2 Preliminaries

We let m, n, and r range over $\mathbb {N} = \{0, 1, 2, \dots \}$ and $\rho $ , $\lambda $ , and $\mu $ be ordinals. The main objects of this paper are kinds of ordered valued differential fields; all fields in this paper are assumed to be of characteristic $0$ . A valued field is a field K equipped with a surjective map $v \colon K^{\times } \to \Gamma $ , where $\Gamma $ is a (totally) ordered abelian group, satisfying for $f, g \in K^{\times }$ :

1. (V1) $v(fg) = v(f)+v(g)$ ;

2. (V2) $v(f+g) \geqslant \min \{v(f), v(g)\}$ whenever $f+g \neq 0$ .

We also impose the condition that $v(\mathbb {Q}^{\times })=\{0\}$ , i.e., that K has equicharacteristic $0$ . A differential field is a field K equipped with a derivation , which satisfies for $f, g \in K$ :

1. (D1) ;

2. (D2) .

Let K be a valued field. We add a new symbol $\infty $ to $\Gamma $ and extend the addition and ordering to by $\infty +\gamma =\gamma +\infty =\infty $ and $\infty>\gamma $ for all $\gamma \in \Gamma $ . This allows us to extend v to K by setting . We often use the following more intuitive notation:

$$\begin{align*}\begin{array}{lc} f \preccurlyeq g\ \Leftrightarrow\ v(f)\geqslant v(g),\qquad f \prec g\ \Leftrightarrow\ v(f)>v(g),\\ f \asymp g\ \Leftrightarrow\ v(f)=v(g),\qquad f\sim g\ \Leftrightarrow\ f-g \prec g. \end{array}\end{align*}$$

The relation $\preccurlyeq $ is called a dominance relation. Both $\asymp $ and $\sim $ are equivalence relations on K and $K^{\times }$ respectively, with a consequence of (V2) being that if $f \sim g$ , then $f \asymp g$ . We set and call it the valuation ring of K. It has a (unique) maximal ideal , and we call the residue field of K, usually denoted by $\boldsymbol k$ . We also let $\overline {a}$ or $\operatorname {\mathrm {res}}(a)$ denote the image of $a \in \mathcal O$ under the natural map to $\boldsymbol k$ . For another valued field L, we denote these objects by $\mathcal O_L$ , $\Gamma _L$ , $\boldsymbol k_L$ , etc.

Let K be a differential field. For $f \in K$ , we often write $f'$ for if the derivation is clear from context and set if $f \neq 0$ , the logarithmic derivative of f. We say that K has exponential integration if $(K^{\times })^{\dagger } = K$ . The field of constants of K is . For another differential field L, we denote this object by $C_L$ . We let be the ring of differential polynomials over K and set . For $P \in K\{Y\}^{\neq }$ , the order of P is the smallest r such that $P \in K[Y, Y', \dots , Y^{(r)}]$ . We extend the derivation of K to $K\{Y\}$ in the natural way. If L is a differential field extension of K and $a \in L$ , then $K \langle a \rangle $ denotes the differential subfield of L generated by a over K. If K is additionally a valued field, we extend v to $K\{Y\}$ by setting $v(P)$ to be the minimum valuation of the coefficients of P and thus also extend the relations $\preccurlyeq $ , $\prec $ , $\asymp $ , and $\sim $ to $K\{Y\}$ .

Relating the valuation and the derivation, we impose throughout most of this paper the condition that K has small derivation, which means that . If K has small derivation, [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 4.4.2], so induces a derivation on $\boldsymbol k$ , and we always construe $\boldsymbol k$ as a differential field with this induced derivation. In contrast to the main results of [Reference Aschenbrenner, van den Dries and van der Hoeven5], we are interested in the case that it is nontrivial. We call $\boldsymbol k$ linearly surjective if for all $a_0, \dots , a_r \in \boldsymbol k$ with $a_r \neq 0$ , the equation $1 + a_0y + a_1y' + \dots + a_r y^{(r)} = 0$ has a solution in $\boldsymbol k$ . We call K differential-henselian ( $\operatorname {d}$ -henselian for short) if K has small derivation and:

1. (DH1) $\boldsymbol k$ is linearly surjective;

2. (DH2) whenever $P \in \mathcal O\{Y\}$ of order r satisfies

   $$\begin{align*}P(0)\ \prec\ 1\qquad \text{and} \qquad \sum_{n=0}^r \frac{\partial P}{\partial Y^{(n)}}(0)Y^{(n)}\ \asymp\ 1,\end{align*}$$
   there is $y \prec 1$ in K with $P(y) = 0$ .

Differential-henselianity, a generalization of henselianity to valued differential fields with small derivation, was introduced in [Reference Scanlon21] and studied more systematically in [Reference Aschenbrenner, van den Dries and van der Hoeven5].

Here is another relation between the valuation and derivation fundamental to this setting. We call K asymptotic if $f \prec g \iff f' \prec g'$ for all nonzero . In the rest of this paragraph, suppose that K is asymptotic. Note that $C \subseteq \mathcal O$ . Also, if $f, g \in K^{\times }$ satisfy $f \sim g \not \asymp 1$ , then $f' \sim g'$ and $f^{\dagger } \sim g^{\dagger }$ . Similarly, for $g \in K^{\times }$ with $g \not \asymp 1$ , $v(g^{\dagger })$ and $v(g')$ depend only on $vg$ and not on g, so for $\gamma = vg$ we set and ; note that $\gamma ^{\dagger }=\gamma '-\gamma $ . For any ordered abelian group G, set , , and . Thus logarithmic differentiation induces a map

$$ \begin{align*} \psi \colon \Gamma^{\neq} &\to \Gamma\\ \gamma &\mapsto \gamma^{\dagger}. \end{align*} $$

We call $(\Gamma , \psi )$ the asymptotic couple of K; such structures were introduced by Rosenlicht [Reference Rosenlicht19], and more about them can be found in [Reference Aschenbrenner, van den Dries and van der Hoeven5, Sections 6.5 and 9.2]. We set , and if we need to indicate the dependence on K, we denote this by $\Psi _K$ . When convenient, we extend $\psi $ to a map $\psi \colon \Gamma _{\infty } \to \Gamma _{\infty }$ by setting .

Here are two additional properties an asymptotic K can have. We say that K is H-asymptotic or of H-type if, for all $f, g \in K^{\times }$ satisfying $f \preccurlyeq g \prec 1$ , we have $f^{\dagger } \succcurlyeq g^{\dagger }$ . We say that K has gap $0$ if it has small derivation and $f^{\dagger } \succ 1$ for all $f \in K^{\times }$ with $f \prec 1$ . Properties of K are reflected in its asymptotic couple, but conversely some properties of K are determined by $(\Gamma , \psi )$ . For instance, K has small derivation if and only if $(\Gamma ^>)' \subseteq \Gamma ^>$ , and K has gap $0$ if and only if $(\Gamma ^>)' \subseteq \Gamma ^>$ and $\Psi \subseteq \Gamma ^<$ . There can be at most one $\beta \in \Gamma $ with $\Psi < \beta < (\Gamma ^>)'$ [Reference Aschenbrenner, van den Dries and van der Hoeven5, Corollary 9.2.4], so K having gap $0$ means that $\beta =0$ separates $\Psi $ and $(\Gamma ^>)'$ in this way. Similarly, K is of H-type just in case $\psi (\alpha )\leqslant \psi (\beta )$ whenever $\alpha \leqslant \beta <0$ in $\Gamma $ . For more on asymptotic fields, see [Reference Aschenbrenner, van den Dries and van der Hoeven5, Chapter 9].

By the above, all asymptotic fields with gap $0$ are clearly pre- $\operatorname {d}$ -valued, where we say that K is pre-differential-valued (pre- $\operatorname {d}$ -valued for short) if:

1. (PDV) for all $f, g \in K^{\times }$ with $f \preccurlyeq 1$ and $g \prec 1$ , we have $f' \prec g^{\dagger }$ .

More on pre- $\operatorname {d}$ -valued fields can be found in [Reference Aschenbrenner, van den Dries and van der Hoeven5, Section 10.1], including a characterization of them as those K that satisfy $C \subseteq \mathcal O$ and a valuation-theoretic analog of l’Hôpital’s Rule [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 10.1.4]. In the rest of this paragraph, let K be pre- $\operatorname {d}$ -valued. If K has small derivation and the derivation induced on $\boldsymbol k$ is nontrivial (for instance, if K is $\operatorname {d}$ -henselian), then K has gap $0$ . Having gap $0$ is inherited by valued differential subfields, and thus if K has a $\operatorname {d}$ -henselian valued differential field extension, then K has gap $0$ . These remarks apply in particular to pre-H-fields with gap $0$ , which we now define.

