# Interpretable Fields in Various Valued Fields

Yatir Halevi  The Fields Institute for Research in Mathematical Sciences,
Toronto, Canada<EMAIL_ADDRESS>,  Assaf Hasson  Department of
Mathematics, Ben Gurion University of the Negev, Be’er-Sheva 84105, Israel
<EMAIL_ADDRESS>and  Ya’acov Peterzil  Department of Mathematics,
University of Haifa, Haifa, Israel<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathcal{K}}=(K,v,\ldots)$ be a dp-minimal expansion of a non-trivially
valued field of characteristic $0$ and ${\mathscr{F}}$ an infinite field
interpretable in ${\mathcal{K}}$.

Assume that ${\mathcal{K}}$ is one of the following: (i) $V$-minimal, (ii)
power bounded $T$-convex, or (iii) $P$-minimal (assuming additionally in (iii)
generic differentiability of definable functions). Then ${\mathscr{F}}$ is
definably isomorphic to a finite extension $K$ or, in cases (i) and (ii), its
residue field. In particular, every infinite field interpretable in
${\mathbb{Q}}_{p}$ is definably isomorphic to a finite extension of
${\mathbb{Q}}_{p}$, answering a question of Pillay’s.

Using Johnson’s work on dp-minimal fields and the machinery developed here, we
conclude that if ${\mathcal{K}}$ is an infinite dp-minimal pure field then
every field definable in ${\mathcal{K}}$ is definably isomorphic to a finite
extension of $K$.

The proof avoids elimination of imaginaries in ${\mathcal{K}}$ replacing it
with a reduction of the problem to certain distinguished quotients of $K$.

The first author was supported by the Kreitman foundation fellowship and the
Fields Institute for Research in Mathematical Sciences. The third author was
partially supported by Israel Science Foundation grant number 290/19

###### Contents

  1. 1 Introduction
    1. 1.1 Strategy of the proof and structure of the paper
  2. 2 Notation and preliminaries
    1. 2.1 Model theory
    2. 2.2 Valued fields
    3. 2.3 The main examples
      1. 2.3.1 P-minimal valued fields
      2. 2.3.2 C-minimal valued fields
      3. 2.3.3 V-minimal valued fields
      4. 2.3.4 $T$-convex valued fields
  3. 3 Simon-Walsberg uniformities and their properties
    1. 3.1 Preliminaries on definable uniform structures
    2. 3.2 Local analysis in SW-uniformities
  4. 4 Fields locally strongly internal to various dp-minimal fields
    1. 4.1 Strong internality, and the main technical lemma
    2. 4.2 The subgroup of infinitesimals
    3. 4.3 Differentiability in SW-uniform fields
    4. 4.4 Strong internality to SW-fields
    5. 4.5 1-h-minimal valued fields
    6. 4.6 Strong internality to strongly minimal fields
    7. 4.7 From a finite-to-finite correspondence to strong internality
  5. 5 Reduction to the distinguished sorts
    1. 5.1 Inter-algebraicity with the distinguished sorts
      1. 5.1.1 The C-minimal case
      2. 5.1.2 The P-minimal case
      3. 5.1.3 The weakly o-minimal case
    2. 5.2 Strong internality to the distinguished sorts
  6. 6 Eliminating the sorts $K/{\mathcal{O}}$ and $\Gamma$
    1. 6.1 Opaque Equivalence Relations
    2. 6.2 Definable functions in $K/{\mathcal{O}}$.
    3. 6.3 1-h-minimality, V-minimality and $T$-convex structures
    4. 6.4 Eliminating $K/{\mathcal{O}}$
    5. 6.5 Eliminating the sort $\Gamma$
  7. 7 Classifying interpretable fields
    1. 7.1 Concluding remarks
  8. A

##  1\. Introduction

We consider families of valued fields of characteristic $0$, such as
algebraically closed valued fields, real closed valued fields, or p-adically
closed fields. The goal of this work is to classify infinite fields
interpretable in certain dp-minimal expansions of such a field $K$, namely
fields whose universe is given as a quotient $X/E$ of a definable set
$X\subseteq K^{n}$ by a definable equivalence relation $E$.

The main result of our paper is:

######  Theorem 1.

[Theorem 7.1] Let ${\mathcal{K}}=(K,v,\dots)$ be a dp-minimal valued field
with residue field k and let ${\mathscr{F}}$ be an infinite field
interpretable in ${\mathcal{K}}$. Then:

  1. (1)

If ${\mathcal{K}}$ is a $V$-minimal valued field then ${\mathscr{F}}$ is
definably isomorphic to $K$ or k.

  2. (2)

If ${\mathcal{K}}$ is a power bounded $T$-convex valued field then
${\mathscr{F}}$ is definably isomorphic to one of $K$, $K(\sqrt{-1})$, k or
$\textbf{k}(\sqrt{-1})$. In particular, the result holds if ${\mathcal{K}}$ is
a real closed valued field.

  3. (3)

If ${\mathcal{K}}=(K,v,\dots)$ is a P-minimal valued field with the property
that any definable function is differentiable outside of nowhere dense subset
of its domain, then ${\mathscr{F}}$ is definably isomorphic to a finite
extension of $K$. In particular, the result holds if ${\mathcal{K}}$ is
$p$-adically closed.

This type of classification of definable quotients originates in Poizat’s
model theoretic consideration of the Borel-Tits theorem, [41]. In the setting
of (pure) algebraically closed valued fields, a study of interepretable groups
and fields was carried out by Hrushovski and Rideau-Kikuchi in [23]. The
result for V-minimal fields generalizes the characteristic $0$ case of [23,
Theorem 6.23].

In [39] Pillay showed that every infinite field definable in $\mathbb{Q}_{p}$
is definably isomorphic to a finite extension of $\mathbb{Q}_{p}$, and asked
whether the same is true for interpretable fields. The above solution to his
question was part of the motivation for this work.111A positive answer to
Pillay’s question on interpretable fields in $\mathbb{Q}_{p}$ was announced
also by E. Alouf, A. Fornasiero and J. de la Nuez Gonzalez.

Theorem 1 classifies in particular fields _definable_ in ${\mathcal{K}}$. In
that regard it generalises the analogous result of Bays and the third author
for real closed valued fields, [1].

Using Johnson’s theorem on dp-minimal fields (see Fact 2.4 below), and the
machinery developed here, we also prove:

######  Theorem 2.

[Corollary 4.26] Let ${\mathcal{K}}=(K,+,\cdot)$ be a pure dp-minimal field of
characteristic $0$. Then every field definable in ${\mathcal{K}}$ is definably
isomorphic to a finite extension of $K$.

Note that without the purity assumption the result fails, in the case of
strongly minimal fields, [20]. In the o-minimal case, however, purity is
superfluous, as shown by Otero, Pillay and the third author, [37]. In fact, in
the notation of Theorem 2, if ${\mathcal{K}}$ is unstable then both the purity
and characteristic assumptions can be replaced by the requirement that
definable functions are generically differentiable (Proposition 4.21).

###  1.1. Strategy of the proof and structure of the paper

The study of imaginaries in algebraically closed valued fields was initiated
by Holly in [19] and first studied in depth and in full generality, in the
setting of algebraically closed valued fields, by Haskell, Hrushovski and
Macpherson in [11] and [12]. Similar results were later obtained by Mellor for
real closed valued fields, [35], and by Hrushovski, Martin and Rideau-Kikuchi
for p-adically closed fields [22]. The work on interpretable fields in [23]
uses these theorems on elimination of imaginaries, and accomplishes
considerably more than the classification of such fields.

We adopt a different approach circumventing elimination of imaginaries, and
avoiding almost completely the so called _geometric sorts_. In fact, our main
result covers expansions of valued fields by analytic functions where no
elimination of imaginaries results are currently available (see [13]). Our
proof is based on the analysis of dp-minimal subsets of the interpreted field
${\mathscr{F}}$, and as such it borrows ideas from Johnson’s work on fields of
finite dp-rank (see for example [26] and [27]), as well as Otero-Peterzil-
Pillay [37].

The general setting in which we carry out most our work is that of dp-minimal
uniformities, suggested and axiomatized by Simon and Walsberg in [47] (called
here SW-uniformities). Their work yields some sort of cell decomposition,
generic continuity of definable functions and other properties similar to
those of o-minimal structures. We discuss and expand their results in Section
3. Another general setting which fits all three cases in our main theorem is
that of _1-h-minimal valued fields_ as developed, in characteristic $0$, in
[3, 4] by Cluckers, Halupczok, Rideau-Kikuchi and Vermeulen.

In Section 4, under the additional assumption that definable functions in
${\mathcal{K}}=(K,v\dots)$ are generically differentiable we use ideas of [34]
and [37] in the o-minimal setting to show that if ${\mathscr{F}}$ is an
infinite field _definable_ in ${\mathcal{K}}$ then ${\mathscr{F}}$ can be
definably embedded into a subring of $M_{n}(K)$, the ring of $n\times n$
matrices over $K$, for some $n\in{\mathbb{N}}$. One of the main technical
results of our work shows that in fact, to obtain the same result it suffices
that ${\mathscr{F}}$ is _locally strongly internal to $K$_, namely, that there
is an infinite subset of ${\mathscr{F}}$ in definable bijection with a subset
of $K^{m}$ (some $m$).

This observation allows us, in the cases covered by Theorem 1, to circumvent
elimination of imaginaries and generalise the result to fields interpretable
in ${\mathcal{K}}$. In Section 5 we prove, using a reduction to unary
imaginaries, that any interpretable field is (almost) locally strongly
internal to one of four distinguished sorts: the valued field, $K$, the value
group, $\Gamma$, the residue field, k, or the closed balls of radius $0$,
$K/{\mathcal{O}}$. Our results on definable fields are used to obtain the main
theorem when ${\mathscr{F}}$ is locally strongly internal to $K$ or to k.
Under the assumptions of Theorem 1, local strong internality to $\Gamma$ can
be eliminated by known results from o-minimality (the $P$-minimal case being
simpler). Section 6.4 is dedicated to eliminating the case of
$K/{\mathcal{O}}$. In the final section of the paper we combine all the
results obtained in this work to prove Theorem 1.

Remark The current article generalizes and replaces two previous preprints,
[16], [10].

Acknowledgments We thank Immanuel Halupczok, Ehud Hrushovski, Itay Kaplan and
Dugald Macpherson for several discussions during the work on the article. We
thank Pablo Cubides Kovacsics and the model theory group in Düssseldorf for
their careful reading of previous preprints.

##  2\. Notation and preliminaries

###  2.1. Model theory

We use standard model theoretic notation see e.g. [48, 46]. Lower case Latin
letters $a,b,c$ are used to denote elements and tuples, capital letters
$A,B,C$ are used for sets. Abusing notation we write $a\in A$ for tuples when
the length of the tuple is immaterial or understood from the context.

We use $\mathcal{M},\mathcal{K}$, etc. to denote structures whose universe is
$M$, $K$, respectively. We allow multi-sorted structures but from Section 5 on
we focus on one-sorted structures expanding a value field, where all other
sorts are taken from $\mathcal{M}^{eq}$ (i.e. $\mathcal{M}$ together with a
sort for every quotient $M^{n}/E$, where $E$ is a $\emptyset$-definable
equivalence relation, and a symbol for the projection $M^{n}\to M^{n}/E$). In
such a setting by an interpretable set, we mean a set definable in
$\mathcal{M}^{eq}$, whereas by a definable set we mean a definable subset of
$M^{n}$ for some $n$ (possibly with parameters). We may also refer to
definable subsets in $S$ for some (imaginary) sort $S$ of ${\mathcal{M}}$, by
which we mean a subset of $S^{n}$ (for some $n$) definable in ${\mathcal{M}}$,
possibly with parameters.

A _monster model_ for a first order theory $T$ is a large sufficiently
saturated and sufficiently homogeneous model containing all sets and models
(as elementary substructures) we will encounter. All subsets and models are
_small_ , i.e. of cardinality smaller than the saturation level of the
monster. When ${\mathcal{M}}$ is $\kappa$-saturated subsets $A\subseteq M$ are
always assumed to be of cardinality $<\kappa$.

Throughout the paper we apply consequences of dp-minimality mostly as a black
box. However, since the actual definition is used in a couple of places, we
remind:

######  Definition 2.1.

Let $T$ be a complete theory with some monster model $\mathbb{U}$, $P$ a
partial type over a set $A\subseteq\mathbb{U}$ and $\kappa$ a cardinal. We say
that $\mathrm{dp\text{-}rk}(P)<\kappa$ if for every family $\langle
I_{t}:t<\kappa\rangle$ of $A$-mutually indiscernible sequences and $b\models
P$, there is $t<\kappa$ such that $I_{t}$ is indiscernible over $Ab$. We say
that $\mathrm{dp\text{-}rk}(P)=\kappa$ if
$\mathrm{dp\text{-}rk}(P)<\kappa^{+}$ but not
$\mathrm{dp\text{-}rk}(P)<\kappa$. For any $a\in\mathbb{U}$ and small set $A$,
$\mathrm{dp\text{-}rk}(a/A)$ is defined to be
$\mathrm{dp\text{-}rk}(\operatorname{tp}(a/A))$.

A (one-sorted) structure has _NIP_ if $\mathrm{dp\text{-}rk}(x=x)<|T|^{+}$, it
is _dp-minimal_ if $\mathrm{dp\text{-}rk}(x=x)=1$ and _dp-finite_ (or _of
finite dp-rank_) if $\mathrm{dp\text{-}rk}(x=x)=n$ for some
$n\in{\mathbb{N}}$. See [46, Section 4.2].

It follows from sub-additivity of dp-rank below that any set interpretable in
a dp-minimal sort has finite dp-rank. We will use this fact throughout the
paper without further reference.

