# A model complete theory of transexponential pre-$H$-fields

Nigel Pynn-Coates  Department of Mathematics, The Ohio State University,
Columbus, OH, United States<EMAIL_ADDRESS>
###### Abstract.

The theory of differential-henselian, real closed pre-$H$-fields that have
exponential integration and closed ordered differential residue field has
quantifier elimination and is the model completion of the theory of
pre-$H$-fields with gap $0$. From quantifier elimination, we deduce that this
theory is distal, so has NIP. Moreover, we establish a two-sorted quantifier
elimination result when the theory of the residue field has quantifier
elimination in a language expanding the language of ordered differential
rings, which yields a weak Ax–Kochen/Ershov principle.

##  1\. Introduction

Pre-$H$-fields and $H$-fields are kinds of ordered valued differential fields
introduced by M. Aschenbrenner and L. van den Dries in [2]; all Hardy fields
are pre-$H$-fields and all Hardy fields containing $\mathbb{R}$ are
$H$-fields. Together with J. van der Hoeven in [3], they showed that a certain
theory $T^{\operatorname{nl}}$ of $H$-fields is the model companion of the
theory of pre-$H$-fields in the language
$\\{+,-,\cdot,0,1,\der,\preccurlyeq,\leqslant\\}$, where is interpreted as a
derivation and $\preccurlyeq$ as a binary relation encoding a valuation; the
latter functions in this paper as a coarse notion of size whereby elements
$f\succ 1$ are thought of as infinite and elements $f\prec 1$ as
infinitesimal. Moreover, $T^{\operatorname{nl}}$ admits quantifier elimination
with the addition of a function symbol for field inversion and two unary
predicates identifying the parameters for which two second-order differential
equations have solutions. In addition,
$T^{\operatorname{nl}}_{\operatorname{small}}=T^{\operatorname{nl}}+``\text{small
derivation''}$ axiomatizes the theory of $\mathbb{T}$, the ordered (valued)
differential field of logarithmic-exponential transseries, and is the model
companion of the theory of $H$-fields with small derivation. Here, “small
derivation” means that derivatives of infinitesimals are infinitesimal; see
the next section for precise definitions of this and other notions. All Hardy
fields have small derivation.

By the results above, every pre-$H$-field extends to a model of
$T^{\operatorname{nl}}$ and every $H$-field with small derivation extends to a
model of $T^{\operatorname{nl}}_{\operatorname{small}}$. However, not every
pre-$H$-field with small derivation extends to a model of
$T^{\operatorname{nl}}_{\operatorname{small}}$ and those that do not must have
gap $0$, where a pre-$H$-field has gap $0$ if it has small derivation and the
logarithmic derivatives of infinite elements are infinite. It follows that in
such structures, infinite elements are transexponential in some sense and the
valuation yields a coarser notion of rate of growth than in Hardy fields. This
paper focuses on pre-$H$-fields with gap $0$.

To obtain an example of a pre-$H$-field with gap $0$, start by taking an
$\aleph_{0}$-saturated elementary extension $\mathbb{T}^{*}$ of $\mathbb{T}$.
Then enlarging the valuation ring (the subring of non-infinite elements) so
that it is the set of elements bounded in absolute value by some finite
iterate of the exponential yields a pre-$H$-field with gap $0$ (see Appendix
A). The saturation ensures that $\mathbb{T}^{*}$ contains a transexponential
element and thus this enlarged valuation ring is a proper subring. Another
example comes from considering the functional equation $f(x+1)=e^{f(x)}$. It
has a solution lying in a Hardy field [4], and any solution is clearly
transexponential, so performing the same enlargement of the valuation ring of
this Hardy field also yields a pre-$H$-field with gap $0$.

The goal of this paper is to find a model companion for the theory of
pre-$H$-fields with gap $0$, or, equivalently, to axiomatize the class of
existentially closed pre-$H$-fields with gap $0$. In pre-$H$-fields with small
derivation, the derivation and ordering induce a derivation and ordering,
respectively, on the residue field, which is the quotient of the valuation
ring by its maximal ideal of infinitesimal elements. One major distinction
between the theory considered here and $T^{\operatorname{nl}}$ is that this
derivation induced on the residue field can be nontrivial, and always is in
existentially closed pre-$H$-fields with gap $0$. Conversely, whenever a
pre-$H$-field with small derivation has nontrivial induced derivation on its
residue field, it must have gap $0$. Thus it is reasonable to expect that an
existentially closed pre-$H$-field with gap $0$ has an ordered differential
residue field that is existentially closed; this class is axiomatized by the
theory of closed ordered differential fields introduced by M. Singer [12]. Our
main result is the following, in the language
$\\{+,-,\cdot,0,1,\der,\preccurlyeq,\leqslant\\}$.

######  Theorem 8.2.

The theory of differential-henselian, real closed pre-$H$-fields with
exponential integration and closed ordered differential residue field has
quantifier elimination.

Differential-henselianity generalizes the notion of henselianity for valued
fields to the setting of valued differential fields with small derivation,
while having exponential integration means that for each $f$ there is $z\neq
0$ that behaves like $e^{\int\\!f}$ in the sense that $\der(z)/z=f$. By
showing that every pre-$H$-field with gap $0$ extends to a model of the theory
above, we obtain the desired result.

######  Corollary 8.4.

The theory of differential-henselian, real closed pre-$H$-fields with
exponential integration and closed ordered differential residue field is the
model completion of the theory of pre-$H$-fields with gap $0$.

It also follows from quantifier elimination that this theory is complete and
decidable, which is Corollary 8.5. Finally, we show that it is tame in the
following sense.

######  Theorem 8.6.

The theory of differential-henselian, real closed pre-$H$-fields with
exponential integration and closed ordered differential residue field is
distal, and hence has NIP.

Consider the example of a pre-$H$-field with gap $0$ obtained by enlarging the
valuation ring of an elementary extension $\mathbb{T}^{*}$ of $\mathbb{T}$.
The residue field of this structure can be equipped with a valuation induced
by the valuation of $\mathbb{T}^{*}$, making it elementarily equivalent to
$\mathbb{T}$ as an ordered valued differential field (see Appendix A). Thus we
consider theories in which the residue field has structure beyond its ordered
differential field structure.

Fix a language $\mathcal{L}_{\operatorname{res}}$ expanding
$\\{+,-,\cdot,0,1,\der,\leqslant\\}$ and an
$\mathcal{L}_{\operatorname{res}}$-theory
$T^{\operatorname{dhl}}_{\operatorname{res}}$ of ordered differential fields.
Let $T^{\operatorname{dhl}}$ be the theory whose models are structures
$(K,\bm{k};\pi)$ such that $K$ is a differential-henselian, real closed
pre-$H$-field with exponential integration in the language
$\\{+,-,\cdot,\iota,0,1,\der,\preccurlyeq,\leqslant\\}$, where $\iota$ is
interpreted by field inversion; $\bm{k}\models
T^{\operatorname{dhl}}_{\operatorname{res}}$; and $\pi\colon K\to\bm{k}$ is a
map inducing an isomorphism of ordered differential fields between the residue
field of $K$ and $\bm{k}$. Then:

######  Theorem 8.8.

If $T^{\operatorname{dhl}}_{\operatorname{res}}$ has quantifier elimination,
then so does $T^{\operatorname{dhl}}$.

A two-sorted model companion result is established in Corollary 8.10. Finally,
Theorem 8.8 yields the following weak Ax–Kochen/Ershov principle.

######  Theorem 8.11.

Suppose that $K_{1}$ and $K_{2}$ are pre-$H$-fields that are differential-
henselian, real closed, and have exponential integration, and let $\bm{k}_{1}$
and $\bm{k}_{2}$ be their ordered differential residue fields. Then
$K_{1}\equiv K_{2}$ if and only if $\bm{k}_{1}\equiv\bm{k}_{2}$.

###  1.A. Outline

After some preliminary definitions and remarks, we show how to extend
embeddings of ordered valued differential fields by first extending the
residue field in §3. Associated to each pre-$H$-field with gap $0$ is an
$H$-asymptotic couple with gap $0$, and in §4 we study them as structures in
their own right. We isolate the model completion of the theory of
$H$-asymptotic couples with gap $0$ and prove that this theory has quantifier
elimination in Theorem 4.1. Since models include the asymptotic couples of
real closed pre-$H$-fields with gap $0$ and exponential integration, Theorem
4.1 gets used in §5 via Corollary 4.2. The main result in that section is
Theorem 5.2, which is a strengthening of [7, Theorem 3.6] under additional
hypotheses. Section 6 deals with extending the constant field for use in the
next section. Section 7 builds towards Theorem 7.16, which shows the existence
of differential-Hensel-Liouville closures; these are extensions that are
differential-henselian, real closed, and have exponential integration, and
that satisfy a semi-universal property. We use Theorem 5.2 to prove that
differential-Hensel-Liouville closures are unique in Corollary 7.18. Finally,
§8 contains the main results advertised above.

##  2\. Preliminaries

We let $d$, $m$, $n$, 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 $f+g\neq 0$.

A _differential field_ is a field $K$ equipped with a _derivation_ $\der\colon
K\to K$, which satisfies for $f,g\in K$:

  1. (D1)

$\der(f+g)=\der(f)+\der(g)$;

  2. (D2)

$\der(fg)=f\der(g)+g\der(f)$.

Let $K$ be a valued field. We add a new symbol $\infty$ to $\Gamma$ and extend
the addition and ordering to $\Gamma_{\infty}\coloneqq\Gamma\cup\\{\infty\\}$
by $\infty+\gamma=\gamma+\infty=\infty$ and $\infty>\gamma$ for all
$\gamma\in\Gamma$. This allows us to extend $v$ to $K$ by setting
$v(0)\coloneqq\infty$. We often use the following more intuitive notation:

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

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
$\mathcal{O}\coloneqq\\{f\in K:f\preccurlyeq 1\\}$ and call it the _valuation
ring_ of $K$. It has a (unique) maximal ideal $\cao\coloneqq\\{f\in K:f\prec
1\\}$, and we call $\operatorname{res}(K)\coloneqq\mathcal{O}/\cao$ the
_residue field_ of $K$, often denoted by $\bm{k}$. From our assumption that
$\bm{k}$ has characteristic $0$, we see that $\mathbb{Q}\subseteq\mathcal{O}$.
We also let $\overline{a}$ or $\operatorname{res}(a)$ denote the image of
$a\in\mathcal{O}$ under the map to $\bm{k}$. For another valued field $L$, we
denote these objects by $\mathcal{O}_{L}$, $\Gamma_{L}$, $\bm{k}_{L}$, etc.

Let $K$ be a differential field. For $f\in K$, we often write $f^{\prime}$ for
$\der(f)$ if the derivation is clear from the context and set
$f^{\dagger}\coloneqq f^{\prime}/f$ if $f\neq 0$. We say that $K$ has
_exponential integration_ if $(K^{\times})^{\dagger}=K$. The _field of
constants_ of $K$ is $C\coloneqq\\{f\in K:f^{\prime}=0\\}$. For another
differential field $L$, we denote this object by $C_{L}$. We let
$K\\{Y\\}\coloneqq K[Y,Y^{\prime},Y^{\prime\prime},\dots]$ be the ring of
differential polynomials over $K$ and set $K\\{Y\\}^{\neq}\coloneqq
K\\{Y\\}\setminus\\{0\\}$. Let $P$ range over $K\\{Y\\}^{\neq}$. The _order_
of $P$ is the smallest $r$ such that $P\in K[Y,Y^{\prime},\dots,Y^{(r)}]$. For
$\bm{i}=(i_{0},\dots,i_{r})\in\mathbb{N}^{1+r}$, we set $Y^{\bm{i}}\coloneqq
Y^{i_{0}}(Y^{\prime})^{i_{1}}\dots(Y^{(r)})^{i_{r}}$. If $P$ has order at most
$r$, then we decompose $P$ as $\sum_{\bm{i}}P_{\bm{i}}Y^{\bm{i}}$, where
$\bm{i}$ ranges over $\mathbb{N}^{1+r}$. Letting $|\bm{i}|\coloneqq
i_{0}+\dots+i_{r}$, we note that
$P_{d}=\sum_{|\bm{i}|=d}P_{\bm{i}}Y^{\bm{i}}$, where $P_{d}$ denotes the
homogeneous part of $P$ of degree $d$. We extend the derivation of $K$ to
$K\\{Y\\}$ in the natural way, and we also extend $v$ to $K\\{Y\\}$ by setting
$v(P)$ to be the minimum valuation of the coefficients of $P$. The relations
$\preccurlyeq$, $\prec$, $\asymp$, and $\sim$ are also extended to $K\\{Y\\}$
in the corresponding way, and the image of $P\in\mathcal{O}\\{Y\\}$ under the
map to $\bm{k}\\{Y\\}$ is also denoted by $\overline{P}$.

Now suppose that $K$ is a valued differential field with nontrivial valuation
and derivation; we continue to assume this throughout the paper. Relating the
valuation and the derivation, we impose throughout most of this paper the
condition that $K$ has _small derivation_ , which means that
$\der\cao\subseteq\cao$. In this case, $\der\mathcal{O}\subseteq\mathcal{O}$
[3, Lemma 4.4.2], so induces a derivation on $\bm{k}$, and is continuous with
respect to the valuation topology on $K$ [3, Lemma 4.4.6]. If $K$ has small
derivation, we always construe $\bm{k}$ as a differential field with this
induced derivation and are typically interested in the case that it is
nontrivial, as happens when $\bm{k}$ is _linearly surjective_ : for all
$a_{0},\dots,a_{r}\in\bm{k}$, the equation
$a_{0}+a_{1}y+a_{2}y^{\prime}+\dots+a_{r}y^{(r)}=0$ has a solution in
$\bm{k}$. We say that $K$ is _differential-henselian_ (_$\operatorname{d}$
-henselian_ for short) if $K$ has small derivation and:

  1. (DH1)

$\bm{k}$ is linearly surjective;

  2. (DH2)

whenever $P\in\mathcal{O}\\{Y\\}$ satisfies $P_{0}\prec 1$ and $P_{1}\asymp
1$, there is $y\prec 1$ with $P(y)=0$.

Differential-henselianity was introduced by T. Scanlon in [9, 10] and studied
more systematically in [3]. It is closely connected to the notion of
differential-algebraic maximality, for which we first need to discuss certain
kinds of extensions. Given an extension $L$ of $K$, we identify $\Gamma$ with
a subgroup of $\Gamma_{L}$ and $\bm{k}$ with a subfield of $\bm{k}_{L}$ in the
obvious way. Here and throughout we use the word _extension_ in the following
way: 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. Where there is particular danger of confusion, we make explicit
the kind of extension meant. The words “embedding,” “isomorphic,” and
“isomorphism” are used similarly.

We say that an extension $L$ of $K$ is _immediate_ if $\Gamma_{L}=\Gamma$ and
$\bm{k}_{L}=\bm{k}$; if $K$ and $L$ have small derivation, then $\bm{k}$ is
naturally a differential subfield of $\bm{k}_{L}$. In an immediate extension
$L$ of $K$, every element of $L\setminus K$ is the pseudolimit of a
pseudocauchy sequence (“pc-sequence” for short) in $K$ that has no pseudolimit
in $K$ (called _divergent_ in $K$); we use this only in Lemma 8.1. For a
definition and basic facts about pc-sequences, see [3, §2.2]. Divergent pc-
sequences in $K$ can be of $\operatorname{d}$-algebraic or
$\operatorname{d}$-transcendental type over $K$, and this comes up in Lemmas
5.1 and 8.1; for more on these two notions, see [3, §4.4 and §6.9]. If $K$ has
small derivation, then we call it _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. If $K$ has small derivation and the
derivation on $\bm{k}$ is nontrivial, then $K$ is
$\operatorname{d}$-algebraically maximal if and only if it has no divergent
pc-sequence of $\operatorname{d}$-algebraic type over $K$ by [3, Lemma 6.9.3].
Relating this to $\operatorname{d}$-henselianity, any
$\operatorname{d}$-algebraically maximal valued differential field with small
derivation and linearly surjective differential residue field is also
$\operatorname{d}$-henselian [3, Theorem 7.0.1]; an earlier case is in [9].

The converse fails in general but holds for asymptotic (valued differential)
fields [7, Theorem 3.6]. We say that $K$ is _asymptotic_ if $f\prec g\iff
f^{\prime}\prec g^{\prime}$ for all nonzero $f,g\in\cao$. Note that if $K$ is
asymptotic, then $C\subseteq\mathcal{O}$. If $K$ is asymptotic, then 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
$f^{\dagger}\succcurlyeq g^{\dagger}$. In the rest of this paragraph, suppose
that $K$ is asymptotic. We say that $K$ has _gap $0$_ if it has small
derivation and $f^{\dagger}\succ 1$ for all $f\in K^{\times}$ with $f\prec 1$.
For $g\in K^{\times}$ with $g\not\asymp 1$, $v(g^{\dagger})$ and
$v(g^{\prime})$ depend only on $vg$ and not on $g$, so with $\gamma\coloneqq
vg$ we set $\gamma^{\dagger}\coloneqq v(g^{\dagger})$ and
$\gamma^{\prime}\coloneqq v(g^{\prime})$; note that
$\gamma^{\dagger}=\gamma^{\prime}-\gamma$. Thus setting
$\Gamma^{\neq}\coloneqq\Gamma\setminus\\{0\\}$, logarithmic differentiation
induces a map

$\displaystyle\psi\colon\Gamma^{\neq}$ $\displaystyle\to\Gamma$
$\displaystyle\gamma$ $\displaystyle\mapsto\gamma^{\dagger}.$

We call $(\Gamma,\psi)$ the _asymptotic couple_ of $K$; such structures were
introduced by M. Rosenlicht [8]. The map $\psi$ is a valuation on $\Gamma$ in
the sense of [3, §2.2], and we set $\Psi\coloneqq\psi(\Gamma^{\neq})$. Note
that $K$ being $H$-asymptotic just says that the valuation $\psi$ is convex
with respect to the ordering of $\Gamma$, and $K$ having small derivation or
gap $0$ are also properties of its asymptotic couple. In fact, having such a
map $\psi$ on the value group satisfying certain axioms described in §4 is
equivalent to being asymptotic [3, Proposition 9.1.3]. We extend $\psi$ to a
map $\psi\colon\Gamma_{\infty}\to\Gamma_{\infty}$ by setting
$\psi(0)=\psi(\infty)\coloneqq\infty$.

Even stronger than being asymptotic is being pre-$\operatorname{d}$-valued. We
say that $K$ is _pre-differential-valued_ (_pre- $\operatorname{d}$-valued_
for short) if:

  1. (PD1)

$C\subseteq\mathcal{O}$;

  2. (PD2)

for all $f,g\in K^{\times}$, if $f\preccurlyeq g\prec 1$, then
$\frac{f}{g}-\frac{f^{\prime}}{g^{\prime}}\prec 1$.