This paper is primarily concerned with certain ordered pre- $\operatorname {d}$ -valued fields called pre-H-fields. Here, K is an ordered valued differential field if, in addition to its valuation and derivation, K is equipped with a (total) ordering $\leqslant $ making it an ordered field. Relating the ordering, valuation, and derivation, we call K a pre- H-field if:

1. (PH1) K is pre- $\operatorname {d}$ -valued;

2. (PH2) $\mathcal O$ is convex (with respect to $\leqslant $ );

3. (PH3) for all $f \in K$ , if $f> \mathcal O$ , then $f'>0$ .

Condition (PH2) holds if and only if is convex, which holds if and only if . Hence if (PH2) holds, then $\leqslant $ induces an ordering on $\boldsymbol k$ making it an ordered field. We thus construe the residue fields of pre-H-fields with small derivation as ordered differential fields. By [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 10.5.2(ii)], pre-H-fields are of H-type.

We now discuss extensions of valued differential fields and ordered valued differential fields. Given an extension L of K, we identify $\Gamma $ with an ordered subgroup of $\Gamma _L$ and $\boldsymbol k$ with a subfield of $\boldsymbol k_L$ in the obvious way. Here and in general we use the word extension as follows: if F is a valued differential field, “extension of F” means “valued differential field extension of F”; if F is an ordered valued differential field, “extension of F” means “ordered valued differential field extension of F”; etc. “Embedding,” “isomorphic,” and “isomorphism” are used similarly. Where there is particular danger of confusion, we are explicit.

An important class of extensions are the immediate extensions: We say that an extension L of K is immediate if $\Gamma _L = \Gamma $ and $\boldsymbol k_L = \boldsymbol k$ ; equivalently, for every $b \in L^{\times }$ there is $a \in K^{\times }$ such that $b \sim a$ . If K is a pre-H-field and L is an immediate valued differential field extension of K that is asymptotic, then L can be given an ordering making it a pre-H-field extension of K; in fact, this is the unique ordering with respect to which $\mathcal O_L$ is convex, and hence any valued differential field embedding of L into a pre-H-field extension M of K is automatically an ordered valued differential field embedding [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 10.5.8].

3 Extensions controlled by the residue field

In this section we prove Lemma 3.1, which yields ordered variants of results from [Reference Aschenbrenner, van den Dries and van der Hoeven5, Sections 6.3 and 7.1] for use in Section 7. In the first lemma, K need not be equipped with a derivation.

Proof. Suppose that L is equipped with an ordering making it an ordered field extension of K with convex valuation ring such that the induced ordering on $\boldsymbol k_L$ agrees with the given one. Let $a \in L^{\times }$ . Then $a=su$ with $s \in K^>$ and $u \asymp 1$ in L. Since $u>0 \iff \overline {u}>0$ , there is at most one such ordering on L, and this also shows how to define the ordering on L: $a>0 \iff \overline {u}>0$ . This is independent of the choice of s and u: If $a=tu_1$ with $t \in K^>$ and $u_1 \asymp 1$ in L, then $u_1 = st^{-1}u$ and $\overline {u_1} = \overline {st^{-1}}\cdot \overline {u}$ , so $\overline {u_1}>0 \iff \overline {u}>0$ . Obviously, $a>0$ or $-a>0$ .

Next, assume that $a, b \in L^{>}$ ; we will show that $a+b>0$ and $ab>0$ . Then $a=su_1$ and $b=tu_2$ with $s, t \in K^>$ and $u_1, u_2 \asymp 1$ in $L^>$ . Without loss of generality, $s \preccurlyeq t$ , so $a+b=t(st^{-1}u_1+u_2)$ and

$$\begin{align*}\overline{st^{-1}u_1+u_2}\ =\ \overline{st^{-1}}\cdot\overline{u_1}+\overline{u_2} > 0. \end{align*}$$

Thus $st^{-1}u_1+u_2 \asymp 1$ and $a+b>0$ . Also, $ab = stu_1u_2$ with $\overline {u_1u_2}=\overline {u_1}\cdot \overline {u_2}>0$ , so $ab>0$ . Similarly, $a^2>0$ . Thus we have defined an ordering on L making it an ordered field extension of K. Obviously, if $a \prec 1$ , then $-1<a<1$ , so the valuation ring of L is convex with respect to this ordering, and by construction it induces the given ordering on $\boldsymbol k_L$ .

Finally, suppose that K is a pre-H-field with gap $0$ . Let $a,b \in L^{\times }$ with $a \preccurlyeq 1$ and $b \not \asymp 1$ . Since L has small derivation, we have $a' \preccurlyeq 1$ . Write $b=su$ with $s \in K^>$ and $u \asymp 1$ in L, so then $b^{\dagger } = s^{\dagger }+u^{\dagger }\sim s^{\dagger } \succ 1$ , since K has gap $0$ . Thus $a' \prec b^{\dagger }$ , showing that L is pre- $\operatorname {d}$ -valued. It has the same asymptotic couple as K, so still has gap $0$ . To see that L is a pre-H-field, it remains to check (PH3): If $b>\mathcal O_L$ , then likewise $b'\sim s'u> 0$ .

For the $\operatorname {d}$ -transcendental analog, which likewise follows from [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 6.3.1] and Lemma 3.1, we construe the fraction field $K \langle Y \rangle $ of $K\{Y\}$ as a valued differential field extension of K by extending and the map $P \mapsto v(P)$ to $K \langle Y \rangle $ in the obvious way. Then $\Gamma _{K \langle Y \rangle }=\Gamma $ , and $K\langle Y\rangle $ has small derivation if K does, in which case $\overline {Y}$ is $\operatorname {d}$ -transcendental over $\boldsymbol k$ and $\operatorname {\mathrm {res}}(K\langle Y\rangle ) = \boldsymbol k\langle \overline {Y}\rangle $ (see [Reference Aschenbrenner, van den Dries and van der Hoeven5, Section 6.3]).

Similarly, we have an ordered variant of [Reference Aschenbrenner, van den Dries and van der Hoeven5, Corollary 7.1.5].

4 Extensions controlled by the asymptotic couple

Towards our quantifier elimination and model completion results for pre-H-fields with gap $0$ , in this section we study extensions controlled by the asymptotic couple. We begin by studying H-asymptotic couples with gap $0$ in their own right and finding the model completion of this theory, which has quantifier elimination.

4.1 Asymptotic couples with small derivation

The material in this subsection is based on [Reference Aschenbrenner, van den Dries and van der Hoeven6], which, apart from its new results, revisits quantifier elimination for the theory of closed H-asymptotic couples from [Reference Aschenbrenner and van den Dries3], introducing several new lemmas that simplify the arguments. For convenience, we use [Reference Aschenbrenner, van den Dries and van der Hoeven5, Sections 6.5 and 9.2] as a reference instead of the original sources [Reference Aschenbrenner and van den Dries3, Reference Aschenbrenner and van den Dries4, Reference Rosenlicht18, Reference Rosenlicht19, Reference Rosenlicht20] for asymptotic couples.

In contrast with the results of [Reference Aschenbrenner and van den Dries3, Reference Aschenbrenner, van den Dries and van der Hoeven6], here we do not need to expand the language by a predicate for the $\Psi $ -set or by functions for divisibility by nonzero natural numbers. Additionally, those authors work over an arbitrary ordered scalar field $\boldsymbol k$ , but here we work over $\mathbb Q$ for concreteness (the results of this section hold in that setting in the language $\mathcal L_{\operatorname {ac}}$ , described later, expanded by functions for scalar multiplication).

4.1.1 Preliminaries

We suspend in this subsection the convention that $\Gamma $ is the value group of K. Instead, $(\Gamma , \psi )$ is an H-asymptotic couple, which means that $\Gamma $ is an ordered abelian group and $\psi \colon \Gamma ^{\neq } \to \Gamma $ is a map satisfying, for all $\gamma , \delta \in \Gamma ^{\neq }$ :

1. (AC1) if $\gamma +\delta \neq 0$ , then $\psi (\gamma +\delta ) \geqslant \min \{\psi (\gamma ),\psi (\delta )\}$ ;

2. (AC2) $\psi (k\gamma ) = \psi (\gamma )$ for all $k \in \mathbb {Z}^{\neq }$ ;

3. (AC3) if $\gamma>0$ , then $\gamma +\psi (\gamma )>\psi (\delta )$ ;

4. (HC) if $0<\gamma \leqslant \delta $ , then $\psi (\gamma ) \geqslant \psi (\delta )$ .