We refer the reader to [46, Section 4] for the basic properties of dp-rank out
which we emphasize sub-additivity:

######  Fact 2.2.

[29] For every $a,b\in{\mathcal{M}}$ and small set $A$

$\mathrm{dp\text{-}rk}(a,b/A)\leq\mathrm{dp\text{-}rk}(a/bA)+\mathrm{dp\text{-}rk}(b/A).$

A consequence of dp-minimality is that dp-rank is local ([45, Theorem 0.3(2)])
in the sense that
$\mathrm{dp\text{-}rk}(a/A)=\min\\{\mathrm{dp\text{-}rk}(\varphi):\varphi\in\operatorname{tp}(a/A)\\}$.
We make use of this fact at some points.

######  Definition 2.3.

For a partial type $P$ over $A$ and some $A$-definable function $f$, with
$P\vdash\operatorname{dom}(f)$, we let $f_{*}(P)$ be the partial type
$\\{f(X):X\in P\\}$.

The choice of working with dp-rank and dp-minimal structures permits us to
study a rich variety of examples. It is also influenced by the foundational
result of Johnson on arbitrary dp-minimal fields. While most of our work does
not rely on his theorem (with the exception of Corollary 4.26), we refer to it
several times in this work:

######  Fact 2.4 ([25]).

Let ${\mathcal{K}}=(K,+,\cdot,\dots)$ be a dp-minimal field. Then $K$ is
either algebraically closed, real closed or ${\mathcal{K}}$ admits a definable
henselian valation.

###  2.2. Valued fields

A valued field $(K,v)$ is a field $K$ together with a surjective group
homomorphims $v:K^{\times}\to\Gamma$, where $\Gamma$ is an ordered abelian
group (we set $v(0)=\infty$, where $\infty$ has the obvious properties),
satisfying the non-archimedean inequality:

$v(x+y)\geq\min\\{v(x),v(y)\\},$

for any $x,y\in K$. The group $\Gamma$ is the _value group_ of the valued
field $K$.

For $\gamma\in\Gamma$ and $a\in K$ we let

$B_{>\gamma}(a)=\\{x\in K:v(x-a)>\gamma\\}$

and

$B_{\geq\gamma}(a)=\\{x\in K:v(x-a)\geq\gamma\\}$

be the open and closed balls of radius $\gamma$ centered at $a$, respectively.
For us, a ball is always infinite, i.e. $\gamma\neq\infty$.

The closed ball $B_{\geq 0}(0)$ is a ring called the _valuation ring of $K$_,
which we denote here by $\mathcal{O}$. It is a local ring with maximal ideal
$\textbf{m}:=B_{>0}(0)$. The quotient $\textbf{k}:={\mathcal{O}}/\textbf{m}$
is the _residue field_. We refer to [8] for the basics of valuation theory.

Some texts on valuation theory use multiplicative notation (and reverse
ordering) for $\Gamma$, and $|x|$ instead of $v(x)$. Thus for example,
${\mathcal{O}}$ becomes $\\{x\in K:|x|\leq 1\\}$ in this notation. We will
comment on the multiplicative writing when we find it more intuitive (e.g.
when we discuss Taylor approximations of functions).

Throughout, ${\mathcal{K}}=(K,+,\cdot,\dots)$ will denote a dp-minimal
expansion of an infinite field $K$. If ${\mathcal{K}}$ expands a valued field
we let $\Gamma$ and k denote the imaginary sorts whose universes are the value
group and the residue field respectively.

The imaginary sort $K/{\mathcal{O}}$ of all closed balls of radius $0$ will
play an important role in Section 6.4. For ease of reference we use the
following _ad hoc_ definition:

######  Definition 2.5.

Given a structure $\mathcal{K}$ expanding a valued field $K$, the
_distinguished sorts of ${\mathcal{K}}$_ are $K$, $\Gamma$, k and
$K/{\mathcal{O}}$.

We also refer at several points to the sort
$\mathrm{RV}=K^{\times}/(1+\textbf{m})$ and the associated exact sequence of
abelian groups:

$1\to\textbf{k}^{\times}\to\mathrm{RV}\to\Gamma\to 0.$

###  2.3. The main examples

For the reader’s convenience we remind the definitions of those theories of
valued fields appearing in the statement of Theorem 1.

####  2.3.1. P-minimal valued fields

The notion of $P$-minimality was introduced by Haskell and Macpherson, [15]:

######  Definition 2.6.

An expansion of a $p$-adically closed valued field
${\mathcal{K}}:=(K,v,\dots)$ is _P-minimal_ if $\Gamma$ is a
$\mathbb{Z}$-group and in every structure ${\mathcal{L}}\equiv{\mathcal{K}}$
every definable subset of $L$ is quantifier-free definable in the Macintyre
language of valued fields. I.e., every such set can be written as a
disjunction of sets of the form

$\\{x\in L:\gamma_{1}<v(x-a)<\gamma_{2}\wedge P_{n}(\lambda\cdot(x-a))\\},$

where $P_{n}$ is the $n$-th power predicate, $\lambda\in L$ and
$\gamma_{1},\gamma_{2}\in\Gamma_{L}\cup\\{\infty,-\infty\\}$, $n\in\mathbb{N}$
and $a\in L$.

By [7, Section 6], P-minimal valued fields are dp-minimal.

######  Remark 2.7.

It follows directly that every definable subset of $\Gamma^{n}$,
$n\in\mathbb{N}$, in a $P$-minimal field is definable in Presburger
Arithmetic, and that any such bounded subset of $\Gamma$ has an infimum.

####  2.3.2. C-minimal valued fields

Let $(K,v)$ be a valued field.

######  Definition 2.8.

A _Swiss cheese_ is a set of the form $b\setminus\bigcup_{i}b_{i}$, where $b$
is a ball and the $b_{i}$ are finitely many subballs of $b$ (where all balls
may be either closed or open).

A Swiss cheese $b\setminus\bigcup_{i}b_{i}$ is _nested_ in another Swiss
cheese $d\setminus\bigcup_{i}d_{i}$ if there exists $i$ such that $b=d_{i}$.

######  Fact 2.9 (Holly, [18]).

Any definable subset of an ACVF has a unique decomposition as a finite
disjoint union of non-nested Swiss cheeses.

######  Definition 2.10.

An expansion ${\mathcal{K}}=(K,v,\dots)$ of a valued field is _C-minimal_ if
in every ${\mathcal{K}}^{\prime}\equiv{\mathcal{K}}$, every definable subset
of $K^{\prime}$ is a finite boolean combination of balls.222Macpherson and
Steinhorn’s original definition of a C-minimal field is quite different, but
the two definitions coincide for valued fields, see [32].

It follows that every definable $X\subseteq K$ has a unique decomposition as a
finite disjoint union of non-nested Swiss cheeses.

Haskel and Macpherson showed, [14], that every C-minimal valued field is an
algebraically closed valued field. In addition, $\Gamma$ is o-minimal and k is
strongly minimal so in particular, the residue field is algebraically closed,
and the value group is divisible. By [7, Corollary 4.3], any C-minimal field
is dp-minimal.

####  2.3.3. V-minimal valued fields

Although we only use the following as a black box, we introduce here the
notion of V-minimality, from Hrushovoski-Kazhdan’s [21]:

######  Definition 2.11.

A C-minimal theory expanding ACVF0,0 is V-minimal if for every
${\mathcal{K}}=(K,v,\dots)\models T$,

(1) Every definable relation on RVn, $n\in\mathbb{N}$, is definable in the
language of valued fields.

(2) For every definable chain of closed balls $W$, we have $\bigcap
W\neq\emptyset.$

(3) For every $A\subseteq K$, if $Y$ is an $A$-definable finite set of closed
balls then for each $b\in Y$, $\operatorname{acl}(A)\cap b\neq\emptyset$.

####  2.3.4. $T$-convex valued fields

We now recall the notion of a $T$-convex theory, due to van den Dries and
Lewenberg, [51], [49].

######  Definition 2.12.

Let $T$ be an o-minimal expansion of a real closed field in a signature
${\mathcal{L}}$. A _T-convex valued field_ is an expansion of $T$ by a
predicate for a convex valuation ring ${\mathcal{O}}$ that is closed under all
$\emptyset$-${\mathcal{L}}$-definable continuous functions.

The resulting theory, $T_{conv}$, is called power bounded if $T$ is, namely if
every ${\mathcal{L}}$-definable function from $(0,\infty)$ to $K$ is
eventually bounded by a definable ${\mathcal{L}}$-automorphism of $K^{>0}$
(such automorphisms are called power functions).

Recall that a linearly ordered structure ${\mathcal{M}}=(M,<,\cdots)$ is
_weakly o-minimal_ if every definable subset of $M$ is a finite union of
convex sets. By [7, Corollary 3.3], weakly o-minimal structures are dp-
minimal.

T-convex power-bounded valued fields satisfy the various properties used here:

######  Fact 2.13.

  1. (1)

Every T-convex valued field is weakly o-minimal [51, Corollary 3.14].

  2. (2)

The residue field of any T-convex valued field is o-minimal [49, Theorem A].

  3. (3)

The value group of any T-convex power-bounded valued field is a pure ordered
vector space over the ordered field of powers of $K$ [49, Theorem B], hence it
is o-minimal. See also [49, Proposition 4.3].

##  3\. Simon-Walsberg uniformities and their properties

###  3.1. Preliminaries on definable uniform structures

One of the challenges of this article was to find a proper framework which
fits a variety of dp-minimal expansions of valued fields. Simon and Walsberg
in [47] (and in somewhat greater generality Dolich and Goodrick, [6]) provide
an elegant setting of dp-minimal uniformities which suits well our needs. We
present here the basic definitions and properties and develop further some
local properties of such uniformities.

######  Assumption.

Throughout entire Section $3$ we assume that ${\mathcal{M}}$ is
$|T|^{+}$-saturated, where $T=\mathrm{Th}({\mathcal{M}})$.

######  Definition 3.1.

A definable set $D$ in a (possibly multi-sorted) structure ${\mathcal{M}}$ has
a _definable uniform structure_ , or _a definable uniformity_ , if there is a
formula $\theta(x,y,z)$ such that $\mathscr{B}=\\{\theta(D^{2},t):t\in T\\}$
satisfies the following:

  1. (1)

the intersection of the elements of $\mathscr{B}$ is $\\{(x,x):x\in D\\}$;

  2. (2)

if $U\in\mathscr{B}$ and $(x,y)\in U$ then $(y,x)\in U$;

  3. (3)

for all $U,V\in\mathscr{B}$ there is a $W\in\mathscr{B}$ such that $W\subseteq
U\cap V$;

  4. (4)

for all $U\in\mathscr{B}$ there exists $V\in\mathscr{B}$ such that

$\\{(x,z)\in D^{2}:(\exists y\in D)\left((x,y)\in V,(y,z)\in
V\right)\\}\subseteq U.$

A definable uniform structure induces a definable topology on $D$ whose basic
open sets are $U[x]:=\\{y:(x,y)\in U\\}$, as $U$ ranges over $\mathscr{B}$.

For the purposes of the present work the main source of examples of such
structures are expansions of definable (abelian) groups equipped with a
definable neighbourhood basis $\mathscr{B}_{0}$ at $0$ and the associated
uniformity

$\mathscr{B}=\\{(x,y)\in G:xy^{-1}\in U\\}_{U\in\mathscr{B}_{0}}.$

Based on [47], we define:

######  Definition 3.2.

A set $D$, equipped with a definable uniformity, is called _an SW-uniform
structure_ (or an SW-uniformity) if:

  * •

$D$ is dp-minimal,

  * •

$D$ has no isolated points,

  * •

every infinite definable subset of $D$ has nonempty interior.

Throughout this section, whenever we say that $D$ is an SW-uniformity, we
tacitly assume that $D$ is a definable set in some ambient (multi-sorted)
structure. Note that any SW-uniformity is unstable.

The following examples emphasis the relevance of this setting to the present
work:

######  Example 3.3.

  1. (1)

A dp-minimal expansion of a divisible ordered abelian group is an SW-
uniformity [44].

  2. (2)

In particular, every weakly o-minimal expansion of an ordered group is an SW-
uniformity [31, Theorem 5.1].

  3. (3)

A dp-minimal expansion of a non-trivially valued field is an SW-uniformity.
Indeed, it has no isolated points since the field topology is not trivial.
Every infinite definable subset has non empty interior by, e.g., [24,
Proposition 3.6].

  4. (4)

Johnson in his work [25] defines a topology on every dp-minimal expansion of a
non strongly minimal field $K$. This topology, which he calls the canonical
topology, has as a base the family of sets ${\bf B}=\\{X-X:X\subseteq K\mbox{
definable and infinite}\\}$, [25, Theorem 6.5]. It turns out to yield an SW-
uniform structure, see also [47, Proposition 1.1]. The converse is true as
well: Assume that ${\mathcal{K}}$ is a dp-minimal expansion of a topological
field, with a definable basis for its topology $\tau$ making it into an SW-
uniform structure. Then $\tau$ equals the canonical topology of Johnson.
Indeed, since every infinite definable $X\subseteq K$ has a non-empty interior
it follows that the family ${\bf B}$ forms a base for $\tau$.

  5. (5)

In the sequel (Lemma 5.13) we prove that if ${\mathcal{K}}=(K,v,\dots)$ is a
dp-minimal valued field with a dense value group then (under an additional
technical assumption) $K/{\mathcal{O}}$ admits the structure of an SW-
uniformity.

