# Un principe d’Ax-Kochen-Ershov imaginaire

Martin Hils  Institut für Mathematische Logik und Grundlagenforschung,
Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster,
Germany<EMAIL_ADDRESS>and  Silvain Rideau-Kikuchi  Université de Paris
and Sorbonne Université, CNRS, IMJ-PRG, F-75006 Paris, France.
<EMAIL_ADDRESS>
###### Abstract.

We study interpretable sets in henselian and $\sigma$-henselian valued fields
with value group elementarily equivalent to $\mathbb{Q}$ or $\mathbb{Z}$. Our
first result is an Ax-Kochen-Ershov type principle for weak elimination of
imaginaries in finitely ramified characteristic zero henselian fields —
relative to value group imaginaries and residual linear imaginaries. We extend
this result to the valued difference context and show, in particular, that
existentially closed equicharacteristic zero multiplicative difference valued
fields eliminate imaginaries in the geometric sorts.

On the way, we establish some auxiliary results on separated pairs of
characteristic zero henselian fields and on imaginaries in linear structures
which are also of independent interest.

###### Key words and phrases:

Model Theory, Valued Fields, Classification of Imaginaries, Non-standard
Frobenius Automorphism, Separated pairs, Linear structures

###### 2020 Mathematics Subject Classification:

Primary: 03C45; Secondary: 03C10, 03C60, 12J10, 12L12

MH was partially supported by the German Research Foundation (DFG) via CRC
878, HI 2004/1-1 (part of the French-German ANR-DFG project GeoMod) and under
Germany’s Excellence Strategy EXC 2044-390685587, ‘Mathematics Münster:
Dynamics-Geometry-Structure’.

SRK was partially supported by GeoMod AAPG2019 (ANR-DFG), Geometric and
Combinatorial Configurations in Model Theory.

Both authors were partially supported by ValCoMo (ANR-13-BS01-0006). A
substantial part of this work was carried out when both authors participated
in the thematic trimester “Model theory, combinatorics and valued fields” at
the “Institut Henri Poincaré” in Paris. We are grateful to the IHP and its
staff for hosting this event and providing us with this opportunity.

##  1\. Introduction

In his seminal work “Une théorie de Galois imaginaire” [Poi83], Poizat
introduced the idea that the classification of certain abstract constructions
of model theory — namely interpretable sets or Shelah’s imaginaries — could
play an important role in our comprehension of specific structures. The
classification of definable sets, in the guise of quantifier elimination
results, has historically been used as a central ingredient in many
applications of model theory. But the development of more sophisticated model
theoretic tools, in particular stability theory, naturally took place in the
larger category of quotients of definable sets by definable equivalence
relations, _i.e._ interpretable sets. Shelah concretised this idea with his
$\mathrm{eq}$ construction that formally makes every interpretable set
definable.

However, these interpretable sets immediately escape the realm of well
understood and classified objects, complicating the possibility of applying
those new tools in specific examples, in particular from algebra. Poizat’s
idea was that these interpretable sets should also be classified, and he did
so in algebraically closed fields and in differentially closed fields. In both
cases, he showed that they are all definably isomorphic to definable sets,
_i.e._ the categories of definable and interpretable sets are equivalent — we
say that these structures _eliminate imaginaries_. This property later became
an essential feature in model-theoretic applications, _e.g._ to diophantine
geometry and algebraic dynamics.

The question of elimination of imaginaries also has a very geometric flavour:
given a definable family of sets $X\subseteq Y\times Z$, one wishes to find a
definable function $f:Z\to W$ such that for all $z_{1},z_{2}\in Z$,
$X_{z_{1}}:=\\{y\in Z:(y,z_{1})\in X\\}=X_{z_{2}}$ if and only if
$f(y_{1})=f(y_{2})$ — in other words, one wishes to find a canonical
parametrisation of this family where each set appears exactly once. We refer
the reader to [TZ12, Section 8.4] for details on these notions and
constructions.

Elimination of imaginaries results were then proved for numerous structures,
but it was not until work of Haskell, Hrushovski and Macpherson [HHM06] that
valued fields entered the picture. However, in this case the situation is more
complex. Indeed, the field itself does not eliminate imaginaries, as both the
value group and the residue field are interpretable but not isomorphic to a
definable set. Nevertheless, one can add certain well understood interpretable
sets, the _geometric sorts_. These sorts consist of the field $\mathbf{K}$
and, for all $n\in\mathbb{Z}_{>0}$, of the space
$\mathbf{S}_{n}:=\mathrm{GL}_{n}(\mathbf{K})/\mathrm{GL}_{n}(\mathcal{O})$ of
free rank $n$ $\mathcal{O}$-submodules of $\mathbf{K}^{n}$, where
$\mathcal{O}$ denotes the valuation ring, and of the space
$\mathbf{T}_{n}:=\bigcup_{s\in\mathbf{S}_{n}}s/\mathfrak{m}s$ where
$\mathfrak{m}\subseteq\mathcal{O}$ is the unique maximal ideal — _cf._ Section
2.1 for a precise definition of the geometric language. The main result of
[HHM06] states that the theory $\mathrm{ACVF}$ of algebraically closed non-
trivially valued fields eliminates imaginaries in the geometric sorts: given a
definable family of sets $X\subseteq Y\times Z$, there exists a definable
function $f:Z\to W$, with $W$ a product of geometric sorts, such that for all
$z_{1},z_{2}\in Z$, $X_{z_{1}}=X_{z_{2}}$ if and only if $f(z_{1})=f(z_{2})$ —
equivalently the category of sets interpretable in an algebraically closed
valued field is equivalent to the category of sets definable in its geometric
sorts. One may not overestimate the impact of this result, as it opened the
way for the development of geometric model theory in the context of valued
fields. A beautiful illustration of the power of these new methods is the work
by Hrushovski and Loeser on topological tameness in non-archimedean geometry
[HL16].

This result was later extended to other valued fields: real closed valued
fields [Mel06], separably closed valued fields of finite imperfection degree
[HKR18]. or $p$-adic fields and their ultraproducts [HMR18] — which allowed to
uniformise and extend Denef’s result on the rationality of certain zeta
functions to interpretable sets. The question remained whether a general
principle underlined all these results and such a principle was conjectured in
the early 2000’s by Hrushovski. The present paper establishes such a principle
for a large class of henselian fields, which covers most of the examples
considered in applications, and extends it to valued fields with operators.

At this level of generality, one cannot expect elimination in the geometric
sorts. Indeed, the residue field and the value group can be arbitrary and
might not themselves eliminate imaginaries as is the case in all the results
cited above. However, a fundamental idea of the model theory of valued fields,
the so-called Ax-Kochen-Ershov principle, is that the model theory of a
henselian equicharactersitic zero field should be controlled by its value
group and residue field. This principle takes its name from the result of Ax
and Kochen [AK65] and independently Ershov [Ers65] that this is indeed the
case for elementary equivalence, but this phenomenon has also been observed
with respect to numerous other aspects of valued fields, from model theoretic
tameness (starting with [Del81]) to motivic integration [HK06].

It is thus tempting to conjecture that, beyond the geometric sorts,
imaginaries in equicharacteristic zero henselian fields only arise from the
value group and the residue field. This is essentially true up to non trivial
torsors of the residue field. One is then naturally led to consider the
$\mathbf{k}$-linear imaginaries: given an interpretable set $X$ in the two
sorted theory of vector spaces $(\mathbf{k},\mathbf{V})$, one can consider the
collection of this interpretable set in all $\mathbf{k}$-vector spaces arising
from $\mathcal{O}$-lattices $s\in\mathbf{S}_{n}$:

$\mathbf{T}_{n,X}:=\bigsqcup_{s\in\mathbf{S}_{n}}X^{(\mathbf{k},s/\mathfrak{m}s)}$

Note that if $X=\mathbf{V}$, $\mathbf{T}_{n,X}=\mathbf{T}_{n}$ and if
$X=\\{0\\}$, $\mathbf{T}_{n,X}\cong\mathbf{S}_{n}$. In general, the
$\mathbf{T}_{n,X}$ consist exactly of those interpretable sets that admit
surjections $\mathbf{T}_{n}\to\mathbf{T}_{n,X}\to\mathbf{S}_{n}$. We write
$\mathbf{k}^{\mathrm{leq}}:=\bigsqcup_{n,X}\mathbf{T}_{n,X}$.

Before we state our main results, let us address two technical points. The
first is that our results also work in unramified mixed characteristic, at the
cost of also considering the higher residue rings
$\mathbf{R}_{\ell}:=\mathcal{O}/\ell\mathfrak{m}$, for
$\ell\in\mathbb{Z}_{\geqslant 1}$, which often play a crucial role in this
situation. These rings also come with their linear imaginaries and hence we
define
$\mathbf{T}_{n,\ell,X}:=\bigsqcup_{s\in\mathbf{S}_{n}}X^{(\mathbf{R}_{\ell},s/\ell\mathfrak{m}s)}$
and $\mathbf{R}^{\mathrm{leq}}:=\bigsqcup_{n,\ell,X}\mathbf{T}_{n,\ell,X}$.
The second point is that eliminating imaginaries often splits in two distinct
problems: describing quotients under the action of finite symmetric groups (in
other words finding canonical parameters for finite sets) and classifying
interpretable sets up to one-to-finite correspondences: given a definable
family of sets $X\subseteq Y\times Z$, one wishes to find a one-to-finite
definable correspondence $F:Z\to W$ such that for all $z_{1},z_{2}\in Z$,
$X_{z_{1}}=X_{z_{2}}$ if and only if $F(z_{1})=F(z_{2})$. This latter property
is usually referred to as weak elimination of imaginaries and will be the main
focus of this paper.

Our first main result is the following Ax-Kochen-Ershov principle for
elimination of imaginaries:

######  Theorem A (Theorem 6.1.1).

Let $T$ be a $\mathbf{\Gamma}$-$\mathbf{R}$-enrichment of $\mathrm{Hen}_{0}$
(with or without angular components), such that:

  * ($\mathbf{C}_{\mathbf{\Gamma}}$)

$T$ has definably complete value group;

  * ($\mathbf{FR}$)

for every $\ell\in\mathbb{Z}_{>0}$, the interval $[0,\mathrm{v}(\ell)]$ is
finite and $\mathbf{k}$ is perfect;

  * ($\mathbf{I}_{\mathbf{k}}$)

the residue field $\mathbf{k}$ is infinite;

  * ($\mathbf{E}^{\infty}_{\mathbf{k}})$

the induced theory on $\mathbf{k}$ eliminates $\exists^{\infty}$.

Then $T$ weakly eliminates imaginaries in
$\mathbf{K}\cup\mathbf{\Gamma}^{\mathrm{eq}}\cup\mathbf{R}^{\mathrm{leq}}$.

Angular components are (compatible) mulitplicative morphisms
$\mathrm{ac}_{n}:\mathbf{K}^{\times}\to\mathbf{R}_{n}^{\times}$ extending the
residue map on $\mathcal{O}^{\times}$. They are often considered in the model
theoretic study of valued fields, in particular in the Cluckers-Loeser
treatment of motivic integration [CL08].

It is worth noting that $\mathrm{PRES}=\mathrm{Th}(\mathbb{Z})$ and
$\mathrm{DOAG}=\mathrm{Th}(\mathbb{Q})$ are the only complete theories of pure
ordered abelian groups which are definably complete. As both $\mathrm{PRES}$
and $\mathrm{DOAG}$ eliminate imaginaries, we thus get
$\mathbf{\Gamma}=\mathbf{\Gamma}^{\mathrm{eq}}$ under the assumptions of
Theorem A in case $\mathbf{\Gamma}$ is not enriched.

As a corollary, Theorem A yields that if $F$ is a field of characteristic 0
which eliminates $\exists^{\infty}$, the theories of the valued fields
$F((t))$ and $F((t^{\mathbb{Q}}))$ (with or without angular components) weakly
eliminate imaginaries in the sorts $(\mathbf{K}\cup\mathbf{k}^{\mathrm{leq}})$
— noting that $\mathbf{\Gamma}\cong\mathbf{S}_{1}$ may be identified with a
sort in $\mathbf{k}^{\mathrm{leq}}$. In the particular case that $F$ is (of
characteristic 0 and) algebraically closed, real closed or pseudofinite, using
results of Hrushovski on linear imaginaries, we deduce that $F((t))$ and
$F((t^{\mathbb{Q}}))$ (with or without angular components) eliminate
imaginaries in the geometric sorts, after naming some constants in the
pseudofinite and in the real closed case (see Corollary 6.1.5 for the precise
statement), thus obtaining an absolute elimination result in these cases.

All the results with angular component, even in the algebraically closed
(equicharacteristic zero) case are new. Without angular components, this
provides alternate proofs of Mellor’s result [Mel06] for
$\mathrm{Th}(\mathbb{R}((t^{\mathbb{Q}}))$ and Hrushovski-Martin-Rideau’s
result [HMR18] for $\mathrm{Th}(F((t))$ where $F$ is pseudofinite of
characteristic 0. Independent work of Vicaria [Vic21] also yields the case of
$\mathrm{Th}(\mathbb{C}((t))$, although her work also applies to more general
value groups.

In mixed characteristic, the main example covered by Theorem A is
$W(\mathbb{F}_{p}^{a})$, the fraction field of the ring of Witt vectors with
coefficients in $\mathbb{F}_{p}^{a}$, and more generally $W(F)$ for any
perfect field $F$ of characteristic $p$ which eliminates $\exists^{\infty}$.
Thus, Theorem A provides an important step towards proving that the
imaginaries of $W(\mathbb{F}_{p}^{a})$ are classified by the geometric sorts
as well.

The second main result of this paper concerns valued difference fields, _i.e._
valued fields with automorphism compatible with the valuation. Quantifier
elimination (and hence an Ax-Kochen-Ershov principle for elementary
equivalence) has been proved for various classes. First for isometries in
[Sca03, BMS07, AvdD10], then for _$\omega$ -increasing_ automorphism — for
every $x\in\mathcal{O}$, $\mathrm{v}(\sigma(x))>\mathbb{Z}\cdot\mathrm{v}(x)$
— in [Azg10, Hru]. Both of these contexts were subsumed in later work of
Kushik [Pal12] on _multiplicative_ automorphisms where the automorphism acts
as multiplication by some element of an ordered field (_cf._ Definition 2.3.5
for precise definitions). Finally Durhan and Onay [DO15] proved that these
results hold without any hypothesis on the automorphism.

Our second result focuses on the multiplicative setting where we prove an
absolute elimination of imaginaries result for the respective model-
companions:

######  Theorem B (Theorem 6.2.1).

The theory $\mathrm{VFA}^{\mathrm{mult}}_{0,0}$ eliminates imaginaries in the
geometric sorts.

Since the isometric case and the $\omega$-increasing case correspond,
respectively, to the asymptotic theory of $\mathbb{C}_{p}$ with an isometric
lifting of the Frobenius and to $\mathbb{F}_{p}((t))^{\mathrm{a}}$ with the
Frobenius, an immediate corollary of these results is a uniform elimination of
imaginaries for large $p$ in these structures.

The proofs of both theorems follow the same general strategy and many
technical results are shared between the two. The proof consists of three
largely independent steps (Sections 3 to 5).

In stable theories every type $p$ — a maximal consistent set of definable sets
— is definable, that is, for every definable $X\subseteq Y\times Z$, the set
$\\{z\in Z:X_{z}\in p\\}$ is definable. This was used in many proofs of weak
elimination of imaginaries in the stable context to reduce the problem of
finding canonical parameters for set to finding canonical parameters for
types; which, counter-intuitively maybe, is a simpler problem. In [Hru14],
Hrushovski formalised the idea that even in an unstable context, this
reduction could also prove useful, provided definable were dense: over any
algebraically closed imaginary set of parameters, any definable set contains a
definable type.

The first step of the proof consists in proving such density results. But the
above statement cannot hold in the full generality of henselian
equicharacteristic zero fields since it might already fail in the residue
field. We prove however that, under certain hypotheses, _quantifier free_
definable types are dense, _cf._ Theorem 3.1.1. This result does not apply to
discrete valued fields since the family of intervals contains arbitrarily
large finite sets. In Theorem 3.1.3, we do however prove that the density of
invariant types holds in this context.

The proof improves on similar results in [Rid19] and the general idea is the
same. In arity one, we look for a minimal finite set of balls covering the
given definable set. The general case proceeds by fibration in relative arity
one and by considering germs of functions into the space of (finite sets of)
balls instead of actual balls. This fibration process is where most of the
technical assumptions of Theorem A are used, in particular the elimination of
$\exists^{\infty}$ in the residue field.

In contexts with an absolute elimination of quantifiers, _e.g._ [Rid19,
HKR18], this first density result (and the implicit computation of canonical
bases) suffices to conclude that weak elimination of imaginaries holds. In
equicharacteristic zero henselian (and $\sigma$-henselian) fields, types come
with more information than quantifier free types; an information that mostly
lives in the short exact sequence
$1\to\mathbf{k}^{\times}\to\mathbf{RV}^{\times}:=\mathbf{K}^{\times}/(1+\mathfrak{m})\to\mathbf{\Gamma}^{\times}\to
0$. The second step of our proof, Theorem 4.1.1, consists in showing that
quantifier free invariant types have invariant completions over $\mathbf{RV}$
(and $\mathbf{k}$-vector spaces) — this generalises to mixed characteristic by
considering the higher residue rings.

By quantifier elimination relative to $\mathbf{RV}$, this step reduces to,
given an invariant type, computing canonical generators of the structure
generated by (realisations of) the type in $\mathbf{RV}$. Note that in the
conclusion of Theorem 4.1.1 the types considered are invariant over definable
sets which are of the same size as the model. We do however show various
folklore results implying that this a well behaved notion when these sets are
stably embedded.

The third step consists in studying imaginaries in $\mathbf{RV}$, which is
left as a black box in the previous steps. We show, in the spirit of [HK06,
Section 3.3], that the imaginaries in the short exact sequence
$1\to\mathbf{k}^{\times}\to\mathbf{RV}^{\times}\to\mathbf{\Gamma}^{\times}\to
0$ come essentially from $\mathbf{k}$ and $\mathbf{\Gamma}$. But our result is
in a sense orthogonal to the one of Hrushovski and Kazhdan since we require
$\mathbf{RV}$ to be a pure (in the sense of model theory) extension of
$\mathbf{k}$ and $\mathbf{\Gamma}$, which can be both arbitrarily enriched,
whereas [HK06] has strong hypotheses on $\mathbf{k}$ and $\mathbf{\Gamma}$ and
no hypothesis on $\mathbf{RV}$.

Theorem 5.1.4 is the first version of a series of such reductions of
increasing complexity so as to cover the various cases that we require, the
ultimate version, 5.2.2 allowing controlled torsion in $\mathbf{\Gamma}$,
auxiliary sorts on both the $\mathbf{k}$ and $\mathbf{\Gamma}$ sides and
considering not one but a projective system of short exact sequences.

These three steps put together allow us to prove relative results like Theorem
A. However, absolute results like Theorem B or Corollary 6.1.5 require one
last ingredient: the classification of imaginaries in collections of vector
spaces — linear structures in the terminology of [Hru12]. Our contribution
consists in a twisted version, _cf._ 2.4.8.(4), of Hrushovski’s result on
$\mathrm{ACF}_{0}$-linear structures with flags and roots endowed with an
automorphism (the final step to prove Theorem B) and a version, Proposition
2.4.20, for real closed fields.

The plan of the paper is as follows. In Section 2, we provide some preliminary
results on separated pairs of valued fields (in the sense of Baur), on valued
difference fields and on linear structures. Section 3 is devoted to the proof
of the two density results for definable (resp. invariant) types mentioned
above. The fact that invariant quantifier free types are invariant over
$\mathbf{RV}$ (and $\mathbf{R}_{n}$-modules), is established in Section 4. In
Section 5, we prove the results about imaginaries in certain (enriched) short
exact sequences of modules. Finally, in Section 6, we put everything together
and prove our main results, in particular Theorem A and Theorem B.

##  2\. Preliminaries

###  2.1. The languages of valued fields

Any valued field $(K,v)$ can be considered as a structure in the language
$\mathcal{L}_{\mathrm{div}}$ with one sort $\mathbf{K}$ for the valued field,
the ring language and a binary relation $x|y$ interpreted as $v(x)\leqslant
v(y)$. This language owes its widespread use to the following result,
essentially due to Robinson [Rob77]:

######  Fact 2.1.1.

The $\mathcal{L}_{\mathrm{div}}$-theory $\mathrm{ACVF}$ of algebraically
closed non trivially valued fields eliminates quantifiers.

We will write $\mathcal{L}_{0}:=\mathcal{L}_{\mathrm{div}}$ and throughout
this paper, notations with an index $0$ — like $\mathrm{dcl}_{0}$,
$\mathrm{acl}_{0}$ or $\mathrm{tp}_{0}$ — will refer to the quantifier free
$\mathcal{L}_{0}$-structure; or equivalently the structure induced by any
model of $\mathrm{ACVF}$ containing the valued field under consideration.