Keeping in mind that later $(\Gamma , \psi )$ will be the asymptotic couple of an H-asymptotic field (such as a pre-H-field), we let and for $\gamma \in \Gamma ^{\neq }$ . The map $\gamma \mapsto \gamma '$ for $\gamma \in \Gamma ^{\neq }$ is strictly increasing [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 6.5.4(iii)]. We let and for any ordered abelian group G we set , having earlier defined $G^>$ and $G^<$ . Thus (AC3) says that $\Psi <(\Gamma ^>)'$ .

A fundamental result is that $\Gamma \setminus (\Gamma ^{\neq })'$ has at most one element [Reference Aschenbrenner, van den Dries and van der Hoeven5, Theorem 9.2.1], and moreover exactly one of the following holds [Reference Aschenbrenner, van den Dries and van der Hoeven5, Corollary 9.2.16]:

1. (1) there exists $\beta \in \Gamma $ such that $\Psi <\beta <(\Gamma ^>)'$ , in which case $(\Gamma , \psi )$ has gap $\beta $ ;

2. (2) there exists $\beta \in \Gamma $ such that $\max \Psi =\beta $ , in which case $(\Gamma , \psi )$ has max $\beta $ ;

3. (3) $(\Gamma ^{\neq })'=\Gamma $ , in which case $(\Gamma , \psi )$ has asymptotic integration.

We are primarily concerned with H-asymptotic couples having gap $0$ , although we also consider the case of max $0$ in this subsection. The material on H-asymptotic couples with max $0$ is only used in one later theorem that itself is not used in the main results, but fits naturally with that of the gap $0$ case. In contrast, the main results of [Reference Aschenbrenner and van den Dries3, Reference Aschenbrenner, van den Dries and van der Hoeven6] concern asymptotic couples with asymptotic integration, such as the asymptotic couple of $\mathbb {T}$ . In adapting their arguments, we try to highlight how the substitution of “gap $0$ ” or “max $0$ ” for “asymptotic integration” (or similar changes) alters the proofs.

Note that if $(\Gamma , \psi )$ is the asymptotic couple of an asymptotic field K, then K has gap $0$ (in the sense of Section 2) if and only if $(\Gamma , \psi )$ has gap $0$ . It follows from [Reference Aschenbrenner, van den Dries and van der Hoeven5, Theorem 9.2.1 and Corollary 9.2.4] that $\sup \Psi =0 \iff (\Gamma ^>)' = \Gamma ^>$ and that $\sup \Psi =0 \notin \Psi $ if and only if $(\Gamma , \psi )$ has gap $0$ . Thus $\sup \Psi =0$ if and only if $(\Gamma , \psi )$ has gap $0$ or max $0$ .

It follows from (AC2) and (HC) that $\psi $ is constant on archimedean classes of $\Gamma $ . For $\gamma \in \Gamma $ , we let denote its archimedean class, and set , ordering it by $[\delta ]<[\gamma ]$ if $n|\delta |<|\gamma |$ for all n, where $\gamma , \delta \in \Gamma $ . The map $\psi $ extends uniquely to the divisible hull $\mathbb {Q}\Gamma $ of $\Gamma $ , defined by $\psi (q\gamma )=\psi (\gamma )$ for $\gamma \in \Gamma ^{\neq }$ and $q \in \mathbb {Q}^{\times }$ (use [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 6.5.3] to get (AC3)), and in this way we always construe $\mathbb {Q}\Gamma $ as an H-asymptotic couple $(\mathbb {Q}\Gamma , \psi )$ extending $(\Gamma , \psi )$ . It satisfies $\psi ((\mathbb {Q}\Gamma )^{\neq }) = \psi (\Gamma ^{\neq })$ , so if $(\Gamma , \psi )$ has gap $0$ (respectively, max $0$ ), then so does $(\mathbb {Q}\Gamma , \psi )$ . We also use that if $(\Gamma , \psi )$ has gap $0$ , then for every $\gamma \in \Gamma ^{\neq }$ , we have $[\gamma ]>[\gamma ^{\dagger }]$ by [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 9.2.10(iv)] and so $[\gamma ]=[\gamma ']$ ; in particular, if $\gamma <0$ , then $\gamma <\gamma ^{\dagger }<0$ .

We call $(\Gamma , \psi )$ gap-closed if $\Gamma $ is nontrivial and divisible, and $\Psi =\Gamma ^<$ . The goal of this section is Theorem 4.8, which states that the theory of gap-closed H-asymptotic couples has quantifier elimination and is the model completion of the theory of H-asymptotic couples with gap $0$ . Likewise, the theory of max-closed H-asymptotic couples, where $(\Gamma , \psi )$ is max-closed if $\Gamma $ is divisible and $\Psi =\Gamma ^{\leqslant }$ , has quantifier elimination and is the model completion of the theory of H-asymptotic couples $(\Gamma , \psi )$ with $\sup \Psi =0$ . The language for these results is the language $\mathcal L_{\operatorname {ac}}=\{+, -, \leqslant , 0, \infty , \psi \}$ of asymptotic couples. The underlying set of $(\Gamma , \psi )$ in this language is , and we interpret $\infty $ in the following way: for all $\gamma \in \Gamma $ , and $\gamma <\infty $ ; ; ; . The other symbols have the expected interpretations.

One of the new extension lemmas of [Reference Aschenbrenner, van den Dries and van der Hoeven6] is [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 2.7]. Here is a variant that avoids assuming asymptotic integration by making a stronger cofinality assumption.

Lemma 4.1. Suppose that $\Psi $ is downward closed in $\Gamma $ . Let $(\Gamma _1, \psi _1)$ and $(\Gamma _*, \psi _*)$ be H-asymptotic couples extending $(\Gamma , \psi )$ such that $\Gamma ^<$ is cofinal in $\Gamma _1^<$ . Suppose that $\gamma _1 \in \Gamma _1 \setminus \Gamma $ and $\gamma _* \in \Gamma _* \setminus \Gamma $ realize the same cut in $\Gamma $ and $\gamma _1^{\dagger } \notin \Gamma $ . Then $\gamma _*^{\dagger } \notin \Gamma $ and $\gamma _*^{\dagger }$ realizes the same cut in $\Gamma $ as $\gamma _1^{\dagger }$ .

Proof. Let $\alpha \in \Gamma ^{\neq }$ . We first show that:

$$\begin{align*}\gamma_1^{\dagger} < \alpha^{\dagger} \implies \gamma_*^{\dagger} < \alpha^{\dagger} \qquad \text{and} \qquad \gamma_1^{\dagger}> \alpha^{\dagger} \implies \gamma_*^{\dagger} > \alpha^{\dagger}. \end{align*}$$

The first implication follows exactly as in [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 2.7], which uses that $\Psi $ is downward closed but not that $(\Gamma , \psi )$ has asymptotic integration. The second implication has a hidden use of asymptotic integration, so here are the modifications.

Suppose that $\gamma _1^{\dagger }>\alpha ^{\dagger }$ , so by the cofinality assumption, there is $\delta \in \Gamma $ with $\gamma _1^{\dagger }> \delta > \alpha ^{\dagger }$ . By the reasoning in the first implication (see [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 2.7]), it suffices to show that $\delta \in \Psi $ ; asymptotic integration is used in [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 2.7] to get this. By the stronger cofinality assumption here, take $\beta \in \Gamma ^{\neq }$ with $|\gamma _1|>|\beta |$ , so $\delta <\gamma _1^{\dagger }< \beta ^{\dagger } \in \Psi $ . Thus $\delta \in \Psi $ , since $\Psi $ is downward closed.

Finally, the last paragraph also shows that $\gamma _*^{\dagger } \notin \Gamma $ , since $\gamma _{*}^{\dagger }\leqslant \beta ^{\dagger } \in \Psi $ and $\Psi $ is downward closed, and therefore it realizes the same cut as $\gamma _1^{\dagger }$ by the displayed implications.

Alternatively, we can weaken the cofinality assumption from $\Gamma ^<$ being cofinal in $\Gamma _1^<$ to $\Gamma ^<$ being cofinal in $(\Gamma + \mathbb {Q}\gamma _1^{\dagger })^<$ (as in [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 2.7]) in the case that $\sup \Psi _1=0$ ; the only change in the proof is that now $\delta \in \Psi $ follows from $\delta <\gamma _1^{\dagger } \leqslant 0$ .

Another of the new lemmas is the following, [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 2.8], which we also use in Section 4.2.

