3.1 Observations on bounded Hardy fields

In the rest of this section, H is a Hardy field. If H is bounded, then there is a $\phi\in\mathcal C$ with $\phi>_{\mathrm{e}}0$ and $g\prec\phi$ for all $g\in H$ , so $\varepsilon:=1/\phi\in\mathcal C^\times$ satisfies $\varepsilon>_{\mathrm{e}} 0$ and $\varepsilon\prec h$ for all $h\in H^\times$ . A germ $y\in\mathcal{C}$ is said to be H-hardian if it lies in a Hardy field extension of H, and hardian if it lies in some Hardy field (equivalently, it is $ {\mathbb Q} $ -hardian). For $r\in\{\infty,\omega\}$ , if $H\subseteq\mathcal{C}^r$ and $y\in\mathcal{C}^r$ is H-hardian, then $H\langle y\rangle\subseteq\mathcal{C}^r$ ; see [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24b, Section 4]. By [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24b, Lemmas 5.18, 5.19] we have the following.

Proof. Let $P\in H\{Y\}^{\neq}$ , so $P(f)\in H\langle f\rangle^\times$ . It suffices to show that then ${P(f) \sim P(g)}$ . By Taylor expansion [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, p. 210], with $r:=\operatorname{order} P$ and $\boldsymbol{i}$ ranging over $ {\mathbb N} ^{1+r}$ :

$$P(g) - P(f) = \sum_{\lvert\boldsymbol{i}\rvert\geqslant 1} P_{(\boldsymbol{i})}(f)(g-f)^{\boldsymbol{i}}\quad\text{where $P_{(\boldsymbol{i})}=\dfrac{P^{(\boldsymbol{i})}}{\boldsymbol{i}!}\in H\{Y\}$.}$$

If $\lvert\boldsymbol{i}\rvert\geqslant 1$ , then $(g-f)^{\boldsymbol{i}}\prec h$ for all $h\in H\langle f\rangle^{\times}$ , and hence $P_{(\boldsymbol{i})}(f)(g-f)^{\boldsymbol{i}} \prec P(f)$ . Therefore, ${P(g)-P(f)\prec P(f)}$ as required.

With Corollary 2.8 we now obtain analytic ‘copies’ of certain H-hardian germs.

Recall from [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 10.6] that an H-field L is said to be Liouville closed if it is real closed and for all $f,g\in L$ there exists $y\in L^\times$ with $y'+fy=g$ . If $H\supseteq {\mathbb R} $ , then our Hardy field H is an H-field, and H has a smallest Liouville closed Hardy field extension ${\operatorname{Li}}(H)$ . (See [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24b, Section 4].) We can now also strengthen [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Theorem 5.1] as follows.

Thus, maximal Hardy fields, maximal $\mathcal{C}^\infty$ -Hardy fields, and maximal $\mathcal{C}^{\omega}$ -Hardy fields are unbounded. (See also [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24b, Corollary 5.23 and succeeding remarks].) The cofinality of a totally ordered set S (that is, the smallest ordinal isomorphic to a cofinal subset of S) is denoted by $\operatorname{cf}(S)$ ; likewise, $\operatorname{ci}(S)$ denotes the coinitiality of S; cf. [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 2.1]. As [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Theorem 5.1] gave rise to [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Corollary 5.2], so Corollary 3.5 yields the following.

Corollary 3.6. If H is a maximal analytic Hardy field, then $\operatorname{cf}(H) \gt \omega$ , and thus

$$\operatorname{ci}(H) = \operatorname{cf}(H^{\lt a}) = \operatorname{ci}(H^{>a})> \omega\ \text{ for all $a\in H$.}$$

Likewise with ‘smooth’ in place of ‘analytic’.

Call a subset F of $\mathcal{C}$ cofinal if, for each $\phi\in\mathcal C$ , there exists $f\in F$ with $\phi\leqslant f$ . If $F_1,F_2\subseteq\mathcal{C}$ and for all $f_1\in F_1$ there is an $f_2\in F_2$ with $f_1\leqslant f_2$ , and $F_1$ is cofinal, then $F_2$ is cofinal. Clearly each cofinal subset of $\mathcal{C}$ is unbounded. The following strengthens [Reference SjödinSjö70, Theorem 7].

Proof. Put $\mathfrak c:=2^{\aleph_0}$ , and let $\alpha$ , $\alpha'$ , $\beta$ range over ordinals $<\mathfrak c$ . Choose an enumeration $(\phi_\alpha)_{\alpha<\mathfrak c}$ of $\mathcal{C}$ . Suppose $((H_\alpha,h_\alpha))_{\alpha<\beta}$ is a family of bounded analytic Hardy fields $H_\alpha$ , each with an element $h_\alpha\in H_\alpha$ , such that

(3.1) \begin{equation}\alpha<\alpha'<\beta\ \Rightarrow\ H_{\alpha}\subseteq H_{\alpha'}\quad\text{and}\quad\alpha<\beta\ \Rightarrow\ \phi_\alpha <_{\mathrm{e}} h_\alpha.\end{equation}

Then $H:=\bigcup_{\alpha<\beta}H_\alpha$ is an analytic Hardy field, and H is bounded, as the union of countably many bounded subsets of $\mathcal C$ . By Lemma 3.1, $H^*:={\operatorname{Li}}(H( {\mathbb R} ))$ is also bounded. Take $\phi\in\mathcal{C}$ with $\phi>_{\mathrm{e}} H^*$ and $\phi\geqslant\phi_\beta$ . Corollary 3.5 yields an $H^*$ -hardian $h_\beta\in\mathcal{C}^\omega$ with $h_\beta>_{\mathrm{e}}\phi$ . Then the analytic Hardy field $H_\beta:=H^*\langle h_\beta\rangle$ is bounded by Lemma 3.1, contains $H_\alpha$ for all $\alpha<\beta$ , and $\phi_\beta<_{\mathrm{e}}h_\beta$ .

