⊂ (ℝ(t)^{r})^{F} ⊂ ℝ(t)^{rF}, Lemma 3.2 yields that v(ℝ(t)^{r})^{F} = vℝ(t)^{r}. Further, (ℝ(t)^{r})^{F}v = ℝ = ℝ(t)^{r}v. Hence also vℝ(t, a)^{r} = vℝ(t)^{r} and ℝ(t, a)^{r}v = ℝ(t)^{r}v. That is, the extension (ℝ(t, a)^{r}|ℝ(t)^{r}, v) is immediate. Hence by Lemma 2.5, a is a limit of a pseudo Cauchy sequence without limit in (ℝ(t)^{r}, v). Set z := a + y. Then v(z − a) = vy > vℝ(t)^{r}, and Lemma 2.4 shows that z is also a limit of this pseudo Cauchy sequence. Hence by Lemma 2.6, the extension (ℝ(t)^{r}(z)|ℝ(t)^{r}, v) is immediate. It follows from Lemma 2.9 that also (ℝ(t, z)^{r}|ℝ(t)^{r}, v) is immediate. On the other hand, a ∈ (ℝ(t, z)^{r})^{F} and consequently, y ∈ (ℝ(t, z)^{r})^{F} with vy > vℝ(t)^{r} = vℝ(t, z)^{r}. So (F, w) = (ℝ(t, z)^{r}, v) is our desired example.

Throughout this section, we will assume that F satisfies conditions (PADE) and (COMP).

Lemma 4.2 Let x_{i} ∈ M be such that the values vx_{i}, i ∈ I are Q-linearly independent over vR. Further, let w be any convex valuation which is trivial on R. Assume that there is a subset I_{w} ⊂ I such that wx_{i} = 0 for all i ∈ I_{w} and that the values wx_{i}, i ∈ I \ I_{w} are Q-linearly independent. Then

wR(x_{i} | i ∈ I)^{rQF} = ⨁_{i∈I\I_{w}} Qwx_{i} and wR(x_{i} | i ∈ I)^{hF} = ⨁_{i∈I\I_{w}} ℤwx_{i} .

Further,

R(x_{i} | i ∈ I)^{rQF}w = R(x_{i} | i ∈ I_{w})^{rQF} = (R(x_{i} | i ∈ I)w)^{rQF}
R(x_{i} | i ∈ I)^{hF}w = R(x_{i} | i ∈ I_{w})^{hF} = (R(x_{i} | i ∈ I)w)^{hF} .

Proof: We set L := R(x_{i} | i ∈ I) and K := R(x_{i} | i ∈ I_{w}). By Corollary 2.3, vL = vR ⊕ ⨁_{i∈I} ℤvx_{i}, vK = vR ⊕ ⨁_{i∈I_{w}} ℤvx_{i}, wL = ⨁_{i∈I\I_{w}} ℤwx_{i} and Lw = K. From Lemma 3.2 we infer that vL^{rQF} = vR ⊕ ⨁_{i∈I} Qvx_{i} = Q ⊗_{ℤ} vL and that vK^{rQF} = vR ⊕ ⨁_{i∈I_{w}} Qvx_{i} = Q ⊗_{ℤ} vK (recall that vM and vR are Q-vector spaces). The former implies that wL^{rQF} = Q ⊗_{ℤ} wL, which in view of the Q-linearly independence of the wx_{i} implies our assertion on the value groups for the rQF-closure.

We prove the assertions of our lemma for the hF-closure. The proof for the residue field of the rQF-closure is analogous. If our assertions are not true, then there is some b ∈ L^{hF} such that wb ∉ ⨁_{i∈I\I_{w}} ℤwx_{i} or bw ∉ K^{hF}. But b is already contained in some subfield R(x_{1}, …, x_{n})^{hF} ⊂ L^{hF}, where x_{1}, …, x_{n} are suitably chosen from the x_{i}’s. So we see that it suffices to prove our lemma in the case of a finite set I = {1, …, n}.

As usual, we let H_{w} denote the convex subgroup of vM associated with w. Since vK is contained in H_{w} and since Q is archimedean, we find that also vK^{rQF} = Q ⊗_{ℤ} vK ⊂ H_{w}. That is, w is trivial on K^{rQF} and thus also on K^{hF}. Therefore, K^{hF} ⊂ L^{hF}w. We will show that equality holds.

First assume that wL is archimedean. Then wL^{rQF} = Q⊗_{ℤ} wL is archimedean, and so is wL^{hF} ⊂ wL^{rQF}. Set F_{0} := K^{hF}(x_{i} | i ∈ I \ I_{w}). Then L^{hF} = K^{hF}(x_{i} | i ∈ I \ I_{w})^{hF} = F_{0}^{hF}, and by Lemma 2.2, F_{0}w = K^{hF} and wF_{0} = ⨁_{i∈I\I_{w}} ℤwx_{i} = wL. By Zorn’s Lemma, we find a maximal subfield F of F_{0}^{hF} containing F_{0} and embeddable in the completion of (F_{0}, w). Since wF = wF_{0} is archimedean and F w = F_{0}w is F-closed, we can apply condition (COMP) to see that F is F-closed. From Lemma 2.8 we infer that F must be equal to its henselization, i.e., it is henselian. Therefore, F = F_{0}^{hF} = L^{hF}, showing that wR(x_{i} | i ∈ I)^{hF} = wL^{hF} = wF = ⨁_{i∈I\I_{w}} ℤwx_{i} and R(x_{i} | i ∈ I)^{hF}w = L^{hF}w = F w = K^{hF}. (For the rQF-closure, one takes F to be a maximal subfield of F_{0}^{rQF} containing F_{0}^{rQ} and embeddable in the completion of (F_{0}^{rQ}, w), and uses Lemma 2.9 in the place of Lemma 2.8.)

Now let wL be non-archimedean. Since it is finitely generated, it has finite rank. So we can proceed by induction on the rank. Let H be the largest proper convex subgroup of wL. Since H is finitely generated, we can choose a system α_{1}, …, α_{ℓ} ∈ L of rationally independent generators of H. We take w′ to be a convex valuation on M whose restriction to L is the valuation associated with H. Since wL is finitely generated, we can choose a system α_{ℓ+1}, …, α_{m} ∈ L of rationally independent generators of wL/H. It follows that α_{i}, ℓ < i ≤ m, are rationally independent over H, and that α_{i}, 1 ≤ i ≤ m, form a rationally independent system of generators of wL. W.l.o.g., we may assume that there is some m′ ≤ n such that I \I_{w} = {1, …, m′}. Then also wx_{i}, 1 ≤ i ≤ m′, form a rationally independent set of generators of wL, and we find that m = m′. Therefore, there is an invertible matrix (μ_{ij}), μ_{ij} ∈ ℤ, such that α_{i} = ∑_{j=1}^{m} μ_{ij}wx_{j} . We set

y_{i} := ∏_{j=1}^{m} x_{j}^{μ_{ij}} ∈ K(x_{1}, …, x_{m}) (1 ≤ i ≤ m)

so that wy_{i} = α_{i}. If (ν_{ij}) denotes the inverse of (μ_{ij}), then x_{i} = ∑_{j=1}^{m} ν_{ij}y_{j} ∈ K(x_{1}, …, x_{n}). This implies that

L = K(x_{1}, …, x_{n}) = K(y_{1}, …, y_{m}) .