Lemma 4.2. Suppose that $(\Gamma _1, \psi _1)$ is an H-asymptotic couple extending $(\Gamma , \psi )$ , and let $\gamma _1 \in \Gamma _1 \setminus \Gamma $ and $\alpha \in \Gamma $ . Suppose that $\gamma _1$ and satisfy $\gamma _1^{\dagger } \notin \Gamma $ and $\beta ^{\dagger } \notin \Psi $ , and that $|\gamma _1|>|\gamma |$ for some $\gamma \in \Gamma ^{\neq }$ . Then $\gamma _1^{\dagger } < \beta ^{\dagger }$ .

4.1.2 Embedding lemmas

We now turn to the proof of quantifier elimination for gap-closed and max-closed H-asymptotic couples, which we handle uniformly as much as possible. To that end, suppose that $(\Gamma , \psi )$ is a divisible H-asymptotic couple with $\sup \Psi =0$ , and let $(\Gamma _1, \psi _1)$ and $(\Gamma _*, \psi _*)$ be divisible H-asymptotic couples extending $(\Gamma , \psi )$ such that $(\Gamma _*, \psi _*)$ is $|\Gamma |^+$ -saturated. Let $\gamma _1 \in \Gamma _1 \setminus \Gamma $ and $(\Gamma \langle \gamma _1 \rangle , \psi _1)$ be the divisible H-asymptotic couple generated by $\Gamma \cup \{\gamma _1\}$ in $(\Gamma _1, \psi _1)$ .

The first two lemmas are proved similarly to [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemmas 3.4 and 3.5], so we only sketch the proofs, omitting some details that are the same. The third lemma is particular to the case of gap $0$ . For convenience, we set , so $\Gamma ^{\dagger } = \Psi \cup \{\infty \}$ .

In the next lemma, the $\sup \Psi =0$ assumption simplifies the case distinctions needed in [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 3.4], and the assumption of asymptotic integration there is used only in Cases 1–2 of that lemma, which do not occur here. We also do not need the “H-cuts” of that paper for quantifier elimination since the $\Psi $ -set in a gap-closed or a max-closed H-asymptotic couple is already quantifier-free definable without parameters.

Lemma 4.3. Suppose that $(\Gamma + \mathbb Q\gamma _1)^{\dagger } = \Gamma ^{\dagger }$ . Then $(\Gamma \langle \gamma _1 \rangle , \psi _1)$ can be embedded into $(\Gamma _*, \psi _*)$ over $\Gamma $ .

Proof. From $(\Gamma +\mathbb Q\gamma _1)^{\dagger } = \Gamma ^{\dagger }$ , we get $\Gamma \langle \gamma _1 \rangle = \Gamma +\mathbb Q\gamma _1$ .

In case $[\Gamma +\mathbb Q\gamma _1] = [\Gamma ]$ , realizing in $\Gamma _*$ the cut in $\Gamma $ realized by $\gamma _1$ yields an embedding $i \colon \Gamma +\mathbb Q\gamma _1 \to \Gamma _*$ of ordered $\mathbb {Q}$ -vector spaces fixing $\Gamma $ , and the assumption $[\Gamma +\mathbb Q\gamma _1] = [\Gamma ]$ ensures that it is an embedding of H-asymptotic couples (see Case 3 of [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 3.4]).

In case $[\Gamma +\mathbb Q\gamma _1] \neq [\Gamma ]$ but $\Gamma ^<$ is cofinal in $(\Gamma +\mathbb Q\gamma _1)^<$ , take $\beta \in \Gamma _1 \setminus \Gamma $ with $\beta>0$ and $[\beta ] \notin [\Gamma ]$ . Letting D ₁ be the cut in $\Gamma $ realized by $\beta $ and , it follows from the cofinality assumption that D ₁ has no greatest element and D ₂ has no least element. Then saturation gives $\beta _* \in \Gamma _*$ realizing the same cut as $\beta $ in $\Gamma $ and with $\beta ^{\dagger } = \beta _*^{\dagger }$ , yielding an embedding $i \colon \Gamma +\mathbb Q\gamma _1 \to \Gamma _*$ of H-asymptotic couples fixing $\Gamma $ (see Cases 4–6 of [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 3.4]).

In case $\Gamma ^<$ is not cofinal in $(\Gamma +\mathbb Q\gamma _1)^<$ , take $\beta \in \Gamma +\mathbb Q\gamma _1$ satisfying $0<\beta <\Gamma ^>$ . In this case, $(\Gamma , \psi )$ must have max $0$ , since if $(\Gamma , \psi )$ has gap $0$ , then $\Psi <\psi _1(\beta ) \in \Psi $ , a contradiction. By saturation, take $\beta _* \in \Gamma _*$ with $0<\beta _*<\Gamma ^>$ and $\beta _*^{\dagger }=\beta ^{\dagger }=0$ , so we get an embedding $i \colon \Gamma +\mathbb Q\gamma _1 \to \Gamma _*$ of H-asymptotic couples as before.

In [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 3.5], the cofinality of $\Gamma ^<$ in $\Gamma _1^<$ follows from asymptotic integration; In the max $0$ case it is also automatic, but needs to be added as an assumption in the gap $0$ case, (i) in the next lemma.

Lemma 4.4. Suppose that $(\Gamma +\mathbb Q\gamma )^{\dagger } \neq \Gamma ^{\dagger }$ for all $\gamma \in \Gamma _1 \setminus \Gamma $ . Also, suppose that either:

1. (i) $(\Gamma , \psi )$ is gap-closed, $(\Gamma _1, \psi _1)$ has gap $0$ , and $\Gamma ^<$ is cofinal in $\Gamma _1^<$ ; or

2. (ii) $(\Gamma , \psi )$ is max-closed and $(\Gamma _1,\psi _1)$ has max $0$ .

Then $(\Gamma \langle \gamma _1\rangle , \psi _1)$ can be embedded into $(\Gamma _*, \psi _*)$ over $\Gamma $ .

Proof. Note that in case (ii), the cofinality assumption comes for free, for if $\gamma \in \Gamma _1 \setminus \Gamma $ satisfies $0<\gamma <\Gamma ^>$ , then $\gamma ^{\dagger } = 0$ and so $(\Gamma +\mathbb Q\gamma )^{\dagger } = \Gamma ^{\dagger }$ , a contradiction.

Take $\alpha _1 \in \Gamma $ such that $(\gamma _1-\alpha _1)^{\dagger } \notin \Gamma ^{\dagger }$ . Then $(\gamma _1-\alpha _1)^{\dagger } \notin \Gamma $ , since $(\gamma _1-\alpha _1)^{\dagger }<0$ and in case (i) and case (ii), $\Psi =\Gamma ^{<}$ and $\Psi =\Gamma ^{\leqslant }$ , respectively. (In [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 3.5], this step uses asymptotic integration.)

The proof now proceeds exactly as in [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 3.5], substituting Lemma 4.1 for [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 2.7] (and $\mathbb {Q}$ for $\boldsymbol k$ ); here is a sketch. Let $n \geqslant 1$ . Continue this procedure to construct sequences $\alpha _1, \alpha _2, \dots $ in $\Gamma $ and $\beta _1, \beta _2, \dots $ in $\Gamma \langle \gamma _1 \rangle \setminus \Gamma $ with $\beta _1 = \gamma _1-\alpha _1$ and $\beta _{n+1} = \beta _n^{\dagger }-\alpha _{n+1}$ . Then $\beta _n^{\dagger }<\beta _{n+1}^{\dagger }$ by Lemma 4.2 (with $\beta _n$ in place of $\gamma _1$ and $\beta _{n+1}$ in place of $\beta $ ), so $[\beta _n]>[\beta _{n+1}]$ , so

$$\begin{align*}\Gamma\langle \gamma_1 \rangle\ =\ \Gamma \oplus \mathbb Q\beta_1 \oplus \mathbb Q\beta_2 \oplus \cdots. \end{align*}$$

Saturation gives $\gamma _* \in \Gamma _* \setminus \Gamma $ realizing the same cut in $\Gamma $ as $\gamma _1$ , so use the same $\alpha _n$ sequence to likewise define $\beta _{*n} \in (\Gamma _*)_{\infty }$ for $n \geqslant 1$ . Inductively construct an increasing sequence of $\mathbb {Q}$ -vector space embeddings

$$\begin{align*}i_n \colon \Gamma + \mathbb Q\beta_1 + \dots + \mathbb Q\beta_n \to \Gamma_* \end{align*}$$

fixing $\Gamma $ and sending $\beta _n$ to $\beta _{*n}$ in two steps: Given that $\beta _n$ and $\beta _{*n}$ realize the same cut in $\Gamma $ , use Lemma 4.1 to get that $\beta _n^{\dagger }$ and $\beta _{*n}^{\dagger }$ realize the same cut in $\Gamma $ . Then argue with archimedean classes, using Lemma 4.2 in $(\Gamma _*,\psi _*)$ now, to extend the embedding to $i_{n+1}$ . The union of these maps is the desired embedding of H-asymptotic couples.