Now, transfinite recursion yields a family $((H_\alpha,h_\alpha))_{\alpha<\mathfrak c}$ , where $H_\alpha$ is a bounded analytic Hardy field and $h_\alpha\in H_\alpha$ such that (3.1) holds with $\mathfrak c$ in place of $\beta$ . Then $\bigcup_{\alpha<\mathfrak c} H_\alpha$ is a cofinal analytic Hardy field.

See Corollary 5.14 for a strengthening of Corollary 3.7.

3.2 Hardy fields of countable cofinality

Hardy fields of countable cofinality are bounded. For later use we study here such Hardy fields in more detail. Given a valued differential field K, let C denote its constant field and $\Gamma$ its value group.

In the next two lemmas, $\Gamma$ is an ordered abelian group. We recall from [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 2.4] that the archimedean class of $\alpha\in\Gamma$ is

$$[\alpha] := \{ \beta\in\Gamma:\ \text{$\lvert\alpha\rvert\leqslant n\lvert\beta\rvert$ and $\lvert\beta\rvert\leqslant n\lvert\alpha\rvert$ for some $n\geqslant 1$} \}.$$

We write $[\alpha]_\Gamma$ instead of $[\alpha]$ if we want to stress the dependence on $\Gamma$ . We equip the set $[\Gamma]=\{[\alpha]:\alpha\in\Gamma\}$ with the ordering satisfying $[\alpha] \leqslant [\beta]$ if and only if $\lvert\alpha\rvert\leqslant n\lvert\beta\rvert$ for some $n \geqslant 1$ . If $\Delta$ is an ordered subgroup of $\Gamma$ , then for each $\delta\in\Delta$ we have $[\delta]_\Delta=[\delta]_\Gamma\cap\Delta$ , and we have an embedding ${[\delta]_\Delta\mapsto[\delta]_\Gamma\colon [\Delta] \to [\Gamma]}$ of ordered sets via which we identify $[\Delta]$ with an ordered subset of $[\Gamma]$ .

Let G be an abelian group, with divisible hull $ {\mathbb Q} G= {\mathbb Q} \otimes_{ {\mathbb Z} }G$ . Then $\operatorname{rank}_{ {\mathbb Q} } G:=\dim_{ {\mathbb Q} } {\mathbb Q} G$ (a cardinal) is the rational rank of G. (Note, in [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 1.7] we defined the rational rank of G to be $\infty$ if the $ {\mathbb Q} $ -linear space $ {\mathbb Q} G$ is not finitely generated.)

A valued differential field K has small derivation if, for all $f\in K$ : $f\prec 1\Rightarrow f'\prec 1$ , and very small derivation if for all $f\in K$ : $f\preccurlyeq 1\Rightarrow f'\prec 1$ . For more on this, see [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 4.4] and [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24d, Section 13], respectively.

By [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 3.1.10] the hypothesis on $\operatorname{rank}_{ {\mathbb Q} }(\Gamma_L/\Gamma)$ in Lemma 3.11 is satisfied if $\operatorname{trdeg}(L|K)$ is countable. Hence this lemma yields the following result.

In [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Corollary 3.13] we showed that if $H\supseteq {\mathbb R} $ and $H^{> {\mathbb R} }$ has countable coinitiality, and $H^{\operatorname{da}}$ is the d-closure of H in a maximal Hardy field extension of H, then $(H^{\operatorname{da}})^{> {\mathbb R} }$ also has countable coinitiality. The property of H having countable cofinality is equally robust.

There is an immediate consequence as follows.

We precede the proof of Theorem 3.13 by a few lemmas. In Lemmas 3.15 and 3.16 we let K be a pre-d-valued field of H-type with asymptotic couple $(\Gamma,\psi)$ , where ${\Gamma\ne\{0\}}$ . By [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 10.3.1], K has a d-valued extension $\operatorname{dv}(K)$ of H-type, the d-valued hull of K, such that any embedding of K into any d-valued field L of H-type extends uniquely to an embedding $\operatorname{dv}(K) \to L$ .

For the proof of the next lemma we recall that ${\lvert\Gamma\setminus (\Gamma^{\ne})'\rvert\leqslant 1}$ , and for $\beta\in\Gamma$ we have $\beta\in \Gamma\setminus (\Gamma^{\ne})'$ if and only if $\beta=\max\Psi$ or $\Psi<\beta<(\Gamma^>)'$ , by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 9.2.1 and 9.2.16]. We say that K has asymptotic integration if $\Gamma=(\Gamma^{\ne})'$ and K is grounded if $\Psi$ has a largest element. A gap in K is a $\beta\in\Gamma$ such that $\Psi<\beta<(\Gamma^>)'$ . (Thus, there is at most one gap in K, and K has asymptotic integration or is grounded if and only if it has no gap.) For all this, see [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 9.1 and 9.2].

