Where do closed $H$-couples come from?

We recall from [Reference Aschenbrenner, van den Dries and van der Hoeven2, Chapter 10] that an $H$-field is an ordered differential field $K$ with constant field $C$ such that:

1. (H1) $a'>0$ for all $a\in K$ with $a>C$;

2. (H2) $\mathcal {O}=C+ o$, where $\mathcal {O}$ is the convex hull of $C$ in the ordered field $K$, and $ o$ is the maximal ideal of the valuation ring $\mathcal {O}$.

Let $K$ be an $H$-field, and let $\mathcal {O}$ and $o$ be as in (H2). Thus, $K$ is a valued field with valuation ring $\mathcal {O}$. Let $v\colon K^\times \to \Gamma$ be the associated valuation. The value group $\Gamma =v(K^\times )$ is made into an $H$-couple $(\Gamma,\,\psi )$—the $H$-couple of $K$—by $\psi (vf):=v(f^{\dagger} )$ for $f\in K^\times$ with $vf\ne 0$. We call $K$ Liouville closed if it is real closed and for all $a\in K$ there exists $b\in K$ with $a=b'$ and also a $b\in K^\times$ such that $a=b^{\dagger}$.

If $K$ is Liouville closed, its $H$-couple is closed as is easily verified. We recall from [Reference Aschenbrenner, van den Dries and van der Hoeven2] that $\mathbb {T}$ is a Liouville closed $H$-field.

Ordered vector spaces

Throughout we let ${\boldsymbol {k}}$, ${\boldsymbol {k}}_0$, and ${\boldsymbol {k}}^*$ be ordered fields. Recall that an ordered vector space over ${\boldsymbol {k}}$ is an ordered abelian group $\Gamma$ with a scalar multiplication ${\boldsymbol {k}}\times \Gamma \to \Gamma$ that makes $\Gamma$ into a vector space over ${\boldsymbol {k}}$ such that $c\gamma >0$ for all $c\in {\boldsymbol {k}}^{>}$ and $\gamma \in \Gamma ^{>}$. Let $\Gamma$ be an ordered vector space over ${\boldsymbol {k}}$. Then any ${\boldsymbol {k}}$-linear subspace of $\Gamma$ is considered as an ordered vector space over ${\boldsymbol {k}}$ in the obvious way. We shall need the following easy result about $\Gamma$:

The ${\boldsymbol {k}}$-archimedean class of $\alpha \in \Gamma$ is

\[ [\alpha]_{{\boldsymbol{k}}}\ :=\ \big\{\gamma\in \Gamma:\, \lvert\gamma\rvert \leqslant c\lvert\alpha\rvert\text{ and }\lvert\alpha\rvert \leqslant c\lvert\gamma\rvert\text{ for some }c\in {\boldsymbol{k}}^{>}\big\}. \]

Let $[\Gamma ]_{{\boldsymbol {k}}}$ be the set of ${\boldsymbol {k}}$-archimedean classes. Then $[\Gamma ]_{{\boldsymbol {k}}}$ is a partition of $\Gamma$, and we linearly order $[\Gamma ]_{{\boldsymbol {k}}}$ by

\begin{align*} [\alpha]_{{\boldsymbol{k}}}<[\beta]_{{\boldsymbol{k}}} & \quad:\Longleftrightarrow\quad c\lvert\alpha\rvert<\lvert\beta\rvert\text{ for all }c\in {\boldsymbol{k}}^{>} \\ & \quad \Longleftrightarrow\quad [\alpha]_{{\boldsymbol{k}}}\neq [\beta]_{{\boldsymbol{k}}}\text{ and }\lvert\alpha\rvert<\lvert\beta\rvert. \end{align*}

Thus, $[0]_{{\boldsymbol {k}}}=\{0\}$ is the smallest ${\boldsymbol {k}}$-archimedean class. For $\alpha,\,\beta \in \Gamma$, $c\in {\boldsymbol {k}}^\times$ we have $[c\alpha ]_{{\boldsymbol {k}}}=[\alpha ]_{{\boldsymbol {k}}}$ and $[\alpha +\beta ]_{{\boldsymbol {k}}} \leqslant \max ([\alpha ]_{{\boldsymbol {k}}},\,[\beta ]_{{\boldsymbol {k}}})$, with equality if $[\alpha ]_{{\boldsymbol {k}}}\neq [\beta ]_{{\boldsymbol {k}}}$.