In case (i) of the previous lemma, we added a cofinality assumption that was not present in [Reference Aschenbrenner, van den Dries and van der Hoeven6, Lemma 3.5]. The next lemma handles the non-cofinal case, which is particular to gap $0$ .

Lemma 4.5. Suppose that $\Gamma ^< < \gamma _1 <0$ , and $(\Gamma _1,\psi _1)$ and $(\Gamma _*,\psi _*)$ have gap $0$ . Then $(\Gamma \langle \gamma _1\rangle , \psi _1)$ can be embedded into $(\Gamma _*, \psi _*)$ over $\Gamma $ .

Proof. Set and for all n. We have

$$\begin{align*}[\gamma_1]>[\gamma_1^{\langle 1 \rangle}]>[\gamma_1^{\langle 2 \rangle}]>\cdots, \end{align*}$$

and so

$$\begin{align*}\Gamma^< < \gamma_1 < \gamma_1^{\langle 1 \rangle} < \gamma_1^{\langle 2 \rangle}<\dots<0 \qquad \text{and} \qquad [\gamma_1^{\langle n \rangle}] \notin [\Gamma]\ \text{for all}\ n. \end{align*}$$

Hence the family $(\gamma _1^{\langle n \rangle })_{n \in \mathbb N}$ is $\mathbb Q$ -linearly independent over $\Gamma $ and

$$\begin{align*}\Gamma\langle \gamma_1 \rangle\ =\ \Gamma \oplus \mathbb Q\gamma_1 \oplus \mathbb Q\gamma_1^{\langle 1 \rangle} \oplus \mathbb Q\gamma_1^{\langle 2 \rangle} \oplus \cdots. \end{align*}$$

By saturation, we may take $\gamma _* \in \Gamma _* \setminus \Gamma $ with $\Gamma ^< < \gamma _* < 0$ . The above holds in $\Gamma _*$ with $\gamma _*$ replacing $\gamma _1$ (and $\gamma _*^{\langle n \rangle }$ defined analogously), so we obtain an embedding of $(\Gamma \langle \gamma _1 \rangle , \psi _1)$ into $(\Gamma _*, \psi _*)$ over $\Gamma $ that sends $\gamma _1$ to $\gamma _*$ .

4.1.3 Quantifier elimination

We call an H-asymptotic couple $(\Gamma _1, \psi _1)$ extending $(\Gamma , \psi )$ a gap-closure of $(\Gamma , \psi )$ if it is gap-closed and it embeds over $(\Gamma , \psi )$ into every gap-closed H-asymptotic couple extending $(\Gamma , \psi )$ . Similarly, we call an H-asymptotic couple $(\Gamma _1, \psi _1)$ extending $(\Gamma , \psi )$ a max-closure of $(\Gamma , \psi )$ if it is max-closed and it embeds over $(\Gamma , \psi )$ into every max-closed H-asymptotic couple extending $(\Gamma , \psi )$ . By the embedding lemmas of the previous subsection and a standard quantifier elimination test (see for example [Reference Aschenbrenner, van den Dries and van der Hoeven5, Corollary B.11.11]), quantifier elimination for gap-closed and max-closed H-asymptotic couples reduces to the existence of gap-closures and max-closures, respectively.

For that, we need one more embedding lemma, Let $\Psi ^{\downarrow }$ be the downward closure of $\Psi $ in $\Gamma $ .

Lemma 4.6. Let $\beta \in \Psi ^{\downarrow } \setminus \Psi $ or $\beta $ be a gap in $(\Gamma , \psi )$ . Then there is an H-asymptotic couple $(\Gamma \oplus \mathbb Z\alpha , \psi ^\alpha )$ extending $(\Gamma , \psi )$ such that:

1. (i) $\alpha>0$ and $\psi ^\alpha (\alpha )=\beta $ ;

2. (ii) given an embedding i of $(\Gamma , \psi )$ into an H-asymptotic couple $(\Gamma ^{*}, \psi ^{*})$ and $\alpha ^{*} \in \Gamma ^{*}$ with $\alpha ^{*}>0$ and $\psi ^{*}(\alpha ^{*})=i(\beta )$ , there is a unique extension of i to an embedding $j \colon (\Gamma \oplus \mathbb Z\alpha , \psi ^\alpha ) \to (\Gamma ^{*}, \psi ^{*})$ with $j(\alpha )=\alpha ^{*}$ .

Proof. This follows from [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 9.8.7] with $C=\{[\gamma ] : \gamma \in \Gamma ^{\neq },\ \psi (\gamma )>\beta \}$ , but here is an outline. Define the ordered abelian group $\Gamma \oplus \mathbb Z\alpha $ so that

$$\begin{align*}\{0\} \cup \{ \gamma \in \Gamma^> : \psi(\gamma)>\beta \}\ <\ \alpha\ <\ \{ \gamma \in \Gamma^> : \psi(\gamma)<\beta \}, \end{align*}$$

and thus $[\alpha ] \notin [\Gamma ^{\neq }]$ . Extend $\psi $ to $\psi ^{\alpha } \colon (\Gamma \oplus \mathbb Z\alpha )^{\neq } \to \Gamma $ by , where $\gamma \in \Gamma $ and $k \in \mathbb Z^{\neq }$ . To verify that this makes $(\Gamma \oplus \mathbb Z\alpha , \psi ^\alpha )$ an H-asymptotic couple involves tedious case distinctions. The axioms (AC1), (AC2), and (HC) are straightforward, while (AC3) is more subtle and uses [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 6.5.4]. The universal property of $(\Gamma \oplus \mathbb Z\alpha , \psi ^\alpha )$ is easy.

Corollary 4.7. Every H-asymptotic couple $(\Gamma , \psi )$ with gap $0$ has a gap-closure. Every H-asymptotic couple $(\Gamma , \psi )$ with $\sup \Psi =0$ has a max-closure.

Proof. If $(\Gamma ,\psi )$ has max $0$ , then by applying Lemma 4.6 for $\beta \in \Psi ^{\downarrow } \setminus \Psi $ , taking the divisible hull, and repeating these procedures, we get a max-closure of $(\Gamma , \psi )$ . If $(\Gamma ,\psi )$ has gap $0$ and $\Gamma \neq \{0\}$ , the same procedure yields a gap-closure of $(\Gamma , \psi )$ . To obtain a max-closure of $(\Gamma ,\psi )$ when it has gap $0$ , first apply Lemma 4.6 with $\beta =0$ and then repeat the previous argument.

It remains to show that the trivial H-asymptotic couple $(\{0\},\psi )$ , where $\psi \colon \emptyset \to \{0\}$ is the empty function, has a gap-closure. Let be the ordered $\mathbb {Q}$ -vector space satisfying $\gamma _n<0$ and $[\gamma _n]>[\gamma _{n+1}]$ for all n, and equip it with the function $\psi _0$ defined by $\psi _0(q_1\gamma _{n_1}+\dots +q_m\gamma _{n_m})=\gamma _{n_1+1}$ for all $m \geqslant 1$ , $n_1>\dots >n_m$ , and $q_1, \dots , q_m \in \mathbb {Q}^{\times }$ . Then by the proof of Lemma 4.5, $(\Gamma _0, \psi _0)$ is an H-asymptotic couple with gap $0$ that embeds into every nontrivial H-asymptotic couple with gap  $0$ . Hence, a gap-closure of $(\Gamma _0,\psi _0)$ is a gap-closure of $(\{0\},\psi )$ .

Next we establish quantifier elimination, first proving it in , the language $\mathcal L_{\operatorname {ac}}$ expanded by unary function symbols $\operatorname {div}_n$ interpreted as division by n with . As noted already, unlike in [Reference Aschenbrenner and van den Dries3, Reference Aschenbrenner, van den Dries and van der Hoeven6], we do not need the predicate for the $\Psi $ -set, since it is already quantifier-free definable without parameters.

Theorem 4.8. The theory of gap-closed H-asymptotic couples has quantifier elimination, and it is the model completion of the theory of H-asymptotic couples with gap $0$ . The theory of max-closed H-asymptotic couples has quantifier elimination, and it is the model completion of the theory of H-asymptotic couples $(\Gamma , \psi )$ with $\sup \Psi =0$ .