Proof. Set $L:=K(y)$ and $M:=\operatorname{dv}(L)$ . Lemma 3.15 allows us to replace K by its d-valued hull inside M to arrange that K is d-valued. Using [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 10.5.15 and the remark preceding 4.6.16], we replace K by $K(C_M)$ to arrange L to also be d-valued with $C=C_L$ . Finally, replacing K by its algebraic closure inside an algebraic closure of L, we arrange K to be algebraically closed. We may assume $y\notin K$ , so y is transcendental over K. Then

$$S := \{ v(s-a') : a\in K\} \subseteq \Gamma.$$

Assume for now that S has a maximum $\beta$ . Then $\beta\notin (\Gamma^{\neq})'$ by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 10.2.5(i)], so $\beta=\max\Psi$ or $\beta$ is a gap in K. If $\beta=\max\Psi$ , then $\Gamma_{L}=\Gamma+ {\mathbb Z} \alpha$ with $\Gamma^{\lt} \lt n\alpha \lt 0$ for all $n\geqslant 1$ , so $\Gamma$ is cofinal in $\Gamma_{L}$ . Suppose $\beta$ is a gap in K. Take $a\in K$ with $\beta=v(s-a')$ and set $z:= y-a$ , so $z'=s-a'$ . We arrange $z\not\asymp 1$ by replacing a with $a+c$ for suitable $c\in C_L=C$ . If $z\prec 1$ , then [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 10.2.1 and its proof] gives $\Gamma_{L}=\Gamma+ {\mathbb Z} \alpha$ with $0 \lt n\alpha \lt \Gamma^{>}$ for all $n\geqslant 1$ , so $\Gamma$ is cofinal in $\Gamma_{L}$ . If $z\succ 1$ , then [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 10.2.2 and its proof] gives likewise that $\Gamma$ is cofinal in $\Gamma_{L}$ .

If S does not have a largest element, then L is an immediate extension of K: this holds by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 10.2.6] if $S<(\Gamma^>)'$ ; otherwise, take $a\in K$ with $v(s-a')\in(\Gamma^>)'$ and $y-a\not\asymp 1$ , and apply [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 10.2.4 and 10.2.5(iii)] to $s-a'$ , $y-a$ in place of s, y, respectively.

Lemmas 3.8 and 3.16 and [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24b, Proposition 4.2(iv)] yield the following.

We can now give the proof of Theorem 3.13. First, replacing E, H by E(x), H(x), respectively, and using the last lemma, we arrange $x\in H$ . Let S be a well-ordered cofinal subset of H of order type $\operatorname{cf}(H)$ . For $\phi\in \mathcal{C}$ , define $\exp_n(\phi)\in\mathcal{C}$ by recursion: $\exp_0(\phi):=\phi$ and $\exp_{n+1}(\phi):=\mathrm{e}^{\exp_n(\phi)}$ . Then $\lvert S\rvert=\operatorname{cf}(H)$ , and $\widetilde S:=\bigcup_n \exp_n(S)$ is a cofinal subset of E, by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24b, Lemma 5.1]. Thus, $\operatorname{cf}(E)=\operatorname{cf}(\widetilde S)\leqslant\lvert\widetilde S\rvert=\lvert S\rvert=\operatorname{cf}(H)$ as claimed. If $\exp(H)\subseteq H$ , then $\widetilde S\subseteq H$ , hence $\operatorname{cf}(E)=\operatorname{cf}(H)$ .

The following corollary of Lemma 3.15 is not used later. If K is a pre-H-field, then by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 10.5.13] there is a unique field ordering on $\operatorname{dv}(K)$ making it a pre-H-field extension of K. Equipped with this ordering, $\operatorname{dv}(K)$ is an H-field, the H-field hull of K, which embeds uniquely over K into any H-field extension of K, notation: H(K) (not to be confused with the Hardy field $H( {\mathbb R} )$ generated over the Hardy field H by $ {\mathbb R} $ ).

For use in § 6 we include the following cofinality result, which is immediate from Lemma 3.9 and [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 10.4.5(i)].

Lemma 3.19. Let K be a d-valued field of H-type with divisible asymptotic couple $(\Gamma,\psi)$ , $\Gamma\neq\{0\}$ , and let $s\in K$ be such that

$$S:=\{ v(s-a^\dagger): a\in K^\times\} \lt (\Gamma^>)'$$

and S has no largest element. Let f be an element of an H-asymptotic field extension of K, transcendental over K, with $f^\dagger = s$ . Then $[\Gamma]=[\Gamma_{K(f)}]$ , so $\Gamma$ is cofinal in $\Gamma_{K(f)}$ .

4.1 Revisiting pseudoconvergence in Hardy fields

Let $H\supseteq {\mathbb R} (x)$ be real closed with asymptotic integration and $(f_\rho)$ a pc-sequence in H of d-transcendental type over H (cf. [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 4.4]) with pseudolimit f in a Hardy field extension of H. Then the valued field extension $H\langle f\rangle\supseteq H$ is immediate by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 11.4.7 and 11.4.13]. In [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a] we only considered pc-sequences of countable length, but here we do not assume $(f_{\rho})$ has countable length (to be exploited in the proof of Theorem B). We begin by deriving a sufficient condition on $y\in \mathcal{C}^{\lt \infty}$ to be H-hardian with ${f_\rho\leadsto y}$ . This will enable us to find such y in $\mathcal{C}^{\omega}$ . (Another possible use is to find such y with oscillating $y-f$ , so that $H\langle y \rangle $ and $H\langle f \rangle $ are ‘incompatible’ Hardy field extensions of H.) To simplify notation, set $t:= x^{-1}$ . Let $\phi\in H^\times$ . Recall from [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 11.1] that $\phi$ is said to be active in H if $\phi\succcurlyeq h^\dagger$ for some $h\in H^\times$ , $h\not\asymp 1$ . Denoting by $\partial$ the derivation of the differential ring $\mathcal{C}^{\lt \infty}$ , we let $(\mathcal{C}^{\lt \infty})^\phi$ be the ring $\mathcal{C}^{\lt \infty}$ equipped with the derivation $\delta:=\phi^{-1}\partial$ and $H^\phi$ be the ordered valued field H equipped with the restriction of $\delta$ to H; we then have a ring isomorphism $P\mapsto P^\phi\colon \mathcal{C}^{\lt \infty}\{Y\}\to(\mathcal{C}^{\lt \infty})^\phi\{Y\}$ with $P(y)=P^\phi(y)$ for each $y\in\mathcal{C}^{\lt \infty}$ . We first observe the following.