Equivalently, $K$ is pre-$\operatorname{d}$-valued if, for all $f,g\in
K^{\times}$ with $f\preccurlyeq 1$ and $g\prec 1$, we have $f^{\prime}\prec
g^{\dagger}$ [3, Lemma 10.1.4].

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, it is also equipped with a (total) ordering $\leqslant$ making it
an ordered field (in the sense that the ordering is preserved by addition and
by multiplication by positive elements). If $\mathcal{O}$ is convex with
respect to the ordering (equivalently, $\cao$ is convex; see [3, Lemma
3.5.11]), then $\leqslant$ induces an ordering on $\bm{k}$ 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 $f^{\prime}>0$.

It follows that if $K$ is a pre-$H$-field and $f,g\in K^{\times}$, then
$f^{\dagger}<g^{\dagger}$ whenever $f\prec g$ [3, Lemma 10.5.2(i)]. By part
(ii) of the same lemma, pre-$H$-fields are $H$-asymptotic. Since
pre-$H$-fields are pre-$\operatorname{d}$-valued, if $K$ is a pre-$H$-field
with small derivation and the derivation induced on its residue field is
nontrivial, then $K$ must have gap $0$.

We now discuss how to extend orderings, valuations, and derivations to various
extensions. If $K$ is an ordered valued differential field with convex
valuation ring, we equip its real closure $K^{\operatorname{rc}}$ with the
unique derivation extending that of $K$ (see [3, Lemma 1.9.2]) and the unique
valuation extending that of $K$ whose valuation ring is convex (see [3,
Corollary 3.5.18]), and always construe $K^{\operatorname{rc}}$ as an ordered
valued differential field in this way. Then $\Gamma_{K^{\operatorname{rc}}}$
is the divisible hull $\mathbb{Q}\Gamma$ of $\Gamma$,
$\bm{k}_{K^{\operatorname{rc}}}$ is the real closure of $\bm{k}$, and
$C_{K^{\operatorname{rc}}}$ is the real closure of $C$. If $K$ is a
pre-$H$-field, then so is $K^{\operatorname{rc}}$ by [3, Proposition 10.5.4];
if $K$ is a pre-$H$-field with gap $0$, then so is $K^{\operatorname{rc}}$ by
the same proposition and the remarks after [3, Lemma 6.5.3]. 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 (see [3, Lemma 10.5.8]).

##  3\. Extensions controlled by the residue field

Here are ordered variants of [3, Theorem 6.3.2] and [3, Lemma 6.3.1]. Let
$F\in K\\{Y\\}^{\neq}$ have order $r$ and set $d\coloneqq\deg_{Y^{(r)}}F$.
Decomposing $F(Y)=\sum_{n=0}^{d}G_{n}(Y)\cdot(Y^{(r)})^{n}$ with $G_{n}\in
K[Y,\dots,Y^{(r-1)}]$ for $n=0,\dots,d$, the _initial_ of $F$ is $G_{d}$.
Below we let $I$ be the initial of $F$. First we recall [3, Theorem 6.3.2]:

######  Theorem 3.1 ([3, 6.3.2]).

Suppose that $K$ has small derivation. Let $K\langle a\rangle$ be a
differential field extension of $K$ such that $a$ has minimal annihilator $F$
as above satisfying $F\asymp 1$ and $I\asymp 1$, and such that $\overline{F}$
is irreducible in $\bm{k}\\{Y\\}$. Then there is a unique valuation $v\colon
K\langle a\rangle^{\times}\to\Gamma$ extending that of $K$ such that:

  1. (i)

$K\langle a\rangle$ has small derivation;

  2. (ii)

$a\preccurlyeq 1$;

  3. (iii)

$\overline{a}$ has minimal annihilator $\overline{F}$ over $\bm{k}$.

It is given by $P(a)/Q(a)\mapsto vP-vQ\in\Gamma$, for $P\in
K[Y,\dots,Y^{(r)}]^{\neq}$ with $\deg_{Y^{(r)}}P<d$ and $Q\in
K[Y,\dots,Y^{(r-1)}]^{\neq}$. The residue field of $K\langle a\rangle$ is
$\bm{k}\langle\overline{a}\rangle$.

Below, we equip $K\langle a\rangle$ with this valuation.

######  Lemma 3.2.

Suppose that $K$ is an ordered valued differential field with small derivation
and convex valuation ring. Let $K\langle a\rangle$ be as above. Suppose that
$\bm{k}\langle\overline{a}\rangle$ is an _ordered_ differential field
extension of $\bm{k}$. Then there exists a unique ordering on $K\langle
a\rangle$ making it an ordered field extension of $K$ with convex valuation
ring such that the induced ordering on $\bm{k}\langle\overline{a}\rangle$
agrees with the given one. If $K$ is a pre-$H$-field with gap $0$, then so is
$K\langle a\rangle$.

###### Proof.

Suppose that $K\langle a\rangle$ is equipped with an ordering making it an
ordered field extension of $K$ with convex valuation ring. Let $P\in
K[Y,\dots,Y^{(r)}]^{\neq}$ and $\deg_{Y^{(r)}}P<d$. By scaling $P$ by an
element of $K^{>}$, we may assume that $v(P)=0$, and thus
$\overline{P}(\overline{a})\neq 0$. We have
$P(a)>0\iff\overline{P}(\overline{a})>0$, which shows that there is at most
one ordering on $K\langle a\rangle$ making it an ordered field extension of
$K$ with convex valuation ring such that the induced ordering on
$\bm{k}\langle\overline{a}\rangle$ agrees with the given one.

To construct such an ordering, let $b\in K\langle a\rangle^{\times}$, so
$b=P(a)/Q(a)$ for $P\in K[Y,\dots,Y^{(r)}]^{\neq}$ with $\deg_{Y^{(r)}}P<d$
and $Q\in K[Y,\dots,Y^{(r-1)}]^{\neq}$. By scaling $b$ by an element of
$K^{>}$, it suffices to define $b>0$ when $b\asymp 1$. Similarly, we may
assume that $P\asymp Q\asymp 1$. Then the condition
$\overline{P}(\overline{a})/\overline{Q}(\overline{a})>0$ in
$\bm{k}\langle\overline{a}\rangle$ depends only on $b$ and not on the choice
of $P$ and $Q$, so we can define
$b>0:\Leftrightarrow\overline{P}(\overline{a})/\overline{Q}(\overline{a})>0$.
Then $b>0$ or $-b>0$.

Next, assume that $b,c\in K\langle a\rangle^{\times}$ and $b,c>0$; we show
that $b+c>0$ and $bc>0$. We have $b=sP(a)/Q(a)$ and $c=tG(a)/H(a)$ with
$s,t\in K^{>}$, $P$ and $Q$ as above, and $G\in K[Y,\dots,Y^{(r)}]^{\neq}$ and
$H\in K[Y,\dots,Y^{(r-1)}]^{\neq}$ such that $\deg_{Y^{(r)}}G<d$ and $G\asymp
H\asymp 1$. First we show that $b+c>0$. Without loss of generality,
$s\preccurlyeq t$, so we have $b+c=t(st^{-1}HP+QG)(a)/QH(a)$ with
$\deg_{Y^{(r)}}(st^{-1}HP+QG)<d$. Then

$\frac{\overline{(st^{-1}HP+QG)}(\overline{a})}{\overline{QH}(\overline{a})}\
=\
\overline{st^{-1}}\frac{\overline{P}(\overline{a})}{\overline{Q}(\overline{a})}+\frac{\overline{G}(\overline{a})}{\overline{H}(\overline{a})}\
>\ 0,$

so $b+c>0$. Now we show that $bc>0$; we may assume that $s=t=1$. If $G\in
K[Y,\dots,Y^{(r-1)}]$, then $bc=PG(a)/QH(a)$ with $\deg_{Y^{(r)}}PG<d$ and

$\frac{\overline{PG}(\overline{a})}{\overline{QH}(\overline{a})}\ =\
\frac{\overline{P}(\overline{a})}{\overline{Q}(\overline{a})}\cdot\frac{\overline{G}(\overline{a})}{\overline{H}(\overline{a})}\
>\ 0.$

It therefore suffices to consider the case that $b=P(a)$ and $c=G(a)$. By
division with remainder in $\mathcal{O}[Y,\dots,Y^{(r)}]$ we have
$I^{m}PG=BF+R$ with $B,R\in\mathcal{O}[Y,\dots,Y^{(r)}]$ and
$\deg_{Y^{(r)}}R<d$, and thus $bc=R(a)/I^{m}(a)$. But
$\overline{R}(\overline{a})/\overline{I^{m}}(\overline{a})=\overline{P}(\overline{a})\cdot\overline{G}(\overline{a})>0$,
so $bc>0$. Thus we have defined an ordering on $K\langle a\rangle$ making it
an ordered field extension of $K$. An easy calculation shows that if $b\prec
1$, then $-1<b<1$, so the valuation ring of $K\langle a\rangle$ is convex with
respect to this ordering (see [3, Lemma 3.5.11]), and by construction it
induces the given ordering on $\bm{k}\langle\overline{a}\rangle$.

Finally, suppose that $K$ is a pre-$H$-field with gap $0$. As a valued
differential field extension of $K$ with small derivation and the same value
group, $K\langle a\rangle$ is pre-$\operatorname{d}$-valued [3, Lemma 10.1.9],
so it has the same asymptotic couple as $K$ and thus has gap $0$. By [3, Lemma
10.5.5] (with $T=K^{\times}$), $K\langle a\rangle$ is in fact a pre-$H$-field.
∎

Recall that the gaussian valuation on $K\langle Y\rangle$ is defined by
setting $v(P)$, for $P\in K\\{Y\\}^{\neq}$, to be the minimum valuation of the
coefficients of $P$; for more details, see [3, §4.5 and §6.3].

######  Lemma 3.3.

Suppose that $K$ is an ordered valued differential field with small derivation
and convex valuation ring. Consider $K\langle Y\rangle$ with the gaussian
valuation. Suppose that $\bm{k}\langle\overline{Y}\rangle$ is an _ordered_
differential field extension of $\bm{k}$. Then there exists a unique ordering
on $K\langle Y\rangle$ making it an ordered field extension of $K$ with convex
valuation ring such that the induced ordering on
$\bm{k}\langle\overline{Y}\rangle$ agrees with the given one. If $K$ is a
pre-$H$-field with gap $0$, then so is $K\langle Y\rangle$.

###### Proof.

The proof is very similar to that of the previous lemma, but easier. ∎

######  Corollary 3.4.

Suppose that $K$ is an ordered valued differential field with small derivation
and convex valuation ring. Let $\bm{k}_{L}$ be an ordered differential field
extension of $\bm{k}$. Then $K$ has an ordered valued differential field
extension $L$ with the following properties:

  1. (i)

$\Gamma_{L}=\Gamma$;

  2. (ii)

$L$ has small derivation;

  3. (iii)

$\mathcal{O}_{L}$ is convex;

  4. (iv)

$\operatorname{res}(L)\cong\bm{k}_{L}$ over $\bm{k}$ (as ordered differential
fields);

  5. (v)

for any ordered valued differential field extension $M$ of $K$ with convex
valuation ring that is $\operatorname{d}$-henselian, every embedding
$\operatorname{res}(L)\to\bm{k}_{M}$ over $\bm{k}$ is induced by an embedding
$L\to M$ over $K$.

Moreover, if $K$ is a pre-$H$-field with gap $0$, then so is $L$.

###### Proof.

First, note that we can reduce to the case that $\bm{k}_{L}=\bm{k}\langle
y\rangle$.

Suppose that $y$ is $\operatorname{d}$-transcendental over $\bm{k}$. Set
$L\coloneqq K\langle Y\rangle$, equipped with the gaussian valuation and the
ordering from Lemma 3.3 so that
$\operatorname{res}(L)=\bm{k}\langle\overline{Y}\rangle\cong\bm{k}\langle
y\rangle$ over $\bm{k}$. Let $M$ be an ordered valued differential field
extension of $K$ with convex valuation ring, and suppose that $M$ is
$\operatorname{d}$-henselian and
$i\colon\bm{k}\langle\overline{Y}\rangle\to\bm{k}_{M}$ is an embedding over
$\bm{k}$. Take $b\in M$ with $b\asymp 1$ and $\overline{b}=i(\overline{Y})$.
Then [3, Lemma 6.3.1] provides a valued differential field embedding $L\to M$
over $K$ sending $Y$ to $b$; this is an ordered field embedding by the
uniqueness in Lemma 3.3.

Now suppose that $y$ is $\operatorname{d}$-algebraic over $\bm{k}$. Take
$F\in\mathcal{O}\\{Y\\}$ so that $\overline{F}\in\bm{k}\\{Y\\}$ is a minimal
annihilator of $y$ over $\bm{k}$ and $F$ and $\overline{F}$ have the same
order $r$, degree in $Y^{(r)}$, and total degree. Note that $F\asymp I\asymp
S\asymp 1$, where $I$ is the initial of $F$ and $S\coloneqq\partial F/\partial
Y^{(r)}$ is the separant of $F$. Take a differential field extension
$L\coloneqq K\langle a\rangle$ of $K$ such that $a$ has minimal annihilator
$F$ over $K$. We equip $L$ with the valuation extending that of $K$ from
Theorem 3.1, so $L$ has small derivation and $a\preccurlyeq 1$, and the
ordering from Lemma 3.2, making it an ordered field extension of $K$ with
convex valuation ring and
$\operatorname{res}(L)=\bm{k}\langle\overline{a}\rangle\cong\bm{k}\langle
y\rangle$ over $\bm{k}$. Let $M$ be an ordered valued differential field
extension of $K$ with convex valuation ring, and suppose that $M$ is
$\operatorname{d}$-henselian and
$i\colon\bm{k}\langle\overline{a}\rangle\to\bm{k}_{M}$ is an embedding over
$\bm{k}$. Let $z\in M$ with $z\preccurlyeq 1$ and
$\overline{z}=i(\overline{a})$. By the minimality of $\overline{F}$, we have
$\overline{S}\big{(}i(\overline{a})\big{)}\neq 0$, so $S(z)\asymp 1$. In
particular, $(F_{+z})_{1}\asymp 1$, so by the $\operatorname{d}$-henselianity
of $M$, there is $b\in M$ with $F(b)=0$, $b\preccurlyeq 1$, and
$\overline{b}=i(\overline{a})$. Note that then $F$ is a minimal annihilator of
$b$ over $K$ by the minimality of $\overline{F}$. Hence by Theorem 3.1 and
Lemma 3.2 we may embed $L$ into $M$ over $K$ sending $a$ to $b$. ∎

##  4\. Asymptotic couples with small derivation

Towards our quantifier elimination and model completion results for
pre-$H$-fields with gap $0$, we first study their associated asymptotic
couples and prove quantifier elimination and model completion results for the
theory of such structures. We suspend in this section the convention that
$\Gamma$ is the value group of $K$. Instead, throughout the section
$(\Gamma,\psi)$ 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 $\psi(\gamma)\geqslant\psi(\delta)$.

It follows from (AC2) and (HC) that $\psi$ is constant on archimedean classes
of $\Gamma$. For $\gamma\in\Gamma$, we let
$[\gamma]\coloneqq\\{\delta\in\Gamma:|\delta|\leqslant n|\gamma|\ \text{and}\
|\gamma|\leqslant n|\delta|\ \text{for some}\ n\\}$ denote its archimedean
class, and set $[\Gamma]\coloneqq\\{[\gamma]:\gamma\in\Gamma\\}$, ordering it
in the natural way. 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 $q\in\mathbb{Q}^{\times}$ (for uniqueness see [3,
Lemma 6.5.3]), and in this way we always construe $\mathbb{Q}\Gamma$ as an
$H$-asymptotic couple $(\mathbb{Q}\Gamma,\psi)$ extending $(\Gamma,\psi)$; it
satisfies $\psi(\mathbb{Q}\Gamma^{\neq})=\psi(\Gamma^{\neq})$.

Keeping in mind that in later sections $(\Gamma,\psi)$ will be the asymptotic
couple of an $H$-asymptotic field (such as a pre-$H$-field), we let
$\gamma^{\dagger}\coloneqq\psi(\gamma)$ and
$\gamma^{\prime}\coloneqq\gamma^{\dagger}+\gamma$ for
$\gamma\in\Gamma^{\neq}$. We let $\Psi\coloneqq\psi(\Gamma^{\neq})$ and let
$\Psi^{\downarrow}$ be the downward closure of $\Psi$ in $\Gamma$. For any
ordered abelian group $G$ we set $G^{<}\coloneqq\\{g\in G:g<0\\}$ and likewise
with $G^{>}$. Thus (AC3) says that $\Psi<(\Gamma^{>})^{\prime}$. For
$\beta\in\Gamma$, we say that $(\Gamma,\psi)$ has _gap $\beta$_ if
$\Psi<\beta<(\Gamma^{>})^{\prime}$ and _max $\beta$_ if $\max\Psi=\beta$. Note
that if $(\Gamma,\psi)$ is the asymptotic couple of an asymptotic field $K$,
then $K$ has gap $0$ (in the sense of §2) if and only if $(\Gamma,\psi)$ has
gap $0$. It follows easily from [3, Theorem 9.2.1 and Lemma 9.2.9] that
$\sup\Psi=0\iff(\Gamma^{>})^{\prime}=\Gamma^{>}$ 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 $0$.

We are concerned primarily with $H$-asymptotic couples having gap $0$, but
using similar techniques we prove analogous results for asymptotic couples
with max $0$, although we do not use them in the main results of the paper.
Before stating the quantifier elimination and model completion results, we
specify the language
$\mathcal{L}_{\operatorname{ac}}=\\{+,-,\leqslant,0,\infty,\psi\\}$ of
asymptotic couples. The underlying set of an $H$-asymptotic couple
$(\Gamma,\psi)$ in this language is
$\Gamma_{\infty}\coloneqq\Gamma\cup\\{\infty\\}$, and we interpret $\infty$ in
the following way: for all $\gamma\in\Gamma$,
$\infty+\gamma=\gamma+\infty\coloneqq\infty$ and $\gamma<\infty$;
$\infty+\infty\coloneqq\infty$; $-\infty\coloneqq\infty$;
$\psi(0)=\psi(\infty)\coloneqq\infty$. The other symbols have the expected
interpretation.

######  Theorem 4.1.

The theory of nontrivial divisible $H$-asymptotic couples $(\Gamma,\psi)$ with
$\Psi=\Gamma^{<}$ has quantifier elimination, and is the model completion of
the theory of $H$-asymptotic couples with gap $0$.