Given a valued field, seen as an $\mathcal{L}_{0}$-structure, we will denote
by $\mathcal{O}:=\\{x\in\mathbf{K}:v(x)\geqslant 0\\}$ its valuation ring,
$\mathfrak{m}:=\\{x\in\mathbf{K}:v(x)>0\\}$ its maximal ideal,
$\mathbf{\Gamma}:=\mathbf{K}/\mathcal{O}^{\times}$ its value group,
$\mathbf{k}:=\mathcal{O}/\mathfrak{m}$ its residue field and
$\mathrm{v}:\mathbf{K}\to\mathbf{\Gamma}$ and
$\mathrm{res}:\mathcal{O}\to\mathbf{k}$ the canonical projections — we will
write $\mathbf{\Gamma}^{\times}$ for
$\mathrm{v}(\mathbf{K}^{\times})\subseteq\mathbf{\Gamma}$. More generally, for
every $n\in\mathbb{Z}_{>0}$, we write
$\mathbf{R}_{n}:=\mathcal{O}/n\mathfrak{m}$. Let also
$\mathrm{res}_{n}:\mathcal{O}\to\mathbf{R}_{n}$ be the canonical projection,
$\mathbf{R}_{\infty}$ the (pro-definable) set $\varprojlim_{n}\mathbf{R}_{n}$
and $\mathrm{res}_{\infty}:\mathcal{O}\to\mathbf{R}_{\infty}$ the natural map.
Note that, working in a sufficiently saturated model,
$\mathbf{R}_{\infty}\cong\mathcal{O}/\mathfrak{m}_{\infty}$, where
$\mathfrak{m}_{\infty}:=\\{x\in\mathbf{K}:\mathrm{v}_{\infty}(x)>\Delta_{\infty}\\}$
and $\Delta_{\infty}\leqslant\mathbf{\Gamma}$ is the convex subgroup generated
by $\mathrm{v}(\mathrm{char}(\mathbf{k}))$, in mixed characteristic, and
$\Delta_{\infty}=0$, otherwise. It is a valuation ring whose fraction field is
naturally identified with the residue field $\mathbf{k}_{\infty}$ associated
to the the (equicharacteristic) valuation
$\mathrm{v}_{\infty}:\mathbf{K}\to\mathbf{\Gamma}\to\mathbf{\Gamma}/\Delta_{\infty}$.
We also define $\mathbf{R}:=\bigsqcup_{n>0}\mathbf{R}_{n}$.

Although most of the present paper is rather insensitive to the choice of
language for valued fields — or, rather, we work in
$\mathcal{L}_{0}^{\mathrm{eq}}$ — we will at times need to work in certain
languages tailored for specific elimination results. The first of them is the
Haskell-Hrushovski-Macpherson geometric language. For every
$n\in\mathbb{Z}_{>0}$, let
$\mathbf{S}_{n}\cong\mathrm{GL}_{n}(\mathbf{K})/\mathrm{GL}_{n}(\mathcal{O})$
be the (interpretable) set of rank $n$ free sub-$\mathcal{O}$-modules of
$\mathbf{K}^{n}$, and
$\mathbf{T}_{n}:=\bigcup_{s\in\mathbf{S}_{n}}s/\mathfrak{m}s$. Let
$\mathbf{S}:=\bigcup_{n}\mathbf{S}_{n}$,
$\mathbf{T}:=\bigcup_{n}\mathbf{T}_{n}$ and
$\mathbfcal{G}:=\mathbf{K}\cup\mathbf{S}\cup\mathbf{T}$. Note that
$\mathrm{GL}_{n}(K)$ naturally acts transitively on $\mathbf{T}_{n}$ over its
action on $\mathbf{S}_{n}$. We also denote by
$s_{n}:\mathrm{GL}_{n}(\mathbf{K})\to\mathbf{S}_{n}$,
$t_{n}:\mathrm{GL}_{n}(\mathbf{K})\to\mathbf{T}_{n}$ and
$\tau_{n}:\mathbf{T}_{n}\to\mathbf{S}_{n}$ the canonical projections.

These interpretable sets (and the so called _geometric language_ of which they
are the sorts) were introduced to classify imaginaries in $\mathrm{ACVF}$:

######  Fact 2.1.2 ([HHM06, Theorem 1.0.1]).

The theory $\mathrm{ACVF}$ eliminates imaginaries in $\mathbfcal{G}$.

The second language that we will use allows for a description of definable
sets in certain henselian fields. For every $n\in\mathbb{Z}_{>0}$, let
$\mathbf{RV}_{n}:=\mathbf{K}/(1+n\mathfrak{m})$. Let
$\mathrm{rv}_{n}:\mathbf{K}\to\mathbf{RV}_{n}$ and
$\mathrm{rv}_{n,m}:\mathbf{RV}_{n}\to\mathbf{RV}_{m}$ denote the canonical
projections. The set $\mathbf{RV}_{n}$ is naturally endowed with a
multiplication and the trace of addition which we denote, in Krasner’s
hyperfield manner:
$\zeta\oplus\xi:=\\{\mathrm{rv}(x+y):\mathrm{rv}_{n}(x)=\zeta\text{ and
}\mathrm{rv}_{n}(y)=\xi\\}\subseteq\mathbf{RV}_{n}$. We say that
$\zeta\oplus\xi$ is well-defined when $\zeta\oplus\xi=\\{\chi\\}$ is a
singleton, and we often write $\zeta\oplus\xi=\chi$ in that case.

Note that for any two disjoint balls $b_{1}$ and $b_{2}$, in some valued field
$(K,v)$, and any $a_{i},c_{i}\in b_{i}$,
$\mathrm{rv}_{1}(a_{1}-a_{2})=\mathrm{rv}_{1}(c_{1}-c_{2})$. We will denote by
$\mathrm{rv}_{1}(b_{1}-b_{2})$ this common value. If $b_{1}\cap
b_{2}\neq\emptyset$, by convention, $\mathrm{rv}_{1}(b_{1}-b_{2})=0$.

We denote by $\mathbf{RV}_{\infty}$ the (pro-definable) set
$\varprojlim_{n}\mathbf{RV}_{n}$ and
$\mathrm{rv}_{\infty}:\mathbf{K}\to\mathbf{RV}_{\infty}$ denotes the natural
map. Note that
$\mathbf{RV}_{\infty}\cong\mathbf{K}/(1+\mathfrak{m}_{\infty})$. We also
denote $\mathbf{RV}:=\bigsqcup_{n}\mathbf{RV}_{n}$.

Let $\mathcal{L}_{\mathbf{RV}}$ be the language with sorts $\mathbf{K}$ and
$\mathbf{RV}_{n}$, for all $n\in\mathbb{Z}_{>0}$, the ring structure on
$\mathbf{K}$, multiplication and $\oplus$ on each $\mathbf{RV}_{n}$ and the
maps $\mathrm{rv}_{n}$ and $\mathrm{rv}_{n,m}$. Let
$\mathcal{L}_{\mathbf{RV}_{1}}$ be its restriction to the sorts $\mathbf{K}$
and $\mathbf{RV}_{1}$.

We say that a valued field is:

  * •

_algebraically maximal_ if it does not admit non trivial immediate algebraic
extensions;

  * •

_Kaplansky_ if $\mathrm{v}(\mathbf{K})$ is $p$-divisible and any finite
extension of $\mathbf{k}$ has degree prime to $p$, where
$p=\mathrm{char}(\mathbf{k})$ if it is positive and $p=1$ otherwise;

  * •

_finitely ramified_ if for any $n\in\mathbb{Z}_{>0}$ the interval
$[0,\mathrm{v}(n)]$ is finite.

Note that a finitely ramified field is algebraically maximal if and only if it
is henselian.

The following quantifier elimination results are due, respectively, to Basarab
[Bas91, Theorem A] in characteristic zero and Delon [Del82, Théorème 3.1] in
positive characteristic:

######  Fact 2.1.3.

  * •

Let $\mathcal{L}$ be an $\mathbf{RV}$-enrichment of
$\mathcal{L}_{\mathbf{RV}}$ and $T$ an $\mathcal{L}$-theory containing the
theory $\mathrm{Hen}_{0}$ of henselian characteristic zero fields. Then $T$
eliminates field quantifiers.

  * •

Let $\mathcal{L}$ be an $\mathbf{RV}_{1}$-enrichment of
$\mathcal{L}_{\mathbf{RV}_{1}}$ and $T$ an $\mathcal{L}$-theory containing the
theory of equicharacteristic $p$ algebraically maximal Kaplansky fields, for
some fixed $p>0$. Then $T$ eliminates field quantifiers.

######  Convention 2.1.4.

Throughout this paper, if $M$ is an $\mathcal{L}$-structure, $X$ is
$\mathcal{L}(M)$-definable and $A\subseteq M$, then $X(A)$ denotes $X\cap A$.
There are too many structures at play to not be explicit as to which definable
or algebraic closures we want to consider.

###  2.2. Separated pairs of valued fields

In this section, we will gather some results about separated pairs of valued
fields, in particular concerning pure stable embeddedness of the residue field
and value group pairs in specific contexts. In equicharacteristic zero, most
of the results below follow from work of Leloup [Lel90]; and from work of
Rioux [Rio17], in unramified mixed characteristic.

Recall that an extension $L/K$ of valued fields is called _separated_ if every
finite-dimensional $K$-vector subspace of $L$ admits a $K$-_valuation basis_ ,
i.e., a $K$-basis $(b_{1},\ldots,b_{n})$ which is _valuation independent_ over
$K$: for any $a_{1},\ldots,a_{n}\in K$ one has $\mathrm{v}(\sum
a_{i}b_{i})=\min\mathrm{v}(a_{i}b_{i})$. Also, for fields $K\subseteq
L,K^{\prime}\subseteq U$, we write $L\downfree^{\mathrm{ld}}_{K}K^{\prime}$ if
$L$ and $K^{\prime}$ are linearly disjoint over $K$.

######  Definition 2.2.1.

Let $K\subseteq L,K^{\prime}\subseteq U$ be valued fields.

  * •

We say $L$ and $K^{\prime}$ are _$\mathbf{\Gamma}\mathbf{k}$ -independent_
over $K$, denoted by $L\downfree^{\mathbf{\Gamma}\mathbf{k}}_{K}K^{\prime}$,
if
$\mathrm{res}(L)\downfree^{\mathrm{ld}}_{\mathrm{res}(K)}\mathrm{res}(K^{\prime})$
and $\mathrm{v}(L)\cap\mathrm{v}(K^{\prime})=\mathrm{v}(K)$.

  * •

Assume that $L/K$ is separated. Then $L$ is said to be _valuatively disjoint_
from $K^{\prime}$ over $K$, denoted by $L\downfree^{vd}_{K}K^{\prime}$, if
whenever a tuple $(b_{1},\ldots,b_{n})$ from $L$ is valuation independent over
$K$, it is valuation independent over $K^{\prime}$.

######  Fact 2.2.2.

Let $K\subseteq L,K^{\prime}\subseteq U$ be valued fields with $L/K$ separated
and $L\downfree^{\mathbf{\Gamma}\mathbf{k}}_{K}K^{\prime}$. Set
$L^{\prime}:=LK^{\prime}$. Then the following hold:

  1. (1)

$L\downfree^{vd}_{K}K^{\prime}$ — in particular,
$L\downfree^{\mathrm{ld}}_{K}K^{\prime}$;

  2. (2)

$L^{\prime}/K^{\prime}$ is separated;

  3. (3)

$\mathrm{res}(L^{\prime})=\mathrm{res}(L)\mathrm{res}(K^{\prime})$ and
$\mathrm{v}(L^{\prime})=\mathrm{v}(L)+\mathrm{v}(K^{\prime})$;

  4. (4)

If $L_{1}\subseteq U$ and $f:L\cong L_{1}\subseteq U$ is an isomorphism over
$K\cup k_{L}\cup\Gamma_{L}$, then $f$ extends (uniquely) to an isomorphism
$f^{\prime}:L^{\prime}\cong L_{1}K^{\prime}$ over $K^{\prime}$.

###### Proof.

This is shown by adapting the proof of the result for $K$ maximally valued
from [HHM08, Proposition 12.11]. ∎

####  2.2.1. Reduction to $\mathbf{RV}$

Most of this paper will be concerned with characteristic zero finitely
ramified fields, however, for future reference, we will state and prove
certain results, mostly regarding pairs, in all characteristics, as the
arguments are essentially identical.

######  Notation.

Assume that $\mathcal{L}$ is a multisorted language,
$\mathcal{L}_{\mathbf{P}}$ is the associated language of pairs and $M$ is a
pair of $\mathcal{L}$-structures considered as an
$\mathcal{L}_{\mathbf{P}}$-structure. If $\mathbf{D}$ is a sort in
$\mathcal{L}$, we write $\mathbf{P}\mathbf{D}(M)$ for the set of elements in
$M$ of sort $\mathbf{D}$ which lie in the substructure singled out by
$\mathbf{P}$. We denote $\mathbf{P}(M)$ the whole $\mathcal{L}$-substructure
singled out by $\mathbf{P}$.

Let $T^{\star}$ be a theory of separated pairs $(M,\mathbf{P}(M))$ in the
language $\mathcal{L}_{\mathbf{RV},\mathbf{P}}^{\mathrm{hyb}}$ consisting of
$\mathcal{L}_{\mathbf{RV}_{1},\mathbf{P}}$ enriched with
$\mathcal{L}_{\mathbf{RV}}$ on $\mathbf{P}$. We assume that $M$ eliminates
fields quantifiers in $\mathcal{L}_{\mathbf{RV}_{1}}$ and $\mathbf{P}(M)$
eliminates fields quantifiers in $\mathcal{L}_{\mathbf{RV}}$. Note that we do
not assume that $\mathbf{P}\mathbf{RV}_{1}$ is stably embedded in
$\mathbf{RV}_{1}$.

By a hybrid $\mathbf{RV}$-structure, we mean a structure (elementarily
equivalent to) $(\mathbf{RV}_{1}(M),\mathbf{P}\mathbf{RV}(M))$, where
$M\models T^{\star}$ — with the restriction of the
$\mathcal{L}_{\mathbf{RV},\mathbf{P}}^{\mathrm{hyb}}$-structure. We also
denote $\mathbf{RV}^{\mathrm{hyb}}$ the set of sorts
$\mathbf{RV}_{1}\cup\bigcup_{n}\mathbf{P}\mathbf{RV}_{n}$.

######  Lemma 2.2.3.

Let $M\preccurlyeq N$ be hybrid $\mathbf{RV}$-structures. Then
$\mathbf{k}(M)\downfree^{\mathrm{ld}}_{\mathbf{P}\mathbf{k}(M)}\mathbf{P}\mathbf{k}(N)$
and
$\mathbf{\Gamma}(M)\cap\mathbf{P}\mathbf{\Gamma}(N)=\mathbf{P}\mathbf{\Gamma}(M)$.

###### Proof.

Immediate from the elementarity of the extension. ∎

Let $M$ be a hybrid $\mathbf{RV}$-structure. If $M$ is of mixed characteristic
we assume that $\mathbf{P}(M)$ is finitely ramified with perfect residue
field. Then $\mathbf{P}\mathbf{R}_{\infty}$ is a complete mixed characteristic
discrete valuation ring with perfect residue field $\mathbf{P}\mathbf{k}$. It
is therefore a finite extension of $\mathrm{W}(\mathbf{P}\mathbf{k})$ of
degree $\mathrm{v}(p)$, where $\mathrm{W}(l)$ denotes the fraction field of
the ring of Witt vectors over $l$, _cf._ [Ser68, Ch. II]. In the lemma that
follows (and more generally when we refer to “the relevant constants”), in
mixed characteristic, we require constants for Witt-coefficients (in
$\mathbf{P}\mathbf{k}$) of the minimal polynomial of some choice of
uniformizer in $\mathbf{P}\mathbf{R}_{\infty}$.

######  Lemma 2.2.4.

Assume that $\mathbf{RV}_{1}(M)$ is divisible, or that
$\mathbf{P}\mathbf{\Gamma}(M)$ is a pure subgroup of $\mathbf{\Gamma}(M)$.
Then in any $\mathbf{k}$-$\mathbf{\Gamma}$-enrichment of $M$, $\mathbf{k}$ and
$\mathbf{\Gamma}$ are purely stably embedded and orthogonal. Moreover, the
theory of $M$ is determined by the theories of the $\mathbf{k}$-pair and of
the $\mathbf{\Gamma}$-pair.

###### Proof.

We may assume that $M$ is $\aleph_{1}$-saturated. Then the short exact
sequence of abelian groups

$1\rightarrow\mathbf{P}\mathbf{R}_{\infty}^{\times}(M)\rightarrow\mathbf{P}\mathbf{RV}_{\infty}^{\times}(M)\rightarrow\mathbf{P}\mathbf{\Gamma}^{\times}(M)\rightarrow
0$

is split and induces coherent splittings of the sequences
$1\rightarrow\mathbf{P}\mathbf{R}_{n}^{\times}(M)\rightarrow\mathbf{P}\mathbf{RV}_{n}^{\times}(M)\rightarrow\mathbf{P}\mathbf{\Gamma}^{\times}(M)\rightarrow
0$. It follows form the assumptions that the splitting of
$1\rightarrow\mathbf{P}\mathbf{k}^{\times}(M)\rightarrow\mathbf{P}\mathbf{RV}_{1}^{\times}(M)\rightarrow\mathbf{P}\mathbf{\Gamma}^{\times}(M)\rightarrow
0$ extends to a splitting of the sequence
$1\rightarrow\mathbf{k}^{\times}(M)\rightarrow\mathbf{RV}_{1}^{\times}(M)\rightarrow\mathbf{\Gamma}^{\times}(M)\rightarrow
0$.

Note that the additional structure on $\mathbf{RV}^{\mathrm{hyb}}$, beyond the
abelian structure, is given by $\oplus$ and some
$\mathbf{k}$-$\mathbf{\Gamma}$-enrichment. But $\oplus$ can be defined using
the ring structure on $\mathbf{k}$ and the $\mathbf{P}\mathbf{R}_{n}$. Since
the $\mathbf{P}\mathbf{R}_{n}$ are $\emptyset$-interpretable in
$\mathbf{P}\mathbf{k}$, it follows that, if we add the sections,
$\mathbf{RV}^{\mathrm{hyb}}$ is a $\mathbf{k}$-$\mathbf{\Gamma}$-enrichment of
the product of $\mathbf{k}$ and $\mathbf{\Gamma}$. The result follows. ∎

Let now $M,N\models T^{\star}$, where we suppose that $N$ is $\lvert
M\rvert^{+}$-saturated, and let $A\leqslant M$ and $f:A\to N$ be some
embedding.

######  Definition 2.2.5.

We say that:

  1. (1)

$A$ is _good_ if $\mathbf{P}\mathbf{K}(A)\leqslant\mathbf{K}(A)$ is a
separated extension of valued fields with

$\mathbf{K}(A)\downfree^{\mathbf{\Gamma}\mathbf{k}}_{\mathbf{P}\mathbf{K}(A)}\mathbf{P}\mathbf{K}(M),$

  2. (2)

$f$ is _good_ if $A\leqslant M$ and $f(A)\leqslant N$ are good and
$f_{\mathbf{RV}^{\mathrm{hyb}}}$ is elementary — for the
${\left.\mathcal{L}_{\mathbf{RV},\mathbf{P}}^{\mathrm{hyb}}\right|_{\mathbf{RV}^{\mathrm{hyb}}}}$-structure.

######  Proposition 2.2.6.

Assume $f$ is a good embedding. Then $f$ extends to a good embedding $g:M\to
N$.

###### Proof.

We proceed step by step.

Step 1. _We may extend $f$ to a good map defined on
$A\cup\mathbf{RV}^{\mathrm{hyb}}(M)$ — and thus assume that
$\mathbf{RV}^{\mathrm{hyb}}(A)=\mathbf{RV}^{\mathrm{hyb}}(M)$._

Indeed, this follows from the fact that $f_{\mathbf{RV}^{\mathrm{hyb}}}$ is
elementary and that the only structure involving both $\mathbf{K}$ and
$\mathbf{RV}^{\mathrm{hyb}}$ are maps from $\mathbf{K}$ to
$\mathbf{RV}^{\mathrm{hyb}}$.

Step 2. _We may extend $f$ to a good map defined on (the substructure
generated by) $A\cup\mathbf{P}(M)$ — and thus assume that
$\mathbf{P}\mathbf{K}(A)=\mathbf{P}\mathbf{K}(M)$._

Indeed, by $\mathbf{K}$-quantifier elimination in the
$\mathcal{L}_{\mathbf{RV}}$-theory of the small valued field $\mathbf{P}(M)$,
the map ${\left.f\right|_{\mathbf{P}\mathbf{K}\cup\mathbf{P}\mathbf{RV}}}$
extends to an (elementary) $\mathcal{L}_{\mathbf{RV}}$-embedding
$g:\mathbf{P}(M)\rightarrow\mathbf{P}(N)$. As
$\mathbf{K}(A)\downfree^{\mathbf{\Gamma}\mathbf{k}}_{\mathbf{P}\mathbf{K}(A)}\mathbf{P}\mathbf{K}(M)$
and $\mathbf{RV}^{\mathrm{hyb}}(A)=\mathbf{RV}^{\mathrm{hyb}}(M)$, by 2.2.2,
$f\cup g$ induces a good embedding of $A\cup\mathbf{P}(M)$ into $N$.