The rank of wK(y_{1}, …, y_{ℓ}) = H is smaller than that of wL. Hence by induction hypothesis,

wK(y_{1}, …, y_{ℓ})^{hF} = ⨁_{1≤i≤ℓ} ℤwy_{i} and K(y_{1}, …, y_{ℓ})^{hF}w = K^{hF} .  (26)

On the other hand, the value group w′K(y_{1}, …, y_{m}) = wL/H is archimedean since H was chosen to be the largest proper convex subgroup of wL. By our choice of the elements y_{i}, w′y_{i} = 0 for 1 ≤ i ≤ ℓ, and the values w′y_{ℓ+1}, …, w′y_{m} are rationally independent. Thus, we can replace w by w′ and apply the assertion of our lemma, which is already proved in the archimedean case, to deduce that

w′K(y_{1}, …, y_{m})^{hF} = ⨁_{ℓ<i≤m} ℤw′y_{i} and K(y_{1}, …, y_{m})^{hF}w′ = K(y_{1}, …, y_{ℓ})^{hF} . (27)

Replacing w′ by an equivalent valuation if necessary, we can write

w′K(y_{1}, …, y_{m})^{hF} = wK(y_{1}, …, y_{m})^{hF}/w(K(y_{1}, …, y_{m})^{hF}w′) .

Note that w′y_{i} is the coset of wy_{i} in this quotient group. Therefore we obtain, using also (26) and (27),

wK(y_{1}, …, y_{m})^{hF} = ⨁_{ℓ<i≤m} ℤwy_{i} ⊕ w(K(y_{1}, …, y_{m})^{hF}w′)
= ⨁_{ℓ<i≤m} ℤwy_{i} ⊕ wK(y_{1}, …, y_{ℓ})^{hF} = ⨁_{ℓ<i≤m} ℤwy_{i} ⊕ ⨁_{1≤i≤ℓ} ℤwy_{i}
= ⨁_{1≤i≤m} ℤwy_{i} = H = ⨁_{i∈I\I_{w}} ℤwx_{i} .

It follows that

wR(x_{1}, …, x_{n})^{hF} = wK(x_{1}, …, x_{m})^{hF} = wK(y_{1}, …, y_{m})^{hF} = ⨁_{i∈I\I_{w}} ℤwx_{i} .

Again by (26) and (27),

R(x_{1}, …, x_{n})^{hF}w = K(x_{1}, …, x_{m})^{hF}w = K(y_{1}, …, y_{m})^{hF}w
= (K(y_{1}, …, y_{m})^{hF}w′)w = K(y_{1}, …, y_{ℓ})^{hF}w
= K^{hF} = R(x_{i} | i ∈ I_{w})^{hF} .
□

Let us note that the result of this lemma remains true if the henselization with respect to v is replaced by the henselization with respect to any convex valuation. — The lemma shows in particular that if the values vx_{i}, i ∈ I, are Q-linearly independent, then

vℝ(x_{i} | i ∈ I)^{rQF} = ⨁_{i∈I} Qvx_{i} and vℝ(x_{i} | i ∈ I)^{hF} = ⨁_{i∈I} ℤvx_{i} .  (28)

Corollary 4.3 Take x_{i} ∈ M such that the values vx_{i}, i ∈ I are Q-linearly independent over vR. Further, let w be any convex valuation. Then there exist some index set J_{w} and algebraically independent elements y_{j} ∈ R(x_{i} | i ∈ I), j ∈ J_{w}, such that

R(x_{i} | i ∈ I)^{rQF}w = R(y_{j} | j ∈ J_{w})^{rQF} .

Proof: By Zorn’s Lemma, choose a maximal subset I′_{w} ⊂ I such that the values wx_{i}, i ∈ I′_{w}, are Q-linearly independent. We set J_{w} := I \ I′_{w}. Then for every j ∈ J_{w}, there are i_{1}, …, i_{ℓ} ∈ I′_{w} and q, q_{1}, …, q_{ℓ} ∈ Q, q ≠ 0, such that wy_{j} = 0 for y_{j} := x_{j}^{q} · x_{i_{1}}^{q_{1}} · . . . · x_{i_{ℓ}}^{q_{ℓ}}. Then x_{j}^{q} ∈ R(x_{i}, y_{j} | i ∈ I′_{w} , j ∈ J_{w})^{rQ} =: L and thus, x_{j} ∈ L. Therefore, L^{rQF} = R(x_{i} | i ∈ I)^{rQF}. From Lemma 4.2 it follows that w is trivial on R(y_{j} | j ∈ J_{w})^{rQF}. By Lemma 4.2, L^{rQF}w = R(y_{j} | j ∈ J_{w})^{rQF}. □

For use in Sections 5 and 6, we add the following lemma:

Lemma 4.4 Let x_{i} ∈ M such that x_{i} > 0 and the values vx_{i}, i ∈ I are Q-linearly independent over vR. Then

R(x_{i} | i ∈ I)^{rF} = ⋃_{I_{0}⊂I finite} ⋃_{k∈ℕ} R(x_{i}^{1/k} | i ∈ I_{0})^{hF}  (29)

with

vR(x_{i}^{1/k} | i ∈ I_{0})^{hF} = vR ⊕ ⨁_{i∈I_{0}} ℤ\frac{vx_{i}}{k} ,

finitely generated over vR (as a ℤ-module). Therefore, v_{R}R(x_{i}^{1/k} | i ∈ I_{0})^{hF} is finitely generated.

Proof: The assertion for the value group follows from Lemma 4.2. Let U denote the union on the right hand side of (29). Every field in the union is henselian, so U is henselian. The value group vU is divisible and the residue field Uv = ℝ is real closed. Hence by Lemma 2.9, U is real closed. By construction, U is also F-closed. Since all x_{i}^{1/k} are in the real closed field R(x_{i} | i ∈ I)^{rF}, we find that U is contained in R(x_{i} | i ∈ I)^{rF}. Since this field is the smallest real closed and F-closed field containing R(x_{i} | i ∈ I), it follows that U = R(x_{i} | i ∈ I)^{rF}. □

Now we are able to improve the results from Section 3.3 for the present more special setting.

Lemma 4.5 Assume that w is trivial on R. Set L = LE_{R,F}^{w}(x) if w ≠ v_{R}, and L = R if w = v_{R}. Suppose that K is of the form

L(x_{i} | i ∈ J)^{rQF} log-closed, with x_{i} > 0 and wx_{i}, i ∈ J, Q-linearly independent. (30)

Take any a ∈ K such that exp a ∉ K. Then w exp a is Q-linearly independent over wK, and

wK(exp a)^{rQF} = wK ⊕ Q v exp a .  (31)

Moreover, K(exp a)^{rQF} is again log-closed, and therefore of the form (30). It contains exp b whenever b ∈ K(exp a)^{rQF} and w exp b is Q-linearly dependent over wK(exp a)^{rQF}.