Proof. This is clear for $k=0$ . Suppose $k\geqslant 1$ . The identity (3.1) in [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a] gives

$$\delta^k(z)=\phi^{-k}\sum_{j=1}^kR^k_j(-\phi^\dagger)z^{(j)}\quad\text{where $R^k_j(Z)\in {\mathbb Q} \{Z\}$ for $j=1,\dots,k$.}$$

This yields $\delta^k(z)\prec 1$ in view of $\phi^\dagger\preccurlyeq 1$ and $z^{(j)}\prec t^{2k}\prec \phi^k$ for $j=1,\dots,k$ .

The proof of the next result uses various items from [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17]: for $P_{+h}$ , $P_{\times h}$ see [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 4.3]; for $\operatorname{ddeg}_{\prec {\mathfrak v}}P$ , see [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 6.6]; for $\operatorname{ndeg}_{\prec {\mathfrak v}}P$ , see [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 11.1]; and for Z(H,f), see [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 11.4].

Using $(\phantom{-})^\circ$ as explained in [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24d, Section 8], we have the Hardy field $H\langle f\rangle^\circ$ and the H-field isomorphism $h\mapsto h^\circ\colon H\langle f\rangle^\phi\to H\langle f\rangle^\circ$ . Put $u:=(y-h)/{\mathfrak m}\in \mathcal{C}^{\lt \infty}$ . Then $\operatorname{ddeg}_{\prec{\mathfrak v}^\circ} Q^{\phi\circ}_{+h^\circ}=0$ and $(u^\circ)^{(j)}\preccurlyeq 1$ for ${j=0,\dots,r}$ , hence $Q^{\phi\circ}(y^\circ)\sim Q^{\phi\circ}(f^\circ)$ by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24d, Lemma 11.7] with $H^\circ$ , $H\langle f\rangle^\circ$ , $h^\circ$ , $f^\circ$ , ${\mathfrak m}^\circ$ , $Q^{\phi\circ}$ , ${\mathfrak v}^\circ$ , and $y^\circ$ in place of H, $\widehat H$ , h, $\widehat h$ , ${\mathfrak m}$ , Q, ${\mathfrak v}$ , and y, respectively. This yields $Q(y) \sim Q(f)$ , since $Q^{\phi\circ}(g^\circ)=Q(g)^\circ$ for $g\in\mathcal{C}^{\lt \infty}$ .

Proposition 4.3. Suppose $0\in v(f-H)$ and $y\in \mathcal{C}^{\lt \infty}$ is such that for all ${\mathfrak m}\in H^\times$ with $v{\mathfrak m}\in v(f-H)$ and all i, j, k we have

$$y^{(i)}-f^{(i)}\ \prec\ {\mathfrak m}^j t^k\quad \text{in $\mathcal{C}^{\lt \infty}$.}$$

Then y is H-hardian and there is an isomorphism $H\langle y\rangle \to H\langle f\rangle$ of Hardy fields over H sending y to f (and thus $f_\rho\leadsto y$ ).

This holds for $n=0$ because ${\mathfrak m} z\prec {\mathfrak m}^{j+1}t^k$ for all j, k. Let $n\geqslant 1$ and assume inductively that $z^{(i)}\prec {\mathfrak m}^jt^k$ for $i=0,\dots, n-1$ and all j, k. Now $({\mathfrak m} z)^{(n)}= y^{(n)}-f^{(n)}\prec {\mathfrak m}^jt^k$ for all j, k, and

$$({\mathfrak m} z)^{(n)}\ =\ \ {\mathfrak m}^{(n)}z + \cdots + {\mathfrak m} z^{(n)}.$$

Since ${\mathfrak m}\preccurlyeq 1$ , the smallness of the derivation of H and the inductive assumption gives ${\mathfrak m}^{(n-i)}z^{(i)}\preccurlyeq z^{(i)}\prec{\mathfrak m}^jt^k$ for $i=0,\dots,n-1$ and all j, k, so ${\mathfrak m} z^{(n)} \prec {\mathfrak m}^jt^k$ for all j, k, and thus $z^{(n)}\prec {\mathfrak m}^jt^k$ for all j, k.

If ${\mathfrak m}\preccurlyeq 1$ , then this holds by Claim 1 and Lemma 4.1. In general, take $h_1\in H$ with $f-h_1\prec f-h$ and $f-h_1\preccurlyeq 1$ , and then ${\mathfrak m}_1\in H^\times$ with $f-h_1\asymp{\mathfrak m}_1$ . By the special case just proved, with $h_1$ , ${\mathfrak m}_1$ in place of h, ${\mathfrak m}$ , we have $\delta^k ( ({y-f})/{{\mathfrak m}_1} )\prec 1$ for all k. Now $z = (({y-f})/{{\mathfrak m}_1})({{\mathfrak m}_1}/{{\mathfrak m}})$ and ${{\mathfrak m}_1}/{{\mathfrak m}} \prec 1$ (in H), so the claim follows using the product rule for the derivation of $(\mathcal{C}^{\lt \infty})^\phi$ and the smallness of the derivation of $H^\phi$ .

The following is a more useful variant for the case $0\notin v(f-H)$ .