Step 3. _We may extend $f$ to a good embedding of $M$ into $N$._

Indeed, by $\mathbf{K}$-quantifier elimination in the
$\mathcal{L}_{\mathbf{RV}_{1}}$-theory of the valued field $M$, the map $f$
extends to an (elementary) $\mathcal{L}_{\mathbf{RV}_{1}}$-embedding
$\tilde{f}:M\rightarrow N$. By Lemma 2.2.3, we get
$\tilde{f}(\mathbf{K}(M))\downfree^{\mathbf{\Gamma}\mathbf{k}}_{f(\mathbf{P}\mathbf{K}(M))}\mathbf{P}\mathbf{K}(N)$,
so in particular
$\tilde{f}(\mathbf{K}(M))\downfree^{\mathrm{ld}}_{f(\mathbf{P}\mathbf{K}(M))}\mathbf{P}\mathbf{K}(N)$
by 2.2.2(1), showing that $\tilde{f}$ is an
$\mathcal{L}_{\mathbf{RV}_{1},\mathbf{P}}$-embedding, with image a good
substructure of $N$. Thus $\tilde{f}$ is a good embedding, since $f$ was
already defined on the whole of $\mathbf{RV}^{\mathrm{hyb}}(M)$. ∎

######  Corollary 2.2.7.

The theory $T^{\star}$ is complete relative to $\mathbf{RV}^{\mathrm{hyb}}$,
and $\mathbf{RV}^{\mathrm{hyb}}$ is purely stably embedded in $T^{\star}$,
i.e., the induced structure is that of a hybrid $\mathbf{RV}$-structures.

This even holds resplendently, for any $\mathbf{RV}^{\mathrm{hyb}}$-enrichment
of the pair of valued fields.

###### Proof.

Assume that $M,N\models T^{\star}$ are models with
$\mathbf{RV}^{\mathrm{hyb}}(M)\equiv\mathbf{RV}^{\mathrm{hyb}}(N)$. The
isomorphism of prime substructures is easily seen to be a good embedding. It
follows by Proposition 2.2.6, and a back and forth argument, that it is, in
fact, elementary — _i.e._ $M\equiv N$.

Similarly, if $M\preccurlyeq N$ — in particular a good substructure — and
$f:M\to N$ is an elementary embedding — in particular a good embedding —
inducing the identity on $\mathbf{RV}^{\mathrm{hyb}}(M)$, then it remains a
good embedding — and hence an elementary one — when extended by the identity
on $\mathbf{RV}^{\mathrm{hyb}}(N)$; in other words,
$\mathbf{RV}^{\mathrm{hyb}}$ is stably embedded. Finally any
${\left.\mathcal{L}_{\mathbf{RV},\mathbf{P}}^{\mathrm{hyb}}\right|_{\mathbf{RV}^{\mathrm{hyb}}}}$-elementary
map on $\mathbf{RV}^{\mathrm{hyb}}$ is good and hence
$\mathcal{L}_{\mathbf{RV},\mathbf{P}}^{\mathrm{hyb}}$-elemetary, so
$\mathbf{RV}^{\mathrm{hyb}}$ is pure. ∎

######  Remark 2.2.8.

If $\mathbf{P}\mathbf{RV}$ is (purely) stably embedded in
$\mathbf{RV}^{\mathrm{hyb}}$, we can can also deduce that
$\mathbf{P}\mathbf{K}$ is (purely) stably embedded.

######  Corollary 2.2.9.

Let $M\models T^{\star}$ be of equicharacteristic or of mixed characteristic
with $\mathbf{P}(M)$ finitely ramified with prefect residue field — with
relevant constants. Assume that $\mathbf{RV}_{1}(M)$ is divisible (which is
the case for example if $M\models\mathrm{ACVF}$) or that
$\mathbf{P}\mathbf{\Gamma}(M)$ is a pure subgroup of $\mathbf{\Gamma}(M)$.
Then in any $\mathbf{k}$-$\mathbf{\Gamma}$-enrichment of $M$, the theory of
$M$ is determined by that of $\mathbf{k}$ and $\mathbf{\Gamma}$ and
$\mathbf{k}$ and $\mathbf{\Gamma}$ are purely stably embedded and orthogonal.

###### Proof.

By Corollary 2.2.7 together with Lemma 2.2.4. ∎

######  Remark 2.2.10.

If $\mathbf{P}\mathbf{k}$ (resp. $\mathbf{P}\mathbf{\Gamma}$) is (purely)
stably embedded in $\mathbf{k}$ (resp. $\mathbf{\Gamma}$), we can can also
deduce that $\mathbf{P}\mathbf{K}$ is (purely) stably embedded.

######  Remark 2.2.11.

If we further assume that $\mathbf{P}(M)$ eliminates fields quantifiers in
$\mathcal{L}_{\mathbf{RV}_{1}}$ — for example if it is algebraically closed or
algebraically maximal Kaplansky and of equicharacteristic — then all the above
results can easily be adapted to pairs of
$\mathcal{L}_{\mathbf{RV}_{1}}$-structures (with no need for the rather exotic
hybrid $\mathbf{RV}$-structures).

####  2.2.2. Characteristic 0 Laurent series fields

Let $F$ be a field of characteristic 0, and let $K:=F((t))$. In what follows,
we are interested in the pair of valued fields $(K^{\mathrm{a}},K)$. Let us
first deal with the pair of value groups. Let $\mathcal{L}_{\mathrm{og}}$ be
the language of ordered groups and DOAG be the theory of non trivial divisible
ordered abelian groups. Let also $\mathcal{L}_{\mathrm{Pres}}$ be the language
$\mathcal{L}_{\mathrm{og}}$ enriched with a constant $1$ and unary predicates
for divisibility by integers. Let PRES be the
$\mathcal{L}_{\mathrm{Pres}}$-theory of $\mathbb{Z}$.

######  Lemma 2.2.12.

Let $T_{\mathbb{Q},\mathbb{Z}}$ be the theory of all structures
$(\Gamma,\Delta)$ with $\Gamma\models\mathrm{DOAG}$,
$\Delta\models\mathrm{PRES}$ and such for any $\gamma\in\Gamma$ there is a
largest $\delta=:\lfloor\gamma\rfloor\in\Delta$ with $\delta\leqslant\gamma$,
considered in the language $\mathcal{L}_{\mathbb{Q},\mathbb{Z}}$ given by
$\mathcal{L}_{\mathrm{og},\mathbf{P}}$ together with
$\mathcal{L}_{\mathrm{Pres}}$ on the predicate $\mathbf{P}$ and the function
$\lfloor\cdot\rfloor$. Then $T_{\mathbb{Q},\mathbb{Z}}$ eliminates quantifiers
and is complete, and $\mathbf{P}$ is purely stably embedded.

###### Proof.

Let $M=(\Gamma,\Delta)$ and $M^{\prime}=(\Gamma^{\prime},\Delta^{\prime})$ be
models of $T_{\mathbb{Q},\mathbb{Z}}$, with $M$ countable and $M^{\prime}$
$\aleph_{1}$-saturated. Let $(A,\mathbf{P}(A))\leqslant M$,
$(A^{\prime},\mathbf{P}(A^{\prime}))\leqslant M^{\prime}$ and
$f:(A,\mathbf{P}(A))\cong(A^{\prime},\mathbf{P}(A^{\prime}))$. We claim that
$f$ extends to an embedding of $M$ into $M^{\prime}$. Clearly, we may assume
that $A$ is a subgroup of $\Gamma$.

Step 1. _We may assume that $\mathbf{P}(A)=\Delta$._

Indeed, we may extend $f\upharpoonright_{\mathbf{P}(A)}$ to an
$\mathcal{L}_{\mathrm{Pres}}$-embedding
$f^{\prime}:\Delta\rightarrow\Delta^{\prime}$, since PRES admits quantifier
elimination and $\Delta^{\prime}$ is an $\aleph_{1}$-saturated model of PRES.
As $A\cap\Delta=\mathbf{P}(A)$, $f$ together with $f^{\prime}$ yield an
embedding of abelian groups
$\tilde{f}:A+\Delta\hookrightarrow\Gamma^{\prime}$. The map $\tilde{f}$
preserves the ordering since, up to translation by an element from
$\mathbf{P}(A)$, any element of $A$ lies in $[0,1)$. Now, for any $z\in\Delta$
and $a\in A$ we have $\lfloor a+z\rfloor=\lfloor a\rfloor+z$. So $\tilde{f}$
respects $\lfloor\cdot\rfloor$ (and $\mathbf{P}$) and is thus an
$\mathcal{L}_{\mathbb{Q},\mathbb{Z}}$-embedding.

Step 2. _We may assume that $A$ is divisible._

Indeed, if $n\in\mathbb{Z}_{>0}$ and $a\in A$, letting $z:=\lfloor a\rfloor$
one has $\lfloor\frac{a}{n}\rfloor=\frac{z-i}{n}$, where
$i\in\\{0,\ldots,n-1\\}$ is (the unique element) such that $z-i$ is divisible
by $n$ in $\Delta$. It follows that that
$\tilde{f}(\frac{a}{n}):=\frac{f(a)}{n}$ defines an
$\mathcal{L}_{\mathbb{Q},\mathbb{Z}}$-embedding of the divisible hull of $A$
which extends $f$, as $f(\Delta)$ is relatively divisible in
$\Delta^{\prime}$.

Step 3. _Let $b\in\Gamma\setminus A$. Then $f$ extends to an
$\mathcal{L}_{\mathbb{Q},\mathbb{Z}}$-embedding $\tilde{f}:\langle
Ab\rangle\hookrightarrow M^{\prime}$._

Indeed, replacing $b$ by $b-\lfloor b\rfloor$, we may assume that $0<b<1$. By
quantifier elimination in DOAG and $\aleph_{1}$-saturation of $M^{\prime}$, we
find $b^{\prime}\in\Gamma^{\prime}$ such that $b\mapsto b^{\prime}$ defines an
extension of $f$ to an embedding of ordered abelian groups $\tilde{f}:\langle
Ab\rangle\hookrightarrow\Gamma^{\prime}$. We claim that $\tilde{f}$ is an
$\mathcal{L}_{\mathbb{Q},\mathbb{Z}}$-embedding. It is clear that it is an
$\mathcal{L}_{og,\mathbf{P}}$-embedding. Moreover, given
$z\in\mathbb{Z}\setminus\\{0\\}$ and $a\in A$, setting $\delta=\lfloor
a\rfloor$, one has $\lfloor zb+a\rfloor=\lfloor zb+a-\delta\rfloor+\delta$, so
in order to show that $\tilde{f}$ preserves $\lfloor\cdot\rfloor$, it suffices
to show that $\lfloor zb+a\rfloor$ is uniquely determined by $a$ and the cut
defined by $b$ over $A$ in case $0\leqslant a<1$. This is clear, as
$zb+a\in[-|z|,|z|+1)$ and for any $\delta\in\Delta\cap[-|z|,|z|+1)$ and
$\lozenge\in\\{\leqslant,<,\geqslant,>\\}$ we have

$zb+a\,\lozenge\,\delta\Leftrightarrow\begin{cases}b\,\lozenge\,\frac{\delta-a}{z},\text{
if }z>0\\\ \frac{\delta-a}{z}\,\lozenge\,b,\text{ if }z<0,\end{cases}.$

proving Step 3.

Iterating steps 2 and 3, this proves the claim, yielding quantifier
elimination and also completeness, as $(\mathbb{Q},\mathbb{Z})$ is a
substructure of any model of $T_{\mathbb{Q},\mathbb{Z}}$. ∎

In fact, the proof of Lemma 2.2.12 yields the following more general result.

######  Remark 2.2.13.

Let $\mathcal{L}^{+}\supseteq\mathcal{L}_{\mathrm{Pres}}$ and
$T^{+}\supseteq\mathrm{PRES}$ be a complete $\mathcal{L}^{+}$-theory with
quantifier elimination. Then the corresponding expansion
$T_{\mathbb{Q},\mathbb{Z}}^{+}$ of $T_{\mathbb{Q},\mathbb{Z}}$ is complete,
eliminates quantifiers, and $\mathbf{P}$ is purely stably embedded with
induced structure given by $\mathcal{L}^{+}$.

As $T_{\mathbb{Q},\mathbb{Z}}$ admits the complete model
$(\mathbb{R},\mathbb{Z})$, it is definably complete. Actually, this even holds
resplendently, as shows the following corollary.

######  Corollary 2.2.14.

Assume that the expansion $T^{+}\supseteq\rm{PRES}$ is definably complete.
Then $T_{\mathbb{Q},\mathbb{Z}}^{+}$ is definably complete.

###### Proof.

Let $(\Gamma,\Delta)\models T_{\mathbb{Q},\mathbb{Z}}^{+}$ and let
$D\subseteq\Gamma$ be a definable subset which is bounded and non-empty. Then
$\lfloor D\rfloor$ is a definable subset of $\Delta$ which is non-empty and
bounded. By assumption, it admits a supremum $s$ in $\Delta$, which is then
the maximum of $\lfloor D\rfloor$.

Quantifier elimination in $T_{\mathbb{Q},\mathbb{Z}}^{+}$ entails that the
structure induced on $[s,s+1]$ is $o$-minimal, so $\sup D=\sup D\cap[s,s+1]$
exists in $[s,s+1]\subseteq\Gamma$. ∎

Let us now consider the residue field. By a classical result of Keisler
[Kei64], if $F$ and $F^{\prime}$ are fields such that $F\equiv F^{\prime}$,
then $(F^{\mathrm{a}},F)\equiv(F^{\prime\mathrm{a}},F^{\prime})$. If
$F=F^{\mathrm{a}}$ or $F^{\mathrm{a}}$ is real closed, then the
axiomatization, in $\mathcal{L}_{ring,\mathbf{P}}$, of $(F^{\mathrm{a}},F)$ is
clear, and $\mathbf{P}$ is stably embedded with induced structure given by
$F$. In case $T_{f}$ is a complete theory of fields whose models are neither
algebraically nor real closed, and $F\models T_{f}$, then the models of the
$\mathcal{L}_{ring,\mathbf{P}}$-theory $T_{f,a}$ of $(F^{\mathrm{a}},F)$ are
precisely the pairs $(M,\mathbf{P}(M))$ of fields such that $M=M^{\mathrm{a}}$
and $\mathbf{P}(M)\models T_{f}$. By [HKR18, Theorem 4.7], if one definably
expands the theory, adding relation symbols $\mathbf{ld}_{n}$ and function
symbols $\ell_{n,i}$, the theory $T_{f,a}$ eliminates quantifiers relative to
$\mathbf{P}$, even resplendently on $\mathbf{P}$. This yields in particular
the following.

######  Fact 2.2.15.

For $T_{f}=\mathrm{Th}(F)$ a complete theory of fields (in arbitrary
characteristic), the predicate $\mathbf{P}$ is stably embedded in the
$\mathcal{L}_{ring,\mathbf{P}}$-theory
$T_{f,a}=\mathrm{Th}(F^{\mathrm{a}},F)$, with induced structure given by
$T_{f}$.

This even holds resplendently, for any
$\mathcal{L}\supseteq\mathcal{L}_{ring}$ and any $\mathcal{L}$-expansion of
$F$.

######  Lemma 2.2.16.

Let $F$ be some (enriched) field which eliminates $\exists^{\infty}$. Then the
pair $(F^{\mathrm{a}},F)$ also eliminates $\exists^{\infty}$.

###### Proof.

We may suppose that $F$ is neither algebraically nor real closed, as otherwise
the result is clear. By the relative quantifier elimination result [HKR18,
Theorem 4.7] already mentioned above, if
$(M,\mathbf{P}(M))\equiv(F^{\mathrm{a}},F)$, and
$(M,\mathbf{P}(M))\preccurlyeq(\mathcal{U},\mathbf{P}(\mathcal{U}))$ then for
any $a,b\in\mathcal{U}\setminus(M\mathbf{P}(\mathcal{U}))^{\mathrm{a}}$ we
have $tp(a/M)=tp(b/M)=:p_{gen}(x)$, so any element of $\mathcal{U}$ is the sum
of two realizations of $p_{gen}$. By compactness, it follows that for a
definable subset $D\subseteq M$ we have $D+D=M$ if and only if
$p_{gen}(x)\vdash x\in D$.

Let $\varphi(x,y)$ be a formula with $x$ a single variable. Assume that $c$ is
a tuple from $M$ such that $D=\varphi(M,c)$ is infinite and
$p_{gen}(x)\not\vdash x\in D$. Choose $a\not\in M$ such that
$\models\varphi(a,c)$. It follows that there is
$a^{\prime}\in(\mathrm{dcl}(Ma)\cap\mathbf{P}(\mathcal{U}))\setminus\mathbf{P}(M)$,
so we find an $M$-definable function $f:D\rightarrow\mathbf{P}(M)$ with
infinite image. Using 2.2.15 and the assumption that the theory of $F$
eliminates $\exists^{\infty}$, we conclude. ∎

Fix some characteristic zero field $F$ and let $T^{\star}_{\mathrm{Laur}}$ be
the theory of separated pairs $(M,\mathbf{P}(M))$ with
$(\mathbf{k}(M),\mathbf{P}\mathbf{k}(M))\equiv(F^{\mathrm{a}},F)$ and
$(\mathbf{\Gamma}(M),\mathbf{P}\mathbf{\Gamma}(M))\models
T_{\mathbb{Q},\mathbb{Z}}$.

######  Proposition 2.2.17.

The theory $T^{\star}_{\mathrm{Laur}}$ is complete, the definable sets
$\mathbf{k}$, $\mathbf{\Gamma}$, $\mathbf{RV}$, $\mathbf{P}\mathbf{K}$,
$\mathbf{P}\mathbf{k}$, $\mathbf{P}\mathbf{\Gamma}$ and
$\mathbf{P}\mathbf{RV}$ are all purely stably embedded and $\mathbf{k}$ and
$\mathbf{\Gamma}$ are orthogonal. All these results hold resplendently for
$\mathbf{P}\mathbf{k}$-$\mathbf{P}\mathbf{\Gamma}$-enrichments, and even if
one adds a coherent system of angular components.

###### Proof.

It suffices to combine Corollary 2.2.9, Remark 2.2.10 with 2.2.15 and Lemma
2.2.12. ∎

####  2.2.3. Finitely ramified fields

We will now prove analogous statements in mixed characteristic. Let $K$ be a
complete mixed characteristic $\mathbb{Z}$-valued field with perfect residue
field $F$. We are interested in the pair of valued fields
$(K^{\mathrm{a}},K)$. Let $T^{\star}_{\mathrm{Witt}}$ be the theory of
separated pairs of henselian valued fields — with relevant contants — such
that $(\mathbf{k}(M),\mathbf{P}\mathbf{k}(M))\equiv(F^{\mathrm{a}},F)$ and
$(\mathbf{\Gamma}(M),\mathbf{P}\mathbf{\Gamma}(M))\models
T_{\mathbb{Q},\mathbb{Z}}$.

######  Proposition 2.2.18.

The theory $T^{\star}_{\mathrm{Witt}}$ is complete, the definable sets
$\mathbf{k}$, $\mathbf{\Gamma}$, $\mathbf{RV}^{\mathrm{hyb}}$,
$\mathbf{P}\mathbf{K}$, $\mathbf{P}\mathbf{RV}$, $\mathbf{P}\mathbf{k}$ and
$\mathbf{P}\mathbf{\Gamma}$ are all purely stably embedded, and $\mathbf{k}$
and $\mathbf{\Gamma}$ are orthogonal. All these results hold resplendently for
$\mathbf{P}\mathbf{k}$-$\mathbf{P}\mathbf{\Gamma}$-enrichments, and even if
one adds a coherent system of angular components.

###### Proof.

One may argue as in the proof of Proposition 2.2.17. ∎

####  2.2.4. Divisible value group

Let $F$ be a field. If $\rm{char}(F)=p>0$, assume that $F$ does not admit a
finite extension of degree divisible by $p$ (in particular $F$ is perfect).
Let $K:=F((t^{\mathbb{Q}}))$. In what follows, we are interested in the pair
of valued fields $(K^{\mathrm{a}},K)$.

Let $T^{\star}_{\mathrm{div}}$ be the theory of separated pairs of
equicharacteristic algebraically maximal valued fields $(M,\mathbf{P}(M))$
such that $(\mathbf{k}(M),\mathbf{P}\mathbf{k}(M))\equiv(F^{\mathrm{a}},F)$
and $\mathbf{P}\mathbf{\Gamma}(M)=\mathbf{\Gamma}(M)\models\mathrm{DOAG}$.
Note that the Kaplansky conditions are satisfied in this case.

######  Proposition 2.2.19.

The theory $T^{\star}_{\mathrm{div}}$ is complete. The definable sets
$\mathbf{k}$, $\mathbf{\Gamma}$, $\mathbf{RV}$, $\mathbf{P}\mathbf{K}$,
$\mathbf{P}\mathbf{k}$ and $\mathbf{P}\mathbf{RV}$ are all purely stably
embedded and $\mathbf{k}$ and $\mathbf{\Gamma}$ are orthogonal. All these
results hold resplendently for
$\mathbf{P}\mathbf{k}$-$\mathbf{\Gamma}$-enrichments, and even if one adds a
coherent system of angular components.