Lemma 1.2 takes care of the easy direction (i) $\Rightarrow$ (ii) in Theorem 0.1. The direction (ii) $\Rightarrow$ (iii) is trivial. The harder direction (iii) $\Rightarrow$ (i) uses a generality on expanded vector spaces, to which we now turn.

Let $V$ be a vector space over a field $C$. We consider the two-sorted structure $(V,\,C;\operatorname {sc})$ consisting of the abelian group $V$, the field $C$, and the scalar multiplication $\operatorname {sc}\colon C\times V \to V$ of the vector space $V$. Let $X\subseteq V$. Then we have the expansion $\mathbf {V}=((V,\,X),\,C;\operatorname {sc}\!)$ of $(V,\,C;\operatorname {sc})$. Let $\mathbf {V}^*=((V^*,\, X^*),\, C^*;\operatorname {sc}\!)$ be an elementary extension of $\mathbf {V}$. Let $C^* V$ be the $C^*$-linear subspace of $V^*$ spanned by $V$.

For certain $(V,\,C;\operatorname {sc})$ this will be applied to sets $X\subseteq V$ that are definable in a suitable expansion of $(V,\,C;\operatorname {sc})$, with $X^*$ the corresponding set in an elementary extension of that expansion.

$H$-couples over ordered fields

Ordered vector spaces come into play as follows. Let $K$ be a Liouville closed $H$-field. It has the (ordered) constant field $C$, and the $H$-couple $(\Gamma,\,\psi )$. We have a map $(c,\,\gamma )\mapsto c\gamma \colon C\times \Gamma \to \Gamma$ such that $cvf=vg$ whenever $f,\,g\in K^\times$ and $cf^{\dagger} =g^{\dagger}$. This map makes $\Gamma$ into an ordered vector space over $C$, and $\psi (c\gamma )=\psi (\gamma )$ for all $c\in C^\times$ and $\gamma \in \Gamma ^{\ne }$.

Accordingly, we define an $H$-couple over ${\boldsymbol {k}}$ to be an $H$-couple $(\Gamma,\,\psi )$ where the ordered abelian group $\Gamma$ is also equipped with a map ${\boldsymbol {k}}\times \Gamma \to \Gamma$ making $\Gamma$ into an ordered vector space over ${\boldsymbol {k}}$ such that $\psi (c\gamma )=\psi (\gamma )$ for all $c\in {\boldsymbol {k}}^\times$ and $\gamma \in \Gamma ^{\ne }$. Thus, the $H$-couple of a Liouville closed $H$-field is naturally an $H$-couple over its constant field.

Let $(\Gamma,\,\psi )$ be an $H$-couple over ${\boldsymbol {k}}$. A basic fact is that for distinct $\alpha,\,\beta \in \Gamma ^{\ne }$ we have $[\psi (\alpha )-\psi (\beta )]_{{\boldsymbol {k}}} < [\alpha -\beta ]_{{\boldsymbol {k}}}$, since for all $c\in {\boldsymbol {k}}^{>}$, we have $\psi (\alpha )-\psi (\beta )=\psi (c\alpha )-\psi (c\beta )=o(c(\alpha -\beta ))$, by [Reference Aschenbrenner, van den Dries and van der Hoeven2, 6.5.4(ii)]. Note also that for all $\alpha,\,\beta \in \Gamma ^{\ne }$,

\[ [\alpha]_{{\boldsymbol{k}}}=[\beta]_{{\boldsymbol{k}}}\ \Longrightarrow\ \psi(\alpha)=\psi(\beta). \]

Hahn spaces

These are the ordered Hahn spaces from [Reference Aschenbrenner, van den Dries and van der Hoeven2, § 2.4]: a Hahn space $\Gamma$ over ${\boldsymbol {k}}$ is an ordered vector space over ${\boldsymbol {k}}$ such that for all $\alpha,\,\beta \in \Gamma ^{\neq }$ with $[\alpha ]_{{\boldsymbol {k}}}=[\beta ]_{{\boldsymbol {k}}}$ there exists $c\in {\boldsymbol {k}}^\times$ such that $[\alpha -c\beta ]_{{\boldsymbol {k}}}<[\alpha ]_{{\boldsymbol {k}}}$.