In this paper, we use this theorem 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

$\psi_{\alpha_{1}}(\gamma)\coloneqq\psi(\gamma-\alpha_{1})\qquad\text{and}\qquad\psi_{\alpha_{1},\dots,\alpha_{n}}(\gamma)\coloneqq\psi\big{(}\psi_{\alpha_{1},\dots,\alpha_{n-1}}(\gamma)-\alpha_{n}\big{)}\
\text{for}\ n\geqslant 2.$

######  Corollary 4.2.

Let $(\Gamma,\psi)$ be a nontrivial divisible $H$-asymptotic couple with
$\Psi=\Gamma^{<}$ 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)

$\psi^{*}_{\alpha_{1},\dots,\alpha_{n}}(\gamma^{*})\neq\infty\ (\text{so}\
\psi^{*}_{\alpha_{1},\dots,\alpha_{i}}(\gamma^{*})\neq\infty\ \text{for}\
i=1,\dots,n);$

  2. (ii)

$\gamma^{*}+q_{1}\psi^{*}_{\alpha_{1}}(\gamma^{*})+\dots+q_{n}\psi^{*}_{\alpha_{1},\dots,\alpha_{n}}(\gamma^{*})\in\Gamma\
(\text{in}\ \mathbb{Q}\Gamma^{*}).$

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

###### Proof.

By Theorem 4.1, $(\Gamma,\psi)$ is an existentially closed $H$-asymptotic
couple with gap $0$ (see [3, Lemma B.10.10]), so we have $\gamma\in\Gamma$
with

$\gamma+q_{1}\psi_{\alpha_{1}}(\gamma)+\dots+q_{n}\psi_{\alpha_{1},\dots,\alpha_{n}}(\gamma)\
=\
\gamma^{*}+q_{1}\psi^{*}_{\alpha_{1}}(\gamma^{*})+\dots+q_{n}\psi^{*}_{\alpha_{1},\dots,\alpha_{n}}(\gamma^{*}).$

It remains to use [3, Lemma 9.9.3] to obtain $\gamma^{*}=\gamma\in\Gamma$. ∎

The rest of the section is devoted to proving Theorem 4.1, as well as an
analogue for $H$-asymptotic couples with max $0$. The material in this section
is based on work in progress by Aschenbrenner, van den Dries, and van der
Hoeven, tentatively titled “Revisiting Closed Asymptotic Couples.” This work
improves [1], in which a result similar to Theorem 4.1 is obtained for a
different theory of $H$-asymptotic couples, by introducing several new lemmas
that simplify the arguments. We use these lemmas to adapt their new proof of
quantifier elimination to this setting. Since their work is unpublished, we
quote the results that we use and give their proofs, also due to
Aschenbrenner, van den Dries, and van der Hoeven. Moreover, many of the proofs
of results specific to the setting of gap $0$ or max $0$ are very similar to
proofs of analogous results in their setting. We are indebted to those authors
for allowing the use of their manuscript.

In contrast with the results of [1], 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 $\bm{k}$, but here we work over $\mathbb{Q}$ for
concreteness (the results of this section hold in that setting in the language
$\mathcal{L}_{\operatorname{ac}}$ expanded by functions for scalar
multiplication).

###  4.A. Preliminaries

The first two lemmas are due to Aschenbrenner, van den Dries, and van der
Hoeven, as part of the work in progress described above, while the third
combines [3, Corollary 9.8.8] with another case needed here.

######  Lemma 4.3.

Suppose that $\Psi$ is downward closed. 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 $\Gamma$ with
$\gamma_{1}^{\dagger}\notin\Gamma$. 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 claim that:

$\gamma_{1}^{\dagger}<\alpha^{\dagger}\implies\gamma_{*}^{\dagger}<\alpha^{\dagger}\qquad\text{and}\qquad\gamma_{1}^{\dagger}>\alpha^{\dagger}\implies\gamma_{*}^{\dagger}>\alpha^{\dagger}.$

First, suppose that $\gamma_{1}^{\dagger}<\alpha^{\dagger}$. Then
$|\gamma_{1}|>|\alpha|$, so $|\gamma_{*}|>|\alpha|$, and thus
$\gamma_{*}^{\dagger}\leqslant\alpha^{\dagger}$. Since $\Gamma^{<}$ is cofinal
in $\Gamma_{1}^{<}$, there is $\delta\in\Gamma$ with
$\gamma_{1}^{\dagger}<\delta<\alpha^{\dagger}$. By taking
$\beta\in\Gamma^{\neq}$ with $\beta^{\dagger}=\delta$, since $\Psi$ is
downward closed, and replacing $\alpha$ by $\beta$ in the argument, we obtain
$\gamma_{*}^{\dagger}\leqslant\beta^{\dagger}<\alpha^{\dagger}$.

Now suppose that $\gamma_{1}^{\dagger}>\alpha^{\dagger}$, so we obtain
$\gamma_{*}^{\dagger}\geqslant\alpha^{\dagger}$ in the same way. By the
cofinality assumption, there is $\delta\in\Gamma$ with
$\gamma_{1}^{\dagger}>\delta>\alpha^{\dagger}$. Note that
$\delta\in\Psi^{\downarrow}$ because there is $\beta\in\Gamma^{\neq}$ with
$|\gamma_{1}|\geqslant|\beta|$, so
$\gamma_{1}^{\dagger}<\beta^{\dagger}\in\Psi$. Hence similar reasoning as in
the first case works. This also shows how the lemma follows from the claim. ∎

######  Lemma 4.4.

Suppose that $(\Gamma_{1},\psi_{1})$ is an $H$-asymptotic couple extending
$(\Gamma,\psi)$, and let $\gamma_{1}\in\Gamma_{1}\setminus\Gamma$ and
$\alpha_{1},\alpha_{2}\in\Gamma$. Suppose that
$\beta_{1}\coloneqq\gamma_{1}-\alpha_{1}$ and
$\beta_{2}\coloneqq\beta_{1}^{\dagger}-\alpha_{2}$ satisfy
$\beta_{1}^{\dagger}\notin\Gamma$ and $\beta_{2}^{\dagger}\notin\Psi$, and
that $|\beta_{1}|\geqslant|\gamma|$ for some $\gamma\in\Gamma^{\neq}$. Then
$\beta_{1}^{\dagger}<\beta_{2}^{\dagger}$.

###### Proof.

Take $\gamma\in\Gamma^{\neq}$ with $|\beta_{1}|\geqslant|\gamma|$, so
$\beta_{1}^{\dagger}\leqslant\gamma^{\dagger}$. From
$\beta_{2}^{\dagger}\notin\Psi$ we obtain
$[\beta_{1}^{\dagger}-\alpha_{2}]\notin[\Gamma]$. Thus
$[\beta_{1}^{\dagger}-\gamma^{\dagger}]\geqslant[\beta_{1}^{\dagger}-\alpha_{2}]$,
as otherwise
$[\beta_{1}^{\dagger}-\alpha_{2}]=[\gamma^{\dagger}-\alpha_{2}]\in[\Gamma]$.
Then, using [3, Lemma 6.5.4(i)] for the first inequality, we get

$\beta_{1}^{\dagger}\ =\ \min\\{\beta_{1}^{\dagger},\gamma^{\dagger}\\}\ <\
(\beta_{1}^{\dagger}-\gamma^{\dagger})^{\dagger}\ \leqslant\
(\beta_{1}^{\dagger}-\alpha_{2})^{\dagger}\ =\ \beta_{2}^{\dagger}.\qed$

######  Lemma 4.5.

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 any embedding $i\colon(\Gamma,\psi)\to(\Gamma^{*},\psi^{*})$ and
$\alpha^{*}\in\Gamma^{*}$ with $\alpha^{*}>0$ and
$\psi^{*}(\alpha^{*})=i(\beta)$, there is a unique embedding
$j\colon(\Gamma\oplus\mathbb{Z}\alpha,\psi^{\alpha})\to(\Gamma^{*},\psi^{*})$
extending $i$ with $j(\alpha)=\alpha^{*}$.

###### Proof.

Apply [3, Lemma 9.8.7] with $C=\\{[\gamma]:\gamma\in\Gamma^{\neq},\
\psi(\gamma)>\beta\\}$. ∎

We call $(\Gamma,\psi)$ _gap-closed_ if $\Gamma$ is nontrivial and divisible,
and $\Psi=\Gamma^{<}$. Similarly, we call $(\Gamma,\psi)$ _max-closed_ if
$\Gamma$ is divisible and $\Psi=\Gamma^{\leqslant}$. Then 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 $(\Gamma,\psi)$.

######  Corollary 4.6.

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

###### Proof.

Let $(\Gamma,\psi)$ be an $H$-asymptotic couple. Suppose that
$\sup\Psi=0\notin\Psi$ and $\Gamma\neq\\{0\\}$. By iterating Lemma 4.5, first
construct an $H$-asymptotic couple $(\Gamma_{1},\psi_{1})$ extending
$(\Gamma,\psi)$ such that $(\Gamma_{1},\psi_{1})$ embeds over $(\Gamma,\psi)$
into every gap-closed $H$-asymptotic couple extending $(\Gamma,\psi)$ and that
for every $\gamma\in\Gamma^{<}$, there is $\gamma_{1}\in\Gamma_{1}^{\neq}$
with $\psi_{1}(\gamma_{1})=\gamma$. Second, take the divisible hull
$(\mathbb{Q}\Gamma_{1},\psi_{1})$ of $(\Gamma_{1},\psi_{1})$. Alternating
these procedures yields a gap-closure of $(\Gamma,\psi)$.

Now suppose that $\Gamma=\\{0\\}$. Then let
$\Gamma_{0}\coloneqq\bigoplus_{n}\mathbb{Q}\gamma_{n}$ be the ordered vector
space over $\mathbb{Q}$ 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 same reasoning as in
Lemma 4.9, $(\Gamma_{0},\psi_{0})$ is an $H$-asymptotic couple with gap $0$
that embeds into every gap-closed $H$-asymptotic couple, so a gap-closure of
$(\Gamma_{0},\psi_{0})$ is also a gap-closure of $(\\{0\\},\psi)$.

A similar argument to the first paragraph shows that $(\Gamma,\psi)$ has a
max-closure when $\sup\Psi=0$. ∎

###  4.B. Quantifier elimination with gap 0

We now turn to the proof of quantifier elimination for gap-closed
$H$-asymptotic couples. To that end, suppose that $(\Gamma,\psi)$ is a
divisible $H$-asymptotic couple with gap $0$, and let $(\Gamma_{1},\psi_{1})$
and $(\Gamma_{*},\psi_{*})$ be gap-closed $H$-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 $(\Gamma_{1},\psi_{1})$.
In light of standard quantifier elimination tests, our goal is to embed
$(\Gamma\langle\gamma_{1}\rangle,\psi_{1})$ into $(\Gamma_{*},\psi_{*})$ over
$\Gamma$. The first two lemmas are adapted from the aforementioned work in
progress, while the third is particular to the case of gap $0$ but is proved
similarly.

For convenience, we set $0^{\dagger}\coloneqq\psi(0)=\infty$, so
$\Gamma^{\dagger}=\Psi\cup\\{\infty\\}$.

######  Lemma 4.7.

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
$\Gamma\langle\gamma_{1}\rangle=\Gamma+\mathbb{Q}\gamma_{1}$. Note that there
is no $\beta_{1}\in\Gamma+\mathbb{Q}\gamma_{1}$ with $0<\beta_{1}<\Gamma^{>}$:
otherwise, $\psi_{1}(\beta_{1})>\Psi$, since $\Psi$ has no greatest element,
contradicting that $(\Gamma+\mathbb{Q}\gamma_{1})^{\dagger}=\Gamma^{\dagger}$.

_Case 1: $[\Gamma+\mathbb{Q}\gamma_{1}]=[\Gamma]$._ By saturation, we may take
$\gamma_{*}\in\Gamma_{*}$ realizing the same cut in $\Gamma$ as $\gamma_{1}$.
Then we have an embedding $i\colon\Gamma+\mathbb{Q}\gamma_{1}\to\Gamma_{*}$ of
ordered vector spaces over $\mathbb{Q}$ that is the identity on $\Gamma$ and
satisfies $i(\gamma_{1})=\gamma_{*}$ by [3, Lemma 2.4.16]. Now for
$\gamma\in\Gamma+\mathbb{Q}\gamma_{1}$ we have
$[i(\gamma)]=[\gamma]\in[\Gamma]$, so
$i(\gamma)^{\dagger}=\gamma^{\dagger}\in\Psi\cup\\{\infty\\}$. Hence $i$ is an
embedding of $(\Gamma\langle\gamma_{1}\rangle,\psi_{1})$ into
$(\Gamma_{*},\psi_{*})$ over $\Gamma$.

_Case 2: $[\Gamma+\mathbb{Q}\gamma_{1}]\neq[\Gamma]$._ Take
$\beta_{1}\in\Gamma_{1}\setminus\Gamma$ with $\beta_{1}>0$ and
$[\beta_{1}]\notin[\Gamma]$, so
$[\Gamma\langle\gamma_{1}\rangle]=[\Gamma]\cup\\{[\beta_{1}]\\}$. Let $D$ be
the cut in $\Gamma$ realized by $\beta_{1}$ and $E\coloneqq\Gamma\setminus D$,
so $D<\beta_{1}<E$. First, we claim that $D$ has no greatest element. If it
did have a greatest element $\delta$, then $0<\beta_{1}-\delta<\Gamma^{>}$,
contradicting the comment at the beginning of the proof. Similarly, $E$ has no
least element. Thus by saturation we have $\beta_{*}\in\Gamma_{*}$ realizing
the same cut in $\Gamma$ as $\beta_{1}$ with
$\beta_{*}^{\dagger}=\beta_{1}^{\dagger}$. Then [3, Lemma 2.4.16] yields an
embedding $i\colon\Gamma+\mathbb{Q}\gamma_{1}\to\Gamma_{*}$ of ordered vector
spaces over $\mathbb{Q}$ that is the identity on $\Gamma$ and satisfies
$i(\beta_{1})=\beta_{*}$. This embedding is also an embedding of
$H$-asymptotic couples. ∎

######  Lemma 4.8.

Suppose that $(\Gamma,\psi)$ is gap-closed,
$(\Gamma+\mathbb{Q}\gamma)^{\dagger}\neq\Gamma^{\dagger}$ for all
$\gamma\in\Gamma_{1}\setminus\Gamma$, and $\Gamma^{<}$ is cofinal in
$\Gamma_{1}^{<}$. Then $(\Gamma\langle\gamma_{1}\rangle,\psi_{1})$ can be
embedded into $(\Gamma_{*},\psi_{*})$ over $\Gamma$.

###### Proof.

Take $\alpha_{1}\in\Gamma$ such that
$(\gamma_{1}-\alpha_{1})^{\dagger}\notin\Gamma^{\dagger}$. Since
$(\gamma_{1}-\alpha_{1})^{\dagger}<0$ and $\Psi=\Gamma^{<}$, we deduce that
$(\gamma_{1}-\alpha_{1})^{\dagger}\notin\Gamma$. Let $n\geqslant 1$. We thus
construct sequences $\alpha_{1},\alpha_{2},\dots$ in $\Gamma$ and
$\beta_{1},\beta_{2},\dots$ in $\Gamma\langle\gamma_{1}\rangle\setminus\Gamma$
with $\beta_{1}=\gamma_{1}-\alpha_{1}$,
$\beta_{n+1}=\beta_{n}^{\dagger}-\alpha_{n+1}$, and
$\beta_{n}^{\dagger}\notin\Gamma$. It follows that
$[\beta_{n}]\notin[\Gamma]$, and by Lemma 4.4 we have
$\beta_{n}^{\dagger}<\beta_{n+1}^{\dagger}$ and thus
$[\beta_{n}]>[\beta_{n+1}]$. In particular, the family
$(\beta_{n})_{n\geqslant 1}$ is $\mathbb{Q}$-linearly independent over
$\Gamma$ and

$\Gamma\langle\gamma_{1}\rangle\ =\
\Gamma\oplus\mathbb{Q}\beta_{1}\oplus\mathbb{Q}\beta_{2}\oplus\dots.$

By saturation, we take $\gamma_{*}\in\Gamma_{*}\setminus\Gamma$ realizing the
same cut in $\Gamma$ as $\gamma_{1}$ and define by recursion on $n\geqslant 1$
$\beta_{*n}\in(\Gamma_{*})_{\infty}$ by
$\beta_{*1}\coloneqq\gamma_{*}-\alpha_{1}$ and
$\beta_{*(n+1)}\coloneqq\beta_{*n}^{\dagger}-\alpha_{n+1}$. We assume
inductively that $\beta_{*m}\in\Gamma_{*}\setminus\Gamma$ for $m=1,\dots n$,
and that we have an embedding

$i_{n}\colon\Gamma+\mathbb{Q}\beta_{1}+\dots+\mathbb{Q}\beta_{n}\to\Gamma_{*}$

of ordered vector spaces over $\mathbb{Q}$ that is the identity on $\Gamma$
and satisfies $i_{n}(\beta_{m})=\beta_{*m}$ for $m=1,\dots,n$. Then
$\beta_{n}$ and $\beta_{*n}$ realize the same cut in $\Gamma$, so
$\beta_{*n}^{\dagger}\notin\Gamma$ and $\beta_{*n}^{\dagger}$ realizes the
same cut in $\Gamma$ as $\beta_{n}^{\dagger}$ by Lemma 4.3. Hence
$\beta_{*(n+1)}\in\Gamma_{*}\setminus\Gamma$ and $\beta_{n+1}$ and
$\beta_{*(n+1)}$ realize the same cut in $\Gamma$. We have

$[\Gamma+\mathbb{Q}\beta_{1}+\dots+\mathbb{Q}\beta_{n}]=[\Gamma]\cup\\{[\beta_{1}],\dots,[\beta_{n}]\\}\qquad\text{and}\qquad[\beta_{1}]>\dots>[\beta_{n}]>[\beta_{n+1}].$

Let $D$ be the cut realized by $[\beta_{n+1}]$ in
$[\Gamma+\mathbb{Q}\beta_{1}+\dots+\mathbb{Q}\beta_{n}]$. The comments above
show that $[\beta_{*(n+1)}]$ realizes the image under $i_{n}$ of $D$ in
$[i_{n}(\Gamma+\mathbb{Q}\beta_{1}+\dots+\mathbb{Q}\beta_{n})]$. Thus we may
extend $i_{n}$ to an embedding

$i_{n+1}\colon\Gamma+\mathbb{Q}\beta_{1}+\dots+\mathbb{Q}\beta_{n}+\mathbb{Q}\beta_{n+1}\to\Gamma_{*}$

of ordered vector spaces over $\mathbb{Q}$ that is the identity on $\Gamma$
and satisfies $i_{n+1}(\beta_{n+1})=\beta_{*(n+1)}$. By induction, this yields
a map $i\colon\Gamma\langle\gamma_{1}\rangle\to\Gamma_{*}$ extending each
$i_{n}$, so $i$ is an embedding of $H$-asymptotic couples. ∎

######  Lemma 4.9.

Suppose that $\Gamma^{<}<\gamma_{1}<0$. Then
$(\Gamma\langle\gamma_{1}\rangle,\psi_{1})$ can be embedded into
$(\Gamma_{*},\psi_{*})$ over $\Gamma$.