###### Proof.

One may argue as in the proof of Proposition 2.2.17. ∎

###  2.3. Valued difference fields

Let $(K,v,\sigma)$ be a valued field with an automorphism.

######  Definition 2.3.1.

  1. (1)

For every $P\in K[x_{0},\ldots,x_{n}]$, $a\in K$, $d\in K^{n+1}$ and
$\gamma\in\mathbf{\Gamma}(K)^{\times}$, we say that $(P,a,d,\gamma)$ is in
_$\sigma$ -Hensel configuration_ if

$\mathrm{v}(P(\nabla a))>\min_{i}\mathrm{v}(d_{i})+\sigma^{i}(\gamma),$

where $\nabla a:=(\sigma^{i}(a))_{i\geqslant 0}$, and, for all $x,y\in K$ with
$v(x-a),v(y-a)>\gamma$,

$\mathrm{v}(f(\nabla y)-f(\nabla
x)-d\cdot\nabla(y-x))>\min_{i}\mathrm{v}(d_{i}\sigma^{i}(y-x)).$

  2. (2)

We say that $(K,v,\sigma)$ is _$\sigma$ -henselian_ if for every
$(P,a,d,\gamma)$ in $\sigma$-Hensel configuration, there exists there exists
$c\in K(M)$ such that $P(\nabla c)=0$ and

$\mathrm{v}(c-a)\geqslant\max_{i,d_{i}\neq 0}\mathrm{v}(\sigma^{-i}(P(\nabla
a))d_{i}^{-1}).$

In [Azg10, Pal12, DO15] an _a priori_ weaker notion of $\sigma$-henselian
fields is considered. They are, in fact, equivalent since both allow to prove
2.3.4 and both hold in maximally complete fields:

######  Fact 2.3.2.

Assume that for every linear non constant
$L\in\mathbf{k}(K)[x_{0},\ldots,x_{n}]$ and $c\in\mathbf{k}(K)$, there exists
$a\in\mathbf{k}(K)$ such that $L(\nabla(a))=c$ and either:

  * •

$K$ is maximally complete;

  * •

$K$ is complete (rank one).

Then $(K,v,\sigma)$ is $\sigma$-henselian.

The Newton approximation proof works, _cf._ [Rid17, Proposition 4.14]. Note
that at each step the approximation to the root of a $\sigma$-Hensel
configuration $(P,a,d,\gamma)$ improves by at least $\gamma$, _cf._ [Rid17,
Lemma 4.16], and hence, in rank one, completeness suffices.

######  Remark 2.3.3.

Conversely, if $(K,v,\sigma)$ is $\sigma$-henselian, then for every linear non
constant $L\in\mathbf{k}(K)[x_{0},\ldots,x_{n}]$,
$L(\nabla(\mathbf{k}(K)))=\mathbf{k}(K)$.

Let $\mathcal{L}_{\mathbf{RV}}^{\sigma}$ be the language
$\mathcal{L}_{\mathbf{RV}}$ with two new unary functions
$\sigma_{\mathbf{K}}:\mathbf{K}\to\mathbf{K}$ and
$\sigma_{\mathbf{RV}}:\mathbf{RV}\to\mathbf{RV}$. The expected quantifier
elimination result also holds in characteristic zero $\sigma$-henselian
fields, by [DO15, Theorem 7.3] in equicharacteristic 0, and by [Rid17, p. 41,
Theorem A] in mixed characteristic:

######  Fact 2.3.4.

Let $\mathcal{L}$ be an $\mathbf{RV}$-enrichment of
$\mathcal{L}_{\mathbf{RV}}^{\sigma}$ and $T$ an $\mathcal{L}$-theory
containing the theory $\mathrm{Hen}^{\sigma}_{0}$ of characteristic zero
$\sigma$-henselian valued fields. Then $T$ eliminates fields quantifiers.

Note that it follows from field quantifier elimination that $\mathbf{RV}$ is
stably embedded and its induced structure is that of a
$\mathbf{\Gamma}$-$\mathbf{k}$-enrichment of the short exact sequence of
$\mathbb{Z}[\sigma]$-modules
$1\to\mathbf{k}^{\times}\to\mathbf{RV}^{\times}\to\mathbf{\Gamma}^{\times}\to
0$.

In order to obtain model complete theories, one often restricts the behaviour
of the automorphism on the value group, _e.g._ the class of (existentially
closed) multiplicative difference valued fields introduced in [Pal12]:

######  Definition 2.3.5.

Let $\mathrm{VFA}^{\mathrm{mult}}_{0,0}$ be the theory of $\sigma$-henselian
valued fields such that:

  1. (1)

for every $P\in\mathbb{Z}[\sigma]$, either
$P(\mathbf{\Gamma}_{>0})=\mathbf{\Gamma}_{>0}$,
$P(\mathbf{\Gamma}_{<0})=\mathbf{\Gamma}_{>0}$ or $P(\mathbf{\Gamma})=0$;

  2. (2)

we have $(\mathbf{k},\sigma_{\mathbf{k}})\models\mathrm{ACFA}_{0}$

  3. (3)

the embedding of $\mathbb{Z}[\sigma]$-modules
$\mathbf{k}^{\times}\to\mathbf{RV}$ is pure.

######  Remark 2.3.6.

Two multiplicative behaviours of $\sigma$ are of particular interest.

  1. (1)

The $\omega$-increasing case — _i.e._ for all $x\in\mathcal{O}$ and
$n\in\mathbb{Z}_{>0}$, $\mathrm{v}(\sigma(x))\geqslant n\mathrm{v}(x)$ —
studied in [Azg10]. One then gets the asymptotic theory of
$(\mathbb{F}_{p}(t)^{\mathrm{a}},\mathrm{v}_{t},\phi_{p})$, where $\phi_{p}$
is the Frobenius automorphism.

  2. (2)

The isometric case, studied in [BMS07]. In that case, one gets the asymptotic
theory of $(\mathbb{C}_{p},\mathrm{v}_{p},\sigma_{p})$, where $\sigma_{p}$ is
an isometric lift of the Frobenius automorphism on
$\mathbf{k}(\mathbb{C}_{p})=\mathbb{F}_{p}^{\mathrm{a}}$.

Both characterisations follow (_e.g._ see [CH14]) from the Ax-Kochen-Ershov
principle for $\sigma$-henselian fields and Hrushovski’s deep result that
$\mathrm{ACFA}_{0}$ is the asymptotic theory of
$(\mathbb{F}_{p}^{\mathrm{a}},\phi_{p})$, _cf._ [Hru].

######  Fact 2.3.7.

In $\mathrm{VFA}^{\mathrm{mult}}_{0,0}$, $\mathbf{k}$ is a stably embedded
pure difference field and $\mathbf{\Gamma}$ is a stably embedded $o$-minimal
pure ordered $\mathbb{Z}[\sigma]$-module and they are orthogonal. These
results also hold if one adds a $\sigma$-equivariant angular component.

###  2.4. Linear structures

Let us now recall the results of [Hru12] on linear structures. In the case of
valued difference fields, we will need “twisted” versions of these results. As
it took us a while to get the arguments clear, we decided to spell them out in
detail.

####  2.4.1. Independent amalgamation

We will first recall some material from [Hru12, Section 4]. We fix a complete
stable theory $T$ in some language $\mathcal{L}$, and we assume that $T$
eliminates quantifiers and imaginaries. Let $\mathcal{U}\models T$ be a
monster model. Let $\mathcal{C}_{T}$ be the category of small algebraically
closed subsets of $\mathcal{U}$, with $\mathcal{L}$-embeddings (which are
$\mathcal{L}$-elementary in $T$ by assumption) as morphisms. Let
$\mathbf{3}=\\{0,1,2\\}$, and set
$\mathcal{P}(\mathbf{3})^{-}:=\mathcal{P}(\mathbf{3})\setminus\\{\mathbf{3}\\}$.
We consider $\mathcal{P}(\mathbf{3})^{-}$ and $\mathcal{P}(\mathbf{3})$ as
categories, with inclusion maps as morphisms.

######  Definition 2.4.1.

Let $P$ equal $\mathcal{P}(\mathbf{3})^{-}$ or $\mathcal{P}(\mathbf{3})$.

  1. (1)

A functor $A:P\rightarrow\mathcal{C}_{T}$ is called _independence preserving_
if for any $w,w^{\prime}\in P$ with $w\cup w^{\prime}\in P$ one has
$A(w)\downfree_{A(w\cap w^{\prime})}A(w^{\prime})$ (inside $A(w\cup
w^{\prime})$).111Here and in what follows, if $w_{1}\subseteq w_{2}$, with
inclusion map $\iota$, we often consider $A(w_{1})$ as a subset of $A(w_{2})$,
thus omitting the map $A(\iota)$ in our notation.

  2. (2)

A functor $A:P\rightarrow\mathcal{C}_{T}$ is called _bounded_ if for any
$\emptyset\neq w\in P$ one has $A(w)=\mathrm{acl}(\bigcup_{i\in
w}A(\\{i\\}))$.

  3. (3)

A _3-amalgamation problem_ in $T$ is a bounded independence preserving functor
$A^{-}:\mathcal{P}(\mathbf{3})^{-}\rightarrow\mathcal{C}_{T}$. A _solution_ of
$A^{-}$ is a bounded independence preserving functor
$A:\mathcal{P}(\mathbf{3})\rightarrow\mathcal{C}_{T}$ extending $A^{-}$.

It follows from stability and elimination of imaginaries in $T$ that every
3-amalgamation problem has a solution in $T$, i.e., $T$ has _3-existence_.

######  Definition 2.4.2.

The theory $T$ is said to have _3-uniqueness_ if whenever $A$ and $A^{\prime}$
are solutions of the same 3-amalgamation problem $A^{-}$ in $T$, then $A$ and
$A^{\prime}$ are isomorphic over $A^{-}$, i.e., there is an
$\mathcal{L}$-isomorphism $f:A(\mathbf{3})\cong A^{\prime}(\mathbf{3})$ fixing
$A^{-}(w)$ pointwise for every $w\in\mathcal{P}(\mathbf{3})^{-}$.

######  Remark 2.4.3.

In the terminology of [Hru12], this notion corresponds to 3-uniqueness of $T$
over every parameter set.

Let $\sigma$ be a new unary function symbol,
$\mathcal{L}_{\sigma}:=\mathcal{L}\cup\\{\sigma\\}$. Consider the category
$\widetilde{\mathcal{C}}_{T}$ of $\mathcal{L}_{\sigma}$-structures of the form
$(A,\sigma)$, where $A\in\mathcal{C}_{T}$ and
$\sigma\in\mathrm{Aut}_{\mathcal{L}}(A)$, with
$\mathcal{L}_{\sigma}$-embeddings as morphisms.

######  Definition 2.4.4.

Let $P$ equal $\mathcal{P}(\mathbf{3})^{-}$ or $\mathcal{P}(\mathbf{3})$.

  1. (1)

A functor $A:P\rightarrow\widetilde{\mathcal{C}}_{T}$ is called _independence
preserving_ (_bounded_ , respectively), if it is so when composed with the
forgetful functor from $\widetilde{\mathcal{C}}_{T}$ to $\mathcal{C}_{T}$.

  2. (2)

We say that $\widetilde{\mathcal{C}}_{T}$ has _3-existence_ if every bounded
independence preserving functor
$A^{-}:\mathcal{P}(\mathbf{3})^{-}\rightarrow\widetilde{\mathcal{C}}_{T}$
extends to a bounded independence preserving functor
$A:\mathcal{P}(\mathbf{3})\rightarrow\widetilde{\mathcal{C}}_{T}$.

######  Fact 2.4.5 ([Hru12, Proposition 4.5 and Proposition 4.7]).

For $T$ as above, the following are equivalent:

  1. (1)

$T$ has 3-uniqueness.

  2. (2)

$\widetilde{\mathcal{C}}_{T}$ has 3-existence.

Moreover, assuming in addition that $T\\!A$ exists, the above conditions
imply:

  1. (3)

$T\\!A$ eliminates imaginaries.∎

It is easy to see that if $T\\!A$ exists and it eliminates bounded
hyperimaginaries (e.g., when $T$ is superstable), then (3) is actually
equivalent to (1) and (2). We will not use this in our paper.

####  2.4.2. Twisted independent amalgamation

Let $\tau:\mathcal{L}\cong\mathcal{L}^{\prime}$ be a bijection between two
first order languages (sending sorts to sorts, functions symbols to functions
symbols consistently with their arity, similarly for constants and relations).
Then $\tau$ extends naturally to a bijection between the set of
$\mathcal{L}$-formulas and the set of $\mathcal{L}^{\prime}$-formulas. Given
an $\mathcal{L}$-formula $\varphi$, we denote by $\varphi^{\tau}$ its image
under this map. If $T$ is an $\mathcal{L}$-theory,
$T^{\tau}:=\\{\varphi^{\tau}:\varphi\in T\\}$ is an
$\mathcal{L}^{\prime}$-theory. Of course, up to changing the names of the
symbols using $\tau$, $T^{\tau}$ is the “same” theory as $T$.

If $M$ is an $\mathcal{L}$-structure, we denote by $M^{\tau}$ the
$\mathcal{L}^{\prime}$-structure with base set $M$ and interpretations
$\left(\Sigma^{\tau}\right)^{M^{\tau}}=\Sigma^{M}$, for any symbol
$\Sigma\in\mathcal{L}$. If $N^{\prime}$ is an
$\mathcal{L}^{\prime}$-structure, we call an
$\mathcal{L}^{\prime}$-isomorphism $\sigma:M^{\tau}\cong N^{\prime}$ a _$\tau$
-twisted isomorphism_. Similarly, one defines the notion of a _$\tau$ -twisted
elementary map_ $\sigma:A\rightarrow A^{\prime}$, where $A\subseteq M$ and
$A^{\prime}\subseteq N^{\prime}$, i.e., one requires that for any
$\mathcal{L}$-formula $\varphi(x)$ and any tuple $a$ from $A$ of the right
length, one has $M\models\varphi(a)$ if and only if
$N^{\prime}\models\varphi^{\tau}(\sigma(a))$.

######  Lemma 2.4.6.

Let $T$ be a complete stable $\mathcal{L}$-theory eliminating quantifiers and
imaginaries. Assume that $T$ has 3-uniqueness. Let
$A:\mathcal{P}(\bf{3})\rightarrow\mathcal{C}_{T}$ and
$A^{\prime}:\mathcal{P}(\bf{3})\rightarrow\mathcal{C}_{T^{\tau}}$ be bounded
independence preserving functors.

Then for any coherent system $(\sigma_{w})_{w\in\mathcal{P}(\bf{3})^{-}}$ of
$\tau$-twisted elementary bijections $\sigma_{w}:A(w)\rightarrow
A^{\prime}(w)$ there exists a $\tau$-twisted elementary bijection
$\sigma_{\bf{3}}:A({\bf{3}})\rightarrow A^{\prime}(\bf{3})$ extending
$\sigma_{w}$ for every $w\in\mathcal{P}(\bf{3})^{-}$.

###### Proof.

The result follows from 3-uniqueness of $T^{\tau}$, since we may consider $A$
as a functor to $\mathcal{C}_{T^{\tau}}$, replacing $\mathcal{U}\models T$ by
$\mathcal{U}^{\tau}\models T^{\tau}$. ∎

We now consider the special case where $\mathcal{L}^{\prime}=\mathcal{L}$,
$\tau$ is a permutation of $\mathcal{L}$ and $T$ is a complete
$\mathcal{L}$-theory such that $T=T^{\tau}$. Let $\widetilde{C}^{(\tau)}_{T}$
be the category of $\mathcal{L}_{\sigma}$-structures $(B,\sigma)$ with
$B\in\mathcal{C}_{T}$ and $\sigma:B\rightarrow B$ a $\tau$-twisted elementary
bijection. When $T$ is stable, we use the same terminology as in Definition
2.4.4, for functors $A:P\rightarrow\widetilde{\mathcal{C}}^{(\tau)}_{T}$.
Lemma 2.4.6 then yields:

######  Corollary 2.4.7.

Let $T$ be a complete stable $\mathcal{L}$-theory eliminating quantifiers and
imaginaries, and let $\tau:\mathcal{L}\rightarrow\mathcal{L}$ be a bijection
such that $T^{\tau}=T$. Assume that $T$ has 3-uniqueness. Then
$\widetilde{\mathcal{C}}^{(\tau)}_{T}$ has 3-existence.∎

Given a complete $\mathcal{L}$-theory $T$ and a permutation $\tau$ of
$\mathcal{L}$ such that $T^{\tau}=T$, we let $T^{(\tau)}_{\sigma}$ be the
$\mathcal{L}_{\sigma}$-theory whose models are of the form $(M,\sigma)$, where
$M\models T$ and where $\sigma$ is a $\tau$-twisted automorphism of $M$.

Assume now in addition that $T$ is stable and eliminates quantifiers and
imaginaries. Then $T^{(\tau)}_{\sigma}$ is a $\forall\exists$-theory, and so
it has a model-companion if and only if the e.c. models of
$T^{(\tau)}_{\sigma}$ form an elementary class. If this is the case, denote by
$T^{(\tau)}\\!A$ the model-companion of $T^{(\tau)}_{\sigma}$. Then the models
of $T^{(\tau)}\\!A$ are precisely the e.c. models of of $T^{(\tau)}_{\sigma}$.
The basic results on $T\\!A$, due to Chatzidakis and Pillay ([CP98]),
generalize to this context in a straight forward manner. We obtain for
instance:

######  Fact 2.4.8.

Let $T$ and $\tau$ be as above, and assume that $T^{(\tau)}\\!A$ exists. Then
the following holds:

  1. (1)

If $(M,\sigma)\models T^{(\tau)}\\!A$ and $B\subseteq M$ then

$\mathrm{acl}_{(M,\sigma)}(B)=\mathrm{acl}_{\sigma}(B):=\mathrm{acl}_{M}(\bigcup_{z\in\mathbb{Z}}\sigma^{z}(B)).$

  2. (2)

Quantifier reduction: If $(M_{i},\sigma_{i})\models T^{(\tau)}\\!A$ and
$B_{i}\subseteq M_{i}$ for $i=1,2$, then
$B_{1}\equiv_{\mathcal{L}_{\sigma}}B_{2}$ if and only if there is an
$L_{\sigma}$-isomorphism from $\mathrm{acl}_{\sigma}(B_{1})$ to
$\mathrm{acl}_{\sigma}(B_{2})$ sending $B_{1}$ to $B_{2}$.

  3. (3)

$T^{(\tau)}\\!A$ is simple and

$A\downfree^{T^{(\tau)}\\!A}_{E}B\text{ if and only if
}\mathrm{acl}_{\sigma}(A)\downfree^{T}_{\mathrm{acl}_{\sigma}(E)}\mathrm{acl}_{\sigma}(B).$

If $T$ is superstable, then $T^{(\tau)}\\!A$ is supersimple.

  4. (4)

Assume that $\widetilde{\mathcal{C}}^{(\tau)}$ has 3-existence. (Equivalently,
in $T^{(\tau)}\\!A$, the independence theorem holds over
$\mathrm{acl}_{\sigma}$-closed sets.) Then $T^{(\tau)}\\!A$ eliminates
imaginaries.

###### Proof.

The items (1–3) may be proved as in [CP98]. Item (4) is the analog of [Hru12,
Proposition 4.7], and it follows directly from a formalization of Hrushovski’s
argument by Montenegro and the second author (see [MR21, Proposition 1.17]). ∎

####  2.4.3. Linear imaginaries

Let us now recall some notions from [Hru12, Section 5].

######  Definition 2.4.9.

Let $\mathfrak{t}$ be a theory of fields (possibly with additional structure).
A _$\mathfrak{t}$ -linear structure_ $M$ is an $\mathcal{L}$-structure with a
sort $\mathbf{k}$ for a model of $\mathfrak{t}$, and additional sorts $V_{i}$
($i\in I$) denoting finite-dimensional $\mathbf{k}$-vector spaces, such that
the family $\left(V_{i}\right)_{i\in I}$ is closed under tensor products and
duals. Each $V_{i}$ has (at least) the $\mathbf{k}$-vector space structure.
One assumes that $\mathbf{k}$ is stably embedded in $M$ with induced structure
given by $\mathfrak{t}$.

We now fix such a $\mathfrak{t}$-linear structure $M$.

  1. (1)

$M$ is said to _have flags_ if for any $i$ with $\dim(V_{i})>1$, for some
$j,k$ with $\dim(V_{j})=\dim(V_{i})-1$, there exists a $\emptyset$-definable
exact sequence $0\longrightarrow V_{k}\longrightarrow V_{i}\longrightarrow
V_{j}\longrightarrow 0$. We will call such a short exact sequence a _flag_.

  2. (2)

$M$ is said to _have roots_ if for any one-dimensional $V=V_{i}$, and any
$m\geqslant 2$, there exists a (one-dimensional) $W=V_{j}$ and a
$\emptyset$-definable $\mathbf{k}$-linear isomorphism $f:W^{\otimes m}\cong
V$.

Let us now mention two results from [Hru12]. The proof of the first one is
rather elementary, whereas that of the second one is quite involved.

######  Fact 2.4.10 ([Hru12, Lemma 5.6]).

The theory of an $\mathrm{ACF}$-linear structure with flags (in any
characteristic) eliminates imaginaries.