Proof: By our choice of L, it is a subfield of O_{w} ∩ K, relatively exp-closed in O_{w}^{×}. Also, it is of the form R(x_{i} | i ∈ I)^{rQF} with x_{i} > 0 and vx_{i}, i ∈ I, Q-linearly independent over vR. Since the values wx_{i}, i ∈ J, are Q-linearly independent over {0} = wL, the values vx_{i}, i ∈ I_{w} ∪ J, are Q-linearly independent over vR. With I = I_{w} ∪ J we infer from Lemma 4.2 that Kw = L. Thus, we can apply Lemma 3.3. We obtain that w exp a is Q-linearly independent over wK and that exp b ∈ K(exp a)^{rQF} whenever b ∈ K(exp a)^{rQF} and w exp b is Q-linearly dependent over wK(exp a)^{rQF}. Equation (31) follows by an application of Lemma 4.2 to K(exp a)^{rQF}. Since the values w exp a, wx_{i}, i ∈ J, are Qlinearly independent, the values v exp a, vx_{i}, i ∈ I, are Q-linearly independent over vR. Thus, we can infer from Lemma 3.4 that K(exp a)^{rQF} is log-closed. □

After these preparations, we can determine the residue field of LE_{R,F}(x) with respect to a given convex valuation w of M which is trivial on R. As we know already that LE_{R,F}(x)v_{R} = R, we can assume that w ≠ v_{R}. As usual, we assume that x ∈ M is such that x > R, that is, v_{R}x < 0. We also assume that wx = 0. By our construction, LE_{R,F}^{w}(x) ⊂ LE_{R,F}(x). So we can rerun our construction of LE_{R,F}^{u}(x) = LE_{R,F}(x) starting with K_{0}^{u} = LE_{R,F}^{w}(x). We note that K_{0}^{u} is of the form R(x_{i} | i ∈ I_{w})^{rQF}, where the x_{i}, i ∈ I_{w}, are obtained from the above construction (and thus, their values v_{R}x_{i} ∈ H_{w} are Q-linearly independent). Since K_{0}^{u} ⊂ O_{w}, we have that K_{0}^{u}w = K_{0}^{u} and that wx_{i} = 0 for i ∈ I_{w}. Suppose that while building up LE_{R,F}^{u}(x) from this field by the above construction, we have reached a field K of the form R(x_{i} | i ∈ I)^{rQF} with Kw = K_{0}^{u} and such that the values wx_{i}, i ∈ I \I_{w}, are Q-linearly independent. If a ∈ K, but exp a ∉ K, then Lemma 4.5 shows that w exp a is Q-linearly independent over wK. Therefore, the values w exp a, wx_{i}, i, are Q-linearly independent, and Lemma 4.2 shows that

K(exp a)^{rQF}w = R(exp a, x_{i} | i ∈ I)^{rQF}w = R(x_{i} | i ∈ I_{w})^{rQF} = K_{0}^{u} .

Hence, K(exp a)^{rQF} is again of the same form as K. By induction, it follows that

LE_{R,F}(x)w = LE_{R,F}^{w}(x) if wx = 0.

In our above considerations, the only assumption on x was that it is a positive infinite element. So we can well replace it by log_{m_{0}} x, for arbitrary m_{0} ∈ ℕ. Note that LE_{R,F}(x) = LE_{R,F}(log_{m_{0}} x). If w log_{m_{0}} x = 0, then we find that

LE_{R,F}(x)w = LE_{R,F}^{w}(log_{m_{0}} x) .  (32)

Theorem 4.6 Let w be an arbitrary convex valuation of M, trivial on R but different from v_{R}. Then there is an integer m_{0} ≥ 0 such that w log_{m_{0}} x = 0. With every such m_{0} , equation (32) holds. If wx = 0, then we can choose m_{0} = 0.

Proof: Starting our above construction from K_{0}^{u} = R(log_{m} x | m ≥ 0)^{rQF}, we can write LE_{R,F}(x) = ⋃_{n∈ℕ} K_{n}^{u}. Take any negative element α in the convex subgroup H_{w} of vLE_{R,F}(x) associated with w. Then there is some n ∈ ℕ and a positive a ∈ K_{n}^{u} such that α = va. By Lemma 3.9, v log_{n} a ∈ K_{0}^{u}. Corollary 2.13 tells us that va < v log_{n} a < vR. On the other hand, the values v log_{m} x are not bounded away from the subgroup vR in the value group vR(log_{m} x | m ≥ 0)^{rQF}. So there is some m_{0} such that v log_{n} a < v log_{m_{0}} x < 0. Thus, α < v log_{m_{0}} x < 0, which yields that v log_{m_{0}} x ∈ H_{w}. That is, w log_{m_{0}} x = 0, and equation (32) holds. □

From this theorem together with the uniqueness of LE_{R,F}^{w}(x) (which also works with log_{m_{0}} x in the place of x), we obtain:

Theorem 4.7 Let w be an arbitrary convex valuation of LE_{R,F}(x), and denote its valuation ring by O_{w}. Then there exists a real closed subfield K ⊂ O_{w} which is log-closed and F-closed, relatively exp-closed in O_{w}^{×} and satisfies LE_{R,F}(x)w = K. If w is not the natural valuation, then there is some integer m_{0} ≥ 0 such that K can be chosen to be the uniquely determined smallest subfield of O_{w} which is real closed, log- and F-closed, relatively exp-closed in O_{w}^{×} and contains ℝ(log_{m_{0}} x). If wx = 0, then we can choose m_{0} = 0, so that K contains x.

Further, if F_{1} ⊂ F_{2} ⊂ F_{T} , then the smallest log- and rQF_{2}-closed subfield of O_{w}, relatively exp-closed in O_{w}^{×} and containing R(x), is contained in the smallest log- and rQF_{1}-closed subfield of O_{w}, relatively exp-closed in O_{w}^{×} and containing ℝ(x). Hence:

Corollary 4.8 Suppose that F_{1} ⊂ F_{2} are subsets of F_{T} , both satisfying conditions (PADE) and (COMP). Then for every convex valuation w such that wx = 0,

LE_{R,F_{1}}^{w}(x) ⊂ LE_{R,F_{2}}^{w}(x) .

5 Further applications

In this section we show how our approach can be used to deduce the applications which van den Dries, Macintyre and Marker give in their paper [D–M–M2].

We take M = H(ℝ_{an,exp}) and x to be the germ of the identity function. Recall that this choice yields that H(ℝ_{an,exp}) = LE_{F_{an}}(x) and LE = LE_{F_{LE}}(x).

We deduce Corollary 2.10 of [D–M–M2]:

Corollary 5.1 If f : ℝ → ℝ is definable in ℝ_{an,exp}, then there are c ∈ ℝ and n ∈ ℕ such that f(z) < exp_{n}(z) for all z > c.