Proposition 4.4. Suppose $0\notin v(f-H)$ , and $y\in \mathcal{C}^{\lt \infty}$ is such that

$$y^{(i)}-f^{(i)}\ \prec\ t^k \text{ for all i, k.}$$

Then y is H-hardian and there is an isomorphism $H\langle y\rangle \to H\langle f\rangle$ of Hardy fields over H sending y to f.

Multiplicative conjugation gives a reduction to Proposition 4.4, except when $(f_{\rho})$ is a cauchy sequence, not just a pc-sequence. We shall exploit this several times.

4.2 Constructing analytic pseudolimits in Hardy field extensions

Let $(f_{\rho})$ be a pc-sequence in H. Corollary 3.2 from [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a] says that if $(f_\rho)$ has countable length, then $(f_{\rho})$ pseudoconverges in some Hardy field extension of H. Using Corollary 2.8 and Proposition 4.4 we now deduce smooth and analytic versions of this key fact. We say that an H-field with real closed constant field is closed if it has no proper d-algebraic H-field extension with the same constant field. (This is not how ‘closed’ was introduced in [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a], but it is equivalent to it in view of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 16.0.3 and the proof of 16.4.8].) By [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24d, Corollary 11.20], every maximal Hardy field is a closed H-field; likewise with ‘maximal smooth’ or ‘maximal analytic’ in place of ‘maximal’. Every closed H-field is Liouville closed, by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 10.6.13 and 10.6.14], and every divergent pc-sequence in a closed H-field is of d-transcendental type over it, by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 11.4.8 and 11.4.13].

Suppose $(f_\rho)$ does not have width $\{\infty\}$ . Then take $h\in H^\times$ with ${v(hf-hf_{\rho})} \lt 0$ for all $\rho$ . Take $\varepsilon\in \mathcal{C}$ such that $\varepsilon>_{\mathrm{e}} 0$ and $\varepsilon \prec t^k$ for all k, for example, $\varepsilon= \mathrm{e}^{-x}$ . Now Corollary 2.8 gives $y\in \mathcal{C}^{\omega}$ such that $(hf)^{(i)}-y^{(i)}\prec t^k$ for all k. Then y is H-hardian and $hf_{\rho}\leadsto y$ by Proposition 4.4, hence $h^{-1}y\in \mathcal{C}^{\omega}$ is H-hardian and $f_{\rho}\leadsto h^{-1}y$ .

Arguing as in the proof of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Corollary 4.8], using Corollary 4.6 instead of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Corollary 3.2], yields the following

Recall from [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a,§ 6] that the H-couple $(\Gamma,\psi)$ of H is said to be countably spherically complete if in the valued abelian group $(\Gamma,\psi)$ , every pc-sequence of length $\omega$ in it pseudoconverges in it. In view of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, remark preceding Corollary 8.1], Corollaries 3.6, 4.6, and 4.7 yield a version of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Corollary 8.1] for maximal analytic Hardy fields as follows.

Corollary 4.8. If H is a maximal analytic or maximal smooth Hardy field, then its H-couple $(\Gamma,\psi)$ is countably spherically complete and

$$\operatorname{cf}(\Gamma^{\lt}) = \operatorname{ci}(\Gamma^{>})\ >\ \omega, \quad \operatorname{ci}(\Gamma) = \operatorname{cf}(\Gamma)\ >\ \omega.$$

5.1 Case (b) extensions

Let $H\supseteq {\mathbb R} $ be a Liouville closed Hardy field with H-couple $(\Gamma,\psi)$ over $ {\mathbb R} $ . Suppose $\beta$ in an H-couple $(\Gamma^*,\psi^*)$ over $ {\mathbb R} $ extending $(\Gamma,\psi)$ falls under case (b), that is, with $(\Gamma\langle \beta \rangle , \psi_{\beta})$ the H-couple over $ {\mathbb R} $ generated by $\beta$ over $(\Gamma, \psi)$ in $(\Gamma^*, \psi^*)$ .

(b) We have a sequence $(\alpha_i)$ in $\Gamma$ and a sequence $(\beta_i)$ in $\Gamma^*$ that is $ {\mathbb R} $ -linearly independent over $\Gamma$ , such that $\beta_0=\beta-\alpha_0$ and $\beta_{i+1}=\beta_i^\dagger-\alpha_{i+1}$ for all i, and such that $\Gamma\langle \beta \rangle =\Gamma \oplus \bigoplus_{i=0}^\infty {\mathbb R} \beta_i$ .

Unlike in key parts of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Section 9], we do not assume $\beta$ is of countable type over $\Gamma$ , and this will be exploited in the proof of Theorem B. (Recall from [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Section 8] that an element $\gamma$ of an ordered vector space over $ {\mathbb R} $ extending $\Gamma$ has countable type over $\Gamma$ if $\gamma\notin\Gamma$ and $\operatorname{cf}(\Gamma^{\lt \gamma}),\operatorname{ci}(\Gamma^{>\gamma})\leqslant\omega$ .) By [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Corollary 8.15], $\beta$ also falls under case (b) with the same sequences $(\alpha_i)$ , $(\beta_i)$ when $(\Gamma,\psi)$ and $(\Gamma^*,\psi^*)$ are viewed as H-couples over $ {\mathbb Q} $ ; see [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Section 8] for the relevant definitions.