The following fact follows from [Hru12, Proposition 5.7] in combination with
[Hru12, Proposition 4.3 and Corollary 4.10].

######  Fact 2.4.11.

Let $T$ be the theory of an $\mathrm{ACF}_{0}$-linear structure with flags and
roots. Then $T$ has 3-uniqueness.

Our main interest in linear structures stems from the fact that the
$\mathbf{k}$-internal sets in a given model of $\mathrm{ACVF}$ give rise to
such a structure. For every $M\models\mathrm{ACVF}$ and
$A\subseteq\mathbfcal{G}(M)$, we define

$\mathbf{Lin}_{A}:=\bigsqcup_{\begin{subarray}{c}s\in\mathbf{S}(\mathrm{dcl}_{0}(A))\\\
\ell\in\mathbb{Z}_{>0}\end{subarray}}s/\ell\mathfrak{m}s.$

In equicharacteristic zero, this is exactly the _stable part_
$\bigsqcup_{s\in\mathbf{S}(\mathrm{dcl}_{0}(A))}s/\mathfrak{m}s$ of [HHM08].
In mixed characteristic, however, this is a more complicated structure since
it also consists of (free) $\mathbf{R}_{\ell}$-modules — and this more
complicated structure is actually needed in Section 4. Note that, by 2.1.4 and
our choice of representation of the geometric sorts, $\mathbf{Lin}_{A}(M)$
denotes the set of cosets $c+\ell\mathfrak{m}s$ where
$s\in\mathbf{S}(\mathrm{dcl}_{0}(A))$ has a basis in $M$ and $c\in s(M)$.

######  Lemma 2.4.12.

Let $M\models\mathrm{Hen}_{0,0}$ and $A\subseteq\mathbfcal{G}(M)$. Then
$\mathbf{Lin}_{A}(M)$, with its $\mathcal{L}_{0}(A)$-induced structure, is an
$\mathrm{Th}(\mathbf{k}(M))$-linear structure with flags. Moreover, if
$\mathbf{\Gamma}(M)$ is divisible, then $\mathbf{Lin}_{A}(M)$ has roots.

Note that, by convention, $\mathbf{Lin}_{A}(M)$ consists of the point of the
form $c+\ell\mathfrak{m}s$ where $s\in\mathbf{S}(\mathrm{dcl}_{0}(A))$ has a
basis in $M$ and $c\in s(M)$.

###### Proof.

We may assume that $\mathrm{dcl}_{0}(A)\cap\mathbfcal{G}(M)\subseteq A$. The
fact that the residue field $\mathbf{k}$ is stably embedded in
$\mathrm{Hen}_{0,0}$, with induced structure that of a pure field, is well
known.

Now let $V,W$ be two sorts from $\mathbf{Lin}_{A}(M)$, _i.e._ vector spaces
over $\mathbf{k}$ of the form $V=a/\mathfrak{m}a$, $W=b/\mathfrak{m}b$ for
some $a\in\mathbf{S}_{m}(A)$ and $b\in\mathbf{S}_{n}(A)$ — with bases in $M$.
Then $a\otimes_{\mathcal{O}}b$ is canonically isomorphic to an element $c$
from $\mathbf{S}_{m\cdot n}(A)$, so we may identify $V\otimes_{\mathbf{k}}W$
with $c/\mathfrak{m}c$, which is a sort from $\mathbf{Lin}_{A}(M)$. Similarly,
$\check{a}$ can be identified with $\\{z\in\mathbf{K}^{n}:\forall v\in
a,\,\sum z_{i}v_{i}\in\mathcal{O}\\}\in\mathbf{S}_{m}(A)$, so
$\check{V}\cong\check{a}/\mathfrak{m}\check{a}$ is a sort from
$\mathbf{Lin}_{A}$ as well.

_Flags:_ For $a\in\mathbf{S}_{n}(A)$ define
$a_{1}:=a\cap(\mathbf{K}\times\\{(0,\ldots,0)\\}$. Then the projection onto
the first coordinate identifies $a_{1}$ with an element of
$\mathbf{S}_{1}(A)$. Let $\pi:a\rightarrow\mathbf{K}^{n-1}$ be induced from
the projection on the last $n-1$ coordinates. Then

$0\rightarrow\ker(\pi)=a_{1}\rightarrow a\rightarrow\pi(a)\rightarrow 0$

is an $A$-definable exact sequence of free $\mathcal{O}$-modules, and
$\pi(a)\in\mathbf{S}_{n-1}(A)$ — this follows from the fact that $\pi(a)$ is a
finitely generated torsion free $\mathcal{O}$-submodule of $\mathbf{K}^{n-1}$
of rank $n-1$. Reducing modulo $\mathfrak{m}$, we conclude.

_Roots:_ Assume $\mathbf{\Gamma}(M)$ is divisible. Let $n\geqslant 1$, and let
$V$ be a one-dimensional sort from $\mathbf{Lin}_{A}$. Then
$V=\gamma\mathcal{O}/\gamma\mathfrak{m}$ for some
$\gamma\in\Gamma(\mathrm{dcl}_{0}(A))$. Consider
$V_{n}:=\delta\mathcal{O}/\delta\mathfrak{m}$, for $\delta=\frac{\gamma}{n}$.
The map

$x/\gamma\mathfrak{m}\mapsto y/\delta\mathfrak{m}\otimes\ldots\otimes
y/\delta\mathfrak{m}:V\rightarrow V_{n}^{\otimes n},$

where $y^{n}=x$, is well-defined and an $A$-definable isomorphism of
$\mathbf{k}$-vector spaces defined over $A$. In particular, $\mathbf{Lin}_{A}$
has roots. ∎

The result above holds of the _stable part_
$\bigsqcup_{s\in\mathbf{S}(\mathrm{dcl}_{0}(A))}s/\mathfrak{m}s$ in all
characteristic (provided $\mathbf{k}$ is stably embedded).

######  Corollary 2.4.13.

Let $M\models\mathrm{ACVF}_{0,0}$ and $A\subseteq\mathbfcal{G}(M)$. Then
$\mathbf{Lin}_{A}$ satisfies 3-uniqueness.∎

####  2.4.4. Twisted linear imaginaries

The following result is a special case of Lemma 2.4.15, which we prove below.

######  Remark 2.4.14.

Let $\mathfrak{t}$ be a stable theory of fields, and let $T$ be the theory of
a $\mathfrak{t}$-linear structure, such that $T$ eliminates quantifiers.
Suppose $\mathfrak{t}A$ exists. Then $T\\!A$ exists and is given by
$T_{\sigma}\cup\mathfrak{t}A$. In particular, this holds for
$\mathfrak{t}=\mathrm{ACF}$.

######  Lemma 2.4.15.

Let $\mathfrak{t}$ be a stable theory of fields, and let $T$ be the theory of
a $\mathfrak{t}$-linear structure such that $T$ eliminates quantifiers. Let
$\tau$ be a permutation of the language with $T=T^{\tau}$ such that $\tau$
fixes all the formulas on the sort $\mathbf{k}$. Suppose $\mathfrak{t}A$
exists. Then $T^{(\tau)}_{\sigma}\cup\mathfrak{t}A$ is the model-companion of
$T^{(\tau)}_{\sigma}$. In particular, this holds for
$\mathfrak{t}=\mathrm{ACF}$.

###### Proof.

Let $(M,\sigma)\models T^{(\tau)}_{\sigma}$. Note that for any
$N\succcurlyeq_{\mathcal{L}}M$, we have
$N=\mathrm{dcl}_{\mathcal{L}}(M\mathbf{k}(N))$. Thus, any extension of
$\sigma$ to a $\tau$-twisted automorphism on $N$ is uniquely determined by its
restriction to $\mathbf{k}(N)$. It follows that $(M,\sigma)$ is an e.c. model
of $T^{(\tau)}_{\sigma}$ if and only if
$(\mathbf{k}(M),{\left.\sigma\right|_{\mathbf{k}(M)}})$ is an e.c. model of
$\mathfrak{t}_{\sigma}$. This yields the statement of the lemma. ∎

######  Definition 2.4.16.

Let $\mathbf{k}$ be a stably embedded sort in a theory $T$. An
$A_{0}$-definable set $D$ is said to be _internally $\mathbf{k}$-internal_
(over $A_{0}$), if there is a tuple $d\in D$ and an $A_{0}d$-definable
surjection $f:Y\rightarrow D$, where $Y\subseteq\mathbf{k}^{n}$ for some $n$.

######  Lemma 2.4.17.

Let $\mathbf{k}$ be a stably embedded sort in a theory $T$, and let $D$ be
$A_{0}$-definable and internally $\mathbf{k}$-internal (over $A_{0}$). Then
$\mathbf{k}\cup D$ is stably embedded (over $A_{0}$).

###### Proof.

Let $f$ be an $A_{0}d$-definable surjection as in the definition, with $d\in
D$. Taking the preimage under $f$, one sees that any $\mathcal{U}$-definable
subset $X$ of $\mathbf{k}^{l}\times D^{m}$ is
$\mathbf{k}(\mathcal{U})A_{0}d$-definable, by stable embeddedness of
$\mathbf{k}$. In particular, $X$ is
$A_{0}\mathbf{k}(\mathcal{U})D(\mathcal{U})$-definable, proving stable
embeddedness of $\mathbf{k}\cup D$ (over $A_{0}$). ∎

######  Proposition 2.4.18.

Let $M\models\mathrm{VFA}^{\mathrm{mult}}_{0,0}$ and
$A\subseteq\mathbfcal{G}(M)$. Then $\mathbf{Lin}_{A}$ is stably embedded in
$\mathrm{VFA}^{\mathrm{mult}}_{0,0}$ and its $A$-induced structure eliminates
imaginaries.

###### Proof.

Stable embeddedness follows from stable embeddedness of $\mathbf{k}$ — _cf._
2.3.7 — and the fact that it is internally $\mathbf{k}$-internal (by naming a
basis for every sort).

Now, let $T$ be the theory of $\mathbf{Lin}_{A}(M)$, with its
$\mathcal{L}_{0}(A)$-induced structure. By 2.4.10 and Lemma 2.4.12, $T$
eliminates imaginaries. Let $\tau$ be the permutation of $\mathcal{L}_{0}(A)$
induced by $\sigma$. Then $\tau$ fixes all the formulas on the sort
$\mathbf{k}$, and we have $T^{\tau}=T$. It follows from Corollary 2.4.13 and
2.4.8(4) that $T^{(\tau)}\\!A$ eliminates imaginaries.

Also, $(\mathbf{k}(M),\sigma_{\mathbf{k}})$ is a stably embedded pure model of
$\mathrm{ACFA}$ and hence $\mathbf{Lin}_{A}(M)\models T^{(\tau)}\\!A$. Since
the $A$-induced structure on $\mathbf{Lin}_{A}$ is a definable enrichment of
its $\mathrm{ACF}$-linear structure with a twisted automorphism, elimination
of imaginaries follows — _e.g._ [Hru12, Lemma 5.4]. ∎

####  2.4.5. Real linear imaginaries

We conclude these preliminaries with a study of $\mathrm{RCF}$-linear
structures.

######  Definition 2.4.19.

An $\mathrm{RCF}$-linear structure with flags is said to be _oriented_ if for
every sort $V$ of dimension one, the two half lines are $\emptyset$-definable.

######  Proposition 2.4.20.

Any oriented $\mathrm{RCF}$-linear structure with flags eliminates
imaginaries.

###### Proof.

Let us first prove a few preliminary results. Let $M$ be an oriented
$\mathrm{RCF}$-linear structure with flags. First, note that if $0\to W\to
V\to U\to 0$ is an $\emptyset$-definable flag, then any translate of $W$ in
$V$ is ordered by $a<b$ if $a-b$ is in a fixed half line of $W$.

######  Claim 2.4.21.

$M$ is rigid: for every $A\subseteq M$, $\mathrm{acl}(A)=\mathrm{dcl}(A)$.

###### Proof.

Let $X$ be a finite $A$-definable set. Using tensors, we may assume that $X$
is contained in some sort $V$. We proceed by induction on $\dim(V)$. Let $0\to
W\to V\to U\to 0$ be an $\emptyset$-definable flag for $V$. By induction, we
may assume that $X$ projects to a singleton $a\in U$, _i.e._ $X$ is contained
in a translate of $W$ in $V$ and we can conclude since this translate is
totally ordered. ∎

######  Claim 2.4.22.

Let $X\subseteq a+W\subseteq V$ be definable for some $\emptyset$-definable
flag $0\to W\to V\to U\to 0$ and some $a\in V$. Then $X$ is coded.

###### Proof.

Since $a+W$ is definably isomorphic to $\mathbf{k}$ which is o-minimal, $X$ is
a finite union of points and intervals and hence it is coded by its (finite)
border. ∎

Let $K=\mathbf{k}(M)^{\mathrm{a}}$ and $K\otimes M$ be the structure whose
sorts are the sorts $V$ of $M$ interpreted as $K\otimes_{\mathbf{k}(M)}V(M)$,
with the field structure on $\mathbf{k}$, the $\mathbf{k}$-vector space
structure on each $V$ and the tensor, dual and flag structure. Then $K\otimes
M$ is an $\mathrm{ACF}$-structure with flags and for every $N\preccurlyeq M$,
and $a\in M$, $\mathrm{dcl}(Na)\subseteq\mathrm{acl}_{0}(Na)$ where
$\mathrm{acl}_{0}$ denotes the algebraic closure in $K\otimes M$.

To prove elimination of imaginaries in $M$, it suffices to code every
definable function $f:V\to S$, where $S$ is a sort. Let $0\to W\to V\to U\to
0$ be an $\emptyset$-definable flag for $V$. Let us first assume that the
domain of $f$ is a translate of $W$ in $V$. Let $F$ be the Zariski closure of
the graph of $f$ in $K\otimes M$ — any choice of basis induces a Zariski
topology on $V$, but this topology is invariant under choice of coordinates.
By the above remark, for every $a\in V(A)$, $F_{a}$ is a finite set containing
$f(a)$. By 2.4.10, $F$ has a code in $\mathrm{dcl}_{0}(\ulcorner
f\urcorner)=\ulcorner f\urcorner\cap M$.

Moreover, by compactness and 2.4.21, we find $\ulcorner f\urcorner$-definable
maps $(f_{i})_{i<n}$ such that for all $a\in M$,
$F_{a}(M)=\\{f_{i}(a):i<n\\}$. Let $X_{i}=\\{x:f(x)=f_{i}(x)\\}$. There
remains to code the $X_{i}$, but this is done in 2.4.22.

We now proceed by induction on $\dim(V)$. For every $a\in U$, let $f_{a}$ be
the restriction of $f$ to the fiber above $a$ in the flag. By the above, we
may assume $g(a):=\ulcorner f_{a}\urcorner\in M$. By induction, $g$ is coded.
But $f$ is $\ulcorner g\urcorner$-definable. ∎

######  Proposition 2.4.23.

Let $(K,<,v)$ be an ordered field with a convex valuation and
$A\subseteq\mathbfcal{G}(A)$. Then $\mathbf{Lin}_{A}(K)$ is oriented.

In particular, if $v$ is henselian and $\mathbf{k}(K)\models\mathrm{RCF}$,
then $\mathbf{Lin}_{A}(K)$ is stably embedded and its $A$-induced structure
eliminates imaginaries.

###### Proof.

Dimension one sorts in $\mathbf{Lin}_{A}$ are of the form
$\gamma\mathcal{O}/\gamma\mathfrak{m}$ for some $\gamma\in\mathbf{\Gamma}(K)$.
But this quotient inherits the order on $\gamma\mathcal{O}$; so it is
oriented. The rest of the proposition follows from Lemma 2.4.12 and
Proposition 2.4.20. ∎

######  Remark 2.4.24.

Any $K\equiv\mathbb{R}((t))$ admits exactly two orders, depending on the sign
of a choice of uniformizer. Both orders are definable (using a constant for
the $\mathrm{rv}$ of said uniformizer).

##  3\. $C$-minimal definable generics

We will now consider generalisations of [Rid19, Theorem 8.7].

######  Notation.

Let $\mathcal{L}_{0}=\mathcal{L}_{\mathrm{div}}$ and $T_{0}$ be the
$\mathcal{L}_{0}$-theory $\mathrm{ACVF}$. Let
$\mathcal{L}\supseteq\mathcal{L}_{0}$ and $T$ be a (complete)
$\mathcal{L}$-theory of valued fields. Let $M\models T$ be sufficiently
saturated and homogeneous and $M_{0}=M^{\mathrm{a}}\models T_{0}$. Note that
since $\mathrm{ACVF}$ eliminates quantifiers, we will implicitly assume that
every $\mathcal{L}_{0}$-formula is quantifier free. We will denote by
$\mathrm{S}^{0}_{x}(M)$ the set of (quantifier free)
$\mathcal{L}_{0}(M)$-types (in $M_{0}$) in variables $x$ and whenever
$\Psi(x;t)$ is a set of $\mathcal{L}_{0}$-formulas, $\mathrm{S}^{\Psi}_{x}(M)$
will denote the set of $\Psi$-types over $M$.

###  3.1. Main results

In this section we prove the following two density results :

######  Theorem 3.1.1.

Assume

  * ($\mathbf{C}_{\mathbf{B}}$)

$T$ is definably spherically complete;

  * ($\mathbf{C}_{\mathbf{\Gamma}}$)

$T$ has definably complete value group;

  * ($\mathbf{E}^{\infty}_{\mathbf{k}})$

for any $M\models T$ and $\mathcal{L}(M)$-definable
$(Y_{z})_{z}\subseteq\mathbf{k}$, there exists $n\in\mathbb{Z}_{\geqslant 0}$
such that $|Y_{z}|<\infty$ implies $|Y_{z}|\leqslant n$;

  * ($\mathbf{E}^{\infty}_{\mathbf{\Gamma}})$

for any $M\models T$ and $\mathcal{L}(M)$-definable
$(Y_{z})_{z}\subseteq\mathbf{\Gamma}$, there exists $n\in\mathbb{Z}_{\geqslant
0}$ such that $|Y_{z}|<\infty$ implies $|Y_{z}|\leqslant n$.

Then, for every strict pro-$\mathcal{L}(A)$-definable
$X\subseteq\mathbf{K}^{x}$, with $x$ countable and
$A=\mathrm{acl}^{\mathrm{eq}}(A)\subseteq M^{\mathrm{eq}}\models
T^{\mathrm{eq}}$, there exists an
$\mathcal{L}_{0}(\mathbfcal{G}(A))$-definable $p\in\mathrm{S}^{0}_{x}(M)$
consistent with $X$.

In other terms, there exists $N\succcurlyeq M$ and $a\in X(N)$ such that
$\mathrm{tp}_{0}(a/M)$ is $\mathcal{L}_{0}(\mathbfcal{G}(A))$-definable.
Recall, _e.g._ [HL16, Section 2.2], that a set is strict pro-definable if it
is the limit of a small directed system of definable sets with surjective
transition maps. In other terms, it is a $\star$-definable set whose
projection on any finite set of variables is definable.

###### Proof.

This is a particular case of Proposition 3.5.1 ∎

######  Remark 3.1.2.

  * •

Any (non zero) definably complete ordered abelian group $\Gamma$ is
elementarily equivalent to either $\mathbb{Z}$ or $\mathbb{Q}$. Indeed,
$\Gamma$ is necessarily regular as it clearly cannot have a proper non trivial
definable convex subgroup. We may thus assume it is a subgroup of
$(\mathbb{R},+,<)$. If it is not elementarily equivalent to $\mathbb{Z}$ or
$\mathbb{Q}$, it is a dense non divisible subgroup of $\mathbb{R}$. For any
$\gamma\in\Gamma$ non divisible by $n\in\mathbb{Z}_{>0}$, the cut at
$\gamma/n$ yields a counter-example.

  * •

Hypotheses ($\mathbf{C}_{\mathbf{B}}$) and ($\mathbf{C}_{\mathbf{\Gamma}}$)
are necessary for the conclusion to hold.

  * •

Hypothesis ($\mathbf{E}^{\infty}_{\mathbf{\Gamma}})$ does not allow for
discrete value groups. However, since the conclusion of the theorem does not
hold in characteristic zero non-Archimedean local fields, some hypothesis is
needed beyond ($\mathbf{C}_{\mathbf{B}}$), ($\mathbf{C}_{\mathbf{\Gamma}}$)
and ($\mathbf{E}^{\infty}_{\mathbf{k}})$.

  * •

As Theorem 3.1.3 illustrates, by restricting to a mild class of enrichments of
$\mathrm{ACVF}_{\forall}$, one can trade hypothesis
($\mathbf{E}^{\infty}_{\mathbf{\Gamma}})$ for purely algebraic conditions and
a weaker conclusion.

Let $\mathrm{Hen}_{0}$ be the $\mathcal{L}_{0}$-theory of characteristic zero
henselian valued fields.