Proof: Let f ∈ H(ℝ_{an,exp}) denote the germ of the function f(z). By Lemma 3.10, there is some n ∈ ℕ such that f < exp_{n} x (as elements in the ordered field H(ℝ_{an,exp})). Since this says that the germ of exp_{n} z is bigger than that of f(z), it follows that f(z) < exp_{n}(z) for all large enough z ∈ ℝ. □

From now on, we will not any more distinguish the variable x from the germ x of the identity function. Note that if f is definable in ℝ_{an,exp} and g ∈ H(ℝ_{an,exp}) is the germ of the function g(x), then the element f(g) ∈ H(ℝ_{an,exp}) is defined to be the germ of the function f(g(x)); in this way, f is made into a function on H(ℝ_{an,exp}). In particular, the element f(x) ∈ H(ℝ_{an,exp}) is the germ of the function f(x).

5.1 The Hardy problem

Take two functions f, g : ℝ → ℝ, definable in ℝ_{an,exp}. Assume that exp f(x) is asymptotic to g(x), that is, lim_{x→∞} \frac{exp f(x)}{g(x)} = 1. This is equivalent to lim_{x→∞} f(x) − h(x) = 0, where h : (r, ∞) → ℝ for suitable r ∈ ℝ is the function log g(x), which again is definable in ℝ_{an,exp}. This means that the function f(x)−h(x) is ultimately smaller than every nonzero constant function. Equivalently, its germ f − h in H(ℝ_{an,exp}) is infinitesimal, or in other words, v(f − h) > 0.

As in [D–M–M2], let the function i(x) denote the compositional inverse of the function x log x. Identifying i(x) with its germ, we have that i(x) ∈ H(ℝ_{an,exp}). But by an argument about Liouville extensions of the Hardy field ℝ(x), Corollary 4.6 of [D–M–M2] shows that i(x) ∉ LE. Assume that exp i(x) were asymptotic to a function g(x) which is a composition of semialgebraic functions, exp and log. Through identification with its germ, the latter means that g(x) ∈ LE. Then also h(x) := log g(x) ∈ LE, and v(i(x) − h(x)) > 0. Further, one shows as in [D–M–M2] that there is a convergent power series f(X, Y) such that

i(x) = \frac{x}{log x} (1 + f (\frac{log log x}{log x} , \frac{1}{log x})) .

Now let w be the convex valuation corresponding to the largest convex subgroup not containing vx. This contains v log x. Therefore, w log x = 0 and w\frac{log x}{x} = −wx > 0. With

h̃ := \frac{log x}{x} h(x) − 1 ∈ LE ,

and

f̃:= f (\frac{log log x}{log x} , \frac{1}{log x}) ∈ ℝ(log x, log log x)^{rF_{an}} ⊂ LE_{F_{an}}^{w}(log x) ,  (33)

we find that

v(f̃− h̃) > v \frac{log x}{x} .  (34)

By Lemma 2.1 it follows that w(f̃ − h̃) ≥ w\frac{log x}{x} > 0. Note that wf̃ = 0 according to (33), so it follows that also wh̃ = 0. Hence, by Theorem 4.7 we can choose an element h̃_{w} ∈ LE_{F_{LE}}^{w}(log x) such that w(h̃ − h̃_{w}) > 0. It follows that w(f̃ − h̃_{w}) > 0. By Corollary 4.8, LE_{F_{LE}}^{w}(log x) ⊂ LE_{F_{an}}^{w}(log x). Therefore, f̃ − h̃_{w} ∈ LE_{F_{an}}^{w}(log x). Since wg ∈ {0, ∞} for every g ∈ LE_{F_{an}}^{w}(log x) (as this is a field of representatives for the w-residue field), we find that w(f̃− h̃_{w}) = ∞. In other words,

f̃ = h̃_{w} ∈ LE_{F_{LE}}^{w}(log x) ⊂ LE .

But then i(x) = \frac{x}{log x}(1 + f̃) ∈ LE, a contradiction. This proves that exp i(x) is not asymptotic to any function with germ in LE.

5.2 Undefinable functions

We choose a representation H(ℝ_{an,exp}) = ℝ(x_{i} | i ∈ I)^{rF^{an}} with vx_{i}, i ∈ I, rationally independent, which exists by Theorem 3.11. For the applications, we will assume in addition that x is among the x_{i}.

Lemma 5.2 Take any positive infinitesimal element t in H(ℝ_{an,exp}). Suppose that the element h ∈ H(ℝ_{an,exp}) satisfies

|h − ∑_{n=0}^{m} r_{n}t^{n}| < r′_{m}t^{m} for all m ∈ ℕ,

where r_{n}, r′_{n} ∈ ℝ, r′_{n} > 0. Then ∑_{n=0}^{∞} r_{n}X^{n} converges in ℝ near 0.

Proof: The assertion is trivial if there is some n_{0} such that r_{n} = 0 for all n > n_{0} (which in particular will hold if r′_{n} = 0 for some n). So let us assume that this is not the case. If n_{k} (k ≥ 0) denotes the k + 1-th among the indeces n for which r_{n} ≠ 0, then we set s_{k} = r_{n_{k}} ≠ 0, s′_{k} = r′_{n_{k}} ≠ 0 and z_{k} = t^{n_{k}}. Then by Lemma 2.10, h is a limit of the pseudo Cauchy sequence formed by the partial sums S_{m} = ∑_{k=0}^{m} r_{n_{k}}t^{n_{k}}. Note that vh = vS_{0} = vt^{n_{0}} = n_{0}vt with n_{0} ∈ ℕ.

Let H be the convex subgroup of vH(ℝ_{an,exp}) generated by vt, and w the convex valuation associated with H. Then wt = 0 and wh = 0. By Theorem 4.7, we can choose an element h_{w} ∈ LE_{F_{an}}^{w}(t^{−1}) such that w(h − h_{w}) > 0. That is, v(h − h_{w}) > vt^{n_{m+1}} = v(S_{m+1} − S_{m}) for all m. So Lemma 2.4 shows that h_{w} is a limit of (S_{m})_{m≥0} , too. Since vℝ(t)^{rF_{an}} = ℚvt (cf. Lemma 3.2), this is a Cauchy sequence in (ℝ(t)^{rF^{an}}, v). Since ℚvt is cofinal in vLE_{F_{an}}^{w}(t^{−1}) by our choice of H, it is also a Cauchy sequence in (LE_{F_{an}}^{w}(t^{−1}), v). Hence, h_{w} is the only limit that the sequence admits in this field. If h_{w} ∈ ℝ(t)^{rF^{an}}, then trivially, vℝ(t)^{rF_{an}}(h_{w}) = ℚvt. Otherwise, this follows by Lemma 2.6. Thus by Corollary 3.7 of [D–M–M1], vℝ(h_{w}, t)^{rF_{an}} = ℚvt.