###### Proof.

Set $\gamma_{1}^{\langle 0\rangle}\coloneqq\gamma_{1}$ and
$\gamma_{1}^{\langle n+1\rangle}\coloneqq(\gamma_{1}^{\langle
n\rangle})^{\dagger}$ for all $n$. We have

$[\gamma_{1}]>[\gamma_{1}^{\langle 1\rangle}]>[\gamma_{1}^{\langle
2\rangle}]>\dots$

by [3, Lemma 9.2.10(iv)], and so

$\Gamma^{<}<\gamma_{1}<\gamma_{1}^{\langle 1\rangle}<\gamma_{1}^{\langle
2\rangle}<\dots<0\qquad\text{and}\qquad[\gamma_{1}^{\langle
n\rangle}]\notin[\Gamma]\ \text{for all}\ n.$

Hence the family $(\gamma_{1}^{\langle n\rangle})_{n\in\mathbb{N}}$ is
$\mathbb{Q}$-linearly independent over $\Gamma$ and

$\Gamma\langle\gamma_{1}\rangle\ =\
\Gamma\oplus\mathbb{Q}\gamma_{1}\oplus\mathbb{Q}\gamma_{1}^{\langle
1\rangle}\oplus\mathbb{Q}\gamma_{1}^{\langle 2\rangle}\oplus\dots.$

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 construct an embedding of
$(\Gamma\langle\gamma_{1}\rangle,\psi_{1})$ into $(\Gamma_{*},\psi_{*})$ over
$\Gamma$ that sends $\gamma_{1}$ to $\gamma_{*}$ as in the proof of the
previous lemma. ∎

We can now complete the proof of Theorem 4.1. Recall from the introduction to
this section the language
$\mathcal{L}_{\operatorname{ac}}=\\{+,-,\leqslant,0,\infty,\psi\\}$ of
asymptotic couples, though we first prove quantifier elimination in the
expanded language
$\mathcal{L}_{\operatorname{ac},\operatorname{div}}=\mathcal{L}_{\operatorname{ac}}\cup\\{\operatorname{div}_{n}:n\geqslant
1\\}$, where each unary function symbol $\operatorname{div}_{n}$ is
interpreted as division by $n$ with
$\operatorname{div}_{n}(\infty)\coloneqq\infty$.

######  Theorem 4.1.

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 $0$.

###### Proof.

That the theory gap-closed $H$-asymptotic couples has quantifier elimination
in $\mathcal{L}_{\operatorname{ac},\operatorname{div}}$ follows from Lemmas
4.7, 4.8, and 4.9, and Corollary 4.6 by a standard quantifier elimination
test. (See for example [3, Corollary B.11.11].) To see that it has quantifier
elimination in $\mathcal{L}_{\operatorname{ac}}$, recall from the beginning of
this section how, for an asymptotic couple $(\Gamma,\psi)$, $\psi$ extends
uniquely to the divisible hull $\mathbb{Q}\Gamma$ of $\Gamma$. The desired
result then follows from [3, Corollary B.11.5].

The model completion statement follows from quantifier elimination and
Corollary 4.6. (See for example [3, Corollary B.11.6].) ∎

######  Corollary 4.10.

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

###### Proof.

The $H$-asymptotic couple $(\\{0\\},\psi)$, where
$\psi\colon\emptyset\to\\{0\\}$ is the empty function, embeds into every gap-
closed $H$-asymptotic couple, yielding completeness. (See for example [3,
Corollary B.11.7].) It also has gap $0$, so its gap-closure is the prime model
of this theory. ∎

###  4.C. Quantifier elimination with max 0

We derive similar quantifier elimination and model completion results in the
setting allowing max $0$. The proofs are as in the previous subsection, except
where indicated. This material is only used in one later theorem that itself
is not used in the main results, but this subsection is naturally
complementary to the previous one.

Suppose that $(\Gamma,\psi)$ is a divisible $H$-asymptotic couple with
$\sup\Psi=0$. Let $(\Gamma_{1},\psi_{1})$ and $(\Gamma_{*},\psi_{*})$ be max-
closed $H$-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 $(\Gamma_{1},\psi_{1})$.

######  Lemma 4.11.

Suppose that $\max\Psi=0$ and
$(\Gamma+\mathbb{Q}\gamma_{1})^{\dagger}=\Gamma^{\dagger}$. 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
$\Gamma\langle\gamma_{1}\rangle=\Gamma+\mathbb{Q}\gamma_{1}$.

_Case 1: $[\Gamma+\mathbb{Q}\gamma_{1}]=[\Gamma]$._ As in _Case 1_ of Lemma
4.7.

_Case 2: $[\Gamma+\mathbb{Q}\gamma_{1}]\neq[\Gamma]$ but there does not exist
$\beta_{1}\in\Gamma+\mathbb{Q}\gamma_{1}$ with $0<\beta_{1}<\Gamma^{>}$._ As
in _Case 2_ of Lemma 4.7.

_Case 3: there exists $\beta_{1}\in\Gamma+\mathbb{Q}\gamma_{1}$ with
$0<\beta_{1}<\Gamma^{>}$._ By saturation, take $\beta_{*}\in\Gamma_{*}$ with
$0<\beta_{*}<\Gamma^{>}$, so $\beta_{*}^{\dagger}=\beta_{1}^{\dagger}=0$. The
proof continues as in _Case 2_ of Lemma 4.7 after
“$\beta_{*}^{\dagger}=\beta_{1}^{\dagger}$.” ∎

######  Lemma 4.12.

Suppose that $(\Gamma,\psi)$ is max-closed and
$(\Gamma+\mathbb{Q}\gamma)^{\dagger}\neq\Gamma^{\dagger}$ for all
$\gamma\in\Gamma_{1}\setminus\Gamma$. Then
$(\Gamma\langle\gamma_{1}\rangle,\psi_{1})$ can be embedded into
$(\Gamma_{*},\psi_{*})$ over $\Gamma$.

###### Proof.

If $\gamma\in\Gamma_{1}\setminus\Gamma$ with $0<\gamma<\Gamma^{>}$, then
$\gamma^{\dagger}=0$ and so
$(\Gamma+\mathbb{Q}\gamma)^{\dagger}=\Gamma^{\dagger}$, a contradiction. Hence
there is no such $\gamma$, and thus $\Gamma^{<}$ is cofinal in
$\Gamma_{1}^{<}$. The rest of the proof is as in Lemma 4.8. ∎

Recall from the introduction to this section the language
$\mathcal{L}_{\operatorname{ac}}$ of asymptotic couples.

######  Theorem 4.13.

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 $\sup\Psi=0$.

######  Corollary 4.14.

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

##  5\. Extensions controlled by the asymptotic couple

###  5.A. A maximality theorem

The results and proofs of this section are adapted from [3, §16.1]. This next
lemma and its consequences are where we use the quantifier elimination for
gap-closed asymptotic couples from §4. Note that if $K$ is an $H$-asymptotic
field with exponential integration and gap $0$, then in fact
$\Psi=\Gamma^{<}$, so if additionally $\Gamma$ is divisible then
$(\Gamma,\psi)$ is a gap-closed $H$-asymptotic couple in the sense of the
previous section.

For the next lemma, recall the discussion of pc-sequences from §2.

######  Lemma 5.1.

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 $\bm{k}_{L}=\bm{k}$, 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 [7, Theorem 3.6], so $(a_{\rho})$ is
not of $\operatorname{d}$-algebraic type over $K$ by [3, Lemma 6.9.3]. Hence
$(a_{\rho})$ is of $\operatorname{d}$-transcendental type over $K$, so
$K\langle\ell\rangle$ is an immediate extension of $K$ by [3, Lemma 6.9.1], a
contradiction.

Thus for all $y\in L\setminus K$, the set $v_{L}(y-K)\subseteq\Gamma_{L}$ has
a maximum. If $v_{L}(y-y_{0})=\max v_{L}(y-K)$, then
$v_{L}(y-y_{0})\notin\Gamma$ since $\bm{k}_{L}=\bm{k}$. Otherwise, there would
be $y_{1}\in K$ with $y-y_{0}\sim y_{1}$, contradicting the maximality of
$v_{L}(y-y_{0})$. For convenience, assume below that $L=K\langle f\rangle$.
Set $f_{0}\coloneqq f$, pick $b_{0}\in K$ with $v_{L}(f_{0}-b_{0})=\max
v_{L}(f_{0}-K)$, and set $f_{1}\coloneqq(f_{0}-b_{0})^{\dagger}\in L$. We
claim that $f_{1}\notin K$. Otherwise, there would be $g\in K^{\times}$ with
$(f_{0}-b_{0})^{\dagger}=g^{\dagger}$, so $v_{L}(f_{0}-b_{0})=v(g)$,
contradicting that $v_{L}(f_{0}-b_{0})\notin\Gamma$. By induction we obtain
sequences $(f_{n})$ in $L\setminus K$ and $(b_{n})$ in $K$ such that for all
$n$:

  1. (i)

$v_{L}(f_{n}-b_{n})=\max v_{L}(f_{n}-K)$;

  2. (ii)

$f_{n+1}=(f_{n}-b_{n})^{\dagger}$.

Hence $v_{L}(f_{n}-b_{n})\notin\Gamma$ for all $n$. The result follows from
the next claim:

$v_{L}(f_{0}-b_{0}),\ v_{L}(f_{1}-b_{1}),\ \dots\ \text{are
$\mathbb{Q}$-linearly independent over}\ \Gamma.$

To see this, let $n\geqslant 1$ and take $a_{n}\in K^{\times}$ with
$a_{n}^{\dagger}=b_{n}$, so

$f_{n}-b_{n}\ =\ \left(f_{n-1}-b_{n-1}\right)^{\dagger}-a_{n}^{\dagger}\ =\
\left(\frac{f_{n-1}-b_{n-1}}{a_{n}}\right)^{\dagger},$

and set $\alpha_{n}\coloneqq v(a_{n})\in\Gamma$. Recall the function
$\psi_{L,\alpha_{1},\dots,\alpha_{n}}$ defined before Corollary 4.2, where the
subscript $L$ indicates that it is defined on $\Gamma_{L}$, not just $\Gamma$.
Then we have

$v_{L}(f_{n}-b_{n})\ =\
\psi_{L}\big{(}v_{L}(f_{n-1}-b_{n-1})-\alpha_{n}\big{)},$

so by induction we get

$v_{L}(f_{n}-b_{n})\ =\
\psi_{L,\alpha_{1},\dots,\alpha_{n}}\big{(}v_{L}(f_{0}-b_{0})\big{)}.$

Suppose towards a contradiction that
$v_{L}(f_{0}-b_{0}),\dots,v_{L}(f_{n}-b_{n})$ are $\mathbb{Q}$-linearly
dependent over $\Gamma$, so we have $m<n$ and
$q_{1},\dots,q_{n-m}\in\mathbb{Q}$ such that

$v_{L}(f_{m}-b_{m})+q_{1}v_{L}(f_{m+1}-b_{m+1})+\dots+q_{n-m}v_{L}(f_{n}-b_{n})\in\Gamma.$

With $\gamma\coloneqq v_{L}(f_{m}-b_{m})\in\Gamma_{L}\setminus\Gamma$, this
means

$\gamma+q_{1}\psi_{L,\alpha_{m+1}}(\gamma)+\dots+q_{n-m}\psi_{L,\alpha_{m+1},\dots,\alpha_{n}}(\gamma)\in\Gamma,$

so $v_{L}(f_{m}-b_{m})=\gamma\in\Gamma$ by Corollary 4.2, a contradiction. ∎

The previous lemma yields a maximality theorem that is used in the following
section to prove the minimality of differential-Hensel-Liouville closures, but
which is also of independent interest as a strengthening of [7, Theorem 3.6]
under additional hypotheses.

######  Theorem 5.2.

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.

###### Proof.

Let $L$ be a proper $\operatorname{d}$-algebraic $H$-asymptotic extension of
$K$ with gap $0$ and $\bm{k}_{L}=\bm{k}$. Since $K$ is
$\operatorname{d}$-algebraically maximal [7, Theorem 3.6], there is no $y\in
L\setminus K$ such that $K\langle y\rangle$ is an immediate extension of $K$.
But for $f\in L\setminus K$, the transcendence degree of $K\langle f\rangle$
over $K$ is finite, so the vector space $\mathbb{Q}\Gamma_{K\langle
f\rangle}/\Gamma$ is finite dimensional by the Zariski–Abhyankar inequality
[3, Corollary 3.1.11], contradicting Lemma 5.1. ∎

By quantifier elimination for max-closed $H$-asymptotic couples and the same
arguments, we also obtain the following, which is not used later. Here we say
an asymptotic field $K$ has _max $0$_ if its asymptotic couple does.

######  Theorem 5.3.

If $K$ is a $\operatorname{d}$-henselian $H$-asymptotic field with exponential
integration and max $0$ whose value group is divisible, then $K$ has no proper
$\operatorname{d}$-algebraic $H$-asymptotic extension with max $0$ and the
same residue field.

We now provide more details about the asymptotic couple of $K\langle f\rangle$
for use in the next subsection.

######  Lemma 5.4.

Let $K$, $L$, and $f$ be as in Lemma 5.1, 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 5.1. Set $\beta_{n}\coloneqq
v_{L}(f_{n}-b_{n})-\alpha_{n+1}$. The asymptotic couple $(\Gamma_{K\langle
f\rangle},\psi_{L})$ of $K\langle f\rangle$ has the following properties:

  1. (i)

$\Gamma_{K\langle f\rangle}=\Gamma\oplus\bigoplus_{n}\mathbb{Z}\beta_{n}$
(internal direct sum);

  2. (ii)

$\beta_{n}^{\dagger}\notin\Gamma$ for all $n$, and
$\beta_{m}^{\dagger}\neq\beta_{n}^{\dagger}$ for all $m\neq n$;

  3. (iii)

$\psi_{L}(\Gamma_{K\langle
f\rangle}^{\neq})=\Psi\cup\\{\beta_{n}^{\dagger}:n\in\mathbb{N}\\}$;

  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 $\Gamma_{K}^{<}$ is cofinal in $\Gamma_{K\langle f\rangle}^{<}$, then
$\beta_{0}^{\dagger}<\beta_{1}^{\dagger}<\beta_{2}^{\dagger}<\cdots$.

###### Proof.

Set $\mathfrak{m}_{n}\coloneqq(f_{n}-b_{n})/a_{n+1}$, so
$v_{L}(\mathfrak{m}_{n})=\beta_{n}$. Then

$\displaystyle\mathfrak{m}_{n+1}\ $ $\displaystyle=\
\frac{f_{n+1}-b_{n+1}}{a_{n+2}}\ =\
\frac{(f_{n}-b_{n})^{\dagger}-b_{n+1}}{a_{n+2}}\ =\
\frac{(a_{n+1}\mathfrak{m}_{n})^{\dagger}-b_{n+1}}{a_{n+2}}\ =\
\frac{a_{n+1}^{\dagger}+\mathfrak{m}_{n}^{\dagger}-b_{n+1}}{a_{n+2}}$
$\displaystyle=\ \frac{\mathfrak{m}_{n}^{\dagger}}{a_{n+2}}.$

Hence $\mathfrak{m}_{n}^{\prime}=a_{n+2}\mathfrak{m}_{n}\mathfrak{m}_{n+1}$.
From $f=b_{0}+a_{1}\mathfrak{m}_{0}$ we get
$f^{\prime}=b_{0}^{\prime}+a_{1}^{\prime}\mathfrak{m}_{0}+a_{1}a_{2}\mathfrak{m}_{0}\mathfrak{m}_{1}$,
so induction yields $F_{n}\in K[Y_{0},\dots,Y_{n}]$ with $\deg F_{n}\leqslant
n+1$ and $f^{(n)}=F_{n}(\mathfrak{m}_{0},\dots,\mathfrak{m}_{n})$. Thus for
$P\in K\\{Y\\}^{\neq}$ of order at most $r$ we have $P(f)=\sum_{\bm{i}\in
I}a_{\bm{i}}\mathfrak{m}_{0}^{i_{0}}\dots\mathfrak{m}_{r}^{i_{r}}$, where $I$
is a nonempty finite set of indices
$\bm{i}=(i_{0},\dots,i_{r})\in\mathbb{N}^{1+r}$. Note that by the proof of
Lemma 5.1, the family $(\beta_{n})$ is $\mathbb{Q}$-linearly independent over
$\Gamma$. Hence
$v_{L}\big{(}P(f)\big{)}\in\Gamma+\sum_{n}\mathbb{N}\beta_{n}$, which proves
(i).

By the proof of Lemma 5.1, we also have

$\beta_{n}^{\dagger}\ =\
\psi_{L}\big{(}v_{L}(f_{n}-b_{n})-\alpha_{n+1}\big{)}\ =\
v_{L}(f_{n+1}-b_{n+1})\ =\ \beta_{n+1}+\alpha_{n+2}\ \notin\ \Gamma.$

Thus the family $(\beta_{n}^{\dagger})$ is $\mathbb{Q}$-linearly independent
over $\Gamma$, since the family $(\beta_{n})$ is, proving (ii).

Note that (iii) follows from (i) and (ii). From (ii), we get
$[\beta_{n}]\notin[\Gamma]$ and $[\beta_{m}]\neq[\beta_{n}]$ for all $m\neq
n$, so (iv) now follows from (i). Finally, (v) follows from (ii) and Lemma
4.4. ∎

###  5.B. Further consequences in the ordered setting

Now we develop further the results of the previous subsection in the
pre-$H$-field setting. In this subsection, $K$ and $L$ are pre-$H$-fields with
small derivation. Suppose that $K$ is $\operatorname{d}$-henselian and has
exponential integration, and that $\Gamma$ is divisible. Suppose that $L$ is
an extension of $K$ with $\bm{k}_{L}=\bm{k}$, and that there is no $y\in
L\setminus K$ such that $K\langle y\rangle$ is an immediate extension of $K$.
Let $f\in L\setminus K$ with $\Gamma^{<}$ cofinal in $\Gamma_{K\langle
f\rangle}^{<}$, 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 5.1. As before, we also set $\beta_{n}\coloneqq
v_{L}(f_{n}-b_{n})-\alpha_{n+1}$. Note that since $K$ is a pre-$H$-field with
small derivation and nontrivial induced derivation on $\bm{k}$, it has gap
$0$, and so does $L$.

######  Lemma 5.5.

Suppose $M$ 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
$g_{1}\coloneqq(g-b_{0})^{\dagger}$ realizes the same cut in $K$ as $f_{1}$.

###### Proof.

Let $\alpha\in\Gamma$ and $b\in K$. We first claim that

$v_{L}(f-b)<\alpha\iff v_{M}(g-b)<\alpha\qquad\text{and}\qquad
v_{L}(f-b)>\alpha\iff v_{M}(g-b)>\alpha.$

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 thus
$v_{M}(g-b)\leqslant\alpha$. By the cofinality assumption, take
$\delta\in\Gamma$ with $v_{L}(f-b)<\delta<\alpha$, and then the same argument
yields $v_{M}(g-b)\leqslant\delta<\alpha$. 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 $\bm{k}=\bm{k}_{L}$. From the convexity of $\mathcal{O}_{K\langle
f\rangle}$ we obtain $|u|a/2<|f-b|<2|u|a$, so $|u|a/2<|g-b|<2|u|a$, and thus
$v_{M}(g-b)=va=\alpha$, completing the proof of the claim.