In the next proposition and its corollary we assume $y\in\mathcal{C}^{\lt \infty}$ is H-hardian, $y>0$ , and vy realizes the same cut in $\Gamma$ as $\beta$ . Then by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Remark 8.21] we have a unique isomorphism over $\Gamma$ of the H-couple over $ {\mathbb Q} $ generated by $\Gamma\cup\{\beta\}$ in $(\Gamma^*, \psi^*)$ with the H-couple of the Hardy field $H\langle y \rangle ^{\operatorname{rc}}$ over $ {\mathbb Q} $ sending $\beta$ to vy. Moreover, if $z\in\mathcal{C}^{\lt \infty}$ is also H-hardian with $z>0$ and vz realizes the same cut in $\Gamma$ as $\beta$ , then we have a unique Hardy field isomorphism $H\langle y \rangle \to H\langle z \rangle $ over H sending y to z, by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Proposition 8.20]. The problem here is to find such z in $\mathcal{C}^{\omega}$ (in which case $H\langle z \rangle $ is smooth, respectively analytic, if H is). This can always be done, in view of Corollary 2.8 and the following.

Combining Corollary 2.8 with Proposition 5.1 yields the following.

We can now use [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Theorem 9.2] to obtain an analytic strengthening of this result as follows.

5.2 Proof of Theorem A

First we give an analytic/smooth version of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Lemma 9.1].

Theorem A from the introduction and its smooth version follow from Corollary 4.6 and Lemma 5.4, just as the main theorem in [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a] is derived in the beginning of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Section 9] from the non-smooth analogues of that corollary and lemma. We now use this to characterize gaps in maximal analytic Hardy fields: see Corollary 5.9.

5.3 Characters of gaps in maximal Hardy fields

Let S be an ordered set (as in [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a] this means linearly ordered set) and C a cut in S, that is a downward closed subset of S. We define the character of C (in S) to be the pair $(\alpha,\beta^*)$ where $\alpha:=\operatorname{cf}(C)$ and $\beta^*$ is the set $\beta:=\operatorname{ci}(S\setminus C)$ equipped with the reversed ordering. We then also call C an $(\alpha,\beta^*)$ -cut (in S); see [Reference HarzheimHar05,§ 3.2]. The characters of the cuts $\emptyset$ and S in S are $(0,\operatorname{ci}(S)^*)$ and $({\operatorname{cf}}(S),0)$ , respectively. Note that S is $\eta_1$ if and only if no cut in S has character $(\alpha,\beta^*)$ with ${\alpha,\beta\leqslant\omega}$ . A gap $A<B$ in S is a pair (A,B) of subsets of S such that $A<B$ and there is no $s\in S$ with $A<s<B$ . The character of such a gap $A<B$ is defined to be the character $(\alpha,\beta^*)$ of the cut $A^{\downarrow}=S\setminus B^\uparrow$ in S, and then $A<B$ is also called an $(\alpha,\beta^*)$ -gap in S.

Let G be an ordered abelian group. If $v\colon G\to S_\infty$ is a surjective convex valuation on G (see [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, p. 99]) and $A<B$ is an $(\alpha,\beta^*)$ -gap in S where $\alpha,\beta\geqslant\omega$ , then $(v^{-1}(A)\cap G^<, v^{-1}(B)\cap G^{\lt })$ is a $(\alpha,\beta^*)$ -gap in G. If H is an ordered field, then $\operatorname{cf}(H^<)=\operatorname{ci}(H^>)=\operatorname{cf}(H)$ , and the cuts $H^{\lt h}$ and $H^{\leqslant h}$ ( $h\in H$ ) in H have character $({\operatorname{cf}}(H),1)$ and $(1,\operatorname{cf}(H)^*)$ , respectively.

Now let G be an ordered abelian group. Assume $G\ne \{0\}$ and $G^>$ has no smallest element, so $\operatorname{ci}(G^>)\geqslant\omega$ . A gap $A<B$ in G is said to be cauchy if $A,B\neq\emptyset$ , A has no largest element, B has no smallest element, and for each $\varepsilon\in G^>$ there are $a\in A$ , $b\in B$ with $b-a<\varepsilon$ . If $A<B$ is a cauchy gap in G, then so is $-B<-A$ .

Proof. The first claim follows from Lemma 5.6 and [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 2.4.11]. It also follows from this lemma that if G is complete, then G has no cauchy gap. Conversely, suppose G has no cauchy gap. Let $(a_\rho)$ be a c-sequence in G. For each $\varepsilon\in G^>$ , take $\rho_\varepsilon$ such that $\lvert a_\rho-a_{\rho'}\rvert<\varepsilon$ for all $\rho,\rho'\geqslant\rho_\varepsilon$ , and set

$$A := \{ a_{\rho_\varepsilon}-\varepsilon:\ \varepsilon\in G^>\}, \quad B := \{ a_{\rho_\delta}+\delta:\ \delta\in G^>\}.$$

Then $A<B$ and for all $\varepsilon\in G^{>}$ there are $a\in A$ and $b\in B$ with $b-a<\varepsilon$ . However, $A< B$ is no cauchy gap, so we have $g\in G$ with $A\leqslant g\leqslant B$ . Then $a_{\rho} \to g$ .

The main result of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a], Theorem A, and Corollaries 5.5 and 5.8 now gives us the following.