It follows from the work of Simon and Walsberg that, in SW-uniformities, the
notion of dp-rank coincides with other notions of dimension. For $X\subseteq
D^{n}$, the topological dimension of $X$ is defined to be the maximal $k\leq
n$ such that some projection of $X$ onto $k$ of the coordinates contains an
open set. The $\operatorname{acl}$-dimension of a tuple $a\in M^{n}$ over
$A\subseteq M$ (even when $\operatorname{acl}$ does not satisfy exchange) is
the minimal $k\leq n$ for which there exists a sub-tuple $a^{\prime}\subseteq
a$, such that $a\in\operatorname{acl}(a^{\prime}A)$. Then, the
$\operatorname{acl}$-dimension of an $A$-definable set $X\subseteq M^{n}$ is
defined to be the maximum of the $\operatorname{acl}$-dimension of $a/A$, for
all $a\in X$.

By Theorem ([47, Proposition 2.4]), if $D$ is an SW-uniformity in a
$|T|^{+}$-saturated structure and $X\subseteq D^{n}$ is definable then
$\mathrm{dp\text{-}rk}(X)$ equals the topological dimension and the
$\operatorname{acl}$-dimension of $X$. The locality of dp-rank implies that
the $\operatorname{acl}$-dimension of a tuple $a$ over $A$ equals
$\mathrm{dp\text{-}rk}(a/A)$. Using the definition of topological dimension we
obtain definability of $\mathrm{dp\text{-}rk}$ in parameteres:

######  Fact 3.4.

[47, Corollary 2.5] Let $X\subseteq D^{n+k}$ be $A$-definable. Then for every
$k\in\mathbb{N}$, the set $\\{a\in D^{k}:\mathrm{dp\text{-}rk}(X_{a})=k\\}$ is
definable over $A$, where $X_{a}=\\{b\in D^{n}:(b,a)\in X\\}$.

In addition, definable functions in SW-uniformities are generically
continuous:

######  Fact 3.5.

[47, Proposition 3.7] Let $D$ be an $SW$-uniformity, $f:U\to W$ a definable
function. Then

$C_{f}:=\\{x\in U:f\text{ is continuous at }x\\}$

is definable and $\mathrm{dp\text{-}rk}(U\setminus
C_{f})<\mathrm{dp\text{-}rk}(U)$.

We also need the following results:

######  Fact 3.6.

[47, Lemma 4.6] Let $X\subseteq D^{n}$ be a definable set in an SW-uniformity
with $\mathrm{dp\text{-}rk}(X)=k$. Then there exists a definable subset
$Y\subseteq X$ with $\mathrm{dp\text{-}rk}(Y)<k$ such that for every $a\in
X\setminus Y$ there is a coordinate projection $\pi:X\to D^{k}$ that is a
local homeomorphism at $a$.

For $X\subseteq Y\subseteq D^{n}$, the relative interior of $X$ in $Y$,
$Int_{Y}(X)$, is the set of $a\in X$ such that for some open $V\ni a$ in
$D^{n}$, $V\cap Y\subseteq X$ (hence $V\cap X=V\cap Y$).

######  Fact 3.7.

[47, Corollary 4.4] If $X\subseteq Y\subseteq D^{n}$ are definable in an SW-
uniformity $D$ and $\mathrm{dp\text{-}rk}(X)=\mathrm{dp\text{-}rk}(Y)$ then
$\mathrm{dp\text{-}rk}(Y\setminus Int_{Y}(X))<\mathrm{dp\text{-}rk}(Y)$.

######  Remark 3.8.

It is easy to see that if ${\mathcal{M}}$ is any dp-minimal structure then
every infinite subset of $M^{n}$ has a definable subset of
$\mathrm{dp\text{-}rk}=1$.

For an SW-uniformity $D$, if $X\subseteq D^{n}$ with
$\mathrm{dp\text{-}rk}(X)=k$ then by Fact 3.6, there are definable open
$X_{1},\dots,X_{k}\subseteq D$ and a definable injection of
$X_{1}\times\dots\times X_{k}$ into $X$, possibly over additional parameters.
If $X$ is defined over a model then so are the $X_{i}$ and the injection.

###  3.2. Local analysis in SW-uniformities

The main goal of this section is to prove Proposition 3.12, which generalizes
a similar, useful, result from the theory of o-minimal structures. It says
that given an $A$-definable set $X\subseteq D^{n}$ and $a\in X$ with
$\mathrm{dp\text{-}rk}(a/A)=\mathrm{dp\text{-}rk}(X)$, there are arbitrarily
small neighborhoods $V$ of $a$, defined over $B\supseteq A$, such that
$\mathrm{dp\text{-}rk}(a/B)=\mathrm{dp\text{-}rk}(a/A)$.

If $\operatorname{acl}$ satisfies exchange then the proposition would be quite
easy to prove, as in the o-minimal setting. However, our eventual aim is to
apply it in the SW-structure structure on $K/{\mathcal{O}}$, where exchange
fails.

The first step is due to Simon and Walsberg:

######  Lemma 3.9.

Let $D$ be a definable SW-uniform structure. Let $X\subseteq Z\subseteq D^{n}$
be $\emptyset$-definable sets with
$\mathrm{dp\text{-}rk}(X)=\mathrm{dp\text{-}rk}(Z)$. For every $d\in X$ if
$\mathrm{dp\text{-}rk}(d)=\mathrm{dp\text{-}rk}(X)$ then there exists an open
neighborhood $U\subseteq D^{n}$ of $d$ such that $U\cap X=U\cap Z$.

###### Proof.

By Fact 3.7, $\mathrm{dp\text{-}rk}(Z\setminus
Int_{Z}(X))<\mathrm{dp\text{-}rk}(Z)$ and hence $d\in Int_{Z}(X)$. ∎

The following is probably well known.

######  Lemma 3.10.

Let ${\mathcal{M}}$ be a structure of finite dp-rank,
${\mathcal{N}}\succ{\mathcal{M}}$ an elementary extension and $X,Y$ definable
over ${\mathcal{M}}$ and ${\mathcal{N}}$, respectively. If
$X({\mathcal{M}})\subseteq Y({\mathcal{M}})$ then
$\mathrm{dp\text{-}rk}(X)\leq\mathrm{dp\text{-}rk}(Y)$.

###### Proof.

We will use a finitary version of ict-patterns, see [46, Definition 4.21], in
order to calculate the dp-rank, see [46, Proposition 4.22].

Assume that $\mathrm{dp\text{-}rk}(X)\geq k$, so $\kappa_{ict}(X)>k$ and we
can find formulas $\\{\varphi_{\alpha}(x_{\alpha},y)\\}_{\alpha<k}$, such that
for any integer $n$ there is an array $\langle
a_{i}^{\alpha}:i<n,\,\alpha<k\rangle$ of tuples from $M$ with
$|a_{i}^{\alpha}|=|x_{\alpha}|$, such that for every $\eta:k\to n$ there is a
tuple $b_{\eta}\in X({\mathcal{M}})$ such that

$\varphi_{\alpha}(a_{i}^{\alpha},b_{\eta})\iff\eta(\alpha)=i.$

The same pattern gives, since $X({\mathcal{M}})\subseteq Y({\mathcal{M}})$,
that $\kappa_{ict}(Y)>k$ so $\mathrm{dp\text{-}rk}(Y)\geq k$. ∎

We fix the uniformity on $D$, given by

$\mathscr{B}=\\{U_{t}=\theta(D^{2},t):t\in T\\},$

and consider the $\emptyset$-definable directed partial order defined by
$t_{1}\leq t_{2}$ if $U_{t_{1}}\subseteq U_{t_{2}}$.

######  Lemma 3.11.

Let $D$ be an SW-uniformity (in some structure ${\mathcal{M}}$),
$a=(a_{1},\dots,a_{n})\in D^{n}$ and $A\subseteq D$.

  1. (1)

For every $t_{0}\in T$, there exists $t\leq t_{0}$ such that
$\mathrm{dp\text{-}rk}(a/tA)=\mathrm{dp\text{-}rk}(a/A)$.

  2. (2)

For every $t\in T$, there exists $a_{1}^{\prime}\in
U_{t}[a_{1}]\setminus\\{a_{1}\\}$ such that
$\mathrm{dp\text{-}rk}(a/Ata_{1}^{\prime})=\mathrm{dp\text{-}rk}(a/At)$.

###### Proof.

(1) Let $\Delta$ be the collection of all $A$-formulas $\psi(\bar{x},u)$, for
$\bar{x}=(x_{1},\ldots,x_{n})$ and $u$ a $T$-variable, such that for every
$t\in T$, $\mathrm{dp\text{-}rk}(\psi(D,t))<\mathrm{dp\text{-}rk}(a/A).$

We want to realize the type

$P(s)=\\{s\leq t_{0}\\}\cup\\{\neg\psi(a,s):\psi\in\Delta\\}.$

If $P$ is consistent, and realized by $s_{1}\in T$ then obviously $s_{1}\leq
t_{0}$. If $\mathrm{dp\text{-}rk}(a/s_{1}A)<\mathrm{dp\text{-}rk}(a/A)$ then
by the locality of dp-rank in $D$, we can find some $A$-formula
$\psi(\bar{x},y)$, such that
$\psi(\bar{x},s_{1})\in\operatorname{tp}(a/S_{1}A)$ and
$\mathrm{dp\text{-}rk}(\psi(D,s_{1}))<\mathrm{dp\text{-}rk}(a/A)$. By Fact
3.4, there is a formula $\theta(u)$ over $A$ such that for every $t\in T$,
$\theta(t)$ holds if and only if
$\mathrm{dp\text{-}rk}(\psi(D,t))<\mathrm{dp\text{-}rk}(a/A)$. Consequently,
$\psi(\bar{x},u)\wedge\theta(u)\in\Delta$, contradicting the fact that
$(a,s_{1})$ satisfies it. Therefore a realisation of $P$ must satisfy the
conclusion of the lemma.

Assume toward a contradiction that $P$ is inconsistent. Thus there are
$\varphi_{1},\ldots,\varphi_{r}\in\Delta$, such that

$\forall t(t\leq t_{0}\to\bigvee_{i}\varphi_{i}(a,t)).$

Let $\psi(\bar{x},u):=\bigvee_{i}\varphi_{i}(\bar{x},u)$. Then it is still the
case that $\psi\in\Delta$ and we have $\forall t(t\leq t_{0}\to\psi(a,t)).$
Let

$\chi(\bar{x}):=\exists s\forall t(t\leq s\to\psi(\bar{x},t)).$

Then $\chi$ is a formula over $A$ and as $\chi(a)$ holds then necessarily
$\mathrm{dp\text{-}rk}(\chi)\geq\mathrm{dp\text{-}rk}(a/A)$.

We will reach a contradiction by showing that, in fact,
$\mathrm{dp\text{-}rk}(\chi)<\mathrm{dp\text{-}rk}(a/A)$. Let

$\widehat{\psi}(\bar{x},u):=\psi(\bar{x},u)\wedge(\forall t_{1}\leq
u)\psi(\bar{x},t_{1}).$

It is easy to verify that $\chi(\bar{x})\leftrightarrow\exists
u\widehat{\psi}(\bar{x},u).$

Let ${\mathcal{M}}\prec{\mathcal{M}}^{*}$ be an
$|{\mathcal{M}}|^{+}$-saturated elementary extension and let
$D^{*}=D({\mathcal{M}}^{*})$. By saturation and the fact that $T$ is directed
by $\leq$ there is $t^{*}\in T({\mathcal{M}}^{*})$ such $t^{*}\leq
T({\mathcal{M}})$.

It follows that $\chi(D)\subseteq\widehat{\psi}(D^{*},t^{*})$. Indeed, if
$b\in\chi(D)$ then there is $t\in T({\mathcal{M}})$ such that for all
$t_{1}\in T({\mathcal{M}})$, if $t_{1}\leq t$ then $\psi(b,t_{1})$. This
remains true in ${\mathcal{M}}^{*}$ hence $\widehat{\psi}(b,t^{*})$ holds.

However, since $\psi(\bar{x},u)\in\Delta$ then
$\widehat{\psi}(\bar{x},u)\in\Delta$. As a result, for every $t\in T$
$\mathrm{dp\text{-}rk}(\widehat{\psi}(D,t))<\mathrm{dp\text{-}rk}(a/A)$. In
particular
$\mathrm{dp\text{-}rk}(\widehat{\psi}(D^{*},t^{*}))<\mathrm{dp\text{-}rk}(a/A)$
and by Lemma 3.10,
$\mathrm{dp\text{-}rk}(\chi(D))<\mathrm{dp\text{-}rk}(a/A)$, with the desired
contradiction.

(2) The proof follows similar lines to (1). Let $A_{1}=At$ and let $\Delta$ be
now all formulas $\varphi(\bar{x},y)$ over $A_{1}$, with $y$ in the $D$-sort,
such that for all $y\in D$,
$\mathrm{dp\text{-}rk}(\varphi(D,y))<\mathrm{dp\text{-}rk}(a/A_{1}).$

As in (1) we want to realize the type

$P(y)=\\{y\in U_{t}[a_{1}]\\}\cup\\{y\neq
a_{1}\\}\cup\\{\neg\varphi(a,y):\varphi\in\Delta\\}.$

Assume the type is inconsistent. Then there are
$\varphi_{1},\ldots,\varphi_{r}\in\Delta$ such that

$\forall y((y\in U_{t}[a_{1}]\wedge y\neq
a_{1})\to\bigvee_{i}\varphi_{i}(a,y)).$

Let $\psi(\bar{x},y)$ be the formula $\bigvee_{i}\psi_{i}(\bar{x},y)\wedge
y\neq x_{1}$, then $\psi\in\Delta$. For $X\subseteq D^{k}$ let
$\operatorname{Fr}(X):=\operatorname{cl}(X)\setminus X$ and note that if
$a_{1}$ satisfies $\forall y((y\in U_{t}[a_{1}]\wedge y\neq
a_{1})\to\psi(a,y))$ then, as $D$ has no isolated points,
$a_{1}\in\operatorname{Fr}(\psi(a,D))$ and therefore
$(a,a_{1})\in\operatorname{Fr}(\psi(D^{n+1}))$. So, as
$\mathrm{dp\text{-}rk}(a_{1},a/A_{1})=\mathrm{dp\text{-}rk}(a/A_{1})$, we see
that
$\mathrm{dp\text{-}rk}(a/A_{1})\leq\mathrm{dp\text{-}rk}(\operatorname{Fr}(\psi(D^{n+1})))$

On the other hand, by [47, Proposition 4.3],

$\mathrm{dp\text{-}rk}(\operatorname{Fr}(\psi(D^{n+1})))<\mathrm{dp\text{-}rk}(\psi(D^{n+1})).$

Thus, it must be the case that
$\mathrm{dp\text{-}rk}(\psi(D^{n+1}))\geq\mathrm{dp\text{-}rk}(a/A_{1})+1$.
However, $\psi\in\Delta$ hence for every $b\in\pi_{n+1}(\psi(D^{n+1}))$ (the
projection to the last coordinate), we have
$\mathrm{dp\text{-}rk}(\psi(D^{n},b))<\mathrm{dp\text{-}rk}(a/A_{1})$,
contradicting subadditivity of dp-rank. ∎

We can now prove the main proposition of this section. Note that in the
following $V$ is not necessarily $A$-definable.