######  Theorem 3.1.3 (_cf._ Corollary 3.5.6).

Let $T$ be a $\mathbf{k}$-$\mathbf{\Gamma}$-enrichment of $\mathrm{Hen}_{0}$,
such that:

  * ($\mathbf{C}_{\mathbf{\Gamma}}$)

$T$ has definably complete value group;

  * ($\mathbf{FR}$)

for every $n\in\mathbb{Z}_{>0}$, the interval $[0,\mathrm{v}(n)]$ is finite
and $\mathbf{k}$ is perfect;

  * ($\mathbf{I}_{\mathbf{k}}$)

the residue field $\mathbf{k}$ is infinite;

  * ($\mathbf{E}^{\infty}_{\mathbf{k}})$

the induced theory on $\mathbf{k}$ eliminates $\exists^{\infty}$.

Then, for every strict pro-$\mathcal{L}(A)$-definable
$X\subseteq\mathbf{K}^{x}$, with $A=\mathrm{acl}^{\mathrm{eq}}(A)\subseteq
M^{\mathrm{eq}}\models T^{\mathrm{eq}}$, there exist an
$\mathrm{Aut}(M/\mathbfcal{G}(A))$-invariant $p\in\mathrm{S}^{0}_{x}(M)$
consistent with $X$.

Note that in this setting $\mathbf{k}$ is stably embedded and there is no
ambiguity as to what the induced structure on $\mathbf{k}$ is. So we need not
be as explicit.

######  Remark 3.1.4.

  * •

Contrary to Theorem 3.1.1, Theorem 3.1.3 requires finite ramification in mixed
characteristic. Even if Theorem 3.1.3 does not apply to characteristic zero
non-Archimedean local fields either, _cf._ the stronger [HMR18, Remark 4.7].

  * •

Under the hypotheses of Theorem 3.1.3, locally, we do find definable types:
for any _finite_ set $\Psi(x;t)$ of $\mathcal{L}_{0}$-formulas, we can find an
$\mathcal{L}_{0}(\mathbfcal{G}(A))$-definable $p\in\mathrm{S}^{\Psi}_{x}(M)$
consistent with $X$, cf. Proposition 3.5.4. This local statement does not hold
in characteristic zero non-Archimedean local fields.

  * •

In both theorems, hypothesis ($\mathbf{E}^{\infty}_{\mathbf{k}})$ is an
artefact of our proof. It is necessary to prove certain intermediate results.
However, we do not know if it is necessary to prove either theorems. Moreover,
these theorems are the only reason hypothesis
($\mathbf{E}^{\infty}_{\mathbf{k}})$ appears in the imaginary Ax-Kochen-Ershov
principle, _cf._ Theorem 6.1.1 .

Given these observations, the following questions are quite natural:

######  Question 3.1.5.

  1. (1)

Can the density of either invariant or definable types — i.e. the conclusion
of either theorem — be proved without assuming
($\mathbf{E}^{\infty}_{\mathbf{k}})$?

  2. (2)

Under the hypotheses of Theorem 3.1.3, can we find an
$\mathcal{L}_{0}(\mathbfcal{G}(A))$-definable type $p$?

  3. (3)

Can the hypotheses of Theorem 3.1.3 be weakened to also encompass
characteristic zero non-Archimedean local fields?

###  3.2. The uniform arity one case

Let us first remind some of terminology and results from [Rid19]. We define
$\mathbf{B}$ to be the ($\mathcal{L}_{0}$-definable) set of closed and open
balls in models of $T_{0}$ — the field itself is the open ball of radius
$-\infty$ and points are closed balls of radius $+\infty$. For every
$r\in\mathbb{Z}_{>0}$, $\mathbf{B}^{[r]}$ is the set of finite subsets of
cardinality at most $r$ of $\mathbf{B}$ of the same type (open or closed) and
the same radius and $\mathbf{B}^{[<\infty]}:=\bigcup_{r>0}\mathbf{B}^{[r]}$.

For every finite set of balls $B$, we define $B^{\cup}:=\bigcup_{b\in B}b$ and
for every finite sets of balls $B_{1}$ and $B_{2}$, we write $B_{1}\leqslant
B_{2}$ if $B_{1}^{\cup}\subseteq B_{2}^{\cup}$. For every
$b_{1},b_{2}\in\mathbf{B}$, we also define
$d(b_{1},b_{2}):=\inf\\{\mathrm{v}(x_{1}-x_{2}):x_{i}\in b_{i}\\}$. Note that
this is not a metric on the space of balls since
$d(b_{1},b_{1})=\mathrm{rad}(b_{1})$, the radius of $b_{1}$. Finally, for
$B_{1},B_{2}\in\mathbf{B}^{[r]}$, we define
$D(B_{1},B_{2}):=\\{d(b_{1},b_{2}):b_{i}\in B_{i}\\}$ and we fix an
enumeration $\\{d_{i}(B_{1},B_{2}):0\leqslant i<r^{2}\\}=D(B_{1},B_{2})$,
with, for every $i<j$, $d_{i}(B_{1},B_{2})<d_{j}(B_{1},B_{2})$ or
$d_{i}(B_{1},B_{2})=d_{j}(B_{1},B_{2})=d_{r^{2}-1}(B_{1},B_{2})$. The actual
enumeration does not actually matter as long as it is chosen uniformly.

######  Definition 3.2.1 (_cf._ [Rid19, Definition 6.11]).

Let $\Psi(x;t)$ be a set of $\mathcal{L}_{0}$-formulas, and
$F:=(F_{\lambda})_{\lambda\in\Lambda}:\mathbf{K}^{x}\to\mathbf{B}^{[r]}$ be
$\mathcal{L}_{0}$-definable. We say that $(\Psi,F)$ is a _good presentation_
if, for every $p\in\mathrm{S}^{\Psi}_{x}(M)$:

  1. (1)

the type $p$ decides the following statements:

     * •

$F_{\lambda}(x)\square\bigcup_{i<r}F_{\mu_{i}}(x)$ where
$\lambda,\mu_{i}\in\Lambda(M)$ and
$\square\in\\{=,\subseteq,\subset,\leqslant,<\\}$;

     * •

$F_{\lambda}^{\cup}(x)=F_{\mu_{1}}^{\cup}(x)\cap F_{\mu_{2}}^{\cup}(x)$, where
$\lambda,\mu_{i}\in\Lambda(M)$;

     * •

$F_{\lambda}(x)$ is closed;

     * •

$\mathrm{rad}(F_{\lambda}(x))\square d_{i}(F_{\mu_{1}}(x),F_{\mu_{2}}(x))$,
where $\lambda,\mu_{i}\in\Lambda(M)$, $\square\in\\{=,\leqslant\\}$ and
$i<r^{2}$;

  2. (2)

there exists $\lambda,\mu\in\Lambda(M)$ with $F_{\lambda}(x)=\\{\mathbf{K}\\}$
and $F_{\mu}(x)=\emptyset$;

  3. (3)

$F$ is closed under intersections of realisations over $p$: for every
$\lambda,\mu\in\Lambda(M)$, there exists $\epsilon\in\Lambda(M)$ with
$p(x)\vdash F_{\lambda}^{\cup}(x)\cap
F_{\mu}^{\cup}(x)=F_{\epsilon}^{\cup}(x)$;

  4. (4)

$F$ is closed under complements over $p$: for every
$\lambda,\mu\in\Lambda(M)$, with $p(x)\vdash F_{\mu}(x)\subseteq
F_{\lambda}(x)$, there exists $\epsilon\in\Lambda(M)$ with $p(x)\vdash
F_{\lambda}(x)=F_{\mu}(x)\sqcup F_{\epsilon}(x)$;

  5. (5)

$F$ has large balls over $p$, that is, for every $\lambda,\mu\in\Lambda(M)$
and $i\in\mathbb{Z}_{\geqslant 0}$:

     * •

if $p(x)\vdash F_{\lambda}(x)\neq\mathbf{K}$, there is $\eta\in\Lambda(M)$
such that
$p(x)\vdash\mathrm{rad}(F_{\eta}(x))=d_{i}(F_{\lambda}(x),F_{\mu}(x))\wedge{}$“$F_{\eta}(x)$
is closed”${}\wedge F_{\lambda}(x)\leqslant F_{\eta}(x)$;

     * •

if $p(x)\vdash$ “$F_{\lambda}(x)$ is
open”${}\vee\mathrm{rad}(F_{\lambda}(x))<d_{i}(F_{\lambda}(x),F_{\mu}(x))$,
there is $\eta\in\Lambda(M)$ such that,
$p(x)\vdash\mathrm{rad}(F_{\eta}(x))=d_{i}(F_{\lambda}(x),F_{\mu}(x))\wedge{}$“$F_{\eta}(x)$
is open”${}\wedge F_{\lambda}(x)\leqslant F_{\eta}(x)$.

Let $\Delta(xy;s)$ with $|y|=1$ be a set of $\mathcal{L}_{0}$-formulas and
$G:=(G_{\omega})_{\omega\in\Omega}:\mathbf{K}^{x}\to\mathbf{B}^{[\ell]}$ be
$\mathcal{L}_{0}$-definable. We say that $(\Psi,F)$ is a _good presentation
for $\Delta$_ if $(\Psi,F)$ is a good presentation and every $M$-instance of
$\Delta$ is a Boolean combination of $M$-instances of $\Psi$ and conditions
$y\in F_{\lambda}(x)^{\cup}$ with $\lambda\in\Lambda(M)$. If, moreover, for
every $\omega\in\Omega(M)$, there exists $\lambda\in\Lambda(M)$ such that
$G_{\omega}=F_{\lambda}$. we say that $(\Psi,F)$ is a _good presentation for
$(\Delta,G)$_.

######  Lemma 3.2.2 (cf. [Rid19, Proposition 5.14, 5.15, 5.18 and 6.12]).

Let $\Delta(xy;s)$ be a finite set of $\mathcal{L}_{0}$-formulas with $|y|=1$
and $(G_{\omega})_{\omega\in\Omega}:\mathbf{K}^{x}\to\mathbf{B}^{[\ell]}$ be
$\mathcal{L}_{0}$-definable. Then there exists a finite set of
$\mathcal{L}_{0}$-formulas $\Psi(x;t)$ and an $\mathcal{L}_{0}$-definable
$F:=(F_{\lambda})_{\lambda\in\Lambda}:\mathbf{K}^{x}\to\mathbf{B}^{[r]}$ such
that $(\Psi,F)$ is a good presentation for $(\Delta,G)$.

###### Proof.

Let us sketch the proof. The existence of $\Psi$ and $F$ that decide all
instances of $\Delta$ follows by compactness from the Swiss cheese
decomposition. Enlarging $F$, we may assume it contains $G$ and that condition
(2) holds. At any point, enlarging $\Psi$, we may assume that condition (1)
holds. Condition (3) is obtained by closing $F$ under intersections. Since the
intersection of two balls is either empty or one of these balls, so finitely
many intersections suffice, and uniformly so. Condition (4) is obtained by
considering the finite Boolean algebra generated by the subsets of any given
$F$. They are generated in (uniformly) finitely many steps. Finally, all the
previous properties are preserved when replacing $F$ by all the instances
required by (4). ∎

Fix $(\Psi(x;t),F)$ a good presentation and $p\in\mathrm{S}^{\Psi}_{x}(M)$.

######  Definition 3.2.3.

Let $\lambda\in\Lambda(M)$. We say that $F_{\lambda}$ is _irreducible over
$p$_, if for every $\mu\in\Lambda(M)$, $p(x)\vdash F_{\mu}(x)\subseteq
F_{\lambda}(x)$ implies $p(x)\vdash F_{\mu}(x)=\emptyset\vee
F_{\mu}(x)=F_{\lambda}(x)$.

We define $\Lambda_{p}(M):=\\{\lambda\in\Lambda(M):F_{\lambda}\text{
irreducible over }p\\}$.

######  Lemma 3.2.4 (cf. [Rid19, Lemma 5.17]).

For every $\lambda\in\Lambda(M)$, there exists
$(\mu_{i})_{i<r}\in\Lambda_{p}(M)$ with $p(x)\vdash
F_{\lambda}(x)=\bigcup_{i}F_{\mu_{i}}(x)$.

######  Lemma 3.2.5 (cf. [Rid19, Proposition 5.9]).

For every $\lambda,\mu\in\Lambda_{p}(M)$, we have $p(x)\vdash
F_{\lambda}(x)\cap F_{\mu}(x)=\emptyset\vee F_{\lambda}(x)\leqslant
F_{\mu}(x)\vee F_{\mu}(x)\leqslant F_{\lambda}(x)$

######  Definition 3.2.6.

Let $\pi(x)$ be a partial $\mathcal{L}(M)$-type and $A\subseteq
M^{\mathrm{eq}}$. We say that $\pi$ is _$\mathcal{L}(A)$ -quantifiable over
$\mathcal{L}$_ if, for every $\mathcal{L}$-formula $\varphi(x;t)$, there
exists an $\mathcal{L}(A)$-formula $\theta(t)$ such that $\\{b\in
M^{t}:\pi(x)\vdash\varphi(x;b)\\}=\theta(M)$. When it exists, we write
$\forall_{\\!\pi}x\ \varphi(x;t):=\theta(t)$ and $\exists_{\\!\pi}x\
\varphi(x;t)=\neg(\forall_{\\!p}x\ \neg\varphi(x;t))$.

######  Remark 3.2.7.

If, for some set of $\mathcal{L}_{0}$-formulas $\Delta$, $p$ is a complete
$\mathcal{L}(A)$-quantifiable $\Delta$-type over $M$, then it is
$\mathcal{L}(A)$-definable, as a $\Delta$-type. As we will see in Lemma 3.3.1,
under certain hypotheses on $T$ and $\Delta$, the converse also holds.

######  Definition 3.2.8.

For every $\mathcal{L}(M)$-definable maps $f,g:X\to Y$ and every partial
$\mathcal{L}(M)$-type $p$ concentrating on $X$, we say that $f$ and $g$ have
the same _$p$ -germ_, and we write $[f]_{p}=[g]_{p}$, if $p(x)\vdash
f(x)=g(x)$.

######  Remark 3.2.9.

A good presentation $(\Psi(x;t),F)$ remains a good presentation as $\Psi$
grows, and irreducibility does not change. So, given a set $\Psi(xy;t)$ of
$\mathcal{L}_{0}$-formulas and an $\mathcal{L}_{0}$-definable
$F:=(F_{\lambda})_{\lambda\in\Lambda}:\mathbf{K}^{x}\to\mathbf{B}^{[r]}$, we
say that $(\Psi(xy;t),F)$ is a good presentation if there exists
$\Phi(x;t)\subseteq\Psi$ such that $(\Phi(x;t),F)$ is a good presentation.

######  Notation.

Let us fix now a good presentation $(\Psi(xy;t),F)$, with $|y|=1$, and
$p\in\mathrm{S}^{\Psi}_{xy}(M)$. Let $(\Psi,F)$ be the set of
$\mathcal{L}_{0}$-formulas $\Psi\cup\\{y\in F_{\lambda}^{\cup}(x)\\}$ in
variables $xy$ and parameters $t\lambda$. Let $\mathrm{S}^{\Psi,F}_{xy}(M)$
denote the space of $(\Psi,F)$-types over $M$ (in $M_{0}$). Let
$X\subseteq\mathbf{K}^{xy}$ be $\mathcal{L}(A)$-definable and
$p\in\mathrm{S}^{\Psi}_{xy}(M)$ be consistent with $X$.

######  Definition 3.2.10.

Let $E\subseteq\Lambda_{p}(M)$. We define

$\displaystyle\eta_{E,p}(xy)$ $\displaystyle:=p(xy)$ $\displaystyle\cup\\{y\in
F_{\lambda}^{\cup}(x):\lambda\in\Lambda(M)\wedge\exists\mu\in E\ p(xy)\vdash
F_{\mu}(x)\leqslant F_{\lambda}(x)\\}$ $\displaystyle\cup\\{y\nin
F_{\lambda}^{\cup}(x):\lambda\in\Lambda(M)\wedge\forall\mu\in\Lambda(M)\
F_{\mu}^{\cup}(x)\cap F_{\lambda}^{\cup}(x)\subset F_{\mu}^{\cup}(x)\\}.$

######  Lemma 3.2.11 (cf. [Rid19, Proposition 5.12]).

Let $E\subseteq\Lambda_{p}(M)$. Provided $\eta_{E,p}$ is consistent, we have
$\eta_{E,p}\in\mathrm{S}^{\Psi,F}_{xy}(M)$.

Let us now assume that $p$ is $\mathcal{L}(A)$-quantifiable over
$\mathcal{L}$, where $A=\mathrm{acl}^{\mathrm{eq}}(A)\subseteq
M^{\mathrm{eq}}$. Since $p$ is, in particular, $\mathcal{L}(A)$-definable,
$\Lambda_{p}(M)$ is an $\mathcal{L}(A)$-definable set. If
$\mathcal{E}\subseteq\Lambda_{p}$ is $\mathcal{L}(A)$-definable then, when it
is consistent, $\eta_{\mathcal{E}(M),p}$ is an $\mathcal{L}(A)$-definable
$(\Psi,F)$-type that we denote $\eta_{\mathcal{E},p}$.

We write $\lambda\leqslant_{p}\mu$ whenever $\forall_{\\!p}{xy}\
F_{\lambda}(x)\leqslant F_{\mu}(x)$. Note that $\leqslant_{p}$ is an
$\mathcal{L}(A)$-definable pre-order whose associated equivalence relation is
equality of $p$-germs. Moreover, restricted to $\Lambda_{p}$, by Lemma 3.2.5,
there is a largest element, $\mathbf{K}$, and the $\leqslant_{p}$-upwards
closure of any $\lambda\in\Lambda_{p}\smallsetminus[\emptyset]_{p}$ is totally
ordered: $(\Lambda_{p}\smallsetminus[\emptyset]_{p},\leqslant_{p})$ is a tree.

We can now prove the crucial step in proving Theorems 3.1.1 and 3.1.3: the
relative arity one case.

######  Definition 3.2.12.

For every $\lambda,\mu\in\Lambda_{p}$, let $\lambda\trianglelefteqslant\mu$
hold whenever $\forall_{\\!p}{xy}\ y\in(X_{x}\cap F_{\lambda}^{\cup}(x))\to
y\in F_{\mu}^{\cup}(x)$.

Note that $\trianglelefteqslant$ depends on both $p$ and $X$ but since it will
only be used in the following pages, we opted for lighter notation. The
relation $\trianglelefteqslant$ is an $\mathcal{L}(A)$-definable preorder on
$\Lambda_{p}$ and we denote $\equiv$ the associated equivalence relation.
Since $\leqslant_{p}$ refines $\trianglelefteqslant$ on
$\Theta_{p}:=\Lambda_{p}\smallsetminus(\emptyset/{\equiv})$, this is also a
tree with root $\mathbf{K}/\equiv$ and $\equiv$-classes are
$\leqslant_{p}$-convex.

######  Lemma 3.2.13 (cf. [Rid19, Claim 8.4]).

For every $\lambda\in\Theta_{p}$, if $\eta_{\lambda/{\equiv},p}$ is not
consistent with $X$, then $\lambda/{\equiv}$ has finitely many
$\trianglelefteqslant$-daughters $(\mu_{i}/{\equiv})_{0\leqslant
i<n}\in\mathrm{acl}^{\mathrm{eq}}(A\ulcorner\lambda/{\equiv}\urcorner)$ in
$\Theta_{p}/{\equiv}$. Moreover, $n\geqslant 2$ and $p(xy)\vdash y\in
X_{x}\cap F_{\lambda}^{\cup}(x)\to\bigvee_{i<n}y\in F_{\mu_{i}}^{\cup}(x)$.

###### Proof.

By compactness, there exist $(\nu_{i})_{0\leqslant i<m}\in\Theta_{p}(M)$ such
that $\nu_{i}\triangleleft\lambda$ and $p(xy)\vdash y\in X_{x}\cap
F_{\lambda}^{\cup}(x)\to\bigvee_{i<m}y\in F_{\nu_{i}}^{\cup}(x)$. The
existence of the $\mu_{i}$ now follows from the facts that any
$\mu\trianglelefteqslant\lambda$ is $\trianglelefteqslant$-comparable to one
of the $\nu_{i}$ and that since the $F_{\nu_{i}}$ are irreducible, the subtree
with root $\lambda$ and leaves $(\nu_{i})_{i<m}$ embeds in the lattice of
subsets of $\\{0,\ldots,m-1\\}$, which is finite. Finally, if $n=1$, we would
have $\lambda\trianglelefteqslant\mu_{0}$, contradicting that $\mu_{0}$ is a
daughter of $\lambda$. ∎

Let $\overline{\Theta}_{p}:=\\{\lambda\in\Theta_{p}:\forall_{\\!p}xy$
“$F_{\mu}(x)$ is closed”$\\}$ and, for every $\lambda\in\Lambda$, let
$Y_{\lambda}:=\\{F_{\mu}\in\overline{\Theta}_{p}:\forall_{\\!p}xy\
\mathrm{rad}(F_{\mu}(x))=\mathrm{rad}(F_{\lambda}(x))\\}$.

######  Lemma 3.2.14.

One of the following holds:

  * •

There exists a $\lambda\in\Theta_{p}$ such that $\lambda/{\equiv}\in A$ and
$\eta_{\lambda/\equiv,p}$ is consistent with $X$.

  * •

For every $n\in\mathbb{Z}_{\geqslant 0}$, there exists $\lambda/{\equiv}\in A$
such that $[Y_{\lambda}]_{p}$ is finite of cardinality larger than $n$, where
$[Y_{\lambda}]_{p}:=\\{[F_{\mu}]_{p}:\mu\in Y_{\lambda}\\}$.