In view of (28), we can write vt = ∑_{i∈I} q_{i}vx_{i} with q_{i} ∈ ℚ, only finitely many of them nonzero. Take i_{0} ∈ I with q_{i_{0}} ≠ 0. Then by the rational independence of the values vx_{i},

vt ∉ ∑_{i∈I\{i_{0}}} ℚvx_{i} = vℝ(x_{i} | i ∈ I \ {i_{0}})^{rF^{an}} .

So t ∉ ℝ(x_{i} | i ∈ I \ {i_{0}})^{rF^{an}}. An application of the Exchange Lemma for o-minimal theories ([P–S]) to this model of T_{an} shows that x_{i_{0}} ∈ ℝ(t, x_{i} | i ∈ I \ {i_{0}})^{rF^{an}}. Hence, H(ℝ_{an,exp}) = ℝ(t, x_{i} | i ∈ I \ {i_{0}})^{rF^{an}}. Moreover, the values vt, vx_{i}, i ∈ I \ {i_{0}}, are rationally independent. Now choose {x_{1}, …, x_{ℓ}} ⊂ {x_{i} | i ∈ I \ {i_{0}}} with ℓ minimal such that h_{w} ∈ ℝ(t, x_{1}, …, x_{ℓ})^{rF^{an}}. Suppose that ℓ > 0. Because of the minimality of ℓ, it follows from the Exchange Lemma that x_{ℓ} ∈ ℝ(h_{w}, t, x_{1}, …, x_{ℓ−1})^{rF^{an}} = ℝ(h_{w}, t)^{rF_{an}}(x_{1}, …, x_{ℓ−1})^{rF_{an}}. By Lemma 3.2 and what we have shown for ℝ(h_{w}, t)^{rF_{an}}, we know that vℝ(h_{w}, t)^{rF_{an}}(x_{1}, …, x_{ℓ−1})^{rF^{an}} = ℚvt⊕ℚvx_{1} ⊕…⊕ℚvx_{ℓ−1}. But this group does not contain vx_{ℓ}. This contradiction shows that ℓ = 0, i.e., h_{w} ∈ ℝ(t)^{rF_{an}}.

Now let ℝ⟨t⟩ denote the set of convergent Puiseux series in t, that is, the subset of the completion of ℝ(t)^{r} consisting of all series ∑_{n=n_{0}}^{∞} r_{n}t^{n/k}, where n_{0} ∈ ℤ, k ∈ ℕ, r_{n} ∈ ℝ, and ∑_{n=0}^{∞} r_{n}X^{n} converges near 0. Then ℝ⟨t⟩ is a real closed field such that it f(X_{1}, …, X_{m}) is a power series over ℝ converging near 0 and ε_{1}, …, ε_{m} are infinitesimals in ℝ⟨t⟩, then f(ε_{1}, …, ε_{m}) ∈ ℝ⟨t⟩. This shows that ℝ⟨t⟩ is rF_{an}-closed. By its definition it is clear that the rF_{an}-closure of ℝ(t) in ℝ⟨t⟩ must be equal to ℝ⟨t⟩. By induction along the lines of the proof of Lemma 4.1, one shows that there is a unique isomorphism ℝ(t)^{rF_{an}} ≃ ℝ⟨t⟩ (of valued fields) which is the identity on ℝ(t). Since h_{w} is the limit of the Cauchy sequence (S_{m})_{m≥0}, this isomorphism sends h_{w} to the unique limit ∑_{k=0}^{∞} r_{n_{k}}t^{n_{k}}, which consequently must lie in ℝ⟨t⟩. By definition of ℝ⟨t⟩, ∑_{k=0}^{∞} r_{n_{k}}t^{n_{k}} = ∑_{n=0}^{∞} r_{n}X^{n} must be convergent near 0. □

If a definable function f : (r, +∞) → ℝ has an asymptotic expansion f(x) ∼ ∑ r_{n}f_{n}(x) in the sense of [D–M–M2], then for some C ∈ ℝ, C > 0,

|f(x) − ∑_{n=0}^{m} r_{n}f_{n}(x)| < Cf_{m}(x)

holds in H(ℝ_{an,exp}) for all m ∈ ℕ. Hence if f(x) ∼ ∑ r_{n}x^{−n}, then with t := x^{−1}, it follows from the foregoing lemma that ∑_{n=0}^{m} r_{n}X^{n} is a convergent series. Using the asymptotic expansions as given in [D–M–M2], it follows that the Gamma-function and the functions

∫_{0}^{x} e^{−t^{2}}dt , ∫_{x}^{∞} \frac{e^{−t}}{t} dt , ∫_{0}^{∞} \frac{e^{−t}}{t + x} dt , ∫_{0}^{∞} \frac{e^{−t}}{1 + xt} dt , ∫_{0}^{x} e^{t^{2}}dt , ∫_{0}^{x} e^{e^{t}}dt

on (0, +∞) are not definable in ℝ_{an,exp} .

Lemma 5.3 Suppose that the element h ∈ H(ℝ_{an,exp}) satisfies

|h − ∑_{n=0}^{m} d_{n}| < r′_{m}d_{m} for all m ∈ ℕ,  (35)

where 0 < r′_{n} ∈ ℝ, and the d_{n} are positive monomials such that the values vd_{n} are strictly increasing. Then these values are contained in a finitely generated subgroup of vH(ℝ_{an,exp}).

Proof: From Lemma 4.4 we infer that h ∈ ℝ(x_{i}^{1/k} | i ∈ I_{0})^{hF_{an}} =: K for some k ∈ ℕ and some finite subset I_{0} ⊂ I, and that the value group of this field is the finitely generated subgroup vK = ⨁_{i∈I_{0}} ℤ\frac{vx_{i}}{k} of vH(ℝ_{an,exp}). From the rational independence of the values vx_{i} it follows for every monomial d that vd ∈ vK if and only if d ∈ K.

Suppose that vd_{n} ∉ vK for some n ∈ ℕ, and take n to be the smallest integer with this property. Then d_{j} ∈ K for 1 ≤ j < n. Consequently, h − S_{n−1} ∈ K. But by Lemma 2.10, v(h − S_{n−1}) = vd_{n} ∉ vK, a contradiction. □

For the application to the Riemann zeta function, we run our construction of LE_{F_{an}}(x) with a slight refinement. We choose a ℚ-basis B of ℝ containing the ℚ-linearly independent elements log p, where p runs through all primes. Starting our construction from K_{0}^{u} = ℝ(log_{m} x | m ≥ 0)^{rF_{an}}, we may first adjoin all elements exp(rx) as new x_{i}’s. Indeed, as the elements rx, log_{m} x, r ∈ B, m ≥ 1, are rationally independent over the valuation ring, Lemma 2.14 shows that the values v exp(rx), v log_{m} x, r ∈ B, m ≥ 0, are rationally independent. Hence, the values v exp(rx), r ∈ B, are rationally independent over vK_{0}^{u}. Therefore, for all s ∈ B, exp(sx) ∉ K_{0}^{u}(exp(rx) | r ∈ B \ {s}). So we can assume the elements exp(x log p) to be among the x_{i}.