######  Proposition 3.12.

Let $D$ be an SW-uniformity. For every open $V\subseteq D^{n}$, $a\in V$, and
$A$ a small set of parameters, there exists $B\supseteq A$ and a $B$-definable
open subset $U=U_{1}\times\dots\times U_{n}\subseteq V$ such that $a\in U$ and
$\mathrm{dp\text{-}rk}(a/B)=\mathrm{dp\text{-}rk}(a/A)$.

###### Proof.

Assume that $a=(a_{1},\ldots,a_{n})$. By induction it is sufficient to prove:

  * ($\dagger$)

Given any open set $W\ni a_{1}$ there exists $B\supseteq A$ and a
$B$-definable open $U\subseteq W$ containing $a_{1}$ such that
$\mathrm{dp\text{-}rk}(a/B)=\mathrm{dp\text{-}rk}(a/A)$.

Indeed, given $a=(a_{1},\ldots,a_{n})$, and given $W\ni a$ we may assume that
$W=W_{1}\times\dots\times W_{n}$. Using ($\dagger$), there exists a finite
$b_{1}$, and an $Ab_{1}$-definable $U_{1}\subseteq W_{1}$ containing $a_{1}$
such that $\mathrm{dp\text{-}rk}(a/b_{1}A)=\mathrm{dp\text{-}rk}(a/A)$. Now
replace $A$ with $A_{1}=b_{1}A$ and apply the inductive hypothesis to
$W_{2}\times\dots\times W_{n}$ and $A_{1}$.

So we now turn to proving ($\dagger$). Fix $s_{0}\in T$ such that
$U_{s_{0}}[a_{1}]\subseteq W$ and by the definition of a uniformity, we find
$t_{0}\in T$, such that $\\{(x,z)\in D^{2}:(\exists y\in D)\left((x,y)\in
U_{t_{0}},\,(y,z)\in U_{t_{0}}\right)\\}\subseteq U_{s_{0}}.$ By Lemma 3.11,
there exists $t_{1}\leq t_{0}$ with
$\mathrm{dp\text{-}rk}(a/At_{1})=\mathrm{dp\text{-}rk}(a/A)$ and
$a_{1}^{\prime}\in U_{t_{1}}[a_{1}]$ such that
$\mathrm{dp\text{-}rk}(a/At_{1}a_{1}^{\prime})=\mathrm{dp\text{-}rk}(a/At_{1})=\mathrm{dp\text{-}rk}(a/A)$.
We claim that $U_{t_{1}}[a_{1}^{\prime}]\subseteq U_{s_{0}}[a_{1}]\subseteq
W$.

Indeed, assume that $x\in U_{t_{1}}[a_{1}^{\prime}]$ then
$(a_{1}^{\prime},x)\in U_{t_{1}}$ so $(a_{1}^{\prime},x)\in U_{t_{0}}$, as
$t_{1}\leq t_{0}$. By construction $a_{1}^{\prime}\in U_{t_{1}}[a_{1}]$, hence
$(a_{1},a_{1}^{\prime})\in U_{t_{1}}$, so also $(a_{1},a_{1}^{\prime})\in
U_{t_{0}}$. It follows that $(a_{1},x)\in U_{s_{0}}$ by our choice of $t_{0}$.
But then $x\in U_{s_{0}}[a_{1}]$, as we wanted.

It follows that the set $W_{1}=U_{t_{1}}[a_{1}^{\prime}]$ is a subset of $W$
containing $a_{1}$ (by symmetry of the uniformity) and is defined over
$Ata_{1}^{\prime}$. ∎

We shall also need the following technical corollary.

######  Corollary 3.13.

Let $D$ be an SW-uniformity. For every definable $X\subseteq D^{n}$,
$Y\subseteq X$ a definable subset, $a$ in the relative interior of $Y$ in $X$,
$b\in D^{k}$, and $A$ a small set of parameters, there exists $B\supseteq A$
and a $B$-definable open subset $U=U_{1}\times\dots\times U_{n}\subseteq
D^{n}$ such that $a\in U\cap X\subseteq Y$ and
$\mathrm{dp\text{-}rk}(a,b/B)=\mathrm{dp\text{-}rk}(x,y/A)$.

###### Proof.

Since $a$ is in the relative interior of $Y$ in $X$, there exist an open
subset $V\subseteq D^{n}$ such that $a\in X\cap V\subseteq Y$. Consider the
open set $V^{\prime}=V\times D^{k}$. Now apply Proposition 3.12 to
$V^{\prime}$, $(a,b)$ and $A$. ∎

The next lemma plays an important role in our analysis of infinitesimal
neighbourhoods in Section 4.2. Clause (1) of the lemma is well known for
$n=1$. We thank Itay Kaplan for the general case. As the concepts needed for
the proof are not exactly inline with the rest of the paper we postpone the
proofs to the appendix.

######  Lemma 3.14.

Let ${\mathcal{M}}$ be a structure of finite dp-rank and
$\mathbb{U}\succ{\mathcal{M}}$ a monster model.

  1. (1)

Let $D$ be an SW-uniformity in ${\mathcal{M}}$ and let $b_{1},\dots,b_{n}$ be
some tuples in $\mathbb{U}$. For every ${\mathcal{M}}$-definable $X$, there
exists $a\in X$, with $\mathrm{dp\text{-}rk}(a/M)=\mathrm{dp\text{-}rk}(X)$,
such that
$\mathrm{dp\text{-}rk}(ab_{i}/M)=\mathrm{dp\text{-}rk}(a/M)+\mathrm{dp\text{-}rk}(b_{i}/M)$
for all $1\leq i\leq n$.

  2. (2)

For $A\subseteq\mathbb{U}$ and $a\in{\mathcal{M}}^{n}$, there exists a small
model ${\mathcal{N}}\prec{\mathcal{M}}$, $A\subseteq N$, such that
$\mathrm{dp\text{-}rk}(a/A)=\mathrm{dp\text{-}rk}(a/N)$.

###### Proof.

See Appendix A. ∎

######  Remark 3.15.

The above lemma, too, can be viewed as a partial substitute for exchange.
Indeed, it is straightforward to see that if $\mathrm{dp\text{-}rk}$ is
additive in $D$, i.e. for all tuples $a,b$ from $D$ and $A$ an arbitrary set
of parameters

$\mathrm{dp\text{-}rk}(a,b/A)=\mathrm{dp\text{-}rk}(a/Ab)+\mathrm{dp\text{-}rk}(b/A),$

then Lemma 3.14(1) is true over any parameter set (not only over a model).
Additivity of dp-rank is equivalent, in the context of dp-minimal structures,
to exchange (see e.g., [45, Observation 3.1, Proposition 3.2]).

##  4\. Fields locally strongly internal to various dp-minimal fields

The aim of this section to classify all fields ${\mathscr{F}}$ of finite dp-
rank such that some infinite definable subset of ${\mathscr{F}}$ can be
definably injected into a field ${\mathcal{K}}$ that is either an SW-
uniformity or strongly minimal. We show that under various assumptions such
fields are definably isomorphic to finite extensions of $K$. We work in
somewhat greater generality that will be useful in the sequel.

######  Assumption.

Throughout the entire of Section 4 we assume that ${\mathcal{M}}$ is
$|T|^{+}$-saturated.

###  4.1. Strong internality, and the main technical lemma

######  Definition 4.1.

Let ${\mathcal{M}}$ be any (multi-sorted) structure.

  1. (1)

An $A$-definable set $X$ is strongly internal to (a definable) set $Y$ over
$A$ if there exists an $A$-definable injection $f:X\to Y^{n}$, for some
$n\in\mathbb{N}$. We may omit the reference to $A$ and just say _$X$ is
strongly internal to $Y$._

$X$ is called _locally strongly internal to $Y$ over $A$_ if there exists some
$A$-definable infinite $X^{\prime}\subseteq X$ which is strongly internal to
$Y$ over $A$. Again, we may omit the reference to $A$.

  2. (2)

Following Johnson [26], we define: For $X$ locally strongly internal to $Y$, a
definable set $Z\subseteq X$ is _$Y$ -critical (in $X$)_ if $Z$ is strongly
internal to $Y$ of maximal dp-rank, i.e.,
$\mathrm{dp\text{-}rk}(Z)\geq\mathrm{dp\text{-}rk}(Z^{\prime})$ for all
$Z^{\prime}\subseteq X$ that is strongly internal to $Y$.

We start our investigation by proving an important technical lemma. Roughly,
the lemma states that if a field of finite dp-rank is locally strongly
internal to an SW-uniform structure $D$ then there exists a subset of the
field strongly internal to $D$ and sufficiently closed under the field
operations.

######  Lemma 4.2.

Let $D$ be an SW-uniformity in some (possibly multi-sorted) structure
${\mathcal{M}}$. Let $({\mathscr{F}},+,\cdot)$ be an infinite field of finite
dp-rank definable field in ${\mathcal{M}}$ and assume that ${\mathscr{F}}$ is
locally strongly internal to $D$ over $A$. Let $Y\subseteq{\mathscr{F}}$ be a
$D$-critical set (over $A$) and $I\subseteq Y$ an $A$-definable set with
$\mathrm{dp\text{-}rk}(I)=1$. Let $(b,c,d)\in I\times Y\times Y$ be such that
$\mathrm{dp\text{-}rk}(b,c,d/A)=2\mathrm{dp\text{-}rk}(Y)+1$.

Then there is $B\supseteq A$ and infinite $B$-definable sets $J\subseteq I$
and $S=Y_{1}\times Y_{2}\subseteq Y^{2}$, with $(b,c,d)\in J\times S$ and
$\mathrm{dp\text{-}rk}(b,c,d/B)=\mathrm{dp\text{-}rk}(J\times
S)=2\mathrm{dp\text{-}rk}(Y)+1$, such that for every $(x,y,z)\in J\times S$,
we have $(x-b)y+z\in Y$.

###### Proof.

For simplicity of notation assume $A=\emptyset$. Denote
$\mathrm{dp\text{-}rk}(Y)=n$. Let $(b,c,d)\in I\times Y^{2}$ be as in the
statement. Since dp-rank is preserved under definable bijections, we may
replace $Y,I$ and, correspondingly, $(b,c,d)$ by their images under any
$\emptyset$-definable bijections. As we are considering $I\times Y$ with all
of its induced structure, and $I,Y$ are strongly internal to $D$, witnessed by
$\emptyset$-definable functions, we may identify $I$ and $Y$ (and therefore
also $I\times Y$) and $(b,c,d)$ with definable sets and tuples in $D$.

Note that

$2n+1=\mathrm{dp\text{-}rk}(b,c,d)\leq\mathrm{dp\text{-}rk}(b,d/c)+\mathrm{dp\text{-}rk}(c)\leq\mathrm{dp\text{-}rk}(b,d)+\mathrm{dp\text{-}rk}(c)\leq
2n+1$

so that $\mathrm{dp\text{-}rk}(b,d/c)=n+1$, $\mathrm{dp\text{-}rk}(c)=n$.
Similarly, $\mathrm{dp\text{-}rk}(b,c)=n+1$ and
$\mathrm{dp\text{-}rk}(b/c)=1$.

For $(x,y,z)\in I\times Y\times Y$ consider the function $f_{y}(x,z)=xy-z$.
Let $e=f_{c}(b,d)$.

######  Claim 4.2.1.

$b\notin\operatorname{acl}(c,e)$.

###### Proof.

Assume towards a contradiction that $b\in\operatorname{acl}(c,e)$. By
Proposition 3.12, we can find some $B$ such that
$b\in\operatorname{dcl}(Bc,e)$ and
$\mathrm{dp\text{-}rk}(b,c,d/B)=\mathrm{dp\text{-}rk}(b,c,d)$. Indeed, we can
first find $U\ni b$ such that $b$ is the only realization of
$\operatorname{tp}(b/c,e)$ in $U$ and then apply Proposition 3.12 (for the
tuple $(b,c,d)$ and any open box whose projection onto the $b$-coordinate is
$U$).