We say that an $H$-couple $(\Gamma,\,\psi )$ over ${\boldsymbol {k}}$ is of Hahn type if for all $\alpha,\,\beta \in \Gamma ^{\ne }$ with $\psi (\alpha )=\psi (\beta )$ there exists a scalar $c\in {\boldsymbol {k}}$ such that $\psi (\alpha -c\beta )> \psi (\alpha )$; equivalently, $\Gamma$ is a Hahn space over ${\boldsymbol {k}}$ and for all $\alpha,\,\beta \in \Gamma ^{\ne }$,

\[ \psi(\alpha)=\psi(\beta)\ \Longrightarrow\ [\alpha]_{\boldsymbol{k}}=[\beta]_{\boldsymbol{k}}. \]

Let $K$ be a Liouville closed $H$-field. We made its $H$-couple $(\Gamma,\, \psi )$ into an $H$-couple over its constant field $C$, and as such $(\Gamma,\,\psi )$ is of Hahn type.

Details on the example in the introduction

We consider the Liouville closed $H$-field $\mathbb {T}$ and its element $x$ with $x'=1$. For $z\in \mathbb {T}$ with $z'\notin \mathbb {R}$ we have

\begin{align*} zz'' = (z')^2 & \Longleftrightarrow\ z^{\dagger}{=} (z')^{\dagger}\ \Longleftrightarrow\ (z'/z)^{\dagger}{=} 0\ \Longleftrightarrow\ z'=tz \text{ for some }t\in \mathbb{R}^{{\times}}\\ & \Longleftrightarrow\ z=s\operatorname{e}^{tx} \text{ for some }s,t\in \mathbb{R}^\times. \end{align*}

Considering also the case where $z'\in \mathbb {R}$ we conclude that

\[ \big\{z\in \mathbb{T}:\, zz''=(z')^2\big\} = \big\{s\operatorname{e}^{tx}:\, s,t\in \mathbb{R}\big\}. \]

Next, let $y\in \mathbb {T}^\times$ and suppose $z:=y^{\dagger}$ satisfies $zz''=(z')^2$. Then $y=r\operatorname {e}^{u}$ for some $r\in \mathbb {R}$ and $u\in \mathbb {T}$ with $u'=z$. For $z=s\operatorname {e}^{tx}$ with $s,\,t\in \mathbb {R}$ and $u\in \mathbb {T}$, $u'=z$ we get $u\in \mathbb {R}\operatorname {e}^{tx}+\mathbb {R}$ if $t\ne 0$, and $u\in \mathbb {R} x+\mathbb {R}$ if $t=0$. Hence $y=a\operatorname {e}^{b\operatorname {e}^{cx}}$ or $y=a\operatorname {e}^{bx}$ for some $a,\,b,\,c\in \mathbb {R}$. From $zz''=(z')^2$ we get

\[ y^2y'y^{(3)}-y^2(y^{(2)})^2-y(y')^2y^{(2)}+(y')^4=0. \]

In this way, we get for

\[ G(Y):=Y^2Y'Y^{(3)}-Y^2(Y^{(2)})^2-Y(Y')^2Y^{(2)}+(Y')^4 \]

that its set of zeros in $\mathbb {T}$ is

\[ \big\{a\operatorname{e}^{b\operatorname{e}^{cx}}:\, a,b,c\in \mathbb{R}\big\}\cup \big\{a\operatorname{e}^{bx}:\, a,b\in \mathbb{R}\big\}. \]

It is easy to see that for $0< c < d$ in $\mathbb {R}$ we have $[v(\operatorname {e}^{\operatorname {e}^{cx}})]_{\mathbb {R}} < [v(\operatorname {e}^{\operatorname {e}^{dx}})]_{\mathbb {R}}$, so the set $\big \{vy:\, y\in \mathbb {T}^\times,\,\ G(y)=0\big \}$ is not contained in a finite-dimensional $\mathbb {R}$-linear subspace of $\Gamma _{\mathbb {T}}$.

Embedding lemmas

The lemmas in this subsection are the analogues for $H$-couples over ${\boldsymbol {k}}$ of similar lemmas for $H$-couples in [Reference Aschenbrenner, van den Dries and van der Hoeven2, § 9.8]. The proofs are essentially the same, so we omit them.