Corollary 5.9. Assume that the continuum hypothesis holds. If H is a maximal Hardy field, a maximal analytic Hardy field, or a maximal smooth Hardy field, then the characters of gaps in H are

$$(0,\omega_1^*),\ (\omega_1,0),\ (1,\omega_1^*),\ (\omega_1,1),\ (\omega,\omega_1^*),\ (\omega_1,\omega^*),\mbox{ and }(\omega_1,\omega_1^*).$$

5.4 The number of maximal analytic Hardy fields

We recall some definitions from [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a]. A germ $\phi\in\mathcal{C}$ is said to be overhardian if $\phi$ is hardian and $\phi>_{\mathrm{e}} \exp_n(x)$ for all n; see [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Corollary 5.11]. Let $H\supseteq {\mathbb R} $ be a Hardy field. Then

$$H^{\operatorname{te}} := \{f\in H:\text{$f \gt \exp_n(x)$ for each n}\}$$

denotes the set of overhardian (or transexponential) elements of H. We let $*H^{\operatorname{te}}$ be the set of equivalence classes of the equivalence relation $\sim_{\exp}$ on $H^{\operatorname{te}}$ given by

$$f\sim_{\exp} g\quad:\Longleftrightarrow\quad\text{$f\leqslant\exp_n(g)$ and $g\leqslant\exp_n(f)$ for some n}\quad (f,g\in H^{\operatorname{te}}).$$

Denoting the equivalence class of $f\in H^{\operatorname{te}}$ by $*f$ , we linearly order $H^{\operatorname{te}}$ by

$$*f \lt *g\quad:\Longleftrightarrow\quad \text{$\exp_n(f) \lt g$ for all n} \quad (f,g\in H^{\operatorname{te}}).$$

We now establish analytic and smooth versions of Theorem 7.1 from [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a].

Corollary 7.8 of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a] has an analytic version as follows with a similar proof.

We now improve Corollary 3.7: assuming that the continuum hypothesis holds, there are as many cofinal maximal analytic Hardy fields as there are maximal analytic Hardy fields, by Corollary 5.14.

We now follow the argument after the statement of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Proposition 7.4], with $\mathcal H$ now the set of all analytic Hardy fields $H\supseteq {\mathbb R} \langle\phi\rangle$ such that $\lvert*H^{\operatorname{te}}\rvert \lt \mathfrak{c}$ and $*\phi=\max*H^{\operatorname{te}}$ . For an ordinal $\lambda$ we let $2^{\lambda}$ be the set of functions ${\lambda\to\{0,1\}}$ . With s ranging over $\bigcup_{\lambda<\mathfrak c} 2^\lambda$ , we construct a tree $(H_s)$ in $\mathcal H$ with $\lvert*H^{\operatorname{te}}\rvert\leqslant\lvert{\lambda+1}\rvert$ for $s\in 2^\lambda$ , as follows. For $\lambda=0$ the function s has empty domain and we take $H_s= {\mathbb R} \langle\phi\rangle$ . If $s\in 2^\lambda$ ( $\lambda<\mathfrak c$ ) and $H_s\in \mathcal{H}$ are given with $\lvert*H_s^{\operatorname{te}}\rvert\leqslant \lvert\lambda+1\rvert$ , then [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Lemma 7.2] provides a countable gap P, Q in ${*H_s^{\operatorname{te}}\setminus\{*\phi\}}$ , and we let $H_{s0}, H_{s1}\in \mathcal{H}$ be obtained from $H_s$ as $H_0$ , $H_1$ are obtained from H in the remark above. Suppose $\lambda< \mathfrak{c}$ is an infinite limit ordinal, $s\in 2^\lambda$ , and that for every $\alpha< \lambda$ we are given $H_{s|\alpha}\in \mathcal{H}$ with $H_{s|\alpha}\subseteq H_{s|\beta}$ whenever $\alpha\leqslant \beta \lt \lambda$ . Then we set $H_s:= \bigcup_{\alpha<\lambda} H_{s|\alpha}\in\mathcal{H}$ . Assuming also inductively that $\lvert*H_{s|\alpha}^{\operatorname{te}}\rvert\leqslant \lvert\alpha+1\rvert$ for all $\alpha<\lambda$ , we have $\lvert*H_s^{\operatorname{te}}\rvert\leqslant \lvert\lambda\rvert\cdot \lvert\lambda+1\rvert=\lvert\lambda+1\rvert$ , as desired. This finishes the construction of our tree. Then for each $s\in 2^{\mathfrak c}$ we have an analytic Hardy field $H_s:=\bigcup_{\lambda<\mathfrak{c}} H_{s|\lambda}$ such that if $s, s'\in 2^{\mathfrak c}$ are different, then $H_s$ , $H_{s'}$ have no common Hardy field extension. Hence, $\mathcal H_\phi:=\{H_s:s\in 2^{\mathfrak c}\}$ has the required properties.

Corollary 3.5 gives an overhardian $\phi\in\mathcal{C}^\omega$ . For such $\phi$ and $\mathcal H_\phi$ as in Lemma 5.12, all $H\in\mathcal H_\phi$ are bounded. With Lemma 5.13 we can now improve Corollary 3.7 as follows.

5.5 Maximal analytic Hardy fields approximate maximal Hardy fields

A maximal analytic Hardy field is an $\infty\omega$ -elementary substructure of any maximal Hardy field extension, by Corollary 8.5. Maximal analytic Hardy fields are also very close to maximal Hardy fields in the following way.

To prove this, assume $f\in \mathcal{C}^{\lt \infty}$ is H-hardian, $(f_{\rho})$ is a divergent pc-sequence in H, and $f_{\rho}\leadsto f$ . By [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, 16.0.3 and Section 11.4], $(f_{\rho})$ is of d-transcendental type over H. If the sequence $(v(f-f_{\rho}))$ is cofinal in $\Gamma:=v(H^\times)$ , then $(f_{\rho})$ is a cauchy sequence, and so H is indeed dense in $H\langle f \rangle $ , by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24c, Corollary 4.1.6]. Suppose $(v(f-f_{\rho}))$ is not cofinal in $\Gamma$ . Then we have $h\in H^\times$ such that $0\notin v({hf-H})$ , so Corollary 2.8 and Proposition 4.4 yield an H-hardian pseudolimit of $(hf_{\rho})$ in $\mathcal{C}^{\omega}$ , contradicting the maximality of H. This proves Claim 1.