Proof. That these theories have quantifier elimination in $\mathcal L_{\operatorname {ac}, \operatorname {div}}$ follows from Lemmas 4.3, 4.4, and 4.5, and Corollary 4.7 by a standard quantifier elimination test. To see that they have quantifier elimination in $\mathcal L_{\operatorname {ac}}$ , recall how, for an H-asymptotic couple $(\Gamma , \psi )$ , $\psi $ extends uniquely to the divisible hull $\mathbb {Q}\Gamma $ of $\Gamma $ , preserving the property of having gap $0$ or max $0$ .

The model completion statements follow from quantifier elimination and Corollary 4.7.

Corollary 4.9. The theory of gap-closed H-asymptotic couples is complete and has a prime model. The theory of max-closed H-asymptotic couples is complete and has a prime model.

Proof. The trivial H-asymptotic couple (see the proof of Corollary 4.7), embeds into every H-asymptotic couple, yielding completeness of the two theories. It also has gap $0$ , so its gap-closure and its max-closure are the respective prime models.

As prime models of their respective complete theories, on standard model-theoretic grounds (see for instance [Reference Marker13, Corollary 4.2.16]) the gap-closure and the max-closure of the trivial H-asymptotic couple are unique up to isomorphism. By quantifier elimination and adding constants to the language, similar reasoning applies to the gap-closure and the max-closure of a countable H-asymptotic couple. However, we do not know in general if gap-closures and max-closures are unique up to isomorphism. One possibility is foreclosed: By adapting the arguments of [Reference Aschenbrenner1, Section 3], we can show that if an H-asymptotic couple $(\Gamma , \psi )$ with gap $0$ is not gap-closed, then a gap-closure of $(\Gamma , \psi )$ is not minimal in the sense that it has a proper gap-closed substructure containing $\Gamma $ . The same is true replacing “gap $0$ ” with “ $\sup \Psi =0$ ,” “gap-closed” with “max-closed,” and “gap-closure” with “max-closure.” This nonminimality result has no bearing on the rest of the paper, so we do not present details here. With more work, one could likely obtain further model-theoretic results on such H-asymptotic couples, along the lines of [Reference Aschenbrenner and van den Dries3, Reference Aschenbrenner, van den Dries and van der Hoeven6].

We do use Theorem 4.8 via the following corollary. For $n \geqslant 1$ , $\alpha _1, \dots , \alpha _n \in \Gamma $ , and $\gamma \in \Gamma $ , we define the function $\psi _{\alpha _1,\dots ,\alpha _n} \colon \Gamma _\infty \to \Gamma _\infty $ recursively by

Corollary 4.10. Let $(\Gamma , \psi )$ be a gap-closed H-asymptotic couple and let $(\Gamma ^{*}, \psi ^{*})$ be an H-asymptotic couple extending $(\Gamma , \psi )$ with gap $0$ . Suppose $n \geqslant 1$ , $\alpha _1, \dots , \alpha _n \in \Gamma $ , $q_1, \dots , q_n \in \mathbb {Q}$ , and $\gamma ^{*} \in \Gamma ^{*}$ are such that:

1. (i) $\psi ^{*}_{\alpha _1, \dots , \alpha _n}(\gamma ^{*}) \neq \infty $ (so $\psi ^{*}_{\alpha _1, \dots , \alpha _i}(\gamma ^{*}) \neq \infty $ for $i=1,\dots ,n$ );

2. (ii) $\gamma ^{*} + q_1 \psi ^{*}_{\alpha _1}(\gamma ^{*}) + \dots + q_n \psi ^{*}_{\alpha _1, \dots , \alpha _n}(\gamma ^{*}) \in \Gamma $ (in $\mathbb {Q}\Gamma ^{*}$ ).

Then $\gamma ^{*} \in \Gamma $ .

Proof. By Theorem 4.8, $(\Gamma , \psi )$ is an existentially closed H-asymptotic couple with gap $0$ , so we have $\gamma \in \Gamma $ with

$$\begin{align*}\gamma + q_1 \psi_{\alpha_1}(\gamma) + \dots + q_n \psi_{\alpha_1, \dots, \alpha_n}(\gamma)\ =\ \gamma^{*} + q_1 \psi^{*}_{\alpha_1}(\gamma^{*}) + \dots + q_n \psi^{*}_{\alpha_1, \dots, \alpha_n}(\gamma^{*}). \end{align*}$$

It remains to use [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 9.9.3] to obtain $\gamma ^{*}=\gamma \in \Gamma $ .

The same holds with “max-closed” replacing “gap-closed”and “ $\sup \Psi =0$ ” replacing “gap $0$ .”

4.2 A maximality theorem

The results and proofs of this section are adapted from [Reference Aschenbrenner, van den Dries and van der Hoeven5, Section 16.1]. In the next lemma and its consequences, we use the quantifier elimination for gap-closed asymptotic couples from Section 4.1 to study extensions of certain asymptotic fields whose asymptotic couples are gap-closed. Note that if K is an H-asymptotic field with exponential integration and gap $0$ , then $\Psi = \Gamma ^<$ , so if additionally $\Gamma $ is divisible then $(\Gamma , \psi )$ is gap-closed.

Assumption. In this subsection, K has small derivation.

The next lemma follows from Corollary 4.10 in the same way that [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 16.1.1] follows from [Reference Aschenbrenner, van den Dries and van der Hoeven5, Proposition 9.9.2], except with “ $\operatorname {d}$ -algebraically maximal” replacing “asymptotically $\operatorname {d}$ -algebraically maximal”: K is differential-algebraically maximal ( $\operatorname {d}$ -algebraically maximal for short) if it has no proper differentially algebraic (“ $\operatorname {d}$ -algebraic” for short) immediate extension with small derivation. And instead of [Reference Aschenbrenner, van den Dries and van der Hoeven5, Theorem 14.0.2] (used implicitly in [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 16.1.1]), we need [Reference Pynn-Coates14, Theorem 3.6]. We also need the notion of a pseudocauchy sequence (“pc-sequence” for short); for the definition and basic facts about pc-sequences, see [Reference Aschenbrenner, van den Dries and van der Hoeven5, Section 2.2], and for pc-sequences in valued differential fields, see [Reference Aschenbrenner, van den Dries and van der Hoeven5, Sections 4.4 and 6.9]. In an immediate extension L of K, every element of $L \setminus K$ is the pseudolimit of a pc-sequence in K that has no pseudolimit in K, called divergent in K, and divergent pc-sequences in K can be of $\operatorname {d}$ -algebraic or $\operatorname {d}$ -transcendental type over K. The exact definitions are not really important here, for which see the end of [Reference Aschenbrenner, van den Dries and van der Hoeven5, Section 4.4]. But note that if the derivation on $\boldsymbol k$ is nontrivial, then K is $\operatorname {d}$ -algebraically maximal if and only if there is no divergent pc-sequence in K of $\operatorname {d}$ -algebraic type over K by [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 6.9.3].

Lemma 4.11. Suppose that K is a $\operatorname {d}$ -henselian H-asymptotic field with exponential integration and gap $0$ whose value group is divisible. Let L be an H-asymptotic extension of K with gap $0$ and $\boldsymbol k_L = \boldsymbol k$ , and suppose that there is no $y \in L \setminus K$ such that $K\langle y \rangle $ is an immediate extension of K. Let $f \in L \setminus K$ . Then the vector space $\mathbb {Q}\Gamma _{K\langle f \rangle }/\Gamma $ is infinite dimensional.

Proof. First, we argue that there is no divergent pc-sequence in K with a pseudolimit in L. Towards a contradiction, suppose that $(a_\rho )$ is a divergent pc-sequence in K with pseudolimit $\ell \in L$ . Since K is $\operatorname {d}$ -henselian and asymptotic, it is $\operatorname {d}$ -algebraically maximal [Reference Pynn-Coates14, Theorem 3.6], so $(a_\rho )$ is not of $\operatorname {d}$ -algebraic type over K. Hence $(a_\rho )$ is of $\operatorname {d}$ -transcendental type over K, so $K\langle \ell \rangle $ is an immediate extension of K by [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 6.9.1], a contradiction.