The restriction ζ of the zeta function to (1, +∞) has the asymptotic expansion ζ(x) ∼ ∑ exp(−x log n). Writing n = ∏_{p prime} p^{ν_{p}} with integers ν_{p} ≥ 0, we obtain that

d_{n} := exp(−x log n) = exp(−x log ∏_{p prime} p^{ν_{p}})
= exp(−x ∑_{p prime} ν_{p} log p) = ∏_{p prime}(exp(x log p))^{−ν_{p}} ,

is a monomial. As the sequence log n, n ∈ ℕ, is strictly increasing, Lemma 2.14 shows that also vd_{n}, n ∈ ℕ, is strictly increasing.

If ζ were definable in ℝ_{an,exp} , it would follow from the foregoing lemma that the values v exp(−x log n), and in particular the values v exp(−x log p), p prime, lie in a finitely generated group. But this is not the case since the latter values are rationally independent. This proves that the restriction of the zeta function to (1, +∞) is not definable in ℝ_{an,exp} .

6 Principal parts of elements in LE_{R,F}(x)

Throughout this section, we keep our general assumptions on R and F and also assume that F satisfies (PADE) and (COMP). We work with a representation

LE_{R,F}(x) = R(x_{i} | i ∈ I)^{rR,F} with v_{R}x_{i}, i ∈ I, rationally independent,  (36)

which exists by Theorem 3.11. Note that it follows from our construction that for any two convex valuations w, w′, trivial on R,

LE_{R,F}(x)w = LE_{R,F}^{w}(x) ⊂ LE_{R,F}^{w′}(x) = LE_{R,F}(x)w′ if w is finer than w′. (37)

We note in advance that Lemmas 6.1 and 6.2 as well as Theorem 6.4 do not require any further assumptions on the x_{i} (not even that x is among them). Similarly, we only need (37), without any further information on the structure induced on the residue fields. So the results will hold for every field of the form (36), since for every real closed field, one can construct embeddings of the residue fields such that (37) holds. This can be done by transfinite induction on an arbitrary enumeration of the elements of the field. However, it should not be overlooked that Corollaries 6.5 and 6.6 and the application to the Hardy problem need additional assumptions on the x_{i} and the residue fields.

Let (K, w) be a valued field and a ∈ K. We write a =_{w} ∑_{n=0}^{∞} a_{n} if all a_{n} ∈ K and – either there is some m_{0} such that a = ∑_{n=0}^{m_{0}} a_{n}, and a_{n} = 0 for all n > m_{0} ,

– or the values wa_{n} form a strictly monotone cofinal sequence in wK and a is the (unique) limit of the Cauchy sequence (∑_{n=0}^{m} a_{n})_{m≥0} in (K, w).

Lemma 6.1 Let (K, w) be a valued field and assume that Kw ⊂ K. Further, let wK be archimedean. Suppose that K = Kw (z_{j} | j ∈ J), where the values wz_{j}, j ∈ J, are rationally independent. Let

D := { ∏_{j∈J_{0}} z_{j}^{n_{j}} | J_{0} ⊂ J finite and n_{j} ∈ ℤ for every j ∈ J_{0}}.

Take any a ∈ K. Then there are uniquely determined elements c_{n} ∈ Kw and d_{n} ∈ D such that

a =_{w} ∑_{n=0}^{∞} c_{n}d_{n} .

The same holds for every element a in the henselization or the completion of (K, w).

Proof: Let R denote the subring of K consisting of all finite sums c_{1}d_{1} +…+c_{m}d_{m} with c_{i} ∈ Kw and d_{i} ∈ D. We show that R is w-dense in K, that is, for every a ∈ K and every α ∈ wK there is a′ ∈ R such that w(a − a′) > α. From the rational independence of the values wz_{j} it follows that every two distinct elements d, d′ ∈ D have distinct values. On the other hand, every a ∈ K can be written as a quotient of two polynomials in finitely many of the z_{j}, and therefore also as a quotient \frac{b_{1}d_{1}+...+b_{m}d_{m}}{b′_{1}d′_{1}+...+b′_{ℓ}d′_{ℓ}} where d_{1}, …, d_{m} ∈ D are distinct and d′_{1}, …, d′_{ℓ} ∈ D are distinct, and b_{i}, b′_{i} ∈ Kw. We may assume that b′_{1}d′_{1} is the summand of least value in the denominator. We write

b′_{1}d′_{1} + … + b′_{ℓ}d′_{ℓ} = b′_{1}d′_{1}(1 + d′) with d′ := \frac{b′_{2}d′_{2}}{b′_{1}d′_{1}} + … + \frac{b′_{ℓ}d′_{ℓ}}{b′_{1}d′_{1}} .

Note that \frac{d′_{2}}{d′_{1}} , …, \frac{d′_{ℓ}}{d′_{1}} are elements of D of positive value. Hence, also wd′ > 0. By the geometric expansion,

w (\frac{1}{1 + d′} − ∑_{i=0}^{k}(−d′)^{i}) = (k + 1)wd′

for every integer k ≥ 1. Take α ∈ wK. Since wK is archimedean, we can choose k as big as to obtain that (k + 1)wd′ > α − w(b_{1}d_{1} + … + b_{m}d_{m})(b′_{1}d′_{1})^{−1}. For

a′ := (b_{1}d_{1} + … + b_{m}d_{m})(b′_{1}d′_{1})^{−1} ∑_{i=0}^{k}(−d′)^{i} ∈ R ,

this yields that w(a − a′) > α.

Every valued field (K, w) is w-dense in its completion (by definition). Since wK is archimedean, then the henselization of (K, w) lies in the completion and thus, (K, w) is also w-dense in its henselization. Since density is transitive, we find that R is also w-dense in the henselization and in the completion of (K, w).

Every element of the ring R can be written as a sum c_{1}d_{1} + … + c_{m}d_{m} with distinct d_{i} ∈ D, and such that wd_{1} < wd_{2} < … < wd_{m}. Its value is equal to wd_{1} . Therefore, such a sum can only be equal to 0 if it is trivial. Consequently, the representation of every element as a sum of this form is uniquely determined.

Now we choose α ∈ wK, α > 0. Then the sequence kα, k > 0, is cofinal in the archimedean group wK. For every k, we choose a_{k} ∈ R such that w(a − a_{k}) > kα. For k′ > k > 0, w(a_{k′} − a_{k}) ≥ min{w(a − a_{k′}), w(a − a_{k})} > kα. Thus, the summands of value ≤ kα in the representations of a_{k} and a_{k′} have to be the same. So we take c_{n}d_{n} to be the uniquely determined n-th summand appearing in the representation of all a_{m}, for m large enough. Since distinct elements of D have distinct values, d_{n} and thus also c_{n} is uniquely determined from the element c_{n}d_{n}. □

Lemma 6.2 Take h ∈ LE_{R,F}(x). Then there are convex valuations w, w′, trivial on R, such that:

a) the value group of (LE_{R,F}(x)w′, w) is archimedean,

b) h ∈ LE_{R,F}(x)w′ \ LE_{R,F}(x)w,

c) there are monomials d_{n} ∈ LE_{R,F}(x)w′ and elements c_{n} ∈ LE_{R,F}(x)w such that in (LE_{R,F}(x)w′, w),

h =_{w} ∑_{n=0}^{∞} c_{n}d_{n} ,  (38)

d) the summands c_{n}d_{n} are uniquely determined,

e) the values v_{R}c_{n}d_{n} lie in a finitely generated subgroup of v_{R}LE_{R,F}(x).