By the claim above and the fact that $v_{L}(f-b_{0})\notin\Gamma$, we get
$v_{M}(g-b_{0})\notin\Gamma$. This yields $v_{M}(g-b_{0})=\max v_{M}(g-K)$, as
otherwise we would have $b\in K$ with $v_{M}(g-b)>v_{M}(g-b_{0})$, so
$v_{M}(g-b_{0})=v(b-b_{0})\in\Gamma$. It also follows that
$(g-b_{0})^{\dagger}\notin K$, as otherwise $(g-b_{0})^{\dagger}=b^{\dagger}$
for some $b\in K^{\times}$, so $v_{M}(g-b_{0})=vb\in\Gamma$.

Finally, we show that $(g-b_{0})^{\dagger}$ realizes the same cut in $K$ as
$(f-b_{0})^{\dagger}$. By replacing $f$, $g$, and $b_{0}$ with $-f$, $-g$, and
$-b_{0}$ if necessary, we may assume that $f>b_{0}$, so $g>b_{0}$. First,
suppose 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
$s\coloneqq(f-b_{0})/\phi$. Then we have $s>0$ and
$s^{\dagger}=(f-b_{0})^{\dagger}-h<0$. By [3, Lemma 10.5.2(i)],
$v_{L}(s)\geqslant 0$, but since $v_{L}(f-b_{0})\notin\Gamma$, we get
$v_{L}(s)>0$; in particular, $0<s<1$ (see [3, Lemma 3.5.11]). Similarly,
$h<(g-b_{0})^{\dagger}$ gives $t\coloneqq(g-b_{0})/\phi>0$ and
$t^{\dagger}>0$, so $v_{M}(t)<0$; in particular, $t>1$. Putting this together
yields

$f\ =\ b_{0}+\phi s\ <\ b_{0}+\phi\ \qquad\text{and}\ \qquad b_{0}+\phi\ <\
b_{0}+\phi t\ =\ g,$

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 handled in the same fashion. ∎

######  Proposition 5.6.

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 $f\mapsto g$.

###### Proof.

Define $g_{0}\coloneqq g$ and $g_{n+1}\coloneqq(g_{n}-b_{n})^{\dagger}$ for
all $n$, so by the previous lemma $g_{n}\in M\setminus K$ realizes the same
cut in $K$ as $f_{n}$, and in particular $v_{M}(g_{n}-b_{n})\notin\Gamma$ for
all $n$. Then using the same argument as in the proof of Lemma 5.1, we have
that $v_{M}(g_{0}-b_{0}),v_{M}(g_{1}-b_{1}),\dots$ are $\mathbb{Q}$-linearly
independent over $\Gamma$. Set $\beta_{n}^{*}\coloneqq
v_{M}(g_{n}-b_{n})-\alpha_{n+1}$ and
$\mathfrak{m}_{n}^{*}\coloneqq(g_{n}-b_{n})/a_{n+1}$, so
$v_{M}(\mathfrak{m}_{n}^{*})=\beta_{n}^{*}$ and the family $(\beta_{n}^{*})$
is $\mathbb{Q}$-linearly independent over $\Gamma$. Note that since $f_{n}$
and $g_{n}$ realize the same cut in $K$, so do $\mathfrak{m}_{n}$ and
$\mathfrak{m}_{n}^{*}$, and hence $\beta_{n}$ and $\beta_{n}^{*}$ realize the
same cut in $\Gamma$. From the proof of Lemma 5.4 we have
$F_{n}(Y_{0},\dots,Y_{n})\in K[Y_{0},\dots,Y_{n}]$ with $\deg F_{n}\leqslant
n+1$ and $g^{(n)}=F_{n}(\mathfrak{m}_{0}^{*},\dots,\mathfrak{m}_{n}^{*})$. For
$P\in K\\{Y\\}^{\neq}$ of order at most $r$ we thus get $P(g)=\sum_{\bm{i}\in
I}a_{\bm{i}}\mathfrak{m}_{0}^{*i_{0}}\cdots\mathfrak{m}_{r}^{*i_{r}}$, where
$I$ is the same nonempty finite index set and $a_{\bm{i}}$ are the same
coefficients as in the proof of Lemma 5.4. Since the family $(\beta_{n}^{*})$
is $\mathbb{Q}$-linearly independent over $\Gamma$, we have that
$v_{M}\big{(}P(g)\big{)}\in\Gamma+\sum_{n}\mathbb{N}\beta_{n}^{*}$. The rest
of the proof of Lemma 5.4 now goes through replacing $f_{n}$ with $g_{n}$ and
$\beta_{n}$ with $\beta_{n}^{*}$.

From this we obtain an ordered abelian group isomorphism
$j\colon\Gamma_{K\langle f\rangle}\to\Gamma_{K\langle g\rangle}$ over $\Gamma$
with $\beta_{n}\mapsto\beta_{n}^{*}$. Using the expressions for $P(f)$ and
$P(g)$, we get $j\big{(}v_{L}(P(f))\big{)}=v_{M}(P(g))$ for all $P\in
K\\{Y\\}^{\neq}$, so we have a valued differential field embedding $K\langle
f\rangle\to M$ over $K$ with $f\mapsto g$. By the above and since
$\mathfrak{m}_{n}$ and $\mathfrak{m}_{n}^{*}$ have the same sign, $P(f)>0\iff
P(g)>0$ for all $P\in K\\{Y\\}^{\neq}$, so this is in fact an ordered valued
differential field embedding, as desired. ∎

###  5.C. The non-cofinal case

In the previous subsection we assumed that $\Gamma^{<}$ was cofinal in
$\Gamma_{K\langle f\rangle}^{<}$, and now we turn to the other case. In this
subsection, $K$ and $L$ are pre-$H$-fields with gap $0$ and $L$ is an
extension of $K$.

######  Lemma 5.7.

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 $f_{0}\coloneqq f$ and $f_{n+1}\coloneqq f_{n}^{\dagger}$, and let
$\beta_{n}\coloneqq v_{L}(f_{n})\in\Gamma_{L}$. By [3, Lemma 9.2.10(iv)],

$[\Gamma^{\neq}]>[\beta_{0}]>[\beta_{1}]>[\beta_{2}]>\cdots>[0].$

In particular, $[\beta_{n}]\notin[\Gamma]$ for all $n$ and the family
$(\beta_{n})$ is $\mathbb{Q}$-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 $K$ [3,
Corollary 3.1.11]. By the same argument as in Lemma 5.4 with $f_{n}$ in place
of $\mathfrak{m}_{n}$ (i.e., with $b_{n}=0$ and $a_{n}=1$), one shows that for
any $P\in K\\{Y\\}^{\neq}$ of order at most $r$, we have $P(f)=\sum_{\bm{i}\in
I}a_{\bm{i}}f_{0}^{i_{0}}\dots f_{r}^{i_{r}}$, where $I$ is a nonempty finite
set of indices $\bm{i}=(i_{0},\dots,i_{r})\in\mathbb{N}^{1+r}$. In particular,
$\Gamma_{K\langle f\rangle}=\Gamma\oplus\bigoplus_{n}\mathbb{Z}\beta_{n}$.

Set $g_{0}\coloneqq g$, $g_{n+1}\coloneqq g_{n}^{\dagger}$, and
$\beta_{n}^{*}\coloneqq v_{M}(g_{n})\in\Gamma_{M}$. The same argument yields
that $g$ is $\operatorname{d}$-transcendental over $K$ and
$P(g)=\sum_{\bm{i}\in I}a_{\bm{i}}g_{0}^{i_{0}}\dots g_{r}^{i_{r}}$, where $I$
is the same set of indices as in $P(f)$ and $a_{\bm{i}}$ 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 $P(g)$,
$j\big{(}v_{L}(P(f))\big{)}=v_{M}(P(g))$, 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 by
[3, Lemma 10.5.2(i)], $f_{n}>0$ and $g_{n}>0$ for all $n$, so $P(f)>0\iff
P(g)>0$. ∎

##  6\. Extending the constant field

######  Assumption.

In this section, $K$ is asymptotic with small derivation.

Since $C\subseteq\mathcal{O}$, $C$ maps injectively into $\bm{k}$ under the
residue field map, and hence into $C_{\bm{k}}$. We say that $K$ is _residue
constant closed_ if $K$ is henselian and $C$ maps onto $C_{\bm{k}}$, that is,
$\operatorname{res}(C)=C_{\bm{k}}$. 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 $K$.

######  Proposition 6.1.

Suppose that $K$ is pre-$\operatorname{d}$-valued of $H$-type with
$\sup\Psi=0$. Then $K$ has a residue constant closure that is an immediate
extension of $K$.

###### Proof.

Recall from §4 that $\sup\Psi=0$ is equivalent to
$(\Gamma^{>})^{\prime}=\Gamma^{>}$. Also note that if $L$ is an immediate
asymptotic extension of $K$, then it is $H$-asymptotic, satisfies
$\Psi=\Psi_{L}$, so $\sup\Psi_{L}=0$, and is pre-$\operatorname{d}$-valued by
[3, Corollary 10.1.17].

Build a tower of immediate asymptotic extensions of $K$ as follows. Set
$K_{0}\coloneqq K$. If $K_{\lambda}$ is not henselian, set
$K_{\lambda+1}\coloneqq K_{\lambda}^{\operatorname{h}}$, the henselization of
$K_{\lambda}$, which as an algebraic extension of $K_{\lambda}$ is asymptotic
by [3, Proposition 9.5.3]. If $K_{\lambda}$ is residue constant closed, we are
done. So suppose that $K_{\lambda}$ is henselian but not residue constant
closed and take $u\in K_{\lambda}$ with $u\asymp 1$, $u^{\prime}\prec 1$, and
$u^{\prime}\notin\der\cao_{K_{\lambda}}$. Let $y$ be transcendental over
$K_{\lambda}$ and equip $K_{\lambda+1}\coloneqq K_{\lambda}(y)$ with the
unique derivation extending that of $K_{\lambda}$ such that
$y^{\prime}=u^{\prime}$. Then by [3, Lemma 10.2.5(iii)]
$\\{v(u^{\prime}-a^{\prime}):a\in\cao_{K_{\lambda}}\\}$ has no maximum, so by
[3, Lemma 10.2.4] we can equip $K_{\lambda+1}$ with the unique valuation
making it an $H$-asymptotic extension of $K_{\lambda}$ with $y\not\asymp 1$;
with this valuation, $y\prec 1$ and $K_{\lambda+1}$ is an immediate extension
of $K_{\lambda}$. If $\lambda$ is a limit ordinal, set
$K_{\lambda}\coloneqq\bigcup_{\rho<\lambda}K_{\rho}$. Since each extension is
immediate, by Zorn’s lemma we may take a maximal such tower
$(K_{\lambda})_{\lambda\leqslant\mu}$.

It is clear that $K_{\mu}$ is residue constant closed, and we show that it
also has the desired universal property. Let $M$ be an $H$-asymptotic
extension of $K$ with small derivation that is residue constant closed, and
let $\lambda<\mu$ and $i\colon K_{\lambda}\to M$ be an embedding. It suffices
by induction to extend $i$ uniquely to an embedding $K_{\lambda+1}\to M$. If
$K_{\lambda+1}=K_{\lambda}^{\operatorname{h}}$, then this follows from the
universal property of the henselization. Now suppose that
$K_{\lambda+1}=K_{\lambda}(y)$ with $y$ and $u$ as above. Take the unique
$c\in C_{M}$ with $c\sim i(u)$ and set $z\coloneqq i(u)-c$. Then
$z^{\prime}=i(u)^{\prime}$ and $z\prec 1$, so by the remarks after [3, Lemma
10.2.4], $z$ is transcendental over $i(K_{\lambda})$, and thus mapping
$y\mapsto z$ yields a differential field embedding $K_{\lambda+1}\to M$
extending $i$. By the uniqueness of [3, Lemma 10.2.4], this is a valued
differential field embedding. Finally, if $i$ extends to an embedding with
$y\mapsto z_{1}\in M$, then $i(u)-z_{1}\in C_{M}$ and $i(u)-z_{1}\sim i(u)\sim
c$, so $z_{1}=z$. ∎

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 by [3, Lemma
10.5.8].

######  Lemma 6.2.

Suppose that $K$ is residue constant closed. Then the algebraic closure
$K^{\operatorname{ac}}$ of $K$ is residue constant closed. If $K$ is
additionally an ordered field with convex valuation ring, then
$K^{\operatorname{rc}}$ is residue constant closed.

###### Proof.

First, note that an algebraic extension of a henselian valued field is
henselian [3, Corollary 3.3.12]. Let $u\in K^{\operatorname{ac}}$ with
$u\asymp 1$ and $u^{\prime}\prec 1$; we need to show that there is $c\in
C_{K^{\operatorname{ac}}}=C^{\operatorname{ac}}$ with $c\sim u$. We have that
$\overline{u}\in C_{\operatorname{res}(K^{\operatorname{ac}})}$ is algebraic
over $\operatorname{res}(K)$, so it is algebraic over
$C_{\operatorname{res}(K)}$ [3, Lemma 4.1.2]. Take a monic $P\in C[X]$, say of
degree $n$, such that $\overline{P}\in C_{\operatorname{res}(K)}[X]$ is the
minimum polynomial of $\overline{u}$ over $C_{\operatorname{res}(K)}$. Then
$P=\prod_{i=1}^{n}(X-c_{i})$ with $c_{1},\dots,c_{n}\in
C^{\operatorname{ac}}$, hence
$\overline{P}=\prod_{i=1}^{n}(X-\overline{c_{i}})$, so we have $i$ with
$1\leqslant i\leqslant n$ and $\overline{u}=\overline{c_{i}}$, and thus $u\sim
c_{i}$. The second statement is proved similarly. ∎

##  7\. Differential-Hensel-Liouville closures

In this section we construct differential-Hensel-Liouville closures (Theorem
7.16) in analogy with the Newton-Liouville closures of [3, §14.5] and prove
that they are unique (Corollary 7.18). To do this, we first construct
extensions that are real closed, have exponential integration, and satisfy an
embedding property (Corollary 7.11, Lemma 7.14), in analogy with the Liouville
closures of [3, §10.6]; some preliminaries are adapted from [3, §10.4–10.6].
Combining this with the residue constant closures from the previous section,
we then construct extensions that are residue constant closed, are real
closed, have exponential integration, and satisfy an embedding property
(Theorem 7.15).

######  Assumption.

In this section, $K$ is a pre-$H$-field.

###  7.A. Adjoining exponential integrals

Suppose that $s\in K\setminus(K^{\times})^{\dagger}$ and $f$ is transcendental
over $K$. We give $K(f)$ the unique derivation extending that of $K$ with
$f^{\dagger}=s$. In the first lemma, $K$ need only be an ordered differential
field.

######  Lemma 7.1.

If $K$ is real closed and $K(f)$ can be ordered making it an ordered field
extension of $K$, then $C_{K(f)}=C$.

###### Proof.

This follows from [3, Lemma 4.6.11 and Corollary 4.6.12]. ∎

In the next two lemmas, $K$ is just a valued differential field, and need not
be ordered. The first is based on [3, Lemma 10.4.2].

######  Lemma 7.2.

Suppose that $K$ has small derivation and
$\bm{k}=(\bm{k}^{\times})^{\dagger}$. Let $K(f)$ have a valuation that makes
it an extension of $K$ with $\Gamma_{K(f)}=\Gamma$ and
$\der\mathcal{O}_{K(f)}\subseteq\mathcal{O}_{K(f)}$. Then $s-a^{\dagger}\prec
1$ for some $a\in K^{\times}$.

###### Proof.

Since $vf\in\Gamma$, there is $b\in K^{\times}$ with $g\coloneqq f/b\asymp 1$.
Then $s-b^{\dagger}=g^{\dagger}\asymp g^{\prime}\preccurlyeq 1$. If
$s-b^{\dagger}\prec 1$, set $a\coloneqq b$. If $s-b^{\dagger}\asymp 1$, since
$\bm{k}=(\bm{k}^{\times})^{\dagger}$, we have $u\asymp 1$ in $K^{\times}$ with
$s-b^{\dagger}\sim u^{\dagger}$. Then set $a\coloneqq bu$. ∎

The last part of the argument also yields the following useful fact.

######  Lemma 7.3.

Suppose that $K$ has small derivation and
$\bm{k}=(\bm{k}^{\times})^{\dagger}$. If $s-a^{\dagger}\succcurlyeq 1$ for all
$a\in K^{\times}$, then $s-a^{\dagger}\succ 1$ for all $a\in K^{\times}$.

Now we return to the situation that $K$ is a pre-$H$-field.

######  Lemma 7.4 ([3, Lemma 10.5.18]).

Suppose that $K$ is henselian and $vs\in(\Gamma^{>})^{\prime}$. Then there is
a unique valuation on $K(f)$ making it an $H$-asymptotic extension of $K$ with
$f\sim 1$. With this valuation, $K(f)$ is an immediate extension of $K$, so
there is a unique ordering of $K(f)$ making it a pre-$H$-field extension of
$K$ by [3, Lemma 10.5.8].

Here is a pre-$H$-field version of [3, Lemma 10.5.20] with the same proof.

######  Lemma 7.5.

Suppose that $K$ is real closed, $s<0$, and
$v(s-a^{\dagger})\in\Psi^{\downarrow}$ for all $a\in K^{\times}$. Then there
is a unique pair of a field ordering and a valuation on $L\coloneqq K(f)$
making it a pre-$H$-field extension of $K$ with $f>0$. Moreover, we have:

  1. (i)

$vf\notin\Gamma$, $\Gamma_{L}=\Gamma\oplus\mathbb{Z}vf$, $f\prec 1$;

  2. (ii)

$\Psi$ is cofinal in $\Psi_{L}\coloneqq\psi_{L}(\Gamma_{L}^{\neq})$;

  3. (iii)

a gap in $K$ remains a gap in $L$;

  4. (iv)

if $L$ has a gap not in $\Gamma$, then $[\Gamma_{L}]=[\Gamma]$;

  5. (v)

$\bm{k}_{L}=\bm{k}$.

###  7.B. Exponential integration closures

Let $E$ be a differential field. We call a differential field extension $F$ of
$E$ an _exponential integration extension_ of $E$ (_expint-extension_ for
short) if $C_{F}$ is algebraic over $C_{E}$ and for every $a\in F$ there are
$t_{1},\dots,t_{n}\in F^{\times}$ with $a\in E(t_{1},\dots,t_{n})$ such that
for $i=1,\dots,n$, either $t_{i}$ is algebraic over $E(t_{1},\dots,t_{i-1})$
or $t_{i}^{\dagger}\in E(t_{1},\dots,t_{i-1})$. In particular, any expint-
extension is $\operatorname{d}$-algebraic. The following is routine.