The universal property (ii) determines $(\Gamma +{\boldsymbol {k}}\alpha,\, \psi ^{\alpha })$ up to isomorphism over $(\Gamma,\, \psi )$, and $0< c\alpha < \Gamma ^{>}$ for all $c\in {\boldsymbol {k}}^{>}$; moreover, for all $\gamma \in \Gamma$ and $c\in {\boldsymbol {k}}$ with $\gamma +c\alpha \ne 0$,

(1)\begin{equation} \psi^{\alpha}(\gamma+c \alpha) = \begin{cases} \psi(\gamma), & \text{if }\gamma\neq 0,\\ \beta-\alpha, & \text{otherwise.} \end{cases} \end{equation}

Note also that $[\Gamma + {\boldsymbol {k}}\alpha ]_{{\boldsymbol {k}}}=[\Gamma ]_{{\boldsymbol {k}}} \cup \big \{[\alpha ]_{{\boldsymbol {k}}}\big \}$, so for $\Psi ^{\alpha }:=\psi ^{\alpha }((\Gamma +{\boldsymbol {k}}\alpha )^{\ne })$ we have:

(2)\begin{equation} \Psi^{\alpha} = \Psi\cup\{\beta-\alpha\}, \quad \max\Psi^{\alpha} = \psi^{\alpha}(\alpha) = \beta-\alpha. \end{equation}

Lemma 2.1 goes through with $\alpha < 0$ and $\alpha _1 <0$ in place of $\alpha >0$ and $\alpha _1 > 0$, respectively. In the setting of this modified lemma, we have $\Gamma ^{<} < c\alpha < 0$ for all $c\in {\boldsymbol {k}}^{>}$, (1) goes through for $\gamma \in \Gamma$ and $c\in {\boldsymbol {k}}$ with $\gamma + c\alpha \ne 0$, (2) goes through. So we have two ways to remove a gap. Removal of a gap as above leads by (2) to a grounded $H$-couple over ${\boldsymbol {k}}$, and this is the situation we consider next.

Let $(\Gamma +{\boldsymbol {k}}\alpha,\, \psi ^{\alpha })$ be as in Lemma 2.2. Then $\Gamma ^{<} < c\alpha < 0$ for all $c\in {\boldsymbol {k}}^{>}$, $[\Gamma +{\boldsymbol {k}}\alpha ]_{{\boldsymbol {k}}}=[\Gamma ]_{{\boldsymbol {k}}} \cup \big \{[\alpha ]_{{\boldsymbol {k}}}\big \}$, so (2) holds for $\Psi ^{\alpha }:=\psi ^{\alpha }((\Gamma +{\boldsymbol {k}}\alpha )^{\ne })$. Thus, our new $\Psi$-set $\Psi ^{\alpha }$ still has a maximum, but this maximum is larger than the maximum $\beta$ of the original $\Psi$-set $\Psi$. By iterating this construction indefinitely and taking a union, we obtain an $H$-couple over ${\boldsymbol {k}}$ with asymptotic integration.

Once we have an $H$-couple over ${\boldsymbol {k}}$ with asymptotic integration, we can create an extension with a gap as follows:

Let $(\Gamma,\, \psi )$ and $(\Gamma + {\boldsymbol {k}}\beta,\, \psi _{\beta })$ be as in Lemma 2.3. If $(\Gamma + {\boldsymbol {k}}\alpha,\, \psi _{\alpha })$ is also an $H$-couple over ${\boldsymbol {k}}$ extending $(\Gamma,\, \psi )$ with $\Psi < \alpha < (\Gamma ^{>})'$, then by (ii) we have an isomorphism $(\Gamma + {\boldsymbol {k}}\beta,\, \psi _{\beta })\to (\Gamma + {\boldsymbol {k}}\alpha,\, \psi _{\alpha })$ of $H$-couples over ${\boldsymbol {k}}$ that is the identity on $\Gamma$ and sends $\beta$ to $\alpha$. In this sense, $(\Gamma + {\boldsymbol {k}}\beta,\, \psi _{\beta })$ is unique up to isomorphism over $(\Gamma,\, \psi )$. The construction of $(\Gamma + {\boldsymbol {k}}\beta,\, \psi _{\beta })$ gives the following extra information, with $\Psi _{\beta }$ the set of values of $\psi _{\beta }$ on $(\Gamma + {\boldsymbol {k}}\beta )^{\ne }$:

Recall that a cut in an ordered set $S$ is just a downward closed subset of $S$, and that an element $a$ of an ordered set extending $S$ is said to realize a cut $D$ in $S$ if $D< a< S\setminus D$ (so $a\notin S$).