Let $\varphi(x,e,c)$ be the algebraic formula isolating
$\operatorname{tp}(b/Bc,e)$, in particular, $\varphi(x,e,c)$ implies that
$x\in I$. By the definition of $f_{c}$, $d\in\operatorname{dcl}(b,c,e)$,
therefore $\varphi(x,e,c)\wedge f_{c}(x,y)=e$ is an algebraic formula
isolating $\operatorname{tp}(b,d/Bc,e)$. In fact, $(b,d)$ is the only pair of
elements realizing this type. Hence, $\exists!(x,y)(\varphi(x,e,c)\wedge
f_{c}(x,y)=e)$. By compactness, there is a formula
$\psi(z,c)\in\operatorname{tp}(e/Bc)$ implying
$\exists!(x,y)(\varphi(x,z,c)\wedge f_{c}(x,y)=z)$. In other words, for
$X:=\psi({\mathscr{F}},c)$, there is a $Bc$-definable injective function $F$
from $X$ into $I\times Y$, sending $e$ to $(b,d)$.

The image of $F$ in $I\times Y$ is a $Bc$-definable set containing $(b,d)$ and
since $\mathrm{dp\text{-}rk}(b,d/Bc)=n+1$, we have
$\mathrm{dp\text{-}rk}(\mathrm{Im}(F))=n+1$ and consequently
$\mathrm{dp\text{-}rk}(\operatorname{dom}(F))>n$. As
$\mathrm{dp\text{-}rk}(\mathrm{dom}(F))>\mathrm{dp\text{-}rk}(Y)$ and
$\operatorname{dom}(F)\subseteq{\mathscr{F}}$ this contradicts $Y$ being
$D$-critical. ∎

We conclude that
$b\notin\operatorname{acl}(c,f_{c}(b,d))=\operatorname{acl}(c,e)$ and in
particular the projection of $f_{c}^{-1}(e)\subseteq I\times Y$ on the first
coordinate, call it $I^{\prime}$, is infinite and contains $b$. So
$\mathrm{dp\text{-}rk}(b/ce)=\mathrm{dp\text{-}rk}(I^{\prime})=1$, and by
definition for every $x\in I^{\prime}$ there is $z\in Y$ with $xc-z=bc-d=e$.
Since ${\mathscr{F}}$ is a field, the map from $I^{\prime}$ to $Y$, taking
$x\in I^{\prime}$ to $z=xc-e=(x-b)c+d$, is injective.

By Fact 3.7, $b$ is in the relative interior of $I^{\prime}$ in $I$. We may
now apply Corollary 3.13 to the definable sets $I^{\prime}\subseteq I$ and the
elements $b$ and $(c,d)$, to get a set of parameters $A_{1}$ and an
$A_{1}$-definable subset $I^{\prime\prime}\subseteq I^{\prime}$ containing
$b$, such that $\mathrm{dp\text{-}rk}(b,c,d/A_{1})=1+2n$.

Let

$S=\\{(y,z)\in Y^{2}:(\forall x\in I^{\prime\prime})((x-b)y+z\in Y)\\}.$

Note that $(c,d)\in S$ and that $S$ is $bA_{1}$-definable. Because $(c,d)\in
S$ and $\mathrm{dp\text{-}rk}(c,d/bA_{1})=2n$ clearly
$\mathrm{dp\text{-}rk}(S)=\mathrm{dp\text{-}rk}(Y^{2})=2n$. Also, by Fact 3.7
again, $(c,d)$ is in the relative interior of $S$ in $Y^{2}$.

Now consider $I^{\prime\prime}\times S\subseteq I\times Y^{2}$. By the above,
$\mathrm{dp\text{-}rk}(I^{\prime\prime}\times S)=\mathrm{dp\text{-}rk}(I\times
Y^{2})$, and $(b,c,d)$ is in the relative interior of $I^{\prime\prime}\times
S$ in $I\times Y^{2}$ so Corollary 3.13 applies (with $x=(b,c,d)$). We can
thus find a set $B\supseteq A_{1}$ and $B$-definable subsets $J\subseteq
I^{\prime\prime}$, $Y_{1}\times Y_{2}\subseteq S$, with $(b,c,d)\in J\times
Y_{1}\times Y_{2}$ such that
$\mathrm{dp\text{-}rk}(b,c,d/B)=\mathrm{dp\text{-}rk}(b,c,d)=2n+1$. ∎

The next corollary will help us late on to show that our field ${\mathscr{F}}$
is not locally strongly internal to various sorts. It is an analogue of the
fact that a linear o-minimal structure cannot support a definable field
structure.

######  Corollary 4.3.

Let $(G,\oplus)$ be an SW-uniformity (in ${\mathcal{M}}$) supporting a
definable abelian group structure. Assume that $G$ satisfies the following:
For every definable (partial) $f:G^{m}\to G$, $m\in\mathbb{N}$, whose domain
is open there exists an open definable $R\subseteq\operatorname{dom}(f)$, a
definable group homomorphism $L:G^{n}\to G$ and $e\in G$ such that for every
$y\in R$, $f(y)=L(y)\oplus e$. Then no definable infinite field is locally
strongly internal to $G$.

###### Proof.

Assume towards contradiction that an infinite definable field ${\mathscr{F}}$
is locally strongly internal to $G$ and let $Y\subseteq{\mathscr{F}}$ be
${\mathcal{G}}$-critical in ${\mathscr{F}}$ with $\mathrm{dp\text{-}rk}(Y)=n$.
By Remark 3.8, there is a definable dp-minimal $I\subseteq{\mathscr{F}}$.
Assume that $Y,I$ and the injection of $Y$ into $G^{k}$ for some $k$, are all
defined over $A$.

By Lemma 4.2 (applied after fixing an appropriate tuple $(b,c,d)\in I\times
Y^{2}$), there are definable $J\subseteq I$ and $S\subseteq Y^{2}$ such that
$\mathrm{dp\text{-}rk}(J\times S)=1+2n$ and $f(x,y,z)=(x-b)y+z$ maps $J\times
S$ into $Y$. By strong internality combined with Fact 3.6 we can, after
possibly replacing $I$ and $Y$ with subsets of the same dp-rank (defined,
possibly, over additional parameters) identify $I$ with a subset of $G$ and
$Y$ with a subset of $G^{n}$. Thus, we may assume that $f:G^{1+2n}\to G^{n}$.

By our assumption, we may inductively find an additive (with respect to
$\oplus$) definable function $L:G^{1+2n}\to G^{n}$ and $e\in G^{n}$ such that
$f(x,y,z)$ and $L(x,y,z)\oplus e$ agree on some $X\subseteq J\times S$ with
$\mathrm{dp\text{-}rk}(X)=2n+1$. Fix some $J_{0}\times S_{0}\subseteq X$ with
$\mathrm{dp\text{-}rk}(J_{0})=1$ and $\mathrm{dp\text{-}rk}(S_{0})=2n$.

The map $(y,z)\mapsto f(0,y,z)$ maps $S_{0}$ into $Y$, where here $0=0_{G}$.
Since $\mathrm{dp\text{-}rk}(S_{0})=2n>n=\mathrm{dp\text{-}rk}(Y)$ the map
cannot be injective and hence there are $(y_{1},z_{1})\neq(y_{2},z_{2})$ for
which $f(0,y_{1},z_{1})=f(0,y_{2},z_{2})$.

It follows that $L(0,y_{1},z_{1})=L(0,y_{2},z_{2})$. But then for every $x\in
G$,

$L(x,y_{1},z_{1})=L(0,y_{1},z_{1})\oplus L(x,0,0)=L(x,y_{2},z_{2})$

Thus also $f(x,y_{1},z_{1})=f(x,y_{2},z_{2})$ for any $x\in J_{0}$.

On the other hand, by the definition of $f$ in the field ${\mathscr{F}}$, for
every $(y_{1},z_{1})\neq(y_{2},z_{2})$ there is at most one $x$ such that
$f(x,y_{1},z_{1})=f(x,y_{2},z_{2})$. Contradicting the fact that $J_{0}$ is
infinite. ∎

###  4.2. The subgroup of infinitesimals

######  Assumption.

In addition to our $|T|^{+}$-saturation assumption, throughout this section
${\mathcal{M}}$ is any first order (multi-sorted) structure and $D$ an
${\mathcal{M}}$-definable SW-uniformity. For ease of presentation, assume that
${\mathcal{M}}$ has one distinguished sort whose universe is $M$.

######  Definition 4.4.

For any $Z\subseteq M^{n}$ and definable injective $g:Z\to D^{m}$, let
$\tau_{Z,g}$ be the topology on $Z$ given by $\\{g^{-1}(U):U\subseteq
D^{m}\text{ is $M$-definable open}\\}$

We observe that because $D^{n}$ has a definable basis for its topology so does
$\tau_{Z,g}$ (for any $Z$ and $g$). Moreover, if $Z_{1},Z_{2}\subseteq M^{n}$
and $g_{i}:Z_{i}\to D^{m_{i}}$ are definable injections (for $i=1,2$) then the
topology $\tau_{Z_{1}\times Z_{2},g_{1}\times g_{2}}$ is the product topology
$\tau_{Z_{1},g_{1}}\times\tau_{Z_{2},g_{2}}$.

It follows immediately from Fact 3.5 and the above observation that:

######  Lemma 4.5.

If $g_{i}:Z_{i}\to D^{m_{i}}$ are definable injections ($i=1,2$) and
$f:Z_{1}^{k}\to Z_{2}^{l}$ is a definable (partial) function then the set of
continuity points $C_{f}$ of $f$ with respect to $\tau_{Z_{i},g_{i}}$ is
definable and $\mathrm{dp\text{-}rk}(\operatorname{dom}(f)\setminus
C_{f})<\mathrm{dp\text{-}rk}(\operatorname{dom}(f))$.

We thus have:

######  Lemma 4.6.

Let $Z\subseteq M^{n}$ be definable and $g:Z\to D^{m}$, $h:Z\to D^{k}$ two
$A$-definable injections. Then, $\tau_{Z,g}$ and $\tau_{Z,h}$ agree at every
$z\in Z$ with $\mathrm{dp\text{-}rk}(z/A)=\mathrm{dp\text{-}rk}(Z)$. Namely,
there is a common basis for the $\tau_{Z,g}$-neighbourhoods and the
$\tau_{Z,h}$-neighbourhoods for such a $z\in Z$.

###### Proof.

Apply Lemma 4.5 to $\operatorname{id}:Z\to Z$. ∎

######  Definition 4.7.

For $Z\subseteq M^{n}$ definable, $g:Z\to D^{m}$ a definable injection, and
$d\in Z$, let $\nu_{Z,g}(d)$ be the partial global type given by all definable
$\tau_{Z,g}$-open sets containing $d$. We call it the infinitesimal
neighborhood of $d$ with respect to $\tau_{Z,g}$.

It follows from the above discussion that:

######  Remark 4.8.

If $g_{i}:Z_{i}\to D^{m_{i}}$ (for $i=1,2$) are $A$-definable injections
$(d_{1},d_{2})\in Z_{1}\times Z_{2}$ then
$\nu_{Z_{1},g_{1}}(d_{1})\times\nu_{Z_{2},g_{2}}(d_{2})=\nu_{Z_{1}\times
Z_{2},g_{1}\times g_{2}}(d_{1},d_{2})$.

By Lemma 4.6, we have:

######  Corollary 4.9.

If $Z$ is strongly internal to $D$ over $A$ and $d\in Z$ is such that
$\mathrm{dp\text{-}rk}(d/A)=\mathrm{dp\text{-}rk}(Z)$ then $\nu_{Z,g}(d)$ does
not depend on the choice of the definable injection $g$ (over $A$).

######  Notation 4.10.

If $Z$ is strongly internal to $D$ over $A$ and $d\in Z$ with
$\mathrm{dp\text{-}rk}(d/A)=\mathrm{dp\text{-}rk}(Z)$, we let
$\nu_{Z}(d):=\nu_{Z,g}(d)$ for some (equivalently, any) $A$-definable
injection $g:Z\to D^{m}$ (some $m$). By Corollary 4.9 this is well defined.

We observe also that $\nu_{Z}(d)$ does not depend on $Z$ (but only on its germ
at $d$) in the following sense:

######  Lemma 4.11.

Assume that $Z\subseteq M^{n}$ is strongly internal to $D$ over $A$, witnessed
by $g$, and $Z_{1}\subseteq Z$ is $A$-definable with
$\mathrm{dp\text{-}rk}(Z_{1})=\mathrm{dp\text{-}rk}(Z)$. If $d\in Z_{1}$ is
such that $\mathrm{dp\text{-}rk}(d/A)=\mathrm{dp\text{-}rk}(Z)$ then
$\nu_{Z_{1}}(d)=\nu_{Z}(d)$.

###### Proof.

By Lemma 3.9, the topologies $\tau_{Z,g}$ and $\tau_{Z_{1},g}$ agree on a
neighborhood of $d$, thus $\nu_{Z}(d)=\nu_{Z_{1}}(d)$. ∎

Finally, we note;

######  Lemma 4.12.

Let $Y_{1},Y_{2}$ be definable sets strongly internal to $D$ over $A$. If
$f:Y_{1}\to Y_{2}$ is an $A$-definable partial function,
$\mathrm{dp\text{-}rk}(\operatorname{dom}(f))=\mathrm{dp\text{-}rk}(Y_{1})$,
and $a\in\operatorname{dom}(f)$ with
$\mathrm{dp\text{-}rk}(a/A)=\mathrm{dp\text{-}rk}(\operatorname{dom}(f))$,
then $\nu_{1}(a)\vdash
f^{-1}(\nu_{2}(f(a))):=\\{f^{-1}(U):U\in\nu_{2}(f(a))\\}$. I.e., if
$b\models\nu_{1}(a)$ then $f(b)\models\nu_{2}(f(a))$

###### Proof.

By Lemma 4.5, $f$ is continuous at $a$ with respect to $\tau_{Y_{1},g}$ and
$\tau_{Y_{2},h}$ (for any $A$-definable $g,h$ witnessing the strong
internality of $Y_{1}$, $Y_{2}$, respectively). The conclusion follows. ∎

We now prove a general statement.