######  Lemma 7.6.

Let $E\subseteq F\subseteq M$ be a chain of differential field extensions.

  1. (i)

If $M$ is an expint-extension of $E$, then $M$ is an expint-extension of $F$.

  2. (ii)

If $M$ is an expint-extension of $F$ and $F$ is an expint-extension of $E$,
then $M$ is an expint-extension of $E$.

Minor modifications to the proof of [3, Lemma 10.6.8] yield the following.

######  Lemma 7.7.

If $F$ is an expint-extension of $E$, then $|F|=|E|$.

Now suppose that $E$ is an ordered differential field. We call $E$
_exponential integration closed_ (_expint-closed_ for short) if it is real
closed and has exponential integration. We call an ordered differential field
extension $F$ of $E$ an _exponential integration closure_ (_expint-closure_
for short) of $E$ if it is an expint-extension of $E$ that is expint-closed.

The next observation has the same proof as [3, Lemma 10.6.9].

######  Lemma 7.8.

If $E$ is expint-closed, then $E$ has no proper expint-extension with the same
constants.

######  Assumption.

For the rest of this subsection, suppose that $K$ has gap $0$.

From this assumption it follows that $(\Gamma^{>})^{\prime}=\Gamma^{>}$ and
$\Psi^{\downarrow}=\Gamma^{<}$ (see §4). Recall from §2 how we construe the
real closure of $K$ as a pre-$H$-field extension of $K$ with gap $0$.

######  Definition.

We call a strictly increasing chain $(K_{\lambda})_{\lambda\leqslant\mu}$ of
pre-$H$-fields with gap $0$ an _expint-tower on $K$_ if:

  1. (i)

$K_{0}=K$;

  2. (ii)

if $\lambda$ is a limit ordinal, then
$K_{\lambda}=\bigcup_{\rho<\lambda}K_{\rho}$;

  3. (iii)

if $\lambda<\lambda+1\leqslant\mu$, then either:

    1. (a)

$K_{\lambda}$ is not real closed and $K_{\lambda+1}$ is the real closure of
$K_{\lambda}$; or

    2. (b)

$K_{\lambda}$ is real closed and $K_{\lambda+1}=K_{\lambda}(y_{\lambda})$ with
$y_{\lambda}\notin K_{\lambda}$ satisfying either:

      1. (b1)

$y_{\lambda}^{\dagger}=s_{\lambda}\in K_{\lambda}$ with $y_{\lambda}\sim 1$,
$s_{\lambda}\prec 1$, and $s_{\lambda}\neq a^{\dagger}$ for all $a\in
K_{\lambda}^{\times}$; or

      2. (b2)

$y_{\lambda}^{\dagger}=s_{\lambda}\in K_{\lambda}$ with $s_{\lambda}<0$,
$y_{\lambda}>0$, and $s_{\lambda}-a^{\dagger}\succ 1$ for all $a\in
K_{\lambda}^{\times}$.

Given such a tower, we call $K_{\mu}$ its _top_ and set $C_{\lambda}\coloneqq
C_{K_{\lambda}}$ and $\bm{k}_{\lambda}\coloneqq\bm{k}_{K_{\lambda}}$ for
$\lambda\leqslant\mu$.

######  Lemma 7.9.

Let $(K_{\lambda})_{\lambda\leqslant\mu}$ be an expint-tower on $K$. Then:

  1. (i)

$K_{\mu}$ is an expint-extension of $K$;

  2. (ii)

$C_{\mu}$ is the real closure of $C$ if $\mu>0$;

  3. (iii)

$\bm{k}_{\mu}$ is the real closure of $\bm{k}$ if $\mu>0$;

  4. (iv)

$|K_{\lambda}|=|K|$, and hence $\mu<|K|^{+}$.

###### Proof.

For (i), go by induction on $\lambda\leqslant\mu$. The main thing to check is
the condition on the constant fields. If $\lambda=0$ or $\lambda$ is a limit
ordinal, this is clear. If $K_{\lambda+1}$ is the real closure of
$K_{\lambda}$, then $C_{\lambda+1}$ is the real closure of $C_{\lambda}$. If
$K_{\lambda}$ is real closed and $K_{\lambda+1}$ is as in (b) above, then
$C_{\lambda+1}=C_{\lambda}$ by Lemma 7.1.

For (ii), $C_{1}$ is the real closure of $C$, and then $C_{\lambda}=C_{1}$ for
all $\lambda\geqslant 1$ as in the proof of (i).

For (iii), $\bm{k}_{1}$ is the real closure of $\bm{k}$, and then
$\bm{k}_{\lambda}=\bm{k}_{1}$ for all $\lambda\geqslant 1$ by the uniqueness
of Lemma 7.4 and Lemma 7.5.

Finally, (iv) follows from (i) and Lemma 7.7. ∎

######  Lemma 7.10.

Let $L$ be the top of a maximal expint-tower on $K$ such that $\bm{k}_{L}$ has
exponential integration. Then $L$ is expint-closed, and hence an expint-
closure of $K$.

###### Proof.

Suppose that $L$ is not expint-closed. If $L$ is not real closed, then its
real closure is a proper pre-$H$-field extension of $L$ with gap $0$, so we
could extend the expint-tower. We are left with the case that $L$ is real
closed and we have $s\in L\setminus(L^{\times})^{\dagger}$. In particular, $L$
is henselian and $\Gamma$ is divisible. We may assume that $s<0$. Take $f$
transcendental over $L$ with $f^{\dagger}=s$.

First suppose that $s-a^{\dagger}\prec 1$ for some $a\in L^{\times}$. Then
taking such an $a$ and replacing $f$ and $s$ by $f/a$ and $s-a^{\dagger}$, we
arrange that $s\prec 1$. Giving $L(f)$ the valuation and ordering from Lemma
7.4 makes it a pre-$H$-field extension of $L$ with gap $0$ of type (b1).

Now suppose that $s-a^{\dagger}\succcurlyeq 1$ for all $a\in L^{\times}$. By
Lemma 7.3, $s-a^{\dagger}\succ 1$ for all $a\in L^{\times}$. Then giving
$L(f)$ the ordering and valuation from Lemma 7.5 makes it a pre-$H$-field
extension of $L$ with gap $0$ of type (b2).

Thus $L$ is expint-closed, and hence an expint-closure of $K$ by Lemma 7.9(i).
∎

######  Corollary 7.11.

Suppose that $\bm{k}$ is expint-closed. Then $K$ has an expint-closure that is
a pre-$H$-field extension of $K$ with gap $0$. If $K$ is residue constant
closed, then $K$ has a residue constant closed expint-closure that is a
pre-$H$-field extension of $K$ with gap $0$.

###### Proof.

By Lemma 7.9(iv), Zorn gives a maximal expint-tower
$(K_{\lambda})_{\lambda\leqslant\mu}$ on $K$. Then $\bm{k}_{\mu}=\bm{k}$ by
Lemma 7.9(iii), and hence $K_{\mu}$ is an expint-closure of $K$ by Lemma 7.10.
If $K$ is residue constant closed, then so is $K_{\mu}$ since it is henselian
and $C$ maps onto $C_{\bm{k}}=C_{\bm{k}_{\mu}}$. ∎

######  Lemma 7.12.

Let $M$ be a residue constant closed, expint-closed pre-$H$-field extension of
$K$ with gap $0$. Suppose that $(K_{\lambda})_{\lambda\leqslant\mu}$ is an
expint-tower on $K$ in $M$ (i.e., each $K_{\lambda}$ is a pre-$H$-subfield of
$M$) and maximal in $M$ (i.e., it cannot be extended to an expint-tower
$(K_{\lambda})_{\lambda\leqslant\mu+1}$ on $K$ in $M$) such that
$\bm{k}_{\mu}$ has exponential integration. Then
$(K_{\lambda})_{\lambda\leqslant\mu}$ is a maximal expint-tower on $K$.

###### Proof.

Since $M$ is real closed, $K_{\mu}$ must be real closed by maximality in $M$.
So supposing $(K_{\lambda})_{\lambda\leqslant\mu}$ is not a maximal expint-
tower on $K$, we have $s_{\mu}\in K_{\mu}$ such that $s_{\mu}\neq a^{\dagger}$
for all $a\in K_{\mu}^{\times}$; we may assume that $s_{\mu}<0$. Since $M$ is
expint-closed, we have $y_{\mu}\in M$ with $y_{\mu}^{\dagger}=s_{\mu}$; we may
assume that $y_{\mu}>0$.

First suppose that $s_{\mu}-a^{\dagger}\succcurlyeq 1$ for all $a\in
K_{\mu}^{\times}$, so actually $s_{\mu}-a^{\dagger}\succ 1$ for all $a\in
K_{\mu}^{\times}$ by Lemma 7.3. Thus setting $K_{\mu+1}\coloneqq
K_{\mu}(y_{\mu})$ yields an extension of $(K_{\lambda})_{\lambda\leqslant\mu}$
in $M$ of type (b2).

Now suppose that $s_{\mu}-a^{\dagger}\prec 1$ for some $a\in
K_{\mu}^{\times}$. Taking such an $a$ and replacing $s_{\mu}$ and $y_{\mu}$ by
$s_{\mu}-a^{\dagger}$ and $y_{\mu}/a$, we may assume that $s_{\mu}\prec 1$.
Since $M$ has gap $0$, we have $y_{\mu}\asymp 1$ and so
$y_{\mu}^{\prime}\asymp s_{\mu}\prec 1$. That is, $\overline{y_{\mu}}\in
C_{\operatorname{res}(M)}$, so we have $c\in C_{M}$ with $y_{\mu}\sim c$.
Replacing $y_{\mu}$ by $y_{\mu}/c$, we obtain the desired extension of
$(K_{\lambda})_{\lambda\leqslant\mu}$ in $M$ of type (b1). ∎

This comment is not used later, but in the above lemma, we can replace the
assumption that $M$ is residue constant closed (so
$C_{\operatorname{res}(M)}=\operatorname{res}(C_{M})$) with
$C_{\operatorname{res}(M)}=C_{\operatorname{res}(K)}$. In the final argument,
instead of $c\in C_{M}$ we have $u\in K$ with $u\asymp 1$ and $u^{\prime}\prec
1$, so we also replace $s_{\mu}$ with $s_{\mu}-u^{\dagger}$.

######  Corollary 7.13.

Suppose that $L$ is an expint-closed pre-$H$-field extension $L$ of $K$.

  1. (i)

If $L$ is an expint-closure of $K$, then no proper differential subfield of
$L$ containing $K$ is expint-closed.

  2. (ii)

Suppose that $\bm{k}$ is expint-closed, and that $L$ has gap $0$ and is
residue constant closed. If no proper differential subfield of $L$ containing
$K$ is expint-closed, then $L$ is an expint-closure of $K$.

###### Proof.

For (i), if $L$ is an expint-closure of $K$, then no proper differential
subfield of $L$ containing $K$ is expint-closed by Lemmas 7.6 and 7.8.

For (ii), suppose that no proper differential subfield of $L$ containing $K$
is expint-closed. Take an expint-tower $(K_{\lambda})_{\lambda\leqslant\mu}$
on $K$ in $L$ that is maximal in $L$. Since $\bm{k}$ is real closed,
$\bm{k}_{\mu}=\bm{k}$, and hence $\bm{k}_{\mu}$ has exponential integration.
Then $(K_{\lambda})_{\lambda\leqslant\mu}$ is a maximal expint-tower on $K$ by
Lemma 7.12. By Lemma 7.10, $K_{\mu}$ is expint-closed and hence equal to $L$.
∎

######  Lemma 7.14.

Let $(K_{\lambda})_{\lambda\leqslant\mu}$ be an expint-tower on $K$. Then any
embedding of $K$ into a residue constant closed, expint-closed pre-$H$-field
extension $M$ of $K$ with gap $0$ extends to an embedding of $K_{\mu}$. If $K$
is residue constant closed and $\bm{k}$ is expint-closed, then any two residue
constant closed expint-closures of $K$ that are pre-$H$-field extensions of
$K$ with gap $0$ are isomorphic over $K$.

###### Proof.

Let $M$ be a residue constant closed, expint-closed pre-$H$-field with gap
$0$. We prove that for $\lambda<\mu$ any embedding $K_{\lambda}\to M$ extends
to an embedding $K_{\lambda+1}\to M$, which yields the result by induction.
Suppose that $i\colon K_{\lambda}\to M$ is an embedding. If $K_{\lambda+1}$ is
the real closure of $K_{\lambda}$, then we may extend $i$ to $K_{\lambda+1}$.

So suppose that $K_{\lambda}$ is real closed and we have $s_{\lambda}\in
K_{\lambda}$ and $y_{\lambda}\in K_{\lambda+1}\setminus K_{\lambda}$ with
$K_{\lambda+1}=K_{\lambda}(y_{\lambda})$, $y_{\lambda}^{\dagger}=s_{\lambda}$,
$y_{\lambda}\sim 1$, $s_{\lambda}\prec 1$, and $s_{\lambda}\neq a^{\dagger}$
for all $a\in K_{\lambda}^{\times}$. Take $z\in M$ with
$z^{\dagger}=i(s_{\lambda})$. Hence $z\asymp 1$ and $\overline{z}\in
C_{\operatorname{res}(M)}$, so we have $c\in C_{M}$ with $z\sim c$. By the
uniqueness of Lemma 7.4, we may extend $i$ to an embedding of
$K_{\lambda}(y_{\lambda})$ into $M$ sending $y_{\lambda}$ to $z/c$.

Now suppose that $K_{\lambda}$ is real closed and we have $s_{\lambda}\in
K_{\lambda}$ and $y_{\lambda}\in K_{\lambda+1}\setminus K_{\lambda}$ with
$K_{\lambda+1}=K_{\lambda}(y_{\lambda})$, $y_{\lambda}^{\dagger}=s_{\lambda}$,
$s_{\lambda}<0$, $y_{\lambda}>0$, and $s_{\lambda}-a^{\dagger}\succ 1$ for all
$a\in K^{\times}_{\lambda}$. Take $z\in M$ with $z^{\dagger}=i(s_{\lambda})$;
we may assume that $z>0$. Then by the uniqueness of Lemma 7.5, we can extend
$i$ to an embedding of $K_{\lambda}(y_{\lambda})$ into $M$ sending
$y_{\lambda}$ to $z$.

The second statement follows from the first by Corollary 7.13(i) and the proof
of Corollary 7.11. ∎

Combining these results with Proposition 6.1 yields the following.

######  Theorem 7.15.

Suppose that $\bm{k}$ is expint-closed. Then $K$ has a pre-$H$-field extension
$L$ with gap $0$ such that:

  1. (i)

$L$ is a residue constant closed, expint-closed extension of $K$;

  2. (ii)

$L$ embeds over $K$ into any residue constant closed, expint-closed
pre-$H$-field extension of $K$ with gap $0$;

  3. (iii)

$L$ has no proper differential subfield containing $K$ that is residue
constant closed and expint-closed.

###### Proof.

By Proposition 6.1, let $K_{0}$ be the residue constant closure of $K$. Taking
the top of a maximal expint-tower on $K_{0}$ yields an expint-closure $L$ of
$K_{0}$ that is residue constant closed as in Corollary 7.11. For (ii), let
$M$ be a pre-$H$-field extension of $K$ with gap $0$ that is residue constant
closed and expint-closed. Then $K_{0}$ embeds uniquely into $M$ over $K$, so
by Lemma 7.14 we can extend this to an embedding of $L$. For (iii), suppose
that $L_{0}\supseteq K$ is a differential subfield of $L$ that is residue
constant closed and expint-closed. Then $L_{0}\supseteq K_{0}$, and hence
$L_{0}=L$ by Corollary 7.13(i). ∎

Let $L$ be as above. Then any pre-$H$-field extension of $K$ with gap $0$
satisfying (i) and (ii) is isomorphic to $L$ over $K$ by (iii). Also, $L$ is a
Liouville extension of $K$ in the sense of [3, §10.6],111The definition is
similar to that of expint-extensions in §7.B except that we also allow
$t_{i}^{\prime}\in E(t_{1},\dots,t_{i-1})$. since $K_{0}$ is a Liouville
extension of $K$ by construction and expint-extensions are Liouville
extensions.

###  7.C. Differential-Hensel-Liouville closures

######  Assumption.

We continue to assume in this subsection that the pre-$H$-field $K$ has gap
$0$.

######  Definition.

We call $K$ _differential-Hensel-Liouville closed_ (slightly shorter:
_$\operatorname{d}$ -Hensel-Liouville closed_) if it is
$\operatorname{d}$-henselian and expint-closed. We call a pre-$H$-field
extension $L$ of $K$ a _differential-Hensel-Liouville closure_ (slightly
shorter: _$\operatorname{d}$ -Hensel-Liouville closure_) of $K$ if it is
$\operatorname{d}$-Hensel-Liouville closed and embeds over $K$ into every
$\operatorname{d}$-Hensel-Liouville closed pre-$H$-field extension of $K$.

Note that, if $K$ is $\operatorname{d}$-henselian, then $K$ is also closed
under integration in the sense that is surjective by [3, Lemma 7.1.8], hence
the use of “Liouville” in the terms just defined.

To build $\operatorname{d}$-Hensel-Liouville closures, we use the fact that if
$F$ is an asymptotic valued differential field with small derivation and
linearly surjective differential residue field, then it has a (unique)
_differential-henselization_ (_$\operatorname{d}$ -henselization_ for short)
$F^{\operatorname{dh}}$ by [7, Theorem 3.7]. For such $F$,
$F^{\operatorname{dh}}$ is an immediate asymptotic
$\operatorname{d}$-algebraic extension of $F$ that embeds over $F$ into every
$\operatorname{d}$-henselian asymptotic extension of $F$; if $F$ is a
pre-$H$-field, then $F^{\operatorname{dh}}$ is too and embeds (as an _ordered_
valued differential field) into every $\operatorname{d}$-henselian
pre-$H$-field extension of $F$ by [3, Lemma 10.5.8].

######  Theorem 7.16.

Suppose that $\bm{k}$ is expint-closed and linearly surjective. Then $K$ has a
$\operatorname{d}$-Hensel-Liouville closure $K^{\operatorname{dhl}}$.

###### Proof.

We use below that any $\operatorname{d}$-henselian asymptotic field is residue
constant closed by [3, Lemma 9.4.10]. Define a sequence of pre-$H$-field
extensions of $K$ with gap $0$ as follows. Set $K_{0}\coloneqq K$. For
$n\geqslant 1$, if $n$ is odd, let $K_{n}$ be the
$\operatorname{d}$-henselization of $K_{n-1}$, and if $n$ is even, let $K_{n}$
be the expint-closure of $K_{n-1}$ from Corollary 7.11. Note that
$\bm{k}_{K_{n}}=\bm{k}$ for all $n$. We set
$K^{\operatorname{dhl}}\coloneqq\bigcup_{n}K_{n}$ and show that
$K^{\operatorname{dhl}}$ is a $\operatorname{d}$-Hensel-Liouville closure of
$K$.