Thus for all $y \in L \setminus K$ , the set $v(y-K) \subseteq \Gamma _L$ has a maximum. Take $b_0 \in K$ with $v(f-b_0) = \max v(f-K)$ , so $v(f-b_0) \notin \Gamma $ since $\boldsymbol k_L = \boldsymbol k$ . Set . Then $f_1 \notin K$ , since otherwise, there would be $g \in K^{\times }$ with $(f-b_0)^{\dagger } = g^{\dagger }$ , so $v(f-b_0) = vg \in \Gamma $ , a contradiction. Setting , this leads to sequences $(f_n)$ in $L \setminus K$ and $(b_n)$ in K such that for all n:

1. (1) $v(f_n-b_n) = \max v(f_n-K) \notin \Gamma $ ;

2. (2) $f_{n+1} = (f_n-b_n)^{\dagger }$ .

To finish the proof of the lemma, show that

$$\begin{align*}v(f_0-b_0),\ v(f_1-b_1),\ \dots\ \text{are } \mathbb{Q}\text{-linearly independent over } \Gamma. \end{align*}$$

If $\Gamma = \{ 0 \}$ , then this follows from [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 9.2.10(iv)]. Otherwise, the idea is that by taking for $n \geqslant 1$ an $a_n \in K^{\times }$ with $a_n^{\dagger }=b_n$ and setting , Corollary 4.10 (with one of the $v(f_n-b_n)$ in the role of $\gamma $ ) shows that a $\mathbb {Q}$ -linear dependence is impossible since each $v(f_n-b_n) \notin \Gamma $ .

The next result is a strengthening of [Reference Pynn-Coates14, Theorem 3.6] under additional hypotheses, including that the asymptotic couple of K is gap-closed. It follows from the Zariski–Abhyankar inequality by combining [Reference Pynn-Coates14, Theorem 3.6] and Lemma 4.11 just as [Reference Aschenbrenner, van den Dries and van der Hoeven5, Theorem 16.0.3] combines [Reference Aschenbrenner, van den Dries and van der Hoeven5, Theorem 14.0.2] and [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 16.1.1]. This result is used in Section 6 to prove the minimality of $\operatorname {d}$ -Hensel–Liouville closures.

Theorem 4.12. Suppose that K is a $\operatorname {d}$ -henselian H-asymptotic field with exponential integration and gap $0$ whose value group is divisible. Then K has no proper $\operatorname {d}$ -algebraic H-asymptotic extension with gap $0$ and the same residue field.

By quantifier elimination for max-closed H-asymptotic couples, the above result holds with “max $0$ ” replacing “gap $0$ ,” where we say an asymptotic field K has max $0$ if its asymptotic couple does.

We now summarize some facts about the asymptotic couple of the $K\langle f \rangle $ from Lemma 4.11 that we need in the next subsection.

Lemma 4.13. Let K, L, and f be as in Lemma 4.11, and let the sequences $(f_n)$ , $(b_n)$ , $(a_n)_{n \geqslant 1}$ , and $(\alpha _n)_{n \geqslant 1}$ be as in the proof of Lemma 4.11. Set . The asymptotic couple $(\Gamma _{K\langle f \rangle }, \psi )$ of $K\langle f \rangle $ has the following properties:

1. (i) $(\beta _n)$ is $\mathbb {Q}$ -linearly independent over $\Gamma $ and $\Gamma _{K\langle f \rangle } = \Gamma \oplus \bigoplus _n \mathbb {Z}\beta _n$ (internal direct sum);

2. (ii) $(\beta _n^{\dagger })$ is $\mathbb {Q}$ -linearly independent over $\Gamma $ , so $\beta _n^{\dagger } \notin \Gamma $ and $\beta _m^{\dagger } \neq \beta _n^{\dagger }$ for all $m \neq n$ ;

3. (iii) $\psi (\Gamma _{K\langle f \rangle }^{\neq }) = \Psi \cup \{ \beta _n^{\dagger } : n \in \mathbb {N} \}$ ;

4. (iv) $[\beta _n] \notin [\Gamma ]$ for all n, $[\beta _m] \neq [\beta _n]$ for all $m \neq n$ , and $[\Gamma _{K\langle f \rangle }] = [\Gamma ] \cup \{ [\beta _n] : n \in \mathbb {N} \}$ ;

5. (v) if $\Gamma ^<$ is cofinal in $\Gamma _{K\langle f \rangle }^<$ , then $\beta _0^{\dagger } < \beta _1^{\dagger } < \beta _2^{\dagger } < \cdots $ .

Proof. The $\mathbb {Q}$ -linear independence of $(\beta _n)$ over $\Gamma $ follows from the proof of Lemma 4.11. Items (i)–(iv) are proved exactly as in [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 16.1.2], without modification, but are sketched below. The only difference is (v): In [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 16.1.2], cofinality is established unconditionally. Here, Lemma 4.2 immediately gives (v).

The hardest part is (i), which involves showing that the element $P(f) \in K\langle f \rangle $ , where $P \in K\{Y\}^{\neq }$ , can be expressed as a polynomial in the “monomials” . Since $v(\mathfrak m_n) = \beta _n$ , item (i) then follows from the $\mathbb {Q}$ -linear independence of $(\beta _n)$ over $\Gamma $ . With that done, the $\mathbb {Q}$ -linear independence of $(\beta _n^{\dagger })$ over $\Gamma $ follows from the $\mathbb {Q}$ -linear independence of $(\beta _n)_{n \geqslant 1}$ over $\Gamma $ by noticing that $\beta _n^{\dagger } = \beta _{n+1}+\alpha _{n+2}$ , and the rest of (ii)–(iv) follows.

4.3 Further consequences in the ordered setting

Now we develop further the results of the previous subsection in the pre-H-field setting. If K, L, and f are as before but additionally pre-H-fields, and M is a pre-H-field with $g \in M$ realizing the same cut in K as f, our goal is to embed $K\langle f \rangle $ into M over K by sending f to g. This is accomplished in Proposition 4.15 and Lemma 4.16. The former handles the case that $\Gamma ^<$ is cofinal in $\Gamma _{K\langle f \rangle }^<$ , which follows from the previous subsection by making small adjustments to the arguments for analogous results in [Reference Aschenbrenner, van den Dries and van der Hoeven5, Section 16.1]. The case that $\Gamma ^<$ is not cofinal in $\Gamma _{K\langle f \rangle }^<$ does not occur in that context, and here is handled in Lemma 4.16 by similar ideas.

In the next lemma, the reader may assume without harm that K, L, and f additionally satisfy the assumptions of Lemma 4.11, but the lemma is stated in more generality to clarify which assumptions are used. In particular, it does not assume that K and L have gap $0$ or that K is $\operatorname {d}$ -henselian. Small modifications to the proof of [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 16.1.4] yield the following generalization.

Lemma 4.14. Let K be a pre-H-field with exponential integration and let L be a pre-H-field extension of K with $\boldsymbol k_L = \boldsymbol k$ . Suppose that $f \in L \setminus K$ and $b_0 \in K$ satisfy:

1. (i) $v_L(f-K) \subseteq \Gamma _L$ has a maximum and $v_L(f-b_0) = \max v_L(f-K)$ ;

2. (ii) $\Gamma ^<$ is cofinal in $\Gamma _{K\langle f \rangle }^<$ .

Suppose that M is a pre-H-field extension of K and $g \in M$ realizes the same cut in K as f. Then $v_M(g-b_0) = \max v_M(g-K) \notin \Gamma $ and $(g-b_0)^{\dagger }$ realizes the same cut in K as $(f-b_0)^{\dagger }$ .

Proof. Let $\alpha \in \Gamma $ and $b \in K$ . First we use the convexity of $\mathcal O_{K\langle f \rangle }$ and the cofinality assumption to show that

$$\begin{align*}v_L(f-b)<\alpha \iff v_M(g-b)<\alpha \qquad \text{and} \qquad v_L(f-b)>\alpha \iff v_M(g-b)>\alpha. \end{align*}$$

To see this, take $a \in K^>$ with $va = \alpha $ . Suppose that $v_L(f-b)<\alpha $ , so $|f-b|>a$ . Hence $|g-b|>a$ , and so $v_M(g-b)\leqslant \alpha $ . To get the strict inequality, use the cofinality assumption to take $\delta \in \Gamma $ with $v_L(f-b)<\delta <\alpha $ , and thus $v_M(g-b) \leqslant \delta < \alpha $ . One proves similarly that $v_L(f-b)>\alpha \implies v_M(g-b)>\alpha $ . Finally, consider the case that $v_L(f-b)=\alpha $ . This yields $f-b \sim ua$ for $u \in K$ with $u \asymp 1$ , since $\boldsymbol k = \boldsymbol k_L$ . Convexity yields $|u|a/2 < |f-b| < 2|u|a$ , so $|u|a/2 < |g-b| < 2|u|a$ , and thus $v_M(g-b)=va=\alpha $ , completing the proof of the displayed equivalences. Hence, $v_M(g-b_0) \notin \Gamma $ , since $v_L(f-b_0) \notin \Gamma $ . This in turn yields $v_M(g-b_0)=\max v_M(g-K)$ . It also follows that $(g-b_0)^{\dagger } \notin K$ , as otherwise $(g-b_0)^{\dagger } = b^{\dagger }$ for some $b \in K^{\times }$ , in which case $v_M(g-b_0)=vb \in \Gamma $ .