######  Lemma 4.13.

Let $(H,\cdot)$ be a definable group in ${\mathcal{M}}$ and assume that
$Y_{1},Y_{2},Y_{3}\subseteq H$ are ${\mathcal{M}}$-definable sets, strongly
internal to $D$, all definable over some model
${\mathcal{N}}\prec{\mathcal{M}}$, with
$\mathrm{dp\text{-}rk}(Y_{1})=\mathrm{dp\text{-}rk}(Y_{2})=\mathrm{dp\text{-}rk}(Y_{3})=k$,
and assume that $Y_{1}\cdot Y_{2}\subseteq Y_{3}$. Then

  1. (1)

For every $c\in Y_{1}$ and $d\in Y_{2}$ such that
$\mathrm{dp\text{-}rk}(c,d/N)=2k$, we have
$c^{-1}\cdot\nu_{Y_{1}}(c)=\nu_{Y_{2}}(d)\cdot d^{-1}$, and for every
$d_{1}\in Y_{2}$ with $\mathrm{dp\text{-}rk}(d_{1}/A)=k$ we have
$\nu_{Y_{2}}(d)\cdot d^{-1}=\nu_{Y_{2}}(d_{1})\cdot d_{1}^{-1}$.

  2. (2)

For every $d\in Y_{2}$ such that $\mathrm{dp\text{-}rk}(d/N)=k$, the partial
type $\nu_{Y_{2}}(d)\cdot d^{-1}$ is a type definable subgroup of $H$.

###### Proof.

It is convenient to work in an $|{\mathcal{M}}|^{+}$-saturated elementary
extension $\widehat{\mathcal{M}}$ of ${\mathcal{M}}$ and with the realizations
in $\widehat{\mathcal{M}}$ of the infinitesimal types. We start with some
general observations.

Consider the function $F:Y_{1}\times Y_{2}\to Y_{3}$,
$F(y_{1},y_{2})=y_{1}\cdot y_{2}$. Applying Lemma 4.12 to $F$ (and using
Remark 4.8), we see that
$F(\nu_{Y_{1}}(c)\times\nu_{Y_{2}}(d))\subseteq\nu_{Y_{3}}(c\cdot d)$. Because
the dp-rank is preserved under the definable bijection $(x,y)\mapsto(x,xy)$,
then $\mathrm{dp\text{-}rk}(c,c\cdot
d)=\mathrm{dp\text{-}rk}(c,d)=2k=\mathrm{dp\text{-}rk}(Y_{1}\times Y_{3})$.

Consider also the function $R:Y_{1}\times Y_{3}\to H$, $R(x,z)=x^{-1}\cdot z$.
Since $R(c,c\cdot d)=d\in Y_{2}$, it follows that the set $W=\\{(x,z)\in
Y_{1}\times Y_{3}:R(x,z)\in Y_{2}\\}$ contains $(c,c\cdot d)$ and has
$\mathrm{dp\text{-}rk}=2k$. Thus, by Lemma 4.11, $\nu_{W}(c,c\cdot
d)=\nu_{Y_{1}\times Y_{3}}(c,c\cdot d)$ and by Lemma 4.12 (and Remark 4.8),
$R$ sends (realisations of) $\nu_{Y_{1}}(c)\times\nu_{Y_{3}}(c\cdot d)$ to
(realisations of) $\nu_{Y_{2}}(d)$. It follows that for every
$c_{1}\in\nu_{Y_{1}}(c)$, the function $y\mapsto c_{1}\cdot y$ is a bijection
between $\nu_{Y_{2}}(d)$ and $\nu_{Y_{3}}(c\cdot d)$. Indeed, $F(c_{1},-)$ is
a function from $\nu_{Y_{2}}(d)$ into $\nu_{Y_{3}}(c\cdot d)$, whose inverse
is $R(c_{1},-)$.

Since $F$ maps realisations of $\nu_{Y_{1}}(c)\times\nu_{Y_{2}}(d)$ onto
realisations of $\nu_{Y_{3}}(c\cdot d)$ it follows that for every
$d^{\prime}\in\nu_{Y_{2}}(d)$, the function $x\mapsto x\cdot d^{\prime}$ is a
bijection between realisations of $\nu_{Y_{1}}(c)$ and realisations of
$\nu_{Y_{3}}(c\cdot d)$. In particular,
$c\cdot\nu_{Y_{2}}(d)=\nu_{Y_{1}}(c)\cdot d=\nu_{Y_{3}}(c\cdot d)$ and we can
therefore conclude that $\nu_{Y_{2}}(d)\cdot
d^{-1}=c^{-1}\cdot\nu_{Y_{1}}(c)$, proving the first clause of (1).

Assume next that $d_{0},d_{1}\in Y_{2}$ and
$\mathrm{dp\text{-}rk}(d_{0}/N)=\mathrm{dp\text{-}rk}(d_{1}/N)=k$. By Lemma
3.14(1), we can find $c\in Y_{1}$ such that
$\mathrm{dp\text{-}rk}(c,d_{0}/N)=\mathrm{dp\text{-}rk}(c,d_{1}/N)=2k$. And
then, by what we just saw, $\nu_{Y_{2}}(d_{0})\cdot
d_{0}^{-1}=c^{-1}\cdot\nu_{Y_{1}}(c)=\nu_{Y_{2}}(d_{1})\cdot d_{1}^{-1}$.

To prove (2) we need to show that $\nu_{Y_{2}}(d)\cdot d^{-1}$ is a subgroup
of $H$, for every $d\in Y_{2}$ with $\mathrm{dp\text{-}rk}(d/N)=k$. Given
$a,b\in\nu_{Y_{2}}(d)$, we need to show that $(a\cdot d^{-1})\cdot(b\cdot
d^{-1})^{-1}=a\cdot b^{-1}$ is also in $\nu_{Y_{2}}(d)\cdot d^{-1}$.

By our above observations, the maps $y\mapsto c\cdot y$ is a bijection of
$\nu_{Y_{2}}(d)$ and $\nu_{Y_{3}}(c\cdot d)$ and the map $x\mapsto x\cdot b$
is a bijection of $\nu_{Y_{1}}(c)$ and $\nu_{Y_{3}}(c\cdot d)$. Hence, there
is $c_{1}\in\nu_{Y_{1}}(c)$ such that $c\cdot a=c_{1}\cdot b$.

It follows that $a\cdot b^{-1}=c^{-1}\cdot c_{1}\in
c^{-1}\cdot\nu_{Y_{1}}(c)$. By what we just showed,
$c^{-1}\cdot\nu_{Y_{1}}(c)=\nu_{Y_{2}}(d)\cdot d^{-1}$, hence $a\cdot
b^{-1}\in\nu_{Y_{2}}(d)\cdot d^{-1}$. ∎

We now apply the above to definable fields.

######  Proposition 4.14.

Let $({\mathscr{F}},+,\cdot)$ be a definable field in ${\mathcal{M}}$.

  1. (1)

Assume that ${\mathscr{F}}$ and $D$ are definable over a small model
${\mathcal{N}}_{0}\prec{\mathcal{M}}$ and let $Y\subseteq{\mathscr{F}}$ be
$D$-critical and strongly internal to $D$ over ${\mathcal{N}}_{0}$. Then for
every $d\in Y$ such that
$\mathrm{dp\text{-}rk}(d/N_{0})=\mathrm{dp\text{-}rk}(Y)$, the partial type
$\nu_{Y}(d)-d$ is a subgroup of $({\mathscr{F}},+)$. Moreover, the subgroup is
independent of the choice of $d$, and we denote it $\nu_{Y}$.

  2. (2)

For every $D$-critical $Y_{1},Y_{2}\subseteq{\mathscr{F}}$,
$\nu_{Y_{1}}=\nu_{Y_{2}}$.

  3. (3)

Let $\nu:=\nu_{Y}$ be the partial type associated to some (any) $D$-critical
$Y$, as implied by (2). Then $\nu$ is invariant under multiplication by
scalars from ${\mathscr{F}}$.

###### Proof.

(1) Since $Y$ is strongly internal to $D$ over ${\mathcal{N}}_{0}$, and $D$ is
dp-minimal there exists some ${\mathcal{N}}_{0}$-definable $I\subseteq Y$ that
is dp-minimal (Remark 3.8). We assume that $Y$ is a subset of $D^{k}$ for some
$k$ and let $n=\mathrm{dp\text{-}rk}(Y)$. By Corollary 4.6, for every $d\in Y$
with $\mathrm{dp\text{-}rk}(d/N_{0})=n$, the infinitesimal neighbourhood
$\nu_{Y}(d)$ (and therefore also $\nu_{Y}(d)-d$) does not depend on the choice
of the embedding of $Y$ in $D$ over $N_{0}$.

We prove simultaneously that $\nu_{Y}(d)-d$ is a group and that it does not
depend on the choice of $d$. Our intention is to apply Lemma 4.13 to the
definable group $({\mathscr{F}},+)$. So let $d,d^{\prime}\in Y$ be such that
$\mathrm{dp\text{-}rk}(d/N_{0})=\mathrm{dp\text{-}rk}(d^{\prime}/N_{0})=n$.
Our first observation is that we may assume that
$\mathrm{dp\text{-}rk}(d,d^{\prime}/N)=2n$. Indeed, by Lemma 3.14(1) we can
find $c\in Y$ such that
$\mathrm{dp\text{-}rk}(d,c/N)=\mathrm{dp\text{-}rk}(d^{\prime},c/N)=2n$. So if
we prove the equality of groups for $d,c$ and $d^{\prime},c$ then we would get
that $\nu_{Y}(d)-d=\nu_{Y}(c)-c=\nu_{Y}(d^{\prime})-d^{\prime}$. So from now
on we assume $\mathrm{dp\text{-}rk}(d,d^{\prime}/N)=2n$.

By Lemma 3.14(1), there are $(b,c)\in I\times Y$ such that
$\mathrm{dp\text{-}rk}(b,c,d,d^{\prime}/N_{0})=1+3n$. In particular

$\mathrm{dp\text{-}rk}(b,c,d/N_{0})=\mathrm{dp\text{-}rk}(b,c,d^{\prime}/N_{0})=1+2n.$

We first prove:

######  Claim 4.14.1.

There are relatively open sets $\bar{J}\subseteq I$, containing $b$,
$\bar{Y}_{1}\subseteq Y$ containing $c$, and $Y_{2},Y_{2}^{\prime}\subseteq Y$
containing $d$ and $d^{\prime}$, respectively, such that for every
$(x,y,z)\in\bar{J}\times\bar{Y}_{1}\times(Y_{2}\cup Y_{2}^{\prime})$, we have

$(x-b)\cdot y+z\in Y.$

###### Proof.

Applying Lemma 4.2 to $(b,c,d)$ we obtain $B\supseteq N_{0}$ and $B$-definable
$J\times Y_{1}\times Y_{2}\subseteq I\times Y^{2}$, with $(b,c,d)\in J\times
Y_{1}\times Y_{2}$ such that $\mathrm{dp\text{-}rk}(b,c,d/B)=1+2n$ and such
that the map $(x,y,z)\mapsto(x-b)y+z$ sends $J\times Y_{1}\times Y_{2}$ into
$Y$. Similarly we obtain $B^{\prime}\supseteq N_{0}$ and
$B^{\prime}$-definable $J^{\prime}\times Y^{\prime}_{1}\times
Y^{\prime}_{2}\subseteq I\times Y^{2}$, with $(b,c,d^{\prime})\in
J^{\prime}\times Y^{\prime}_{1}\times Y^{\prime}_{2}$ such that
$\mathrm{dp\text{-}rk}(b,c,d^{\prime}/B^{\prime})=1+2n$ and such that
$(x,y,z)\mapsto(x-b)y+z$ sends $J^{\prime}\times Y^{\prime}_{1}\times
Y^{\prime}_{2}$ into $Y$.

Since $b\in J$ and $\mathrm{dp\text{-}rk}(b/B)=1$, $b$ is in the relative
interior of $J$ in $I$, by Fact 3.7. Likewise, $b\in J^{\prime}$ is in the
relative interior of $J^{\prime}$ in $I$. Hence $b$ is in the relative
interior of $\bar{J}=J\cap J^{\prime}$ in $I$. By similar arguments, $c\in
Y_{1}\cap Y_{1}^{\prime}$ is in the relative interior of
$\bar{Y}_{1}=Y_{1}\cap Y_{1}^{\prime}$ in $Y$ and $d\in Y_{2}$ (respectively
$d^{\prime}\in Y_{2}^{\prime}$) is in the relative interior of $Y_{2}$
(respectively $Y_{2}^{\prime}$) in $Y$. Consequently, $(b,c,d,d^{\prime})$ is
in the relative interior of $\bar{J}\times\bar{Y}\times Y_{2}\times
Y_{2}^{\prime}$ in $I\times Y^{3}$, and by replacing the sets with their
relative interior we may assume that $\bar{J}$ is realtively open in $J$ and
$\bar{Y}_{1}$, $Y_{2}$, $Y_{2}^{\prime}$ are realtively open in $Y$. This ends
the proof of the claim. ∎

By Corollary 3.13 and Lemma 3.14(2) there is some small model
${\mathcal{L}}\supseteq{\mathcal{N}}_{0}$ and ${\mathcal{L}}$-definable
subsets $\widehat{J}\subseteq\bar{J}$, $\widehat{Y}_{1}\subseteq\bar{Y}_{1}$,
$W_{2}\subseteq Y_{2}$ and $W_{2}^{\prime}\subseteq Y_{2}^{\prime}$ such that

$(b,c,d,d^{\prime})\in\widehat{J}\times\widehat{Y}_{1}\times W_{2}\times
W_{2}^{\prime},$

and
$\mathrm{dp\text{-}rk}(b,c,d,d^{\prime}/L)=\mathrm{dp\text{-}rk}(b,c,d,d^{\prime}/N)=3n+1$.
Thus for all $(x,y)\in\widehat{J}\times\widehat{Y}_{1}$ and for every $z\in
W_{2}\cup W_{2}^{\prime}$,

$(x-b)y+z\in Y.$

By the sub-additivity of dp-rank, we must have
$\mathrm{dp\text{-}rk}(c,d,d^{\prime}/Lb)=3n$, so by Lemma 3.14(2), we can
find a small model ${\mathcal{K}}\supseteq Lb$ such that
$\mathrm{dp\text{-}rk}(c,d,d^{\prime}/K)=\mathrm{dp\text{-}rk}(c,d,d^{\prime}/Lb)=3n$.