###### Proof.

Assume that $X$ is consistent with no $\eta_{\lambda/{\equiv},p}$, where
$\lambda/{\equiv}\in A$. Then, by Lemma 3.2.13, $\Theta_{p}/\equiv$ admits an
initial finitely strictly branching discrete tree — that is every, element has
at least two daughters — with every branch infinite. Note that, for every
$\lambda\in\Theta_{p}$ with $\lambda<_{p}\mathbf{K}$, by the large ball
property, there is $\mu\in\Lambda$ with $\lambda\leqslant_{p}\mu$ and
$\forall_{\\!p}{xy}$ “$F_{\mu}(x)$ is
closed”${}\wedge\mathrm{rad}(F_{\mu}(x))=\mathrm{rad}(F_{\lambda}(x))$. We may
assume that $F_{\mu}$ is irreducible over $p$. Then $\lambda=\mu$ or $\lambda$
is the unique $\leqslant_{p}$-daugther of $\mu$. Note also that, by the large
ball property, $\overline{\Theta}_{p}\cap\mathbf{K}/{\equiv}\neq\emptyset$. It
follows that $\overline{\Theta}_{p}/\equiv$ also admits an initial finitely
branching discrete tree, denoted $\Xi_{p}$, with every branch infinite.

Note that, for any two $\mu,\nu\in Y_{\lambda}$, since $\forall_{\\!p}xy\
\mathrm{rad}(F_{\mu}(x))=\mathrm{rad}(F_{\nu}(x))$, we have that
$[F_{\mu}]_{p}=[F_{\nu}]_{p}$ implies $\mu\equiv\nu$, which implies that
$\forall_{\\!p}{xy}\ F_{\mu}^{\cup}(x)\cap F_{\nu}^{\cup}(x)\neq\emptyset$,
which, by irreducibility, implies that $[F_{\mu}]_{p}=[F_{\nu}]_{p}$, so these
three statements are equivalent. In particular, the identity induces a
bijection between $[Y_{\lambda}]_{p}$ and $Y_{\lambda}/{\equiv}$.

We now build, by induction, $\lambda_{i}\in\Lambda$ such that
$Y_{\lambda_{i}}/{\equiv}\subseteq\Xi_{p}$ and
$|Y_{\lambda_{i}}/{\equiv}|=|[Y_{\lambda_{i}}]_{p}|$ is finite and strictly
increasing. Start with any
$\lambda_{0}\in\overline{\Theta}_{p}\cap\mathbf{K}/{\equiv}$. Then
$Y_{\lambda_{0}}=[F_{\lambda}]_{p}$ and
$Y_{\lambda_{0}}/{\equiv}=\lambda_{0}/{\equiv}=\mathbf{K}/{\equiv}$. If
$\lambda_{i}$ is built, let $(\mu_{j})_{j<m}$ enumerate all the
$\trianglelefteqslant$-daughters — in $\Xi_{p}$ — of the elements in
$Y_{\lambda_{i}}/{\equiv}$. Let $j_{0}$ be such that, for all $j$,
$\forall_{\\!p}{xy}\
\mathrm{rad}(F_{\mu_{j_{0}}}(x))\leqslant\mathrm{rad}(F_{\mu_{j}}(x))$. For
every $\nu\in Y_{\mu_{j_{0}}}$, by the large ball property, we find
$\lambda\in Y_{\lambda_{i}}$ such that $\nu\leqslant_{p}\lambda$. Since
$\forall_{\\!p}{xy}\
\mathrm{rad}(F_{\nu}(x))=\mathrm{rad}(F_{\mu_{j_{0}}}(x))\leqslant\mathrm{rad}(F_{\mu_{j}}(x))$,
we cannot have $\nu\triangleleft\mu_{j}$ and hence $\nu/{\equiv}$ is either in
$Y_{\lambda_{i}}/{\equiv}$ or it is one of the $\mu_{j}/{\equiv}$. So
$Y_{\mu_{j_{0}}}/{\equiv}\subseteq\Xi_{p}$ is finite.

Furthermore, for every $\mu_{j}$, by the large ball property, there exists
$\nu\in Y_{\mu_{j_{0}}}$ such that $\mu_{j}\leqslant_{p}\nu$. It follows that,
for every element of $Y_{\lambda_{i}}/{\equiv}$, either it or all of its (more
than one) daughters appear in $Y_{\mu_{j_{0}}}$. In particular, all the
sisters of $\mu_{j_{0}}/{\equiv}$ appear, and hence
$|Y_{\mu_{j_{0}}}/{\equiv}|>|Y_{\lambda_{i}}/{\equiv}|$. Thus, we can choose
$\lambda_{i+1}=\mu_{j_{0}}$. ∎

We can now eliminate the second option in Lemma 3.2.14 by imposing a uniform
bound on the size of finite instances of $(Y_{\lambda})_{\lambda\in\Lambda}$:

######  Corollary 3.2.15.

Assume:

  * ($\mathbf{E}^{\infty}_{p,\overline{F}}$)

for every $\mathcal{L}(M)$-definable family $(Y_{z})_{z}$ of subsets of
$[F_{\Lambda_{p}}]_{p}:=\\{[F_{\lambda}]_{p}:\lambda\in\Lambda_{p}\\}$ such
that for all $z$ and $[F_{\lambda}]_{p},[F_{\mu}]_{p}\in Y_{z}$, $p(xy)\vdash$
“$F_{\mu}(x)$ is
closed”${}\wedge\mathrm{rad}(F_{\lambda}(x))=\mathrm{rad}(F_{\mu}(x))$, there
exists $n\in\mathbb{Z}_{>0}$ such that, for all $z$, $|Y_{z}|<\infty$ implies
$|Y_{z}|\leqslant n$.

Then there exists an $\mathcal{L}(A)$-definable
$\mathcal{E}\subseteq\Lambda_{p}$ such that $\eta_{\mathcal{E},p}$ is
consistent with $X$.∎

However the family $[Y_{\Lambda}]_{p}$ is not any definable family in
$[F_{\Lambda_{p}}]_{p}$. It has certain geometric properties that reflect that
of $X$. In particular, with further hypotheses on $X$, we can dispense with
($\mathbf{E}^{\infty}_{p,\overline{F}}$) altogether.

We now wish to apply the construction above in the pair $(M_{0},M)$ which is
naturally an $\mathcal{L}_{0,P}$ structure enriched with the
$\mathcal{L}$-structure on $M$. To be precise and avoid an unnecessary
conflict of notation:

######  Notation.

Let $\mathcal{L}_{1}$ be some enrichment of $\mathcal{L}_{0}$ and $T_{1}$ some
$\mathcal{L}_{1}$-theory of valued fields. In the following lemma, we apply
the above with $T$ the theory of the pair $M:=(M_{1}^{\mathrm{a}},M_{1})$,
where $M_{1}\models T_{1}$ is sufficiently saturated and homogeneous, in the
language $\mathcal{L}:=\mathcal{L}_{\mathrm{P}}$ consisting of
$\mathcal{L}_{0,P}$ structure enriched with the $\mathcal{L}_{1}$-structure on
$\mathbf{P}$ — so $M_{0}=M_{1}^{\mathrm{a}}$.

Let us now introduce some useful terminology from [CHR]:

######  Definition 3.2.16.

Fix $n\in\mathbb{Z}_{>0}\cap\mathbf{K}^{\times}(M)$. For any ball $b$, we
define $b[n]:=\\{a+n^{-1}(a-c):a,c\in b\\}$. It is a ball of radius
$\mathrm{rad}(b)-\mathrm{v}(n)$ around $b$, open if $b$ is, closed otherwise.
For a set of balls $B$, we set $B[n]:=\\{b[n]:b\in B\\}$.

  1. (1)

An $\mathcal{L}(M_{1})$-definable set $X\subseteq\mathbf{K}(M_{1})$ is _$n$
-prepared_ by some finite set $C\subseteq\mathbf{K}(M_{0})$ if for every ball
$b\in\mathbf{B}(M_{0})$ with $b[n]\cap C=\emptyset$, either $b\cap
X(M_{1})=\emptyset$ or $b\cap X(M_{1})=b(M_{1})$.

  2. (2)

We say that some $\mathcal{L}_{0}(M_{1})$-definable
$G:\mathbf{K}^{n}\to\mathbf{K}^{[r]}$ _$n$ -prepares_
$X\subseteq\mathbf{K}^{n+1}(M_{1})$ if, for every $x\in\mathbf{K}(M_{1})^{n}$,
$G(x)$ $n$-prepares $X_{x}$.

  3. (3)

We say that $X\subseteq\mathbf{K}^{n+1}(M_{1})$ is _$n$ -prepared_ by $F$ if
there exists $\lambda\in\Lambda(M_{1})$ such that $F_{\lambda}$ $n$-prepares
$X$.

######  Remark 3.2.17.

By field quantifier elimination (_cf._ 2.1.3), if $M_{1}$ is a pure henselian
field of characteristic zero, any $\mathcal{L}(M_{1})$-definable
$X\subseteq\mathbf{K}$ is $p^{\ell}$-prepared, for some $\ell$, by the finite
set of roots of polynomials that appear in the (field quantifier free)
definition of $X$, where $p$ is either $1$ or the residue characteristic when
it is positive.

Let also $A_{1}=\mathrm{acl}^{\mathrm{eq}}_{1}\subseteq M_{1}^{\mathrm{eq}}$,
$X\subseteq\mathbf{K}^{xy}(M_{1})$ be $\mathcal{L}_{1}(A_{1})$-definable,
$A=A_{P}=\mathrm{acl}^{\mathrm{eq}}(A)$ and
$p\in\mathrm{S}^{\Psi}_{xy}(M_{0})$ be
$\mathcal{L}_{\mathrm{P}}(A_{P})$-definable and consistent with $X$.

######  Lemma 3.2.18.

Let $\Phi(x;t)\subseteq\Psi$ be such that $(\Phi,F)$ is a good presentation
for $\Psi$. Assume that there is some
$n\in\mathbb{Z}_{>0}\cap\mathbf{K}^{\times}(M)$ such that:

  * ($\mathbf{P}^{F,n}_{X}$)

the set $X$ is $n$-prepared by $F$;

  * ($\mathbf{FR}_{n}$)

the interval $[0,\mathrm{v}(n)]\subseteq\mathbf{\Gamma}(M_{1})$ is finite and
$\mathbf{k}(M_{1})$ is perfect;

  * ($\mathbf{I}_{\mathbf{k}}$)

the residue field $\mathbf{k}(M_{1})$ is infinite.

Then, there exists an $\mathcal{L}_{\mathrm{P}}(A_{P})$-definable
$\mathcal{E}\subseteq\Lambda_{p}(M_{0})$ such that
${\left.\eta_{\mathcal{E},p}\right|_{M_{0}}}$ is consistent with $X$.

###### Proof.

Let $\rho$ be such that $F_{\rho}(x)$ $n$-prepares $X_{x}$ for all $x$. By
Lemma 3.2.14, applied in $M=(M_{1}^{\mathrm{a}},M_{1})$, either the conclusion
of the lemma holds, or we can find $\lambda/{\equiv}\in A$ with
$|[Y_{\lambda}]_{p}|>r$. Then some element of $Y_{\lambda}$, say
$F_{\lambda}$, does not contain any point of $F_{\rho}(x)$. Replacing
$\lambda$ with any $\mu\trianglelefteqslant\lambda$ in $\Xi_{p}$ such that
$|[\mu,\lambda]|\geqslant|[0,\mathrm{v}(n)]|$, we may further assume that
$F_{\rho}(x)\cap F_{\lambda}(x)[n]^{\cup}=\emptyset$ — where, for any
$B\in\mathbf{B}^{[<\infty]}$, $B[n]:=\\{b[n]:b\in B\\}$ — and that
$\lambda\triangleleft\mathbf{K}$.

By Lemma 3.2.13, we have $p(xy)\vdash y\in X_{x}\cap
F_{\lambda}^{\cup}(x)\to\bigvee_{i<n}y\in F_{\mu_{i}}^{\cup}(x)$, where the
$(\mu_{i}/{\equiv})_{i<n}$ are the daughters of $\lambda/{\equiv}$. By
compactness, there exists some $\psi(xy)\in p$ such that
$q:={\left.p\right|_{\Phi}}\vdash\forall y\ \psi(xy)\wedge y\in X_{x}\cap
F_{\lambda}^{\cup}(x)\to\bigvee_{i<n}y\in F_{\mu_{i}}^{\cup}(x)$. Since
$(\Phi,F)$ is a good presentation for $\Psi$, there are $\kappa$ and
$(\mu_{i})_{n\leqslant i<m}\in\Lambda_{p}$ with
$\models\psi(xy)\leftrightarrow y\in
F_{\kappa}^{\cup}(x)\smallsetminus(\bigcup_{n\leqslant
i<m}F_{\mu_{i}}^{\cup}(x))$. In particular, $p(xy)\vdash y\in
F_{\kappa}^{\cup}(x)$. It follows that $\kappa\equiv\mathbf{K}$ and hence
$\lambda\trianglelefteqslant\kappa$. So, we have

$q\vdash F_{\lambda}^{\cup}(x)\cap
X_{x}\subseteq\bigcup_{i}F_{\mu_{i}}^{\cup}(x).$

Since $\lambda\nequiv\emptyset$, there exists $ac\models p$ in some
$N_{1}\succcurlyeq M_{1}$ such that $c\in X_{a}\cap F_{\lambda}^{\cup}(a)$.
Let $b_{0}\in\mathbf{B}(N_{1}^{\mathrm{a}})$ be the ball of $F_{\lambda}(a)$
containing $c$. Since $F_{\rho}(a)\cap b_{0}[n]=\emptyset$, we have $b_{0}\cap
X_{a}(N_{1})=b_{0}(N_{1})$. It follows that
$b_{0}(N_{1})\subseteq\bigcup_{i}F_{\mu_{i}}^{\cup}(a)$. By construction,
$b_{0}(N_{1})$ is not covered by any single ball in
$\bigcup_{i}F_{\mu_{i}}(a)$. So $\\{\mathrm{v}(x-y):x,y\in b_{0}(N_{1})\\}$
has a minimal element (realised by some $x,y$ in distinct balls of
$\bigcup_{i}F_{\mu_{i}}(a)$). Let $b\in\mathbf{B}(N_{1})$ be the smallest
closed ball containing $b_{0}(N_{1})$. Then $b(N_{1})=b_{0}(N_{1})$ is covered
by finitely many of its maximal open subballs, contradicting hypothesis
($\mathbf{I}_{\mathbf{k}}$). ∎

###  3.3. Quantifiable types

To use the above constructions in an induction, we need a number of results on
quantifiable types. The first one is that $\eta_{\mathcal{E},p}$ is itself
quantifiable when $p$ is. For any finite set $B$ of balls, let
$\mathbf{R}_{B}$ be the set of maximal open subballs of the balls $b\in B$ and
$\mathrm{res}_{B}:B^{\cup}\to\mathbf{R}_{B}$ be the projection.

######  Lemma 3.3.1 (cf. [Rid19, Corollary 6.9]).

Let $(\Psi(xy;t),(F_{\lambda}(x))_{\lambda\in\Lambda})$ be a good
presentation. Let $p\in\mathrm{S}^{\Psi}_{xy}(M)$ be
$\mathcal{L}(A)$-quantifiable over $\mathcal{L}$, where $A\subseteq
M^{\mathrm{eq}}$. Assume:

  * ($\mathbf{E}^{\infty}_{p,\mathring{F}}$)

for any $\mathcal{L}(M)$-definable $(Y_{z})_{z}\subseteq[F_{\Lambda}]_{p}$ and
$\lambda\in\Lambda(M)$ such that, for all $z$ and $[F_{\mu}]_{p}\in Y_{z}$,
$p(xy)\vdash F_{\mu}(x)\subseteq\mathbf{R}_{F_{\lambda}(x)}$, there exists
$n\in\mathbb{Z}_{\geqslant 0}$ such that, for all $z$, $|Y_{z}|<\infty$
implies $|Y_{z}|\leqslant n$.

Then, any $\mathcal{L}(A)$-definable $q\in\mathrm{S}_{xy}^{\Psi,F}(M)$
containing $p$ is $\mathcal{L}(A)$-quantifiable over $\mathcal{L}$.

###### Proof.

Let $\mathcal{E}:=\\{\lambda\in\Lambda_{p}:q(xy)\vdash y\in
F_{\lambda}^{\cup}(x)\\}$. Then $\mathcal{E}$ is $\mathcal{L}(A)$-definable
and $q=\eta_{\mathcal{E},p}$. If $\mathcal{E}$ does not have an
$\leqslant_{p}$-minimal element which is closed, for any $\mathcal{L}$-formula
$\varphi(xy;s)$ and $e\in M^{s}$, $q(xy)\vdash\varphi(xy;e)$ if and only if
there exists $\lambda\in\mathcal{E}(M)$ and $\mu\in\Lambda_{p}(M)$ with
$\lambda<_{p}\mathcal{E}$ and $q(xy)\vdash y\in
F_{\lambda}^{\cup}(x)\smallsetminus F_{\mu}^{\cup}(x)\to\varphi(xy;e)$ — cf.
[Rid19, Proposition 6.4]. So let us assume that $\mathcal{E}$ has an
$\leqslant_{p}$-minimal element $\lambda_{0}$ which consists of closed balls.
If $p(xy)\vdash\mathrm{rad}(F_{\lambda_{0}}(x))=+\infty$, then
$q(xy)\vdash\varphi(xy;e)$ if and only if $p(xy)\vdash
F_{\lambda_{0}}^{\cup}(x)\to\varphi(xy;e)$ — cf. [Rid19, Proposition 6.6]. If
$p(xy)\vdash\mathrm{rad}(F_{\lambda_{0}}(x))\neq+\infty$, let

$Y_{s}:=\\{\mu\in\Lambda_{p}:\forall_{\\!p}xy\
F_{\mu}(x)\subseteq\mathbf{R}_{F_{\lambda}(x)}\wedge\exists_{\\!p}xy\
\varphi(xy;s)\wedge y\in F_{\mu}^{\cup}(x)\\}.$

Let $n$ be a uniform bound on the cardinality of finite $[Y_{s}]_{p}$, as in
($\mathbf{E}^{\infty}_{p,\mathring{F}}$).

######  Claim 3.3.2.

For every $e\in M^{s}$, $q$ is consistent with $\varphi(xy;e)$ if and only if,
for every $(\mu_{i})_{i<n}\in\Lambda_{p}$, with
$\forall_{\\!p}xyF_{\mu_{i}}(x)\in\mathbf{R}_{F_{\lambda_{0}(x)}}$,
$\exists_{\\!p}xy\ \varphi(xy;e)\wedge y\in
F_{\lambda_{0}}^{\cup}(x)\smallsetminus\bigcup_{i<n}F_{\mu_{i}}^{\cup}(x)$.

###### Proof.

Assume $q$ is not consistent with $\varphi(xy;e)$, then, by compactness, there
exists $(\mu_{i})_{i<m}\in\Lambda_{p}$ such that $\mu_{i}<_{p}\mathcal{E}$ and
$\forall_{\\!p}xy\ \varphi(xy;e)\wedge y\in
F_{\lambda_{0}}^{\cup}(x)\to\bigvee_{i<m}y\in F_{\mu_{i}}^{\cup}(x)$. By the
large ball property, we may assume $\forall_{\\!p}xy\
F_{\mu_{i}}(x)\in\mathbf{R}_{F_{\lambda_{0}(x)}}$. Choosing a minimal $m$, we
may also assume that, $\exists_{\\!p}xy\ \varphi(xy;e)\wedge y\in
F_{\mu_{i}}^{\cup}(x)$. In particular, $\mu_{i}\in Y_{e}$.

By definition of $Y_{s}$, for every $\mu\in Y_{e}(M)$, we find $ac\models p$
such that $\varphi(ac;e)$ and $c\in F_{\mu}^{\cup}(a)\subseteq
F_{\lambda_{0}}^{\cup}(a)$. So there is an $i$ such that $c\in
F_{\mu_{i}}^{\cup}(a)$. By irreducibility, $F_{\mu}(a)=F_{\mu_{i}}(a)$. It
follows that $[Y_{e}]_{p}$ is finite and thus
$m\leqslant|[Y_{e}]_{p}|\leqslant n$. ∎

Since $q\vdash\varphi(xy;e)$ if and only if $q$ is not consistent with
$\neg\varphi(xy;e)$, 3.3.2 allows us to conclude. ∎

We also need a better understanding of the interpretable set
$[F_{\Lambda}]_{p}$. Note that it is, a priori, $\mathcal{L}(M)$-interpretable
which is exactly the kind of sets elimination of imaginaries aims at
describing. However, if $p$ happens to be the restriction to $M$ of a global
$\mathcal{L}_{0}(M)$-definable type, then $[F_{\Lambda}]_{p}$ naturally embeds
in an $\mathcal{L}_{0}(M)$-interpretable set. The goal of the following lemmas
is to give (necessary) hypotheses under which any definable $p$ verifies that
condition. Valued vector spaces will play an important role:

######  Definition 3.3.3.

Let $(K,\mathrm{v})$ be a valued field and $V$ be a $K$-vector space. A
_valuation on $V$_ is a map $v:V\to X$ where $X$ is an ordered set with a
maximal element $\infty$ and an action $+$ of $\Gamma$, respecting the order,
such that:

  * •

$v(0)=\infty$;

  * •

for all $x,y\in V$, $v(x+y)\geqslant\min\\{v(x),v(y)\\}$;

  * •

for all $a\in K$ and $x\in V$, $v(a\cdot x)=\mathrm{v}(a)+v(x)$.