Proof. By a straightforward analogue of [Reference Aschenbrenner, van den Dries and van der Hoeven2, Lemma 2.4.5] we extend $\Gamma$ to an ordered vector space $\Gamma ^{\alpha }=\Gamma \oplus {\boldsymbol {k}}\alpha$ over ${\boldsymbol {k}}$ with $\alpha >0$ such that $[\alpha ]_{{\boldsymbol {k}}}$ realizes the cut $D$ in $[\Gamma ^{\ne }]_{{\boldsymbol {k}}}$. Then $[\Gamma \oplus {\boldsymbol {k}}\alpha ]_{{\boldsymbol {k}}}=[\Gamma ]_{{\boldsymbol {k}}} \cup \big \{[\alpha ]_{{\boldsymbol {k}}}\big \}$. We extend $\psi$ to $\psi ^{\alpha }\colon (\Gamma ^{\alpha })^{\ne } \to \Gamma$ by

\[ \psi^{\alpha}(\gamma+c\alpha) := \min\!\big\{\psi(\gamma),\beta\big\}\ \text{ for }\gamma\in \Gamma, c\in {\boldsymbol{k}}^\times. \]

Apart from some obvious changes, we now follow the proof of [Reference Aschenbrenner, van den Dries and van der Hoeven2, Lemma 9.8.7]. This gives the desired results, except for the last Claim of the lemma. To prove that claim, let $\beta \in \Psi ^{\downarrow }$, let $\gamma \in \Gamma$ be a gap in $(\Gamma,\,\psi )$, and assume towards a contradiction that $\gamma$ is not a gap in $(\Gamma ^{\alpha },\,\psi ^{\alpha })$. Then $\gamma > \Psi ^\alpha$, so $\gamma =(\delta +c\alpha )'$ with $\delta \in \Gamma$, $c\in {\boldsymbol {k}}^\times$ and $0 <\delta +c\alpha < \Gamma ^{>}$. Then $[\delta +c\alpha ]_{{\boldsymbol {k}}}\notin [\Gamma ]_{{\boldsymbol {k}}}$, so $[\delta +c\alpha ]_{{\boldsymbol {k}}}=[\alpha ]_{{\boldsymbol {k}}}$. As $\Psi$ has no largest element, we get $\Psi < (\delta +c\alpha )^{\dagger} =\alpha ^{\dagger} =\beta$, a contradiction.

The case of Hahn type

In Lemma 2.1 (and in its variant with $\alpha <0$), in Lemma 2.2, and in Lemma 2.5 for $\beta \notin \Psi$, we have:

\[ \text{ if }(\Gamma, \psi)\text{ is of Hahn type, then so is }(\Gamma+{\boldsymbol{k}}\alpha, \psi^{\alpha}). \]

Suppose $(\Gamma,\, \psi )$ and $(\Gamma +{\boldsymbol {k}}\beta,\, \psi _{\beta })$ are as in Lemma 2.3, and $(\Gamma,\,\psi )$ is of Hahn type. We claim that then $(\Gamma +{\boldsymbol {k}}\beta,\, \psi _{\beta })$ is also of Hahn type. To prove this claim, recall from Corollary 2.4 that $\Gamma$ is dense in $\Gamma +{\boldsymbol {k}}\beta$. It follows easily that for non-zero $\alpha _1,\, \alpha _2\in \Gamma +{\boldsymbol {k}}\beta$ with $\psi _{\beta }(\alpha _1)=\psi _{\beta }(\alpha _2)$ we have $[\alpha _1]_{{\boldsymbol {k}}}=[\alpha _2]_{{\boldsymbol {k}}}$. It remains to show that $\Gamma +{\boldsymbol {k}}\beta$ is a Hahn space over ${\boldsymbol {k}}$. So let $\alpha _1,\, \alpha _2\in \Gamma +{\boldsymbol {k}}\beta$ be non-zero with $[\alpha _1]_{{\boldsymbol {k}}}=[\alpha _2]_{{\boldsymbol {k}}}$. By density again, and the fact that $[\Gamma ]_{{\boldsymbol {k}}}=[\Gamma +{\boldsymbol {k}}\beta ]_{{\boldsymbol {k}}}$ has no least element $> [0]_{{\boldsymbol {k}}}$, we have $\gamma _1,\, \gamma _2\in \Gamma$ such that $[\alpha _1-\gamma _1]_{{\boldsymbol {k}}}< [\alpha _1]_{{\boldsymbol {k}}}$ and $[\alpha _2-\gamma _2]_{{\boldsymbol {k}}}< [\alpha _2]_{{\boldsymbol {k}}}$. Take $c\in {\boldsymbol {k}}^\times$ such that $[\gamma _1-c\gamma _2]_{{\boldsymbol {k}}}< [\gamma _1]_{{\boldsymbol {k}}}$. It follows easily that then $[\alpha _1-c\alpha _2]_{{\boldsymbol {k}}}< [\alpha _1]_{{\boldsymbol {k}}}$.