Let $b_{1}\in\widehat{J}({\mathcal{K}})$, with $b_{1}\neq b$, let
$W_{1}=(b_{1}-b)\widehat{Y}_{1}$ and let $c^{\prime}=(b-b_{1})c\in W_{1}$. By
our assumptions, $W_{1}+W_{2}\subseteq Y$ and $W_{1}+W_{2}^{\prime}\subseteq
Y$. Note that $W_{1},W_{2},W_{2}^{\prime}$ are ${\mathcal{K}}$-definable and
since $b_{1}-b$ is invertible (i.e. $x\mapsto(b_{1}-b)x$ is invertible) we
have $\mathrm{dp\text{-}rk}(W_{1})=\mathrm{dp\text{-}rk}(\widehat{Y}_{1})=n$
and also
$\mathrm{dp\text{-}rk}(W_{2})=\mathrm{dp\text{-}rk}(W_{2}^{\prime})=\mathrm{dp\text{-}rk}(Y)=n$.

Finally, $\mathrm{dp\text{-}rk}(d/K)=\mathrm{dp\text{-}rk}(d^{\prime}/K)=n$,
$\mathrm{dp\text{-}rk}(c,d/K)=\mathrm{dp\text{-}rk}(c,d^{\prime}/K)=2n$ and
since $c^{\prime}=(b-b_{1})c$ and $c$ are interdefinable over $K$ (which
contains $b,b_{1}$), it follows that
$\mathrm{dp\text{-}rk}(c^{\prime},d/K)=\mathrm{dp\text{-}rk}(c^{\prime},d^{\prime}/K)=2n$.

We can now apply Lemma 4.13 to both $(c^{\prime},d)$ and
$(c^{\prime},d^{\prime})$, in $W_{1}\times W_{2}$ and $W_{1}\times
W_{2}^{\prime}$, respectively. The assumptions of the lemma are satisfied
(with $W_{1},W_{2}$ in the respective roles of $Y_{1},Y_{2}$, and the
parameter set $K$ replacing $N$). Thus, $\nu_{W_{2}}(d)-d$ is a subgroup of
$({\mathscr{F}},+)$ which equals $\nu_{W_{1}}(c^{\prime})-c^{\prime}$.
Repeating the same argument with $W_{1},W_{2}^{\prime}$ we get that
$\nu_{W_{2}}(d^{\prime})-d^{\prime}$ is a subgroup of $({\mathscr{F}},+)$
which equals $\nu_{W_{1}}(c^{\prime})-c^{\prime}$. It follows that

$\nu_{W_{2}}(d)-d=\nu_{W_{2}^{\prime}}(d^{\prime})-d^{\prime}.$

Finally, since $W_{2},W_{2}^{\prime}\subseteq Y$,
$\mathrm{dp\text{-}rk}(W_{2})=\mathrm{dp\text{-}rk}(W_{2}^{\prime})=\mathrm{dp\text{-}rk}(Y)=n$
and $\mathrm{dp\text{-}rk}(d/K)=\mathrm{dp\text{-}rk}(d^{\prime}/K)=n$, it
follows from Lemma 4.11 that $\nu_{W_{2}}(d)=\nu_{Y}(d)$ and
$\nu_{W_{2}^{\prime}}(d^{\prime})=\nu_{Y}(d)$, and therefore

$\nu_{Y}(d)-d=\nu_{Y}(d^{\prime})-d^{\prime}.$

We conclude that for every $d\in Y$ such that
$\mathrm{dp\text{-}rk}(d/N_{0})=n$, the set $\nu_{Y}(d)-d$ is a subgroup of
$({\mathscr{F}},+)$ and that this group is independent of the choice of $d$.
We denote this subgroup by $\nu_{Y}$.

(2) Let $Y_{1},Y_{2}\subseteq{\mathscr{F}}$ be $D$-critical sets over small
models ${\mathcal{N}}_{0}$ and ${\mathcal{M}}_{0}$, respectively. Let
$Y^{\prime}=Y_{1}\cup Y_{2}$. Note that $Y^{\prime}$ is also $D$-critical and
strongly internal to $D$ over ${\mathcal{L}}$, some small model containing
$N_{0}M_{0}$. For every $d\in Y_{1}$ with $\mathrm{dp\text{-}rk}(d/L)=n$, it
follows from Lemma 4.11 that $\nu_{Y_{1}}(d)=\nu_{Y_{1}\cup Y_{2}}(d)$.
Starting with $d^{\prime}\in Y_{2}$ with
$\mathrm{dp\text{-}rk}(d^{\prime}/L)=n$ we similarly conclude that
$\nu_{Y_{2}}(d)=\nu_{Y_{1}\cup Y_{2}}(d)$. What we have proved in (1) implies
that $\nu_{Y_{1}}=\nu_{Y_{1}\cup Y_{2}}=\nu_{Y_{2}}$.

(3) Let $c\in{\mathscr{F}}$ and let $Y\subseteq{\mathscr{F}}$ be a
$D$-critical set over some small model ${\mathcal{N}}_{0}$. Let
${\mathcal{K}}\supseteq N_{0}c$ be a small model and $d\in Y$ be such that
$\mathrm{dp\text{-}rk}(d/K)=n$. So also $\mathrm{dp\text{-}rk}(c\cdot d/K)=n$.
The function $x\mapsto cx$ sends $Y$ to $cY$, and by Lemma 4.12, it sends
$\nu_{Y}(d)$ onto $\nu_{cY}(cd)$. Hence,

$c(\nu_{Y}(d)-d)=c\nu_{Y}(d)-cd=\nu_{cY}(cd)-cd=\nu.$

∎

######  Remark 4.15.

One can also show that if $\widehat{\mathcal{M}}$ is an $|M|^{+}$-saturated
elementary extension of ${\mathcal{M}}$ then $\nu(\widehat{M})$ is closed
under addition and multiplication. As this result will not be needed, we omit
the proof.

###  4.3. Differentiability in SW-uniform fields

Our goal in this section is, under suitable assumptions, to put a differential
structure on the group of infinitesimal $\nu$ constructed in the previous
subsection. Before doing that, we have to discuss briefly the notion of
differntiability in the context of SW-uniform fields, which are either real
closed or valued.

For a real closed field $L$ and $a=(a_{1},\dots,a_{n})\in L^{n}$, we let
$|a|=\max_{i}\\{|a_{i}|\\}$. If $(L,v)$ is a valued field we let
$v(a)=\min_{i}\\{v(a_{i})\\}$.

######  Definition 4.16.

Let $L$ be either a real closed field or a valued field with valuation $v$.
Given $U\subseteq L^{n}$ open, a map $f:U\to L^{m}$ is _differentiable_ at
$x_{0}\in U$ if there exists a linear map $D_{x_{0}}f:L^{n}\to L^{m}$ such
that:

In the real closed case:

$\lim_{x\to
x_{0}}\frac{|f(x)-f(x_{0})-(D_{x_{0}}f)\cdot(x-x_{0})|}{|x-x_{0}|}=0,$

and in the valued case:

$\lim_{x\to
x_{0}}\left[v(f(x)-f(x_{0})-(D_{x_{0}}f)\cdot(x-x_{0}))-v(x-x_{0})\right]=\infty.$

If $f$ is differentiable at a point $x_{0}$ we say that $x_{0}$ is a
${\mathcal{D}}^{1}$-point of $f$ and that $f$ is ${\mathcal{D}}^{1}$ at
$x_{0}$.

Exactly as in real analysis, if $f$ is differentiable at $x_{0}$ then
$D_{x_{0}}f$ is represented by the Jacobian matrix of $f$ at $x_{0}$. It
follows that, in expansions of both valued fields and real closed fields, if
$f$ is definable, the set of points in its domain where $f$ is
${\mathcal{D}}^{1}$ is definable as well.

The next remark is not needed in the sequel, but simplifies the discussion, as
it shows that the notion of differentiability does not depend on the
valuation, but only on the topology it generates. Thus, if $K$ is a convexly
valued real closed field this notion of differentiability coincides with the
standard definition:

######  Remark 4.17.

Notice that if $f:K\to K$ is a one-variable function then the above definition
of $d:=D_{x_{0}}(f)$ is purely topological, since it requires the limit of the
quotient $\frac{f(x_{0})-f(x_{0})-d}{x-x_{0}}$ to be $0$ in $K$. Thus, if
$v,v^{\prime}$ valuations on $K$ (possibly one of them the absolute value, if
$K$ is real closed) which generate the same topology, then for one variable
functions, $D_{x_{0}}(f)$ is the same when computed with respect to
$v,v^{\prime}$. It follows, for $f:K^{n}\to K^{m}$, that the Jacobian Matrix
of $f$ at $x_{0}$ does not depend on $v$, and as a result $D_{x_{0}}(f)$ does
not depend on $v$ either.

In the classical setting of ${\mathbb{R}}$ or ${\mathbb{C}}$ (and also in
o-minimal expansions of fields) the existence of continuous partial
derivatives (denoted ${\mathcal{C}}^{1}$) is a sufficient condition for
differentiability. We do not know whether this is a sufficient condition in
the more general setting we are interested in.

To address this we need an additional axiom (which is elementary in any
expansion of a real closed or a valued field):

######  A Gen-Dif.

For every definable open $D\subseteq K^{n}$, and definable $f:D\to K$, the set
of points $x\in D$ such that $f$ is not ${\mathcal{D}}^{1}$ at $x$ has empty
interior.

######  Assumption.

From now on, whenever we say a definable field satisfies (Gen-Dif) we
implicitly mean that it is either real closed or supports a (fixed) definable
valuation.

######  Remark 4.18.

As we noted in Example 3.3(4), it follows from Johnson’s work that a dp-
minimal field supports a unique field topology (with definable basis), which
gives rise to an SW-uniformity. Thus, by Example 3.3(3), any two definable
valuations give rise to two will generate the same topology, and if $K$ is a
real closed valued field, then the valuation and the order will generate the
same topology.

We are now ready to prove that under (Gen-Dif) the group $\nu$ of
infinitesimals can be endowed with a structure of a “Lie-group” with respect
to $K$, namely, that $\nu$ has a structure of a differential manifold
respecting the group operations. Conveniently, the manifold structure can be
given by a single chart. The proof follows a strategy due to Maříková [34],
generalising Pillay’s original work [38], where she proved that invariant
groups in o-minimal structures admit a unique definable group topology. In our
proof we replace continuity of functions in her work with differentiability.

######  Proposition 4.19.

Let ${\mathcal{M}}$ be a $|T|^{+}$-saturated multi-sorted structure,
$(K,+,\cdot)$ be a definable SW-uniform field satisfying (Gen-Dif) and let
${\mathscr{F}}$ be a infinite definable field locally strongly internal to
$K$.

Let $Y\subseteq{\mathscr{F}}$ be $K$-critical, $g:Y\to K^{n}$ an injective
$A$-definable map with open image, and let
$\widehat{\mathcal{M}}\succ{\mathcal{M}}$ be an
$|{\mathcal{M}}|^{+}$-saturated elementary extension. Then there exists $c\in
Y$, with $\mathrm{dp\text{-}rk}(c/A)=\mathrm{dp\text{-}rk}(Y)$, such that,

  1. (1)

The map $x\mapsto g(x+c)$ from $\nu(\widehat{M})$ to
$\nu_{g(Y)}(c)(\widehat{M})$ induces on $\nu(\widehat{\mathcal{M}})$ a
${\mathcal{D}}^{1}$-group structure.

  2. (2)

Let $\alpha:({\mathscr{F}},+)\to({\mathscr{F}},+)$ be an
${\mathcal{M}}$-definable endomorphism leaving the type $\nu$ invariant
(namely, $\alpha_{*}(\nu)=\nu)$). Then
$\alpha:\nu(\widehat{\mathcal{M}})\to\nu(\widehat{\mathcal{M}})$ is a
${\mathcal{D}}^{1}$-map with respect to the above differential structure on
$\nu$.

  3. (3)

For every $a\in{\mathscr{F}}$, the function $\lambda_{a}:x\mapsto a\cdot x$ is
differentiable at $0$ with respect to the above ${\mathcal{D}}^{1}$-structure
on $\nu$.

###### Proof.

All group operations appearing in the proof are the restriction to $Y$ of
${\mathscr{F}}$-addition (and subtraction). Let us explain what we mean by
(1): Let $\sigma:\nu(\widehat{M})\to\nu_{g(Y)}(c)(\widehat{M})$ be given by
$x\mapsto g(x+c)$. Consider the push-forward of $x+y$ via $\sigma$. Namely,
the function on $\nu_{g(Y)}(c)\times\nu_{g(Y)}(c)$ given by

$(z,w)\mapsto g((g^{-1}(z)-c)+(g^{-1}(w)-c)+c)=g(g^{-1}(z)+g^{-1}(w)-c).$

Similarly, consider the push-forward of $x\mapsto-x$, given by

$z\mapsto g(-(g^{-1}(z)-c)+c).$

We claim that both functions are ${\mathcal{D}}^{1}$ in $K$, in a neighborhood
of $(g(c),g(c))\in K^{2}$ and $g(c)\in K$, respectively.