Proof: From Lemma 4.4 we infer that h ∈ R(x_{i}^{1/k} | i ∈ I_{0})^{hR,F} =: K for some k ∈ ℕ and some finite subset I_{0} ⊂ I. Since v_{R}K is finitely generated, it has finite rank. That is, there are only finitely many distinct convex valuations on K, trivial on R. Therefore, there are convex valuations w′_{0}, w_{0} on LE_{R,F}(x), trivial on R, such that the value group w_{0}(Kw′_{0}) is archimedean and h ∈ LE_{R,F}(x)w′_{0} \ LE_{R,F}(x)w_{0} .

Every element in v_{R}K is the value of a monomial built up from the elements x_{i}, i ∈ I_{0} . Hence, we can choose monomials z_{1}, …, z_{ℓ} ∈ K such that:

• v_{R}z_{1}, …, v_{R}z_{ℓ_{1}} form a set of rationally independent generators of v_{R}(Kw_{0}),

• w_{0}z_{ℓ_{1}+1} = v_{R}z_{ℓ_{1}+1} + v_{R}(Kw_{0}) , …, w_{0}z_{ℓ_{2}} = v_{R}z_{ℓ_{2}} + v_{R}(Kw_{0}) form a set of rationally independent generators of w_{0}(Kw′_{0}), and

• w′_{0}z_{ℓ_{2}+1} = v_{R}z_{ℓ_{2}+1} + v_{R}(Kw′_{0}) , …, w′_{0}z_{ℓ} = v_{R}z_{ℓ} + v_{R}(Kw′_{0}) form a set of rationally independent generators of w′_{0}K. Similarly as in the proof of Lemma 4.2, one finds that ℓ = |I_{0}| and that R(z_{1}, …, z_{ℓ})^{hF} = R(x_{i}^{1/k} | i ∈ I_{0})^{hF}. From Lemma 4.2 it follows that Kw_{0} = R(z_{1}, …, z_{ℓ_{1}})^{hF} and Kw′_{0} = R(z_{1}, …, z_{ℓ_{2}})^{hF}. Now we have that Kw_{0} ⊂ Kw′_{0} ⊂ K. Since Kw_{0} = K ∩ LE_{R,F}(x)w_{0} and Kw′_{0} = K ∩ LE_{R,F}(x)w′_{0} , we obtain that h ∈ Kw′_{0} \ Kw_{0} . Since w_{0}(Kw′_{0}) = ⨁_{ℓ_{1}<j≤ℓ_{2}} ℤw_{0}z_{j} is archimedean, we can apply Lemma 6.1, where we set J = {ℓ_{1}+1, …, ℓ_{2}}, to obtain the unique representation (38). Here, the d_{n} are monomials built up from z_{ℓ_{1}+1}, …, z_{ℓ_{2}} . Thus, they are also monomials built up from x_{i}, i ∈ I_{0} . Note that the d_{n} depend on our choice of the elements z_{j}, j = ℓ_{1} + 1, …, ℓ_{2} . These in turn are uniquely determined only up to multiplication with monomials with trivial w-value. Thus, the d_{n} are in general not uniquely determined. However, the uniqueness of the summands c_{n}d_{n} can be shown as in the proof of the foregoing lemma. The values v_{R}c_{n}d_{n} lie in the value group v_{R}K, which is finitely generated, according to Lemma 4.4.

It remains to find appropriate valuations w, w′ on LE_{R,F}(x). Since h ∉ Kw_{0} , there is at least one summand c_{n}d_{n} such that w_{0}c_{n}d_{n} ≠ 0. We take w to be the valuation associated with the smallest convex subgroup H of v_{R}LE_{R,F}(x) containing v_{R}c_{n}d_{n}. Then w is the finest convex valuation on LE_{R,F}(x) which coincides with w_{0} on K. Similarly, the valuation w′ associated with the largest convex subgroup H′ of v_{R}LE_{R,F}(x) not containing v_{R}c_{n}d_{n} is the coarsest convex valuation on LE_{R,F}(x) coinciding with w′_{0} on K. Finally, w(LE_{R,F}(x)w′) = H/H′ is archimedean. □

For each monomial d ∈ LE_{R,F}(x) we define w_{d} to be the convex valuation associated with the largest convex subgroup not containing v_{R}d. Then w_{d} is the coarsest convex valuation such that w_{d}d ≠ 0. The residue field LE_{R,F}(x)w_{d} can be thought of as the largest residue field “below d”; it is the largest residue field in which the residue of either d or d^{−1} is 0. Note that if w_{d}d < w_{d}d′ < 0, then the largest convex subgroup not containing v_{R}d must be equal to that not containing v_{R}d′ and therefore, w_{d} = w_{d′}. The following theorem is the intrinsic version of “truncation at 0”.

Theorem 6.3 Take h ∈ LE_{R,F}(x) such that v_{R}h < 0. Then there exist m ∈ ℕ, monomials d_{n} ∈ LE_{R,F}(x), elements c_{n} ∈ LE_{R,F}(x)w_{d_{n}}, 1 ≤ n ≤ m, some r_{h} ∈ R, and h^{+} ∈ LE_{R,F}(x) of value v_{R}h^{+} > 0, such that

h = c_{0}d_{0} + … + c_{m}d_{m} + r_{h} + h^{+} with v_{R}c_{0}d_{0} < … < v_{R}c_{m}d_{m} < 0 ,  (39)

and such that w_{d_{n}}d_{n} < w_{d_{n}}d_{n+1} for all n < m. The summands c_{n}d_{n} and the elements r_{h} and h^{+} are uniquely determined.

Proof: In the representation (38) of h given by Lemma 6.2, there are only finitely many summands c_{0}d_{0}, …, c_{m_{1}}d_{m_{1}} of negative v_{R}-value. Note that the valuation w′ of that lemma coincides with w_{d_{0}} . Therefore, c_{0}, …, c_{m_{1}} ∈ LE_{R,F}(x)w_{d_{0}} . Assume that w_{d_{0}}c_{m_{1}}d_{m_{1}} < 0. Then also w_{d_{0}}c_{n}d_{n} < 0 and thus, w_{d_{0}} = w_{d_{n}} for 0 ≤ n ≤ m_{1}. It follows that c_{n} ∈ LE_{R,F}(x)w_{d_{0}} = LE_{R,F}(x)w_{d_{n}} for 0 ≤ n ≤ m_{1} and w_{d_{n}}d_{n} < w_{d_{n}}d_{n+1} for 0 ≤ n < m_{1} . We have that v_{R}(h − c_{1}d_{1} − … − c_{m_{1}}d_{m_{1}}) ≥ 0. Consequently, there is a unique r_{h} ∈ R such that h^{+} := h − c_{1}d_{1} − … − c_{m_{1}}d_{m_{1}} − r_{h} has value v_{R}h^{+} > 0, and we are done.