New extension lemmas

The three next lemmas will enable in the next section a simpler proof of QE for closed $H$-couples than in [Reference Aschenbrenner and van den Dries1]: in that paper, we needed “properties (A) and (B)” with long and tedious proofs, and here we avoid this.

Proof. Let $\alpha \in \Gamma ^{\ne }$. We claim:

\[ \beta^{\dagger}{<}\alpha^{\dagger}\ \Rightarrow\ \beta_*^{\dagger}{<}\alpha^{\dagger}, \quad \beta^{\dagger}{>}\alpha^{\dagger}\ \Rightarrow\ \beta_*^{\dagger}{>}\alpha^{\dagger}. \]

To prove the first implication, assume $\beta ^{\dagger} <\alpha ^{\dagger}$. Then $|\beta |>|\alpha |$, so $|\beta _*|>|\alpha |$, and thus $\beta _*^{\dagger} \leqslant \alpha ^{\dagger}$. Since $(\Gamma,\,\psi )$ is closed and $\Gamma ^{<}$ is cofinal in $(\Gamma +{\boldsymbol {k}}\beta ^{\dagger} )^{<}$, we can replace in this argument $\alpha$ by some $\gamma \in \Gamma ^{\ne }$ with $\beta ^{\dagger} <\gamma ^{\dagger} <\alpha ^{\dagger}$, to get $\beta _*^{\dagger} \leqslant \gamma ^{\dagger} <\alpha ^{\dagger}$, and thus $\beta _*^{\dagger} <\alpha ^{\dagger}$ as claimed. The second implication follows in the same way.

If $\beta ^{\dagger} < \gamma ^{\dagger}$ for some $\gamma \in \Gamma ^{\ne }$, then $(\Gamma,\,\psi )$ being closed gives the desired conclusion. If $\beta ^{\dagger} >\Psi$, then we use instead $\Psi < \beta ^{\dagger} < (\Gamma ^{>})'$ and $\Psi < \beta _*^{\dagger} < (\Gamma ^{>})'$.

Proof. From $|\beta _0|\geqslant |\alpha |$ with $\alpha \in \Gamma ^{\ne }$ we get $\beta _0^{\dagger} \leqslant \alpha ^{\dagger}$. Also, $\beta _0^{\dagger} -\alpha ^{\dagger} \notin \Gamma$ and ${[\beta _0^{\dagger} -\alpha _1]_{{\boldsymbol {k}}}}\notin [\Gamma ]_{{\boldsymbol {k}}}$, hence $[\beta _0^{\dagger} -\alpha ^{\dagger} ]_{{\boldsymbol {k}}}\geqslant [\beta _0^{\dagger} -\alpha _1]_{{\boldsymbol {k}}}$. In view of [Reference Aschenbrenner, van den Dries and van der Hoeven2, 6.5.4(i)], this gives

\[ \beta_0^{\dagger}{=} \min(\beta_0^{\dagger},\alpha^{\dagger}) < (\beta_0^{\dagger}{-}\alpha^{\dagger})^{\dagger}\ \leqslant\ (\beta_0^{\dagger}{-}\alpha_1)^{\dagger}{=} \beta_1^{\dagger}. \]

The proof of QE

Towards Theorem 3.3 we consider an $H$-triple $(\Gamma,\, \psi,\, P)$ over ${\boldsymbol {k}}$ and closed $H$-triples $(\Gamma _1,\, \psi _1,\, P_1)$ and $(\Gamma _*,\, \psi _*,\, P_*)$ over ${\boldsymbol {k}}$ that extend $(\Gamma,\, \psi,\, P)$, and such that $(\Gamma _*,\, \psi _*,\, P_*)$ is $|\Gamma |^+$-saturated. For $\gamma \in \Gamma _1$ we let $(\Gamma \langle \gamma \rangle,\, \psi _{\gamma })$ be the $H$-couple over ${\boldsymbol {k}}$ generated by $\Gamma \cup \{\gamma \}$ in $(\Gamma _1,\, \psi _1)$, and set $P_{\gamma }:= P_1\cap \Gamma \langle \gamma \rangle$.