Let $L$ be a pre-$H$-field extension of $K$ that is
$\operatorname{d}$-henselian and expint-closed. We show by induction on $n$
that we can extend any embedding $K_{n}\to L$ to an embedding $K_{n+1}\to L$,
so suppose that we have an embedding $i\colon K_{n}\to L$. If $n$ is even,
then $K_{n+1}$ is the $\operatorname{d}$-henselization of $K_{n}$, so we may
extend $i$ to an embedding $K_{n+1}\to L$. If $n$ is odd, then $K_{n}$ is
$\operatorname{d}$-henselian and $K_{n+1}$ is the expint-closure of $K_{n}$,
so we can extend $i$ to an embedding $K_{n+1}\to L$ by Lemma 7.14. ∎

Note that $K^{\operatorname{dhl}}$ is a $\operatorname{d}$-algebraic extension
of $K$ with the same residue field. In the next two results, adapted from [3,
§16.2], we show that $K^{\operatorname{dhl}}$ is the unique, up to isomorphism
over $K$, $\operatorname{d}$-Hensel-Liouville closure of $K$.

######  Lemma 7.17.

Suppose that $\bm{k}$ is expint-closed and linearly surjective. Let $i\colon
K^{\operatorname{dhl}}\to L$ be an embedding into a pre-$H$-field $L$ with gap
$0$ such that
$\operatorname{res}\big{(}i(K^{\operatorname{dhl}})\big{)}=\operatorname{res}(L)$.
Then

$i(K^{\operatorname{dhl}})\ =\ i(K)^{\operatorname{dalg}}\ \coloneqq\ \\{f\in
L:f\ \text{is $\operatorname{d}$-algebraic over}\ i(K)\\}.$

###### Proof.

We have $i(K^{\operatorname{dhl}})\subseteq i(K)^{\operatorname{dalg}}$ since
$K^{\operatorname{dhl}}$ is a $\operatorname{d}$-algebraic extension of $K$.
For the other direction, note that $i(K^{\operatorname{dhl}})$ is a
$\operatorname{d}$-henselian, expint-closed pre-$H$-subfield of
$i(K)^{\operatorname{dalg}}$, so
$i(K^{\operatorname{dhl}})=i(K)^{\operatorname{dalg}}$ by Theorem 5.2. ∎

Hence for $K$ as in the lemma above, any $\operatorname{d}$-algebraic
extension of $K$ that is a $\operatorname{d}$-henselian, expint-closed
pre-$H$-field with the same residue field as $K$ is isomorphic to
$K^{\operatorname{dhl}}$ over $K$, and is thus a $\operatorname{d}$-Hensel-
Liouville closure of $K$.

######  Corollary 7.18.

Suppose that $\bm{k}$ is expint-closed and linearly surjective. Then
$K^{\operatorname{dhl}}$ has no proper differential subfield containing $K$
that is $\operatorname{d}$-Hensel-Liouville closed. Thus any
$\operatorname{d}$-Hensel-Liouville closure of $K$ is isomorphic to
$K^{\operatorname{dhl}}$ over $K$.

###### Proof.

If $L\supseteq K$ is a $\operatorname{d}$-Hensel-Liouville closed differential
subfield of $K^{\operatorname{dhl}}$, then we have an embedding $i\colon
K^{\operatorname{dhl}}\to L$ over $K$. Viewing this as an embedding into
$K^{\operatorname{dhl}}$, by Lemma 7.17 we have
$K^{\operatorname{dhl}}=i(K^{\operatorname{dhl}})$, so
$K^{\operatorname{dhl}}=L$. ∎

##  8\. Main results

###  8.A. Quantifier elimination

We now turn to the proof of quantifier elimination. Recall from [12] the
theory of closed ordered differential fields, which has quantifier elimination
and is the model completion of the theory of ordered differential fields
(where no assumption is made on the interaction between the ordering and the
derivation). Then let $T^{\operatorname{dhl}}_{\operatorname{codf}}$ be the
theory, in the language $\\{+,-,\cdot,0,1,\der,\preccurlyeq,\leqslant\\}$ of
ordered valued differential fields, of $\operatorname{d}$-Hensel-Liouville
closed pre-$H$-fields that have closed ordered differential residue field.

######  Assumption.

In this section, $K$ and $L$ are pre-$H$-fields with small derivation.

In the next results, for an ordered set $S$ we denote the cofinality of $S$ by
$\operatorname{cf}(S)$. Recall also for _Case 3_ the discussion of pc-
sequences in §2.

######  Lemma 8.1.

Suppose that $K$ is $\operatorname{d}$-Hensel-Liouville closed, and let $E$ be
a pre-$H$-subfield of $K$ with $\bm{k}_{E}=\bm{k}$. Suppose that $L$ is
$\operatorname{d}$-Hensel-Liouville closed. Assume that $L$ is
$|K|^{+}$-saturated as an ordered set and
$\operatorname{cf}(\Gamma_{L}^{<})>|\Gamma|$. Then any embedding $E\to L$ can
be extended to an embedding $K\to L$.

###### Proof.

Let $i\colon E\to L$ be an embedding. We may assume that $E\neq K$. It
suffices to show that $i$ can be extended to an embedding $F\to L$ for some
pre-$H$-subfield $F$ of $K$ properly containing $E$.

First, suppose that $\Gamma_{E}^{<}$ is not cofinal in $\Gamma^{<}$ and let
$f\in K^{>}$ with $\Gamma_{E}^{<}<vf<0$. By the cofinality assumption on
$\Gamma_{L}^{<}$, take $g\in L^{>}$ with $\Gamma_{i(E)}^{<}<v_{L}(g)<0$. Then
we extend $i$ to an embedding $E\langle f\rangle\to L$ sending $f\mapsto g$ by
Lemma 5.7.

Now suppose that $\Gamma_{E}^{<}$ is cofinal in $\Gamma^{<}$ and consider the
following three cases.

Case 1: $E$ is not $\operatorname{d}$-Hensel-Liouville closed. From the
assumptions on $K$, we get that $\bm{k}$ is expint-closed and linearly
surjective. Since $\bm{k}_{E}=\bm{k}$, we may extend $i$ to an embedding of
the $\operatorname{d}$-Hensel-Liouville closure of $E$ into $L$ by Theorem
7.16.

Case 2: $E$ is $\operatorname{d}$-Hensel-Liouville closed and $E\langle
y\rangle$ is an immediate extension of $E$ for some $y\in K\setminus E$. Take
such a $y$ and let $(a_{\rho})$ be a divergent pc-sequence in $K$ with
$a_{\rho}\rightsquigarrow y$. Since $E$ is $\operatorname{d}$-henselian, it is
$\operatorname{d}$-algebraically maximal by [7, Theorem 3.6], and so
$(a_{\rho})$ is of $\operatorname{d}$-transcendental type over $E$. By the
saturation assumption on $L$ and [3, Lemma 2.4.2], we have $z\in L$ with
$i(a_{\rho})\rightsquigarrow z$. Then [3, Lemma 6.9.1] yields a valued
differential field embedding $E\langle y\rangle\to L$ sending $y\mapsto z$; by
[3, Lemma 10.5.8], this is also an ordered field embedding.

Case 3: $E$ is $\operatorname{d}$-Hensel-Liouville closed and there is no
$y\in K\setminus E$ with $E\langle y\rangle$ an immediate extension of $E$.
Take any $f\in K\setminus E$. By saturation, take $g\in L$ such that for all
$a\in E$, we have

$a<f\implies i(a)<g\qquad\text{and}\qquad f<a\implies g<i(a).$

Then we can extend $i$ to an embedding $E\langle f\rangle\to L$ with $f\mapsto
g$ by Proposition 5.6. ∎

######  Theorem 8.2.

The theory $T^{\operatorname{dhl}}_{\operatorname{codf}}$ has quantifier
elimination.

###### Proof.

Suppose that $K$ and $L$ are $\operatorname{d}$-Hensel-Liouville closed and
have closed ordered differential residue fields. Suppose further that $L$ is
$|K|^{+}$-saturated as an ordered set,
$\operatorname{cf}(\Gamma_{L}^{<})>|\Gamma|$, and $\bm{k}_{L}$ is
$|\bm{k}|^{+}$-saturated as an ordered differential field. Let $E$ be a
substructure of $K$, so $E$ is a differential subring of $K$ with the induced
dominance relation and ordering. By a standard quantifier elimination test
(see for example [3, Corollary B.11.9]), it suffices to show that any
embedding $i\colon E\to L$ can be extended to an embedding $K\to L$, so let
$i\colon E\to L$ be an embedding.

By extending $i$ to the fraction field of $E$, we may assume that $E$ is a
field. The embedding $i$ induces an embedding
$i_{\operatorname{res}}\colon\bm{k}_{E}\to\bm{k}_{L}$ of ordered differential
fields. Since $\bm{k}_{L}$ is $|\bm{k}|^{+}$-saturated, by quantifier
elimination for closed ordered differential fields [12] we may extend
$i_{\operatorname{res}}$ to an embedding $\bm{k}\to\bm{k}_{L}$. By Corollary
3.4, we can now extend $i$ to an embedding $F\to L$ for a differential
subfield $F$ of $K$ with $\bm{k}_{F}=\bm{k}$. It remains to apply Lemma 8.1. ∎

######  Lemma 8.3.

Every pre-$H$-field with gap $0$ can be extended to a
$\operatorname{d}$-Hensel-Liouville closed pre-$H$-field with closed ordered
differential residue field.

###### Proof.

Suppose we have a pre-$H$-field $K_{0}$ with gap $0$. We first extend its
residue field to a closed ordered differential field, since the theory of
closed ordered differential fields is the model completion of the theory of
ordered differential fields, and apply Corollary 3.4 to obtain a pre-$H$-field
extension $K_{1}$ of $K_{0}$ with gap $0$ whose residue field is a closed
ordered differential field. It follows from their definition that closed
ordered differential fields are expint-closed and linearly surjective, so we
can extend $K_{1}$ to a pre-$H$-field $K_{2}$ with the same residue field that
is $\operatorname{d}$-Hensel-Liouville closed by Theorem 7.16. ∎

######  Corollary 8.4.

The theory $T^{\operatorname{dhl}}_{\operatorname{codf}}$ is the model
completion of the theory of pre-$H$-fields with gap $0$.

###### Proof.

This follows from Theorem 8.2 and Lemma 8.3 by a standard model-theoretic fact
(see for example [3, Corollary B.11.6]). ∎

######  Corollary 8.5.

The theory $T^{\operatorname{dhl}}_{\operatorname{codf}}$ is complete, and
hence decidable.

###### Proof.

The structure $(\mathbb{Z};+,-,\cdot,0,1,{}_{0},\preccurlyeq_{0},\leqslant)$,
where 0 is the trivial derivation (${}_{0}(\mathbb{Z})=\\{0\\}$) and
$\preccurlyeq_{0}$ is the trivial dominance relation ($k\preccurlyeq_{0}l$ for
all $k,l\in\mathbb{Z}^{\neq}$), embeds into every model of
$T^{\operatorname{dhl}}_{\operatorname{codf}}$, so the theory is complete (see
for example [3, Corollary B.11.7]). Decidability then follows from the
recursive axiomatization of $T^{\operatorname{dhl}}_{\operatorname{codf}}$
(see for example [3, Corollary B.6.9]). ∎

We now use quantifier elimination to show that
$T^{\operatorname{dhl}}_{\operatorname{codf}}$ is distal, a notion of model-
theoretic tameness introduced by P. Simon to isolate those NIP theories that
are “purely unstable” [11]. The definition used here, one of several
equivalent formulations, is in terms of indiscernible sequences; first, some
conventions. We use the term “indiscernible” to mean “indiscernible over
$\emptyset$,” and if $B$ is a parameter set, “$B$-indiscernible” to mean
“indiscernible over $B$.” If $I_{0}$ and $I_{1}$ are linearly ordered sets,
then $I_{0}+I_{1}$ denotes their natural concatenation, $I_{0}$ followed by
$I_{1}$. The singleton $\\{\ell\\}$ viewed as a linearly ordered set is
denoted by $(\ell)$.

######  Definition.

Let $T$ be a complete theory in a language $\mathcal{L}$. Then $T$ is _distal_
if in every model $\bm{M}$ of $T$, for any $B\subseteq M$ and infinite
linearly ordered sets $I_{0}$, $I_{1}$, whenever

  1. (i)

$(a_{i})_{i\in I_{0}+(\ell)+I_{1}}$ is indiscernible and

  2. (ii)

$(a_{i})_{i\in I_{0}+I_{1}}$ is $B$-indiscernible,

$(a_{i})_{i\in I_{0}+(\ell)+I_{1}}$ is also $B$-indiscernible.

Recall that the theory RCVF of real closed fields with a nontrivial valuation
whose valuation ring is convex, in the language
$\\{+,-,\cdot,0,1,\preccurlyeq,\leqslant\\}$, is distal: It is weakly
o-minimal by quantifier elimination [5], hence dp-minimal by [6, Corollary
4.3], and thus distal by [11, Lemma 2.10]. We reduce Theorem 8.6 to the
distality of RCVF by “forgetting” the derivation.

######  Theorem 8.6.

The theory $T^{\operatorname{dhl}}_{\operatorname{codf}}$ is distal, and hence
has NIP.

###### Proof.

Let $K$ be a $\operatorname{d}$-Hensel-Liouville closed pre-$H$-field with
closed ordered differential residue field. Let $I_{0}$ and $I_{1}$ be infinite
linearly ordered sets and $B\subseteq K$. Suppose that $(a_{i})_{i\in
I_{0}+(\ell)+I_{1}}$ is indiscernible with $a_{i}\in K^{d}$, $d\geqslant 1$,
and $(a_{i})_{i\in I_{0}+I_{1}}$ is $B$-indiscernible. Let
$\varphi(x_{1},\dots,x_{k},y)$, $k\in\mathbb{N}$, be a formula with
$|x_{1}|=\dots=|x_{k}|=m$ and $|y|=n$. We need to show that for all $b\in
B^{n}$ and $i_{1},\dots,i_{k},j_{1},\dots,j_{k}\in I_{0}+(\ell)+I_{1}$ with
$i_{1}<\dots<i_{k}$ and $j_{1}<\dots<j_{k}$,

$K\models\varphi(a_{i_{1}},\dots,a_{i_{k}},b)\leftrightarrow\varphi(a_{j_{1}},\dots,a_{j_{k}},b).$

For simplicity of notation, we assume that $d=k=m=n=1$. By quantifier
elimination, any formula in the variable $z=(z_{1},\dots,z_{t})$,
$t\in\mathbb{N}$, is equivalent in $K$ to a boolean combination of formulas of
one of the following forms:

$F(z)=0,\quad F(z)>0,\quad F(z)\preccurlyeq G(z),\qquad\text{where}\
F,G\in\mathbb{Z}\\{Z_{1},\dots,Z_{t}\\}.$

In particular, there is a formula $\psi(x_{0},\dots,x_{r},y_{0},\dots,y_{r})$
in the language
$\mathcal{L}_{\textnormal{OR},\preccurlyeq}\coloneqq\\{+,-,\cdot,0,1,\leqslant,\preccurlyeq\\}$
such that, for all $a,b\in K$,

$K\models\varphi(a,b)\leftrightarrow\psi(a,a^{\prime},\dots,a^{(r)},b,b^{\prime},\dots,b^{(r)}).$

Let $B_{r}\coloneqq B\cup\der(B)\cup\dots\cup{}^{r}(B)$. With respect to
$\mathcal{L}_{\textnormal{OR},\preccurlyeq}$, the sequences
$(a_{i},a_{i}^{\prime},\dots,a_{i}^{(r)})_{i\in I_{0}+(\ell)+I_{1}}$ and
$(a_{i},a_{i}^{\prime},\dots,a_{i}^{(r)})_{i\in I_{0}+I_{1}}$ are
indiscernible and $B_{r}$-indiscernible, respectively. As a structure in this
language, $K\models\textnormal{RCVF}$, and RCVF is distal. Thus for any $b\in
B$ and $i,j\in I_{0}+(\ell)+I_{1}$,

$K\models\psi(a_{i},a_{i}^{\prime},\dots,a_{i}^{(r)},b,b^{\prime},\dots,b^{(r)})\leftrightarrow\psi(a_{j},a_{j}^{\prime},\dots,a_{j}^{(r)},b,b^{\prime},\dots,b^{(r)}).\qed$

Another consequence of quantifier elimination is that this theory is o-minimal
at infinity in the following sense, from which it follows that it is locally
o-minimal (see [13]) by taking fractional linear transformations.

######  Corollary 8.7.

Suppose that $K$ is a $\operatorname{d}$-Hensel-Liouville closed pre-$H$-field
with closed ordered differential residue field, and let $X\subseteq K$ be
definable with parameters. Then there exists $a\in K$ such that either
$(a,+\infty)\subseteq X$ or $(a,+\infty)\cap X=\emptyset$.

###### Proof.

By quantifier elimination and a standard model-theoretic argument, it suffices
to show that for every elementary extension $L$ of $K$ with $a,b\in L$
satisfying $a,b>K$, there is an isomorphism $K\langle a\rangle\to K\langle
b\rangle$ over $K$ with $a\mapsto b$, which follows easily from the proofs of
[3, Lemmas 16.6.9 and 16.6.10]. ∎

###  8.B. Quantifier elimination with extra structure on the residue field

In this final subsection, we consider a theory of pre-$H$-fields with gap $0$
where extra structure is allowed on the residue field and prove quantifier
elimination and model companion results similar to those in the previous
subsection. We use this two-sorted quantifier elimination to deduce an
Ax–Kochen/Ershov principle, namely that the theory of a
$\operatorname{d}$-Hensel-Liouville closed pre-$H$-field is determined by the
theory of its ordered differential residue field.

We suspend here the convention that $\bm{k}$ is the residue field of $K$,
likewise with $\bm{k}_{L}$, etc. Now consider the two-sorted structure
$(K,\bm{k};\pi)$, where the language on the sort of $K$ is
$\\{+,-,\cdot,0,1,\der,\preccurlyeq,\leqslant\\}$, the language on the sort of
$\bm{k}$ is
$\mathcal{L}_{\operatorname{res}}\supseteq\\{+,-,\cdot,0,1,\der,\leqslant\\}$,
and $\pi$ is a map $\pi\colon K\to\bm{k}$; call this language $\mathcal{L}$.
We fix an $\mathcal{L}_{\operatorname{res}}$-theory $T_{\operatorname{res}}$
of ordered differential fields and let $T^{\operatorname{gap}}$ be the
$\mathcal{L}$-theory whose models are structures $(K,\bm{k};\pi)$ satisfying:

  1. (i)

$K$ is a pre-$H$-field with gap $0$;

  2. (ii)

$\bm{k}\models T_{\operatorname{res}}$;

  3. (iii)

$\pi|_{\mathcal{O}}$ is a surjective ordered differential ring homomorphism
with kernel $\cao$ and $\pi(K\setminus\mathcal{O})=\\{0\\}$.