As written, the argument in [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 16.1.4] that $(g-b_0)^{\dagger }$ realizes the same cut in K as $(f-b_0)^{\dagger }$ uses that K and L are H-fields, so we give the following modified version. First, we may assume that $f>b_0$ , so $g>b_0$ . Suppose towards a contradiction that we have $h \in K$ with $(f-b_0)^{\dagger }<h$ and $h<(g-b_0)^{\dagger }$ . Take ${\phi \in K^>}$ with $h=\phi ^{\dagger }$ and set . Then we have $s>0$ and $s^{\dagger }=(f-b_0)^{\dagger }-h<0$ . Since L is a pre-H-field, $v_L(s) \geqslant 0$ by [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 10.5.2(i)], but since $v_L(f-b_0) \notin \Gamma $ , we have $v_L(s)>0$ . Thus $0<s<1$ . Similarly, $h<(g-b_0)^{\dagger }$ gives and $t^{\dagger }>0$ , so $v_M(t)<0$ and thus $t>1$ . Putting this together yields

$$\begin{align*}f\ =\ b_0+\phi s\ <\ b_0+\phi\ \qquad \text{and}\ \qquad b_0+\phi\ <\ b_0 + \phi t\ =\ g, \end{align*}$$

contradicting that f and g realize the same cut in K. The other case, that there is $h \in K$ with $(f-b_0)^{\dagger }>h$ and $h>(g-b_0)^{\dagger }$ , is symmetric.

The next embedding lemma follows from Lemma 4.14 in the same way that [Reference Aschenbrenner, van den Dries and van der Hoeven5, Proposition 16.1.5] follows from [Reference Aschenbrenner, van den Dries and van der Hoeven5, Lemma 16.1.4], without modification. The idea is to use Lemma 4.14 to carry out the proof of Lemma 4.13 in M and thereby construct the desired embedding.

Proposition 4.15. Let K, L, and f be as in Lemma 4.11 and suppose that $\Gamma ^<$ is cofinal in $\Gamma _{K\langle f \rangle }^<$ . Additionally, suppose that K and L are pre-H-fields and L is a pre-H-field extension of K with gap $0$ . Suppose that M is a pre-H-field extension of K with gap $0$ and $g \in M$ realizes the same cut in K as f. Then there exists an embedding $K \langle f \rangle \to M$ over K with $f \mapsto g$ .

Above we assumed that $\Gamma ^<$ was cofinal in $\Gamma _{K\langle f \rangle }^<$ , and now we turn to the other case, which does not occur in the H-field setting of [Reference Aschenbrenner, van den Dries and van der Hoeven5, Section 16.1].

Lemma 4.16. Let K be a pre-H-field and L be a pre-H-field extension of K with gap $0$ . Let $f \in L^>$ with $\Gamma ^< < v_L(f) < 0$ . Suppose that M is a pre-H-field extension of K with gap $0$ and $g \in M^>$ satisfies $\Gamma ^< < v_M(g) < 0$ . Then there is an embedding $K\langle f \rangle \to M$ over K with $f \mapsto g$ .

Proof. Set and , and let . Then

$$\begin{align*}[\Gamma^{\neq}]> [\beta_0]>[\beta_1]>[\beta_2]>\cdots>[0]. \end{align*}$$

In particular, $[\beta _n] \notin [\Gamma ]$ for all n and $\beta _0, \beta _1, \beta _2, \dots $ are $\mathbb {Q}$ -linearly independent over $\Gamma $ . Hence the vector space $\mathbb {Q}\Gamma _{K\langle f \rangle }/\Gamma $ is infinite dimensional, so f is $\operatorname {d}$ -transcendental over K. Since $f=f_0$ and $f_n' = f_n f_{n+1}$ , induction yields $F_n \in K[Y_0, \dots , Y_n]$ of (total) degree at most $n+1$ with $f^{(n)}=F_n(f_0, \dots , f_n)$ . Let $P \in K\{Y\}^{\neq }$ of order at most r. Then $P(f)=\sum _{\boldsymbol i \in I} a_{\boldsymbol i} f_0^{i_0} \dots f_r^{i_r}$ , where I is a nonempty finite set of indices $\boldsymbol i=(i_0, \dots , i_r) \in \mathbb {N}^{1+r}$ . In particular, $\Gamma _{K\langle f \rangle } = \Gamma \oplus \bigoplus _{n} \mathbb {Z}\beta _n$ .

Set , , and . The same argument yields that g is $\operatorname {d}$ -transcendental over K and $P(g)=\sum _{\boldsymbol i \in I} a_{\boldsymbol i} g_0^{i_0} \dots g_r^{i_r}$ , where I is the same set of indices as in $P(f)$ and $a_{\boldsymbol i}$ are the same coefficients. Hence $\Gamma _{K\langle g \rangle } = \Gamma \oplus \bigoplus _{n} \mathbb {Z}\beta _n^{*}$ . Thus we have an isomorphism of ordered abelian groups $j \colon \Gamma _{K\langle f \rangle } \to \Gamma _{K\langle g \rangle }$ with $\beta _n \mapsto \beta _n^{*}$ . By the expressions for $P(f)$ and $P(g)$ , we have $j\big (v_L(P(f))\big ) = v_M(P(g))$ , which yields a valued differential field embedding from $K\langle f \rangle \to M$ over K with $f \mapsto g$ . To see that this is an ordered valued differential field embedding, note that $f_n>0$ and $g_n>0$ for all n, so $P(f)>0 \iff P(g)>0$ .

5 Extending the constant field

The results of this short section are used mainly to strengthen the statement of Theorem 6.15 in the next section and play no role in the main results of Section 7. The term “residue constant closed” defined in the next paragraph, however, is used in some key lemmas of Section 6.

Since $C \subseteq \mathcal O$ , C maps injectively into $\boldsymbol k$ under the residue field map, and hence into $C_{\boldsymbol k}$ . We say that K is residue constant closed if K is henselian and C maps onto $C_{\boldsymbol k}$ , that is, $\operatorname {\mathrm {res}}(C)=C_{\boldsymbol k}$ . We say that L is a residue constant closure of K if it is a residue constant closed H-asymptotic extension of K with small derivation that embeds uniquely over K into every residue constant closed H-asymptotic extension M of K with small derivation. Note that if K has a residue constant closure, then it is unique up to unique isomorphism over K.

Note that if K is a pre-H-field with $\sup \Psi =0$ , then as an immediate extension of K any residue constant closure of K embeds uniquely, as an ordered valued differential field, over K into every residue constant closed pre-H-field extension of K with small derivation.

The following result is worth recording but not needed later. In it, we construe an algebraic extension of a residue constant closed K as a valued differential field extension of K by equipping it with the unique derivation and valuation extending those of K (uniqueness of the valuation follows from the henselianity of K); if K is additionally an ordered field, then by henselianity $\mathcal O$ is necessarily convex.

6 Differential-Hensel–Liouville closures

In this section we construct differential-Hensel–Liouville closures (Theorem 6.16) in analogy with the Newton–Liouville closures of [Reference Aschenbrenner, van den Dries and van der Hoeven5, Section 14.5] and prove that they are unique (Corollary 6.18). To do this, we first construct extensions that are real closed, have exponential integration, and satisfy an embedding property (Corollary 6.11, Lemma 6.14), in analogy with the Liouville closures of [Reference Aschenbrenner and van den Dries4]; however, we use [Reference Aschenbrenner, van den Dries and van der Hoeven5, Section 10.6] as a reference and also adapt some preliminaries from [Reference Aschenbrenner, van den Dries and van der Hoeven5, Sections 10.4–10.6]. Combining this with the residue constant closures from the previous section, we record in Theorem 6.15 an improvement that is not needed later.

Suppose momentarily that K has gap $0$ . In several results below, it is assumed that $\boldsymbol k$ has exponential integration. It is worth noting that if K has exponential integration, then so does $\boldsymbol k$ . Hence, the assumption that $\boldsymbol k$ has exponential integration is necessary for K to have an extension with exponential integration and the same ordered differential residue field.