Let $\beta \in \Gamma _1\setminus \Gamma$. Theorem 3.3 follows if we can show that under these assumptions $(\Gamma \langle \beta \rangle,\,\psi _{\beta },\, P_{\beta })$ can be embedded over $\Gamma$ into $(\Gamma _*,\, \psi _*,\, P_*)$. We first do this in a situation that may seem rather special:

Note that Cases 4, 5, 6 in the proof above do not occur if $(\Gamma _1,\,\psi _1)$ is of Hahn type.

In view of Corollary 3.2 and Lemma 3.4, Theorem 3.3 reduces to:

Proof. If $\gamma \in \Gamma _1\setminus \Gamma$ and $\Psi < \gamma < (\Gamma ^{>})'$, then $(\Gamma + {\boldsymbol {k}}\gamma )^{\dagger} = \Gamma ^{\dagger}$, contradicting our assumption. Hence there is no such $\gamma$. It follows that $\Gamma ^{<}$ is cofinal in $\Gamma _1^{<}$.

Take $\alpha _0\in \Gamma$ such that $(\beta -\alpha _0)^{\dagger} \notin \Gamma ^{\dagger}$. Since $(\Gamma,\,\psi )$ is closed, this means ${(\beta -\alpha _0)^{\dagger} \notin \Gamma }$. Next take $\alpha _1\in \Gamma$ with $((\beta -\alpha _0)^{\dagger} -\alpha _1)^{\dagger} \notin \Gamma ^{\dagger}$. Continuing this way, we obtain sequences $\alpha _0,\, \alpha _1,\, \alpha _2,\,\ldots$ in $\Gamma$ and $\beta _0,\, \beta _1,\, \beta _2,\,\ldots$ in $\Gamma \langle \beta \rangle \setminus \Gamma$ with

\[ \beta_0 = \beta-\alpha_0, \qquad \beta_{n+1} = \beta_{n}^{\dagger}{-}\alpha_{n+1} \text{ for all }n, \]

such that $\beta _n^{\dagger} \notin \Gamma$ for all $n$. By Lemma 2.8 we have $\beta _0^{\dagger} < \beta _1^{\dagger} < \beta _2^{\dagger} < \cdots$. It follows that $[\beta _0]_{{\boldsymbol {k}}} >[\beta _1]_{{\boldsymbol {k}}} >[\beta _2]_{{\boldsymbol {k}}} > \cdots$, with $[\beta _n]_{{\boldsymbol {k}}}\notin [\Gamma ]_{{\boldsymbol {k}}}$ for all $n$. In particular, the family $(\beta _n)$ is ${\boldsymbol {k}}$-linearly independent over $\Gamma$, and

\[ \Gamma\langle\beta\rangle = \Gamma\oplus {\boldsymbol{k}}\beta_0 \oplus {\boldsymbol{k}}\beta_1 \oplus {\boldsymbol{k}}\beta_2\oplus \cdots. \]

By saturation we can take $\beta _*\in \Gamma _*\setminus \Gamma$ realizing the same cut in $\Gamma$ as $\beta$. This gives an embedding $e_0\colon \Gamma \oplus {\boldsymbol {k}} \beta \to \Gamma _*$ of ordered vector spaces over ${\boldsymbol {k}}$ that is the identity on $\Gamma$ and sends $\beta$ to $\beta _*$. We define recursively $\beta _{*n}\in (\Gamma _*)_\infty$ by

\[ \beta_{*0}:= \beta_*-\alpha_0, \qquad \beta_{*(n+1)} := \beta_{*n}^{\dagger}{-}\alpha_{n+1}. \]

Assume inductively that $\beta _{*0},\,\ldots,\,\beta _{*n}\in \Gamma _{*}$ and that we have an embedding

\[ e_n\ :\ \Gamma+{\boldsymbol{k}}\beta_0+ \cdots + {\boldsymbol{k}}\beta_n \to \Gamma_{*} \]