Now assume that w_{d_{0}}c_{m_{1}}d_{m_{1}} = 0 (note that w_{d_{0}}c_{n}d_{n} < 0 for n < m_{1} by condition c) of Lemma 6.2). It follows that c_{m_{1}}d_{m_{1}} ∈ LE_{R,F}(x)w_{d_{0}}, and we apply the lemma again to this element in the place of h. We repeat this procedure, thereby descending through the convex valuations of LE_{R,F}(x). But we are actually working with elements inside of the rF-closure of the field K which we used in the proof of Lemma 6.2. Since the value group of K has finite rank, there are only finitely many distinct convex valuations on K. Therefore, after a finite repetition of our procedure, we reach a convex valuation which coincides with v_{R} on K (if the procedure doesn’t stop before). If c_{ℓ}d_{ℓ}, …, c_{m}d_{m} are the summands obtained from Lemma 6.2 at this step, then by their choice we have that w_{d_{ℓ}}c_{n}d_{n} = v_{R}c_{n}d_{n} < 0 for ℓ ≤ n ≤ m, and our procedure will stop here. Now necessarily v_{R}(h − c_{1}d_{1} − … − c_{m}d_{m}) ≥ 0 since otherwise, we would have to obtain a further summand from our procedure. — The uniqueness of the summands c_{n}d_{n} follows from the uniqueness assertion of Lemma 6.2. □

Given the representation (39) of an element h according to this theorem, the uniquely determined finite sum

pp(h) := c_{1}d_{1} + … + c_{m}d_{m}

will be called the principal part of h; we set pp(h) := 0 if v_{R}h ≥ 0. The principal part is uniquely determined, once the set of monomials in LE_{R,F}(x) is fixed. Note that v_{R}(h − pp(h) − r_{h}) > 0 with r_{h} ∈ R.

Theorem 6.4 Let f, g : R → R be ultimately positive R-definable functions. Then f is asymptotic to rg on R for some positive r ∈ R if and only if the germs log f and log g in H(R) have the same principal part.

Proof: We know already that f is asymptotic to rg on R if and only if v_{R}(log f−log rg) > 0. This in turn is equivalent to v_{R}(log f −log g) ≥ 0, since if the latter holds, then there is some r_{0} ∈ R such that v_{R}(log f −log g−r_{0}) > 0, and we set r = exp r_{0} . By the uniqueness of the principal part, v_{R}(log f − log g) ≥ 0 if and only if pp(log f) = pp(log g). □

To apply this theorem in the spirit of the Hardy problem, we take F to be any set of restricted analytic functions, closed under partial derivations. Then by running our construction of Section 3 simultaneously for F and F_{an} , we find index sets I_{F} ⊂ I and elements x_{i} such that LE_{ℝ,F}(x) = ℝ(x_{i} | i ∈ I_{F})^{rF} and LE_{ℝ,F_{an}}(x) = ℝ(x_{i} | i ∈ I)^{rF_{an}}. So the monomials of LE_{ℝ,F}(x) will also be monomials of LE_{ℝ,F_{an}}(x). Moreover, we can take

LE_{ℝ,F}(x)w = LE_{ℝ,F}^{w}(log_{m_{0}} x) ⊂ LE_{ℝ,F_{an}}^{w}(log_{m_{0}} x) = LE_{ℝ,F_{an}}(x)w

for each convex valuation w and suitable m_{0} , according to Theorem 4.6 and Corollary 4.8. Using principal parts determined by this choice of the x_{i} and the residue fields, we get:

Corollary 6.5 Assume that h : ℝ → ℝ is definable in ℝ_{an,exp}. Then exp h is asymptotic to a composition of semialgebraic functions, exp, log and restricted analytic functions in F, if and only if pp(h) ∈ LE_{ℝ,F}(x).

As an example, let us reconsider the Hardy problem. Here we assume in addition that the x_{i} include x (cf. Theorem 3.11). We choose w as in Section 5.1. The representation of i(x) is just i(x) = cx, where c = \frac{1}{log x}(1 + f̃) ∈ H(ℝ_{an,exp})w. Thus, pp(i(x)) = i(x) ∉ LE. Hence by our corollary, exp i(x) is not asymptotic to any element of LE.

Let us give a further application of Theorem 6.3. Denote by L_{F} the language of ordered rings, enriched by symbols for the functions from F. Recall that every generalized power series field ℝ((G)) has a canonical cross-section, sending α ∈ G to the element 1_{α} ∈ ℝ((G)) which has a 1 at α and zeros everywhere else. (1_{α} is the characteristic function of the singleton {α}.)

Corollary 6.6 Take any L_{F}-embedding of LE_{ℝ,F}(x) in some generalized power series field ℝ((G)), and denote by L its image in ℝ((G)). Assume that the restriction of the canonical cross-section of ℝ((G)) to vL is a cross-section π of (L, v), and that L = ℝ(πvL)^{rF}. Then the nonzero elements of the support of each element in L are bounded away from 0.

Proof: For every convex valuation w with associated convex subgroup H_{w} ⊂ G, we have that ℝ((G))w = ℝ((H_{w})). These residue fields satisfy (37).

Let I ⊂ vL be a maximal set of rationally independent values. Set x_{i} := 1_{α} for i = α ∈ I. Then ℝ(x_{i} | i ∈ I)^{r} = ℝ(πvL)^{r} and hence, ℝ(x_{i} | i ∈ I)^{rF} = ℝ(πvL)^{rF} = L by hypothesis. The monomials obtained from the x_{i} are precisely the elements of the form r · 1_{α} with r ∈ ℝ and α ∈ vL. Note that if α < H_{w}, then for every c ∈ ℝ((H_{w})), the support of c r1_{α} is bounded away from 0 by every element β which satisfies α+H_{w} < β < 0. For example, β = α/2 is a good choice.

Take h ∈ L and consider the representation (39) with respect to the monomials x_{i} and the residue fields ℝ((H_{w})). Now support(h) \ {0} is the union of the support of c_{1}d_{1} + … + c_{m}d_{m} and the support of h^{+}. The latter is bounded away from 0 by vh^{+}. The support of c_{1}d_{1} + … + c_{m}d_{m} is the union of the supports of c_{1}d_{1}, …, c_{m}d_{m}. This union is bounded away from 0 by \frac{1}{2}vd_{m}. □

Note that the embeddings of H(ℝ_{an,exp}) and of LE in the logarithmic power series field ℝ((t))^{LE} given in [D–M–M2] satisfy the conditions of the corollary. (Recall that ℝ((t))^{LE} can be viewed as a subfield of a suitable power series field.)

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Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6

email: fvk@math.usask.ca — home page: http://math.usask.ca/˜ fvk/index.html email: skuhlman@math.usask.ca — home page: http://math.usask.ca/˜ skuhlman/index.html