We say that a family $(x_{i})_{i}\in V$ is _separating_ if for every almost
everywhere zero $(a_{i})_{i}\in K$,
$v(\sum_{i}a_{i}x_{i})=\min_{i}(\mathrm{v}(a_{i})+v(x_{i}))$.

The following lemma owes much to Johnson’s computation of the canonical basis
of definable types in $\mathrm{ACVF}$, _cf._ [Joh20, Section 5.2]. Let
$\varphi_{d}(x;yz):=\mathrm{v}(\sum_{|I|<d}y_{I}x^{I})\geqslant\mathrm{v}(\sum_{|I|<d}z_{I}x^{I})$.

######  Proposition 3.3.4.

Assume:

  * ($\mathbf{C}_{V}$)

every $\mathcal{L}(M)$-definable valuation $v$ on $\mathbf{K}^{n}$ has a
separating basis;

  * ($\mathbf{C}_{\mathbf{\Gamma}}$)

$T$ has definably complete value group.

Then, for every $A=\mathrm{dcl}^{\mathrm{eq}}(A)\subseteq M^{\mathrm{eq}}$,
$\mathcal{L}(A)$-definable $p\in\mathrm{S}_{\varphi_{d}}(M)$ and algebraic
extension $K(M)\leqslant L$, any $q\in\mathrm{S}_{\varphi_{d}}(L)$, extending
$p$ and finitely satisfiable in $M\\!$, is
$\mathcal{L}_{0}(\mathbfcal{G}(A))$-definable.

###### Proof.

For every field $F$ with $K=\mathbf{K}(M)\leqslant F\leqslant L$, we define a
valuation $v$ on the $F$-vector space $V_{d}(F):=F[x]_{\leqslant d}$ of
polynomials in variables $x$ over $F$ of degree at most $d$ by
$v(P(x))\leqslant v(Q(x))$ if $\mathrm{v}(P(x))\leqslant\mathrm{v}(Q(x))\in
q$. The valuation $v$ on $V_{d}(K)$ is $\mathcal{L}(A)$-definable. By
hypothesis ($\mathbf{C}_{V}$), it has a separating basis $(P_{i})_{i}\in
V_{d}(K)$.

######  Claim 3.3.5.

$(P_{i})_{i}$ is a separating basis of $V_{d}(L)$ over $L$.

###### Proof.

We may assume that $K\leqslant L$ is finite. By [Joh21, Remark 2.7], the
valuation on $L$ (interpreted in $K$) is then $\mathcal{L}(M)$-definable and
hence, by hypothesis ($\mathbf{C}_{V}$), also has a separating basis
$(c_{j})_{j}\in L$ over $K$. Let $b_{i}=\sum_{j}b_{i,j}c_{j}\in L$, where
$b_{i,j}\in K$. If
$v(\sum_{j}(\sum_{i}b_{i,j}P_{i})c_{j})>\min_{j}v(\sum_{i}b_{i,j}P_{i})+\mathrm{v}(c_{j})$,
since $q$ is finitely satisfiable in $M$, we have
$v(\sum_{j}(\sum_{i}b_{i,j}P_{i}(a))c_{j})>\min_{j}v(\sum_{i}b_{i,j}P_{i}(a))+\mathrm{v}(c_{j})$
for some $a\in M$, contradicting that $(c_{j})_{j}$ is separating over $K$. So

$\displaystyle v(\sum_{i}b_{i}P_{i})$
$\displaystyle=v(\sum_{j}(\sum_{i}b_{i,j}P_{i})c_{j})$
$\displaystyle=\min_{j}v(\sum_{i}b_{i,j}P_{i})+\mathrm{v}(c_{j})$
$\displaystyle=\min_{i,j}\mathrm{v}(b_{i,j})+\mathrm{v}(c_{j})+v(P_{i})$
$\displaystyle\leqslant\min_{i}\mathrm{v}(b_{i})+v(P_{i})$
$\displaystyle\leqslant v(\sum_{i}b_{i}P_{i})\qed$

We now define the $L$-Archimedean equivalence on $v(V_{d}(L))$:
$v(P)\sim_{L}^{\infty}v(Q)$ holds if there exists $c\in L^{\times}$ such that
$-\mathrm{v}(c)+v(P)\leqslant v(Q)\leqslant\mathrm{v}(c)+v(P)$. One can check
that
$|v(V_{d}(L))/\sim_{L}^{\infty}|\leqslant|v(V_{d}(L))/\mathrm{v}(L)|\leqslant\dim_{L}(V_{d}(L))+1<\infty$.
We also define the $L$-infinitesimal equivalence on $v(V_{d}(L))$:
$v(P)\sim_{L}^{0}v(Q)$ holds if for every $\gamma\in\mathrm{v}(L)_{>0}$ we
have $-\gamma+v(P)<v(Q)<\gamma+v(P)$. Note that two elements of the same
$\mathrm{v}(L)$-orbit cannot be $\sim^{0}_{L}$-equivalent unless they are
equal. It follows that $\sim_{L}^{0}$-classes are finite.

Let $C$ be any $K$-Archimedean class and $\overline{C}$ denote its upwards
closure. Then $V_{C}:=v^{-1}(\overline{C})\leqslant V_{d}(K)$ is a
sub-$K$-vector space.

######  Claim 3.3.6 ([Joh20, Observation 4.3]).

$V_{C}$ has a basis of elements in $A$.

###### Proof.

Some coordinatewise projection $V_{C}\subseteq K^{l}\to K^{m}$ restricts to an
isomorphism on $V_{C}$. The preimage of the standard basis of $K^{m}$ then has
the required properties. ∎

Let $C_{0}$ be the successor of $C$ in $V_{d}(K)/\sim_{K}^{\infty}$. Since
$V_{C_{0}}\subset V_{C}$, any basis of $V_{C}$ has an element outside
$V_{C_{0}}$. In particular $(V_{C}\smallsetminus V_{C_{0}})(A)\neq\emptyset$
and we find $\gamma_{C}\in C(A)\neq\emptyset$. Then the whole (finite)
$K$-infinitesimal class of $\gamma_{C}$ is in $A$. Let $i$ be such that
$v(P_{i})\in C$. By ($\mathbf{C}_{\mathbf{\Gamma}}$), the set
$\\{\gamma\in\mathrm{v}(K):\gamma+v(P_{i})\leqslant\gamma_{C}\\}$ has a
supremum $\gamma_{i}$. Multiplying $P_{i}$ by some constant $c\in K$ with
$\mathrm{v}(c)=-\gamma_{i}$, we may assume that $\gamma_{i}=0$ in which case
$v(P_{i})$ is $K$-infinitesimally close to $\gamma_{C}\in A$ and hence is
itself in $A$. Since every $\mathrm{v}(K)$-orbit is contained in some
$K$-Archimedean class, we now have that for any $i$, $v(P_{i})\in A$ and for
any $j$, if $v(P_{i})\sim_{K}^{\infty}v(P_{j})$, then
$v(P_{i})\sim_{K}^{0}v(P_{j})$.

Note that, since $\mathrm{v}(L)$ is in the convex hull of $\mathrm{v}(K)$,
$\sim_{L}^{\infty}$ extends $\sim_{K}^{\infty}$. Also, if $v(K)$ is dense,
then $\sim_{L}^{0}$ extends $\sim_{K}^{0}$. However, if $\mathrm{v}(K)$ is
discrete then $\sim_{K}^{0}$ reduces to equality. In particular, we also have
that, if $v(P_{i})\sim_{L}^{\infty}v(P_{j})$, then
$v(P_{i})\sim_{L}^{0}v(P_{j})$.

For every $i$, $d$, let $M_{i,d}(L)=\\{P\in V_{d}(L):v(P)\geqslant
v(P_{i})\\}$. Note that, by 3.3.5, $\sum\lambda_{j}P_{j}\in M_{i,d}(L)$ if and
only if:

  * •

$\lambda_{j}=0$ for every $j$ with $v(P_{j})<v(P_{i})$ and
$v(P_{j})\nsim_{K}^{\infty}v(P_{i})$;

  * •

$\lambda_{j}\in\mathfrak{m}$ for $j$ with $v(P_{j})<v(P_{i})$ and
$v(P_{j})\sim_{K}^{0}v(P_{i})$;

  * •

$\lambda_{j}\in\mathcal{O}$ for $j$ with $v(P_{i})\leqslant v(P_{j})$ and
$v(P_{j})\sim_{K}^{0}v(P_{i})$.

So $M_{i,d}$ is (quantifier free)
$\mathcal{L}_{0}(\mathbfcal{G}(A))$-definable. Since $q$ is
$\mathcal{L}_{0}(\bigcup_{i,d}\ulcorner M_{i,d}\urcorner)$-definable, it is
indeed $\mathcal{L}_{0}(\mathbfcal{G}(A))$-definable. ∎

######  Remark 3.3.7.

Any definably complete ordered abelian group is elementarily equivalent to
either $\mathbb{Q}$, $\mathbb{Z}$ or $0$.

If $p\in\mathrm{S}^{0}(M)$ the existence (and uniqueness) of such a $q$
follows, on general grounds, from the finite satisfiability of $p$:

######  Lemma 3.3.8.

Let $p\in\mathrm{S}^{0}(M)$ be finitely satisfiable in $M$. Then any two
realisations of $p$ have the same $\mathcal{L}_{0}(\mathrm{acl}_{0}(M))$-type.
In particular, the unique extension of $p$ to $\mathrm{acl}_{0}(M)$ is
finitely satisfiable in $M$.

###### Proof.

Fix any $c\in\mathrm{acl}_{0}(M)$, $\varphi(xy)$ an $\mathcal{L}_{0}$-formula
and $\psi(y)$ an $\mathcal{L}_{0}(M)$-formula that algebrises $c$. Then
$\forall y\ [\psi(y)\to(\varphi(x_{1}y)\leftrightarrow\varphi(x_{2}y)$)]
defines an $\mathcal{L}_{0}(M)$-definable equivalence relation with finitely
many classes. Then the $E$-class of any $a\in N\succcurlyeq M$ realising $p$
has an element $e\in M$. It follows that $p(x)\vdash xEe$. In particular
$p(x)\vdash\varphi(xc)$ whenever $\varphi(ec)$ holds.

Let $q$ be the unique extension of $p$ to $\mathrm{acl}_{0}(M)$. Then $p\vdash
q$ and hence $q$ is finitely satisfiable in $M$. ∎

Following [Bau82], we can prove that ($\mathbf{C}_{V}$) follows from definable
spherical completeness. Up to definability, this is a standard result. But we
include its proof, on the one hand, for the sake of completeness, and, on the
other, to show that the proof can indeed been done definably.

######  Lemma 3.3.9.

Assume:

  * ($\mathbf{C}_{\mathbf{B}}$)

$T$ is definably spherically complete : any $\mathcal{L}(M)$-definable chain
of balls has a non-empty intersection.

Then any (finite dimensional) $\mathcal{L}(M)$-interpretable valued
$\mathbf{K}$-vector space $(V,v)$ has a separating basis.

###### Proof.

Let us proceed by induction on $n+1:=\dim(V)$. In particular, we may assume
that we have found a separating family $(y_{i})_{0\leqslant i<n}\in V$.

######  Claim 3.3.10.

For every $x\in V$, $\\{v(x-\lambda y):\lambda\in\mathbf{K}^{n}\\}$ has a
maximal element.

###### Proof.

For every $\lambda,\mu\in\mathbf{K}^{n}$, we have $v(x-\mu y)\geqslant
v(x-\lambda y)=:\gamma$ if and only if
$\min_{i}\\{\mathrm{v}(\mu_{i}-\lambda_{i})+v(y_{i})\\}=v((\mu-\lambda)y)\geqslant\gamma$.
For every $i<n$ and $\lambda\in\mathbf{K}^{n}$, let
$B_{i,\lambda}:=\\{\mu\in\mathbf{K}:v(x-\lambda_{\neq i}y_{\neq i}-\mu
y_{i})\geqslant v(x-\lambda
y)\\}=\\{\mu\in\mathbf{K}:\mathrm{v}(\mu-\lambda_{i})+v(y_{i})\geqslant
v(x-\lambda y)\\}$. They form a chain for inclusion.

If there is a minimal $B_{i,\lambda_{0}}$, pick any $\lambda_{i}\in
B_{i,\lambda_{0}}$. If there is no minimal $B_{i,\lambda}$, for every
$\gamma\in\mathrm{v}(\mathbf{K})$, let $b_{i,\gamma}$ be the closed ball of
radius $\gamma$ containing some $B_{i,\lambda}$, if it exists, or $\mathbf{K}$
otherwise. Since the chain of $B_{i,\lambda}$ does not have a minimal element,
any $B_{i,\lambda}$ contains a $b_{i,\gamma}$ that itself contains a
$B_{i,\mu}$. By definable spherical completeness, we find
$\lambda_{i}\in\bigcap_{\gamma}b_{i,\gamma}=\bigcap_{\lambda}B_{i,\lambda}$.
Then $\lambda_{i}$ has the property that, for any $\mu\in\mathbf{K}^{n}$,
$v(x-\mu y)\leqslant v(x-\mu_{\neq i}y_{\neq_{i}}-\lambda_{i}y_{i})$. It
follows that $v(x-\lambda y)$ is maximal. ∎

Let $x$ be linearly independent from the $y_{i}$. By the 3.3.10, we may assume
that $v(x)=\max\\{v(x-\lambda y):\lambda\in\mathbf{K}^{n}\\}$. Then, for every
$\mu\in\mathbf{K}$ and $\lambda\in\mathbf{K}^{n}$, we have $v(\mu x+\lambda
y)\leqslant v(\mu x)$ and thus, $v(\mu x+\lambda y)=\min\\{v(\mu x),v(\lambda
y)\\}=\min_{i}\\{v(\mu x),v(\lambda_{i}y_{i})\\}$. ∎

###  3.4. Counting germs

The last ingredient in this section is to reduce the (seemingly horrendous)
hypotheses ($\mathbf{E}^{\infty}_{p,\mathring{F}}$) and
($\mathbf{E}^{\infty}_{p,\overline{F}}$) to something more tractable. We start
by generalizing [Rid19, Section 7]. Let $M\models T$,
$M_{0}:=M^{\mathrm{a}}\models T_{0}$ and $Q$ be either $\mathbf{\Gamma}$ or
$\mathbf{k}$.

######  Lemma 3.4.1.

Let $(f_{\lambda})_{\lambda\in\Lambda}:\mathbf{K}^{n}\to Q$ be an
$\mathcal{L}_{0}$-definable family and $c\in\mathbf{K}(N_{0})^{n}$, where
$N_{0}\succcurlyeq M_{0}$. There exists an $\mathcal{L}_{0}(M)$-definable
family $(g_{\rho})_{\rho\in R}:\mathbf{K}^{n}\to Q^{[<\infty]}$, with
$R\subseteq Q^{m}$, such that for all $\lambda\in\Lambda(M_{0})$, there exists
$\rho\in R(M_{0})$ with $f_{\lambda}(c)\in g_{\rho}(c)$.

In particular, if $p\in\mathrm{S}^{0}(M_{0})$ is
$\mathcal{L}_{0}(M)$-definable, there is a $\mathcal{L}_{0}(M)$-definable
finite-to-one map $\\{[f_{\lambda}]_{p}:\lambda\in\Lambda(M_{0})\\}\to Q^{m}$,
for some $m\in\mathbb{Z}_{>0}$.

###### Proof.

We start with the non-uniform version of the result:

######  Claim 3.4.2.

For every $N_{0}\models\mathrm{ACVF}$, $A\leqslant\mathbf{K}(N_{0})$ and
finite tuple $c\in\mathbf{K}(N_{0})$, there exists a finite tuple $a\in A$,
such that $Q(\mathrm{acl}_{0}(Ac))\subseteq\mathrm{acl}_{0}(Q(A)ac)$.

###### Proof.

If $|c|=1$, we find $a_{0}\in\mathrm{acl}_{0}(A)$ with
$Q(\mathrm{acl}_{0}(Ac))=\mathrm{acl}_{0}(Q(A(c)))\subseteq\mathrm{acl}_{0}(Q(\mathrm{acl}_{0}(A))ca_{0})=\mathrm{acl}_{0}(Q(A)ca_{0})$,
by the characterization of transcendental $1$-types in $\mathrm{ACVF}$. If
$a\in A$ is such that $a_{0}\in\mathrm{acl}_{0}(a)$, we indeed have
$Q(\mathrm{acl}_{0}(Ac))\subseteq\mathrm{acl}_{0}(Q(A)ac)$. If $c=de$ with
$|e|=1$, we proceed by induction:
$Q(\mathrm{acl}_{0}(Ade))\subseteq\mathrm{acl}_{0}(Q(\mathrm{acl}_{0}(Ad))be)\subseteq\mathrm{acl}_{0}(Q(A)acbe)$,
with $a,b\in A$. ∎

By 3.4.2, and compactness in a saturated model of the pair $(N_{0},M_{0})$,
there exists an $\mathcal{L}_{0}(M_{0})$-definable $g$ as above. The union of
its conjugates over $M$ has the same properties and is
$\mathcal{L}_{0}(M)$-definable.

Now, if $p\in\mathrm{S}^{0}(M_{0})$ is $\mathcal{L}_{0}(M)$-definable, then
for any $\lambda\in\Lambda(M_{0})$, let
$Y_{\lambda}:=\\{\rho:\forall_{\\!p}{x}\ f_{\lambda}(x)\in g_{\rho}(x)\\}$ and
$h([f_{\lambda}]_{p}):=\ulcorner Y_{\lambda}\urcorner\in Q^{m}$. Note that
$p(x)\models f_{\mu}(x)\in\bigcap_{\rho\in Y_{\mu}}g_{\rho}(x)$, so $h$ is
finite-to-one. ∎

######  Lemma 3.4.3.

Assume:

  * ($\mathbf{E}^{\infty}_{\mathbf{k}})$

for any $\mathcal{L}(M)$-definable $(Y_{z})_{z}\subseteq\mathbf{k}$, there
exists $n\in\mathbb{Z}_{\geqslant 0}$ such that, for all $z$, $|Y_{z}|<\infty$
implies $|Y_{z}|\leqslant n$;

Then, for every $\mathcal{L}_{0}(M)$-definable $p\in\mathrm{S}^{0}_{x}(M_{0})$
and $\mathcal{L}_{0}$-definable
$(F_{\lambda})_{\lambda\in\Lambda}:\mathbf{K}^{x}\to\mathbf{B}^{[r]}$,
($\mathbf{E}^{\infty}_{p,\mathring{F}}$) holds, uniformly in $\lambda$.

###### Proof.

The core of the proof is the following almost internality result:

######  Claim 3.4.4.

For every $\lambda\in\Lambda(M)$, there exists an
$\mathcal{L}_{0}(M)$-definable finite-to-one map
$g_{\lambda}:X_{\lambda}:=\\{[F_{\mu}]_{p}:\mu\in\Lambda(M_{0})$ and
$p(x)\vdash$ “$F_{\mu}(x)$ are maximal open balls of
$F_{\lambda}(x)$”$\\}\to\mathbf{k}^{m}$, for some $m\in\mathbb{Z}_{>0}$.

###### Proof.

Let $c\models p$. Working in $M(c)^{\mathrm{a}}$ and then taking unions of
Galois conjugates, we may find $G_{i}(c)\in\mathbf{B}^{[<\infty]}$, for
$i:=1,2$, two $\mathcal{L}_{0}(Mc)$-definable sets picking at least one
maximal open balls in each of the ball of $F_{\lambda}$ and such that
$G_{1}(c)\cap G_{2}(c)=\emptyset$. For every $\mu$ with $[F_{\mu}]_{p}\in
X_{\lambda}$, let $f_{\mu}(x):=\\{(b-b_{1})/(b_{2}-b_{1}):b\in F_{\mu}(x),\
b_{i}\in G_{i}(x)$ and $b,b_{1},b_{2}$ are in the same ball of
$F_{\lambda}(x)\\}\in\mathbf{k}^{[<\infty]}$. Note that
$F_{\mu}(c)\in\mathrm{acl}_{0}(f_{\mu}(c)G_{1}(c)G_{2}(c))$. Using symmetric
functions, we identify $\mathbf{k}^{[<\infty]}$ with some $\mathbf{k}^{n}$.

By Lemma 3.4.1, we find an $\mathcal{L}_{0}(M)$-definable finite to one map
$h:\\{[f_{\mu}]_{p}:[F_{\mu}]_{p}\in X_{\lambda}\\}\to\mathbf{k}^{m}$. Then
$[F_{\mu}]_{p}\in\mathrm{acl}_{0}([G_{1}]_{p}[G_{1}]_{p}[f_{\mu}]_{p})\subseteq\mathrm{acl}_{0}(Mh(\mu))$
and