of ordered vector spaces over ${\boldsymbol {k}}$ that is the identity on $\Gamma$ and sends $\beta _i$ to $\beta _{*i}$ for $i=0,\,\ldots,\,n$. Then $\beta _n$ and $\beta _{*n}$ realize the same cut in $\Gamma$, and so $\beta _{*n}^{\dagger} \notin \Gamma$, and $\beta _n^{\dagger}$ and $\beta _{*n}^{\dagger}$ realize the same cut in $\Gamma$ by Lemma 2.7. Hence $\beta _{n+1}$ and $\beta _{*(n+1)}\in \Gamma _*\setminus \Gamma$ realize the same cut in $\Gamma$. Moreover, $\beta ^{\dagger} _{*n} < \beta _{*(n+1)}^{\dagger}$ by Lemma 2.8. We have

\[ [\Gamma+{\boldsymbol{k}}\beta_0+\cdots +{\boldsymbol{k}}\beta_n]_{{\boldsymbol{k}}} = [\Gamma]_{{\boldsymbol{k}}}\cup\big\{[\beta_0]_{{\boldsymbol{k}}},\ldots,[\beta_n]_{{\boldsymbol{k}}}\big\},\quad [\beta_0]_{{\boldsymbol{k}}} >\cdots >[\beta_n]_{{\boldsymbol{k}}} > [\beta_{n+1}]_{{\boldsymbol{k}}}. \]

Let $D$ be the cut realized by $[\beta _{n+1}]_{{\boldsymbol {k}}}$ in $[\Gamma +{\boldsymbol {k}}\beta _0+\cdots +{\boldsymbol {k}}\beta _n]_{{\boldsymbol {k}}}$. Then the above together with $[\beta _{*n}]_{{\boldsymbol {k}}} > [\beta _{*(n+1)}]_{{\boldsymbol {k}}}$ shows that $[\beta _{*(n+1)}]_{{\boldsymbol {k}}}$ realizes the $e_n$-image of the cut $D$ in $[e_n(\Gamma +{\boldsymbol {k}}\beta _0+\cdots +{\boldsymbol {k}}\beta _n)]_{{\boldsymbol {k}}}$. Hence $e_n$ extends to an embedding

\[ e_{n+1}\ :\ \Gamma+{\boldsymbol{k}}\beta_1+ \cdots + {\boldsymbol{k}}\beta_n+{\boldsymbol{k}}\beta_{n+1} \to \Gamma_{*} \]

of ordered vector spaces over ${\boldsymbol {k}}$ that is the identity on $\Gamma$ and sends $\beta _{n+1}$ to $\beta _{*(n+1)}$. This leads to a map $e\colon \Gamma \langle \beta \rangle \to \Gamma _*$ that extends each $e_n$, and is, therefore, an embedding of $H$-couples over ${\boldsymbol {k}}$. Since $P_{\beta }$ is the only $H$-cut in $\Gamma \langle \beta \rangle$, $e$ embeds $(\Gamma \langle \beta \rangle,\, \psi _{\beta },\, P_{\beta })$ into $(\Gamma _*,\, \psi _*,\, P_*)$ over $\Gamma$.

This concludes the proof of Theorem 3.3.

Let $T_{{\boldsymbol {k}}}$ be the $\mathcal {L}_{{\boldsymbol {k}}}$-theory of closed $H$-triples over ${\boldsymbol {k}}$. Let $T_{{\boldsymbol {k}}}^{>}$ be the $\mathcal {L}_{{\boldsymbol {k}}}$-theory whose models are the closed $H$-triples $(\Gamma,\,\psi,\, P)$ over ${\boldsymbol {k}}$ with $0\in P$, equivalently $\Psi \cap \Gamma ^{>}\ne \emptyset$. Let $T_{{\boldsymbol {k}}}^{<}$ be the $\mathcal {L}_{{\boldsymbol {k}}}$-theory whose models are the closed $H$-triples $(\Gamma,\,\psi,\, P)$ over ${\boldsymbol {k}}$ with $0\notin P$, equivalently $\Psi \subseteq \Gamma ^{<}$.

Suppose $K$ is a Liouville closed $H$-field. Then its $H$-couple $(\Gamma,\,\psi )$ is naturally an $H$-couple over its constant field $C$. The case $(\Gamma,\,\psi )\models T_{C}^{>}$ corresponds to the derivation $\partial$ of $K$ being small (that is, $\partial f \prec 1$ for all $f\prec 1$ in $K$), while the case $(\Gamma,\,\psi )\models T_{C}^{<}$ corresponds to this derivation not being small. For example, the usual derivation $\frac {d}{dx}$ of $\mathbb {T}$ is small. The derivation $x^2\frac {d}{dx}$ on $\mathbb {T}$ is not small, but $\mathbb {T}$ with this derivation is still Liouville closed.