Thus $\pi$ induces an isomorphism of ordered differential fields
$\operatorname{res}(K)\cong\bm{k}$; conversely, an isomorphism
$\operatorname{res}(K)\cong\bm{k}$ lifts to a surjective ordered differential
ring homomorphism $\mathcal{O}\to\bm{k}$ with kernel $\cao$.

Suppose $T_{\operatorname{res}}^{\operatorname{dhl}}$ is an
$\mathcal{L}_{\operatorname{res}}$-theory extending the theory
$T^{\operatorname{expint},\operatorname{ls}}_{\operatorname{res}}$ of ordered
differential fields that are expint-closed and linearly surjective; these
conditions are necessary if $T_{\operatorname{res}}^{\operatorname{dhl}}$ is
to be the theory of a residue field of a $\operatorname{d}$-Hensel-Liouville
closed pre-$H$-field. Let $T^{\operatorname{dhl}}$ be the $\mathcal{L}$-theory
whose models are structures $(K,\bm{k};\pi)$ satisfying:

  1. (i)

$K$ is a $\operatorname{d}$-Hensel-Liouville closed pre-$H$-field;

  2. (ii)

$\bm{k}\models T_{\operatorname{res}}^{\operatorname{dhl}}$;

  3. (iii)

$\pi\colon K\to\bm{k}$ is as in $T$.

To obtain quantifier elimination we need to expand $\mathcal{L}$ by a unary
function symbol $\iota$ on the sort of $K$, interpreted as multiplicative
inversion on $K^{\times}$ and $\iota(0)=0$; set
$\mathcal{L}^{\iota}\coloneqq\mathcal{L}\cup\\{\iota\\}$.

######  Theorem 8.8.

If $T^{\operatorname{dhl}}_{\operatorname{res}}$ has quantifier elimination,
then so does $T^{\operatorname{dhl}}$ in $\mathcal{L}^{\iota}$. If
$T^{\operatorname{dhl}}_{\operatorname{res}}$ is model complete, then so is
$T^{\operatorname{dhl}}$ in $\mathcal{L}$.

###### Proof.

Suppose that $T^{\operatorname{dhl}}_{\operatorname{res}}$ has quantifier
elimination, and let $(K,\bm{k};\pi)$ and $(L,\bm{k}_{L};\pi_{L})$ be models
of $T^{\operatorname{dhl}}$ such that $(L,\bm{k}_{L};\pi_{L})$ is
$|K|^{+}$-saturated. Let $(E,\bm{k}_{E};\pi_{E})$ be an
$\mathcal{L}^{\iota}$-substructure of $(K,\bm{k};\pi)$ and
$i\colon(E,\bm{k}_{E};\pi_{E})\to(L,\bm{k}_{L};\pi_{L})$ be an embedding. Thus
$E$ is a pre-$H$-subfield of $K$ and $\bm{k}_{E}$ is an
$\mathcal{L}_{\operatorname{res}}$-substructure of $\bm{k}$, with
$\pi_{E}=\pi|_{E}\colon E\to\bm{k}_{E}$ a (not necessarily surjective) ordered
differential ring homomorphism on $\mathcal{O}_{E}$. It suffices to extend $i$
to an embedding $(K,\bm{k};\pi)\to(L,\bm{k}_{L};\pi_{L})$.

Let $i_{\operatorname{res}}\colon\bm{k}_{E}\to\bm{k}_{L}$ be the restriction
of $i$ to $\bm{k}_{E}$. By quantifier elimination for
$T^{\operatorname{dhl}}_{\operatorname{res}}$ and the
$|\bm{k}|^{+}$-saturation of $\bm{k}_{L}$, we extend $i_{\operatorname{res}}$
to an embedding $i^{*}_{\operatorname{res}}\colon\bm{k}\to\bm{k}_{L}$, which,
by pulling back $i_{\operatorname{res}}^{*}$ via the ordered differential
field isomorphisms induced by $\pi$ and $\pi_{L}$, yields an embedding
$\operatorname{res}(K)\to\operatorname{res}(L)$. Applying Corollary 3.4 with
$\operatorname{res}(K)$ instead of $\bm{k}_{L}$ gives a pre-$H$-subfield $F$
of $K$ that extends $E$, has ordered differential residue field
$\operatorname{res}(F)=\operatorname{res}(K)$, and such that the embedding
$\operatorname{res}(K)\to\operatorname{res}(L)$ is induced by an embedding
$F\to L$ extending $i|_{E}$. Now by Lemma 8.1, this embedding extends further
to an embedding $j\colon K\to L$. Then the map $i^{*}$ that is $j$ on $K$ and
$i_{\operatorname{res}}^{*}$ on $\bm{k}$ is an embedding
$(K,\bm{k};\pi)\to(L,\bm{k}_{L};\pi_{L})$ extending $i$.

The second statement is proved similarly. ∎

######  Lemma 8.9.

Suppose that every model of $T_{\operatorname{res}}$ can be extended to a
model of $T_{\operatorname{res}}^{\operatorname{dhl}}$. Then every model of
$T^{\operatorname{gap}}$ can be extended to a model of
$T^{\operatorname{dhl}}$.

###### Proof.

Let $(K,\bm{k};\pi)\models T^{\operatorname{gap}}$, and extend $\bm{k}$ to a
model $\bm{k}^{*}$ of $T_{\operatorname{res}}^{\operatorname{dhl}}$. Let
$\bm{k}_{L}$ be an ordered differential field extension of
$\operatorname{res}(K)$ such that we have an isomorphism
$i\colon\bm{k}_{L}\to\bm{k}^{*}$ of ordered differential fields extending the
isomorphism $\operatorname{res}(K)\cong\bm{k}$ induced by $\pi$. Then by
applying Corollary 3.4 with $\operatorname{res}(K)$ instead of $\bm{k}$, we
obtain a pre-$H$-field extension $L$ of $K$ with gap $0$ that has ordered
differential residue field isomorphic to $\bm{k}_{L}$ over
$\operatorname{res}(K)$. By composing this isomorphism with $i$, we may assume
that $i$ is an isomorphism $i\colon\operatorname{res}(L)\to\bm{k}^{*}$. By
Theorem 7.16, we extend $L$ to its $\operatorname{d}$-Hensel-Liouville closure
$L^{\operatorname{dhl}}$ with residue field
$\operatorname{res}(L^{\operatorname{dhl}})=\operatorname{res}(L)$. Defining
$\pi^{*}\colon L^{\operatorname{dhl}}\to\bm{k}^{*}$ by $\pi^{*}(f)\coloneqq
i(\operatorname{res}{f})$ for $f\in\mathcal{O}_{L^{\operatorname{dhl}}}$ and
$\pi^{*}(f)=0$ otherwise, we obtain a model
$(L^{\operatorname{dhl}},\bm{k}^{*};\pi^{*})$ of $T^{\operatorname{dhl}}$
extending $(K,\bm{k};\pi)$. ∎

######  Corollary 8.10.

If $T_{\operatorname{res}}^{\operatorname{dhl}}$ is the model companion of
$T_{\operatorname{res}}$, then $T^{\operatorname{dhl}}$ is the model companion
of $T^{\operatorname{gap}}$.

######  Theorem 8.11.

Suppose that $K_{1}$ and $K_{2}$ are $\operatorname{d}$-Hensel-Liouville
closed pre-$H$-fields and let $\bm{k}_{1}$ and $\bm{k}_{2}$ be their ordered
differential residue fields. Then

$K_{1}\equiv K_{2}\ \iff\ \bm{k}_{1}\equiv\bm{k}_{2}.$

###### Proof.

The left-to-right direction is obvious, so suppose that
$\bm{k}_{1}\equiv\bm{k}_{2}$ as ordered differential fields. Construing
$(K_{1},\bm{k}_{1};\pi_{1})$ and $(K_{2},\bm{k}_{2};\pi_{2})$ as models of
$T^{\operatorname{dhl}}$, with
$\mathcal{L}_{\operatorname{res}}=\\{+,-,\cdot,0,1,\der,\leqslant\\}$ and
$T^{\operatorname{dhl}}_{\operatorname{res}}=\operatorname{Th}(\bm{k}_{1})$,
it suffices to show that
$(K_{1},\bm{k}_{1};\pi_{1})\equiv(K_{2},\bm{k}_{2};\pi_{2})$. By expanding the
language $\mathcal{L}_{\operatorname{res}}$, we may assume that
$T^{\operatorname{dhl}}_{\operatorname{res}}$ has quantifier elimination, and
thus so does $T^{\operatorname{dhl}}$ in $\mathcal{L}^{\iota}$ by Theorem 8.8.
Now we assume the Continuum Hypothesis and explain why this is unnecessary
after Corollary 8.12. Then by passing to elementarily equivalent structures,
we arrange that $(K_{1},\bm{k}_{1};\pi_{1})$ and $(K_{2},\bm{k}_{2};\pi_{2})$
are saturated of cardinality $\aleph_{1}$. In particular, $\bm{k}_{1}$ and
$\bm{k}_{2}$ are also saturated of cardinality $\aleph_{1}$, so
$\bm{k}_{1}\cong\bm{k}_{2}$. Finally, consider the structure
$(\mathbb{Q},\bm{k}_{1};\pi)$, where $\mathbb{Q}$ is equipped with the usual
ordered field structure, the trivial derivation, and the trivial dominance
relation, and $\pi$ is the unique field embedding $\mathbb{Q}\to\bm{k}_{1}$.
This structure embeds into both $(K_{1},\bm{k}_{1};\pi_{1})$ and
$(K_{2},\bm{k}_{2};\pi_{2})$, and thus quantifier elimination yields
$(K_{1},\bm{k}_{1};\pi_{1})\equiv(K_{2},\bm{k}_{2};\pi_{2})$. ∎

######  Corollary 8.12.

Let $\mathcal{L}_{\operatorname{res}}=\\{+,-,\cdot,0,1,\der,\leqslant\\}$ and
$T^{\operatorname{dhl}}_{\operatorname{res}}=T^{\operatorname{expint},\operatorname{ls}}_{\operatorname{res}}$.
Then every $\mathcal{L}^{\iota}$-sentence is
$T^{\operatorname{dhl}}$-equivalent to an
$\mathcal{L}_{\operatorname{res}}$-sentence.

###### Proof.

Let $B$ be the boolean algebra of $\mathcal{L}^{\iota}$-sentences modulo
$T^{\operatorname{dhl}}$-equivalence and let $B_{\operatorname{res}}$ be the
boolean subalgebra of $B$ of $\mathcal{L}_{\operatorname{res}}$-sentences
modulo $T^{\operatorname{dhl}}$-equivalence. Recall the correspondence between
complete theories extending $T^{\operatorname{dhl}}$ and elements of $S(B)$,
the space of ultrafilters on $B$. By Theorem 8.11, the restriction map
$S(B)\to S(B_{\operatorname{res}})$ is injective, and thus by the Stone
Representation Theorem, the inclusion map $B_{\operatorname{res}}\to B$ is
surjective. ∎

Corollary 8.12 is clearly equivalent to Theorem 8.11. As
$T^{\operatorname{dhl}}$ has a recursive axiomatization, Corollary 8.12 is an
arithmetic statement, and thus by absoluteness Theorem 8.11 holds without
assuming the Continuum Hypothesis.

## Acknowledgements

Thanks are due to Lou van den Dries for helpful discussions, for comments on a
draft of this paper, and for providing the manuscript, written with M.
Aschenbrenner and J. van der Hoeven, on which §4 is based. Thanks are also due
to Anton Bernshteyn for some suggestions on a draft of this paper and to Chris
Miller for suggesting the Boshernitzan paper. Presentation of some of this
research at DART X was supported by NSF DMS-1952694.

##  Appendix A Motivating example

This appendix elaborates on [3, Example 10.1.7] to provide an example of a
pre-$H$-field with gap $0$ motivating the results of §8.B. Let
$\mathbb{T}^{*}$ be an $\aleph_{0}$-saturated elementary extension of
$\mathbb{T}$. Enlarging the valuation ring $\mathcal{O}_{\mathbb{T}^{*}}$ of
$\mathbb{T}^{*}$ to
$\dot{\mathcal{O}}_{\mathbb{T}^{*}}=\\{f\in\mathbb{T}^{*}:|f|\leqslant\exp^{n}(x)\
\text{for some}\ n\geqslant 1\\}$ yields a $\operatorname{d}$-henselian
pre-$H$-field $(\mathbb{T}^{*},\dot{\mathcal{O}}_{\mathbb{T}^{*}})$
(necessarily with gap $0$); the saturation ensures that
$\dot{\mathcal{O}}_{\mathbb{T}^{*}}$ is a proper subring of $K$, i.e., $K$
contains a _transexponential_ element. Moreover, the residue field of
$(\mathbb{T}^{*},\dot{\mathcal{O}}_{\mathbb{T}^{*}})$ is elementarily
equivalent to $\mathbb{T}$ as an ordered valued differential field. To explain
this, we first review the theory of $\mathbb{T}$.

In the language $\\{+,-,\cdot,0,1,\der,\preccurlyeq,\leqslant\\}$ the theory
of $\mathbb{T}$ is model complete and axiomatized by the theory
$T^{\operatorname{nl}}_{\operatorname{small}}$ of newtonian, $\upomega$-free,
Liouville closed $H$-fields with small derivation. An asymptotic field $K$ is
_differential-valued_ (_$\operatorname{d}$ -valued_ for short) if
$\mathcal{O}=C+\cao$; $\operatorname{d}$-valued fields are
pre-$\operatorname{d}$-valued and $H$-fields are exactly the
$\operatorname{d}$-valued pre-$H$-fields. An $H$-field $K$ is _Liouville
closed_ if it is real closed, has exponential integration, and has
_integration_ in the sense that is surjective. For more on $H$-fields and
related notions, see [3, Chapter 10]. The property of $\upomega$-freeness is
crucial to studying $\mathbb{T}$ but incidental here, so we refer the reader
to [3, §11.7]. Likewise, we do not define newtonianity, a technical cousin of
$\operatorname{d}$-henselianity, and instead refer the reader to [3, Chapter
14].

To describe $(\mathbb{T}^{*},\dot{\mathcal{O}}_{\mathbb{T}^{*}})$ and its
residue field, it is convenient to work with value groups instead of valuation
rings via the notions of coarsening and specialization. Let $K$ be an
$H$-asymptotic field with small derivation and let $\Delta$ be a nontrivial
proper convex subgroup of $\Gamma$ such that
$\psi(\Delta^{\neq})\subseteq\Delta$ and
$\psi(\Gamma\setminus\Delta)\subseteq\Gamma\setminus\Delta$. The _coarsening
of $K$ by $\Delta$_ is the differential field $K$ with the valuation

$\displaystyle v_{\Delta}\colon K^{\times}$ $\displaystyle\to\Gamma/\Delta$
$\displaystyle a$ $\displaystyle\mapsto va+\Delta,$

denoted by $K_{\Delta}$. Its valuation ring is

$\dot{\mathcal{O}}\ \coloneqq\ \\{a\in K:va\geqslant\delta\ \text{for some}\
\delta\in\Delta\\}\ \supseteq\ \mathcal{O}$

with maximal ideal

$\dot{\cao}\ \coloneqq\ \\{a\in K:va>\Delta\\}\ \subseteq\ \cao.$

Then $K_{\Delta}$ has small derivation by [3, Corollary 4.4.4], and it is
$H$-asymptotic with gap $0$ by [3, Corollary 9.2.26 and Lemma 9.2.24].
Moreover, it is pre-$\operatorname{d}$-valued by [3, Corollary 10.1.6]. If $K$
is equipped with an ordering with respect to which $\mathcal{O}$ is convex,
then $\dot{\mathcal{O}}$ remains convex; if $K$ is additionally a
pre-$H$-field, then so is $K_{\Delta}$.

Setting $\dot{a}\coloneqq a+\dot{\cao}$ for $a\in\dot{\mathcal{O}}$, we equip
the differential residue field $\dot{K}\coloneqq\dot{\mathcal{O}}/\dot{\cao}$
of $K_{\Delta}$ with the valuation

$\displaystyle v\colon\dot{K}^{\times}$ $\displaystyle\to\Delta$
$\displaystyle\dot{a}$ $\displaystyle\mapsto va,$

making $\dot{K}$ a valued differential field with small derivation called the
_specialization of $K$ to $\Delta$_. Its valuation ring is
$\mathcal{O}_{\dot{K}}\coloneqq\\{\dot{a}:a\in\mathcal{O}\\}$ with maximal
ideal $\cao_{\dot{K}}\coloneqq\\{\dot{a}:a\in\cao\\}$. The map
$\mathcal{O}\to\mathcal{O}_{\dot{K}}$ given by $a\mapsto\dot{a}$ induces an
isomorphism $\operatorname{res}(K)\cong\operatorname{res}(\dot{K})$ of
differential fields. From $\psi(\Delta^{\neq})\subseteq\Delta$ we get that
$\dot{K}$ is $H$-asymptotic with asymptotic couple
$(\Delta,\psi|_{\Delta^{\neq}})$. Furthermore, if $K$ is
pre-$\operatorname{d}$-valued, then so is $\dot{K}$, and if $K$ is
$\operatorname{d}$-valued, then so is $\dot{K}$ with $C_{\dot{K}}=C$, where we
identify $C$ with a subfield of $C_{\dot{K}}$ via $a\mapsto\dot{a}$ [3, Lemma
10.1.8]. If $K$ is equipped with an ordering with respect to which
$\mathcal{O}$ is convex, then $\mathcal{O}_{\dot{K}}$ is convex with respect
to the induced ordering on $\dot{K}$. Moreover, suppose that $K$ is a
pre-$H$-field. To see that $\dot{K}$ is a pre-$H$-field, it remains to check
(PH3), so let $a\in\dot{\mathcal{O}}$ with $\dot{a}>\mathcal{O}_{\dot{K}}$.
Then $va<0$, so $va\in\Delta^{\neq}$, and hence
$v(a^{\prime})=v(a)+\psi(va)\in\Delta$. But we also have $a^{\prime}>0$, and
so $a^{\prime}>\dot{\cao}$, as desired. If additionally $K$ is an $H$-field,
then so is $\dot{K}$, since $H$-fields are exactly the
$\operatorname{d}$-valued pre-$H$-fields.

Suppose now that $K=\mathbb{T}^{*}$ and set
$\Delta\coloneqq\\{\gamma\in\Gamma:\psi^{n}(\gamma)\geqslant 0\ \text{for
some}\ n\geqslant 1\\}$, a nontrivial proper convex subgroup of $\Gamma$ with
$\psi(\Delta^{\neq})\subseteq\Delta$ and
$\psi(\Gamma\setminus\Delta)\subseteq\Gamma\setminus\Delta$, as in [3, Example