Towards a contradiction, suppose K is a Hardy field extension of H and ${\beta\in \Gamma_K\setminus\Gamma}$ . We arrange that K is Liouville closed. Let $(\Gamma,\psi)$ and $(\Gamma_K, \psi_K)$ be the H-couples of H and K over $ {\mathbb R} $ , respectively, and let $(\Gamma\langle \beta \rangle ,\psi_{\beta})$ be the H-couple over $ {\mathbb R} $ generated by $\beta$ over $(\Gamma,\psi)$ in $(\Gamma_K, \psi_K)$ . There are several cases to consider, and we show that each is impossible. For closed H-couples and H-couples of Hahn type mentioned in the following, see [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH22, p. 536].

First the case that $(\Gamma\langle \beta \rangle ,\psi_{\beta})$ is an immediate extension of $(\Gamma,\psi)$ . Then we have a divergent pc-sequence $(\gamma_{\rho})$ in $(\Gamma,\psi)$ with $\gamma_{\rho}\leadsto \beta$ . As in the beginning of [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Section 8], we take $g_{\rho}\in H$ with $vg_{\rho}=\gamma_{\rho}$ so that $(g_{\rho}^\dagger)$ is a pc-sequence in H, and arguing as in [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH24a, Section 8] (using $H^\dagger=H$ ) we see that $(g_{\rho}^\dagger)$ has no pseudolimit in H (because then $(\gamma_{\rho})$ would have one in $\Gamma$ ). Now take $g\in K$ with $vg=\beta$ . Then $v(g^\dagger-g_{\rho}^\dagger)=(\beta-\gamma_{\rho})^\dagger$ , and the latter is eventually strictly increasing as a function of $\rho$ , and so $g_{\rho}^\dagger\leadsto g^\dagger$ . Moreover, $(v(g^\dagger-g_{\rho}^\dagger))$ is not cofinal in $\Gamma$ . As at the end of the proof of Claim 1, with $g^\dagger$ and $(g_{\rho}^\dagger)$ in the role of f and $(f_{\rho})$ , this contradicts the maximality assumption on H.

Since the H-field H is Liouville closed with constant field $ {\mathbb R} $ , its H-couple $(\Gamma,\psi)$ over $ {\mathbb R} $ is closed. Hence, by [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH22, Proposition 4.1] and the remark following its proof, the vector $\beta$ falls under case (a), case (b), or case $\textrm{(c)}_n$ for a certain n. In case (a) we have $(\Gamma+ {\mathbb R} \beta)^\dagger=\Gamma^\dagger$ and so $\Gamma\langle \beta \rangle =\Gamma+ {\mathbb R} \beta$ ; but $(\Gamma_K, \psi_K)$ is of Hahn type, an immediate extension of $(\Gamma,\psi)$ , and we have just excluded that possibility. Case $\textrm{(c)}_n$ gives an element $\beta_n\in \Gamma\langle \beta \rangle $ with $\beta_n^\dagger\notin \Gamma$ and $\beta_n^\dagger$ falling under case (a), and so this is also impossible.

Finally, suppose $\beta$ falls under case (b). Take $y\in K^{>}$ with $vy=\beta$ . Then Corollary 37 gives an H-hardian $z\in \mathcal{C}^{\omega}$ with vz realizing the same cut in $\Gamma$ as $\beta$ , contradicting the maximality assumption on H. This finishes the proof of Claim 2.

To finish the proof of the theorem, let $f\in \mathcal{C}^{\lt \infty}$ be H-hardian; it suffices to show that H is then dense in $H\langle f \rangle $ . Now, by Claim 2, $H\langle f \rangle $ is an immediate extension of H, and hence H is indeed dense in $H\langle f \rangle $ by Claim 1.

5.6 Dense pairs of closed H-fields

Let $\mathcal L=\{0,1,{-},{+},{\,\cdot\,},{\partial},{\leqslant},{\preccurlyeq}\}$ be the language of ordered valued differential rings; cf. [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17, p. 678]. We view each ordered valued differential field as an $\mathcal L$ -structure in the natural way. We let $\mathcal L^2$ extend $\mathcal L$ by a new unary predicate symbol U. The $\mathcal L^2$ -structures are presented as pairs (K,F), where K is an $\mathcal L$ -structure and U names the subset F of K. Let T be the $\mathcal L$ -theory of closed H-fields with small derivation. Recall from [Reference Aschenbrenner, van den Dries and van der HoevenAvdDvdH17] that T is complete and model-complete. We announce the following result here.

1. (1) $K\models T$ , that is, K is a closed H-field with small derivation;

2. (2) F is the underlying set of a closed H-subfield of K; and

3. (3) $F\neq K$ and F is dense in the ordered field K.

Moreover, each $\mathcal L^2$ -formula $\varphi(x)$ , where $x=(x_1,\dots,x_m)$ , is $T^{\mathrm{d}}$ -equivalent to a boolean combination of formulas of the form

(5.1) \begin{equation}\exists y_1\cdots\exists y_n( U(y_1)\ \&\ \cdots\ \&\ U(y_n)\ \&\ \psi(x,y)),\end{equation}

where $\psi(x,y)$ with $y=(y_1,\dots,y_n)$ is an $\mathcal L$ -formula.

This follows from Fornasiero [Reference FornasieroFor11, Theorems 8.3 and 8.5], with details of how it follows to appear in future work (by AschenbrennerPlease supply full reference details for Aschenbrenner et al. (in preparation) if now available. et al., More on dimension in transseries, in preparation). Note that by this theorem the $\mathcal L^2$ -theory $T^{\mathrm{d}}$ is decidable. Moreover, no pair $(K,F)\models T^{\mathrm{d}}$ induces ‘new structure’ on F.