Valuation theory of exponential Hardy fields I^{∗}

Franz-Viktor and Salma Kuhlmann

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

11. 4. 2002

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 F_{T} we denote the set of all 0-definable functions of P. 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. Theorem B of [D–S2] tells us that T (exp) admits quantifier elimination and a universal axiomatization in the language augmented by a symbol for the inverse function log of exp. This implies that H(R) is equal to the closure of its subfield R(x) under F_{T} , exp and log; here, x denotes the germ of the identity function (cf. [D–M–M1], §5; the arguments also hold in the case where R is a non-archimedean model).

We shall analyze the valuation theoretical structure of this closure by explicitly showing how it can be built up from R(x) (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:

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

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

P_{r} : (0, ∞) ⟶ ℝ
x ↦ x^{r} .

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} (cf. [M]), 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 a subsequent paper, we shall study the residue fields of exponential Hardy fields with respect to arbitrary convex valuations (which are not necessarily T (exp)-convex).

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]. For the general notions and tools from model theory we use in this paper, we refer the reader to [C–K].

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)

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 .  (4)

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 2.2 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,  (5)

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}) ,  (6)

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 (5) 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 2.3 For every z ∈ M,

wz > 0 ⇒ w exp z = 0 ∧ w(exp z − 1) = wz  (7)
vz = 0 ⇒ v exp z = 0 .  (8)

Proof: By Lemma 2.1, wz > 0 implies vz > 0, that is, z is infinitesimal. In particular, |z| < 1, and (TA) holds. Applying (6) of Lemma 2.2 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 (7).

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 (7) 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 (8). □

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 2.4 For every z ∈ M,

wz < 0 ∧ z > 0 ⇒ w exp z ≪ wz ≪ w log z < 0  (9)
wz = 0 ∧ z > 0 ⇒ w log z ≥ 0  (10)
vz ≥ 0 ⇔ v exp z = 0 .  (11)

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 (4), we can replace z by log z to get that wz ≪ w log z < 0. This proves (9).

Now assume that wz = 0 and z > 0. If vz < 0, then by (9), vz < v log z < 0. If vz > 0, then vz^{−1} < 0 and by (9), 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 (9), 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 (10).

Implication “⇒” of (11) follows from (7) with w = v, together with (8). The converse implication follows from (10), 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 2.5 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 .  (12)

Proof: The part “wx ≪ w log x ≪ … ≪ w log_{m} x” follows from (9) 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. □

For further details on the valuation theory of exponential fields, see [KS2], [KS1] 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 functions 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 .  (13)

(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,  (14)

or in other words,

v (\frac{f}{g} − 1) > vR ⟺ f and g are asymptotic on R,  (15)

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. (Later, we will assume that it is a model of T (exp), but we will not distinguish between this model and its reduct to the language of T.) Suppose that K is a submodel (hence an elementary substructure) of M. Take x_{i} ∈ M, i ∈ I. By K⟨x_{i} | i ∈ I⟩ we denote the 0-definable closure of K ∪ {x_{i} | i ∈ I} in M. By our assumption on the language of T, this 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 a universal axiomatization and K⟨x_{i} | i ∈ I⟩ is a substructure of M, it is a model of T. Since T admits quantifier elimination, K⟨x_{i} | i ∈ I⟩ is an elementary substructure of M.

For an arbitrary subfield F ⊂ M, we let F^{Q} denote the smallest subfield of M which contains F and is Q-closed, i.e., closed under the exponents from Q. Further, we let F^{rQF} denote the smallest real closed subfield of M which contains F, is Q-closed, and is F-closed, i.e., closed under all functions on M which are interpretations of function symbols in F. We will say that F is rQF-closed if F = F^{rQF}. Note that real closures can be taken to lie in M since M is real closed.

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 w of a model is T -convex (cf. [D–L], §4).

Lemma 3.1 Assume that R is a submodel of M. If x ∈ M such that wx ∉ wR, then wR⟨x⟩ = wR ⊕ Qwx.

Lemma 3.2 Assume that R is a submodel of M. Take 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} .  (16)

Proof: 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, Lemma 3.1 shows that

wR⟨x_{1}, …, x_{n}⟩ = wR ⊕ ⨁_{i=1}^{n} Qwx_{i} .  (17)

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 wR(x_{1}, …, x_{n})^{Q} is a Q-vector space and contains wx_{1}, …, wx_{n}, we obtain that

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

Note that in view of Theorem B of [D–S2], R is an elementary substructure of M, and (ℝ, +, ·, <, F_{T} , exp) is an elementary substructure of both. However, we will not use this fact in our constructions.

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 2.4, 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 2.3, 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} \}  (18)

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 .  (19)

Moreover, K(exp a)^{rQF} is again log-closed, and therefore of the form (18). 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 (19) 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 (R, v) ⊂ (K, v) is any extension of valued fields and that w is a valuation on K such that v is finer than w, and Kw = R. Take x_{i} ∈ K such that the values vx_{i}, i ∈ I, are Q-linearly independent over vR. Then the values wx_{i}, i ∈ I, are Q-linearly independent.

Proof: From Kw = R it follows that v is the composition of w with the restriction of v to R. 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 assumption, 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 (18). From now on, we always assume that x ∈ M such that x > R, that is, vx < vR and x > 0. By v_{R} we will denote the finest convex valuation on M which is trivial on R.

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 2.5 we know that

vx ≪ v log x ≪ … ≪ v log_{m} x ≪ … < vR .  (20)

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 (20) 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}. □

For F ⊆ F_{T} we denote by LE_{R,F}(x) the smallest subfield of M which contains R(x) and is real closed and closed under F, exp and log. We shall show how to build up LE_{R,F}(x) from R(x). As a preparation for what we will need in a later paper, 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 (18). 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 (18), 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 (18), 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 (18). Their union

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

is rQF-closed and of the form (18). By construction, we have:

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 (18).

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 2.4, 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 2.4, 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 H(R) = LE_{R,F}(x) by what we have remarked in the introduction, and thus 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.  (21)

Moreover,

LE_{R,F}(x)v_{R} = R .  (22)

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 (18). 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 the T (exp)-definable closure of R(x) in LE_{R,F_{T}} (x) is equal to LE_{R,F_{T}} (x), 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 (22). 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. Note that if the latter holds, then it also holds for every integer N′ > N in the place of N. If a denotes the germ of f in H(R), then by (15) the condition is equivalent to

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

Here, N can be chosen such that N + s ≥ 0. Suppose that s < s′ ∈ ℤ. Since a > R we have that va < vR; hence by Corollary 2.5, v log_{N+s} a ≠ v log_{N+s′} a which shows that the above inequality cannot hold for s′ in the place of s. Thus, the level s is uniquely determined (see also [M–M]).

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 one 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 .  (23)

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 (23) 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}} .  (24)

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 (24) 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 function symbols for 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. □

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Mathematical Sciences Group, 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