Let $d\in Y$ be such that
$\mathrm{dp\text{-}rk}(d/A)=\mathrm{dp\text{-}rk}(Y)$. By replacing $Y$ with
$Y-d$, we may assume that $\nu\vdash Y$. Now absorb $A$ and $d$ into the
language. In order to keep notation simple we identify $Y$ with its image
under $g$ (so $g=\operatorname{id}$).

(1) Since $\nu$ is a group, type definable over ${\mathcal{M}}$, there are, by
compactness, ${\mathcal{M}}$-definable open sets $V_{1},V_{0}$, such that
$\nu\vdash V_{1}\subseteq V_{0}\subseteq K^{n}$ and

$\varphi_{4}:=(x,y)\mapsto x+y\text{ maps }V_{1}^{2}\text{ into }V_{0}.$

Similarly, we find $V_{2}\subseteq V_{1}$ such that

$\varphi_{3}:=(x,y,z)\mapsto(-z,x+y)\text{ maps }V_{2}^{3}\text{ into
}V_{1}^{2}.$

We also find $V_{3}\subseteq V_{2}$ such that

$\varphi_{2}:=(x,y,z)\mapsto(x+y,z)\text{ maps }V_{3}^{3}\text{ into
}V_{2}^{2},$

and $V_{4}\subseteq V_{3}$ such that

$\varphi_{1}:=(x,y,z,w)\mapsto(w+x,-y,z)\text{ maps }V_{4}^{4}\text{ into
}V_{3}^{3}.$

We may assume that all the above are $\emptyset$-definable. Let $a,b\in V_{4}$
with
$\mathrm{dp\text{-}rk}(a,b)=2\mathrm{dp\text{-}rk}(Y)=2\mathrm{dp\text{-}rk}(V_{4})$.
By (Gen-Dif), $x+y$ is ${\mathcal{D}}^{1}$ at $(a,b)$ and the function
$y\mapsto-y$ is ${\mathcal{D}}^{1}$ at $a$, hence $\varphi_{1}(x,y,z,b)$ is
${\mathcal{D}}^{1}$ at $(a,a,a)$. Similarly, $\varphi_{2}$ is
${\mathcal{D}}^{1}$ at $(b+a,-a,a)$, $\varphi_{3}(x,y,b)$ at $(b,a)$ and
$\varphi_{4}$ at $(b^{-1},b+a)$. Composing, we obtain
$\varphi_{4}\circ\varphi_{3}\circ\varphi_{2}\circ\varphi_{1}(x,y,z)=x-y+z$, so
we have shown that the map $(x,y,z)\mapsto x-y+z$ is ${\mathcal{D}}^{1}$ at
$(a,a,a)$. In fact, the proof provides an open set, $U\subseteq V_{4}$ which,
by compactness, we may take to be ${\mathcal{M}}$-definable, with $a\in U$,
such that $(x,y,z)\mapsto x-y+z:U^{3}\to V_{0}$ is ${\mathcal{D}}^{1}$.

Our goal is to show that the push-forward of addition restricted to $\nu^{2}$
and of the inverse function restricted to $\nu$ via the map $x\mapsto x+a$ are
${\mathcal{D}}^{1}$ (in the sense of $K$). Namely, we need to prove that the
functions $(x-a)+(y-a)+a$ and $-(x-a)+a$ are ${\mathcal{D}}^{1}$ on
$\nu_{Y}(a)^{2}$ and $\nu_{Y}(a)$, respectively. This follows immediately from
our choice of $U$.

(2) Let $\alpha$ be a an ${\mathcal{M}}$-definable endomorphism of
$({\mathscr{F}},+)$ fixing $\nu$ setwise. Fix $c\in M$ as in (1) and
$\sigma(x)=g(x+c)$. Choose $e\in\nu(\widehat{\mathcal{M}})$ with
$\mathrm{dp\text{-}rk}(e/{\mathcal{M}})=\mathrm{dp\text{-}rk}(\nu)=n$. By
(Gen-Dif), $\alpha$ is ${\mathcal{D}}^{1}$ at $e$ with respect to the
differentiable structure on $\nu(\widehat{\mathcal{M}})$ (i.e.,
$\sigma\alpha\sigma^{-1}$ is ${\mathcal{D}}^{1}$ at $\sigma(e)$). Since
$\alpha$ is a homomorphism and $\nu$ is a ${\mathcal{D}}^{1}$-group, it is
standard to verify that $\alpha$ is a ${\mathcal{D}}^{1}$-function on all of
$\nu$.

(3) By Proposition 4.14, for every $a\in{\mathscr{F}}({\mathcal{M}})$,
$\lambda_{a}$ is a endomorphism of $\nu$, so by (2), it is $\nu$\-
differentiable at $0$, i.e., $\sigma\lambda_{a}\sigma^{-1}$ is
$K$-differentiable at $c=\sigma(0)$. ∎

###  4.4. Strong internality to SW-fields

The next theorem is an important step in our proof of Theorem 1. First, we
need the following easy and well known fact:

######  Remark 4.20.

If ${\mathcal{L}}=(L,+,\cdot,\dots)$ is a dp-minimal expansion of a field,
then $L$ has no definable infinite subfields. Indeed, if $K$ were such a field
then $K$ itself is dp-minimal. And if we had some $u\in L\setminus K$ we could
define $T:K^{2}\to L$ by $(a,b)\mapsto a+bu$. Since $u$ is $K$-linearly
independent of $1$, we get that $T$ is a linear injection, so
$\mathrm{dp\text{-}rk}(T(K^{2}))=2$ which is impossible.

######  Theorem 4.21.

Let ${\mathcal{M}}$ be a multi-sorted structure and $K$ an
${\mathcal{M}}$-definable SW-uniform field satisfying (Gen-Dif). Let
${\mathscr{F}}$ be a definable field of finite dp-rank that is locally
strongly internal to $K$. Then ${\mathscr{F}}$ is definably isomorphic to a
finite extension of $K$.

###### Proof.

By passing to an elementary extension, we may assume that ${\mathcal{M}}$ is
$|T|^{+}$-saturated. Let $Y\subseteq{\mathscr{F}}$ be $K$-critical, assume
that $Y$ (and a corresponding definable injection $g:Y\to K^{n}$) is defined
over some small model ${\mathcal{N}}$. By [47, Proposition 4.6], we may assume
that $g(Y)$ is open. Let $\nu$ be the infinitesimal subgroup of
$({\mathscr{F}},+)$, as given by Proposition 4.14, endowed with its Lie group
structure. It will be convenient to evaluate $\nu$ in some
$|{\mathcal{M}}|^{+}$-saturated $\widehat{\mathcal{M}}\succ{\mathcal{M}}$.

By Proposition 4.19(3), for every $z\in{\mathscr{F}}({\mathcal{M}})$, the
function $\lambda_{z}$ is ${\mathcal{D}}^{1}$ at $0$, with respect to this
differential structure on $\nu$.

To each $z\in{\mathscr{F}}$ we associate, definably, the differential
$D_{0}(\lambda_{z})$ identified with the Jacobian matrix of $\lambda_{z}$ at
$0$, with respect to the differentiable structure on $\nu$ (formally the
Jacobian matrix of $\sigma\lambda_{z}\sigma^{-1}$ at $\sigma(0)$).

We claim that the function $z\mapsto J_{z}$ is a ring homomorphism from
$({\mathscr{F}},+,\cdot)$ into $(M_{n}(K),+,\cdot)$. To see that we shall
apply the chain rule to our differentiable functions (see [50, Chapter 7.1]
for the real closed case and [42, Remark 4.1.ii] for the valuation case333Note
that in the proof of the latter, one must replace the usual operator norm with
the $\inf$-valuation on the matrix space.). We now recall the arguments from
[37, Lemma 4.3]:

To see that field multiplication is sent to matrix multiplication: First, note
that $\lambda_{a\cdot b}=\lambda_{a}\circ\lambda_{b}$, and hence by the chain
rule,

$D_{0}(\lambda_{a\cdot
b})=D_{0}(\lambda_{a}\circ\lambda_{b})=D_{0}(\lambda_{a})\cdot
D_{0}(\lambda_{b}),$

where on the right we have matrix multiplication.

To see that field addition is sent to matrix addition, let $P(x,y)=x+y$ and
note that $\lambda_{a+b}=\lambda_{P(a,b)}=P(\lambda_{a},\lambda_{b})$. It is
easy to see that $D_{(0,0)}(P)=(I_{n},I_{n})$, where $I_{n}$ is the $n\times
n$ identity matrix (since $P(x,0)=P(0,x)=x$). By the chain rule,

$D_{0}(\lambda_{a+b})=D_{0}(\lambda_{P(a,b)})=D_{(0,0)}(P\circ(\lambda_{a},\lambda_{b}))=D_{0}(\lambda_{a})+D_{0}(\lambda_{b}),$

where on the right we have matrix addition.

Thus, the map $z\mapsto D_{0}(\lambda_{z})$ is a ring homomorphism sending
$1\in{\mathscr{F}}$ to the identity matrix $I_{n}$. Since ${\mathscr{F}}$ is a
field, the map is injective so we have definably embedded ${\mathscr{F}}$ into
a definable subring of $M_{n}(K)$.

We may now view ${\mathscr{F}}$ as a definable subfield of $M_{n}(K)$. Let
$K_{0}=\\{xI_{n}:x\in K\\}$, where now we take the usual scalar multiplication
in the algebra of matrices. Note that $K_{0}\cap{\mathscr{F}}$ is an infinite
definable subfield of $K$. Indeed, if the characteristic of $K$ is $0$ then it
follows since both contain $I_{n}$. If the characteristic is positive then it
follows since both contain $\mathbb{F}_{p}^{alg}$ by [30, Corollary 4.5].
Since $K_{0}\cong K$ is dp-minimal, it has no infinite definable subfield by
Remark 4.20, so $K_{0}\cap{\mathscr{F}}=K_{0}$ i.e.
$K_{0}\subseteq{\mathscr{F}}$. Thus ${\mathscr{F}}$ is a finite extension of
$K_{0}$. ∎

###  4.5. 1-h-minimal valued fields

In [3, 4] Cluckers, Halupczok, Rideau-Kikuchi and Vermeulen introduce the
class of _1-h-minimal valued fields_ in characteristic 0, encompassing several
examples of interest, such as:

######  Example 4.22.

[3, Corollaries 6.2.6,6.2.7][3, Theorem 6.3.4][3, Proposition 6.4.2]

  1. (1)

pure henselian valued fields of characteristic $0$.

  2. (2)

finite field extensions of $\mathbb{Q}_{p}$ in the sub-analytic language

  3. (3)

henselian valued fields of characteristic $0$ in the valued field language
expanded by function symbols from a separated Weierstrass system $\mathcal{A}$
and equipped with analytic $\mathcal{A}$-structure.

  4. (4)

$T$-convex valued fields expanding a power-bounded o-minimal field.

  5. (5)

$V$-minimal fields.

The exact definition of 1-h-minimal fields is irrelevant for the present
section (see Section 6.3). What will be relevant for us here is that they
satisfy (Gen-Dif) (see below).

######  Corollary 4.23.

Let ${\mathcal{M}}$ be a multi-sorted structure and ${\mathscr{F}}$ a
definable field of finite dp-rank. If ${\mathscr{F}}$ is locally strongly
internal to a definable dp-minimal 1-h-minimal valued field $(K,v)$ then it is
definably isomorphic to a finite extension of $K$. In particular, this is true
if $K$ is a pure dp-minimal valued field of characteristic $0$.

###### Proof.

Every dp-minimal expansion of a valued field is an SW-uniformity by Example
3.3. Since ${\mathscr{F}}$ is 1-h-minimal, it satisfies (Gen-Dif) by [3,
Theorem 5.1.5] and [4, Proposition 3.1.1]. We may now apply Theorem 4.21 with
${\mathscr{F}}$ the definable field. Since pure henselian valued fields are
1-h-minimal (by Clause (1) of the above example), the conclusion of the
addendum follows. ∎

###  4.6. Strong internality to strongly minimal fields

In Theorem 4.21 we classified, under the additional assumption of generic
differentiability, fields locally strongly internal to SW-uniform fields. Such
fields are clearly unstable. We now turn our attention to fields locally
strongly internal to a strongly minimal field. In the following let
$\operatorname{RM}$ be the Morley rank and $\operatorname{DM}$ be the Morley
degree.

######  Proposition 4.24.

Let ${\mathcal{M}}$ be multi-sorted structure and let $\mathscr{K}$ be a
strongly minimal definable field. If ${\mathscr{F}}$ is a definable field of
finite dp-rank that is locally strongly internal to $\mathscr{K}$ then
${\mathscr{F}}$ is strongly internal to $\mathscr{K}$, so in particular it is
algebraically closed. If $\mathscr{K}$ is a pure field then ${\mathscr{F}}$ is
definably isomorphic to $\mathscr{K}$.

###### Proof.

Since $\mathscr{K}$ is strongly minimal, it is dp-minimal and acl satisfies
exchange. As a result, dp-rank is equal to the acl-dimension that is exactly
Morley rank. Furthermore, by e.g. [48, Lemmas 8.4.10, 8.4.11], since
$\operatorname{acl}(\emptyset)$ is infinite, $\mathscr{K}$ eliminates
imaginaries.

Among all definable subsets of ${\mathscr{F}}$ strongly internal to
$\mathscr{K}$ choose one, call it $Y$, that is $\mathscr{K}$-critical.
Translating $Y$, if needed, we may assume that $0_{\mathscr{F}}\in Y$. Since
$Y$ is in definable bijection with a definable subset of $\mathscr{K}^{n}$
then, after shrinking $Y$, we may assume that $Y$ is of finite Morley rank and
$\mathrm{dM}(Y)=1$. We also need the observation that if
$D_{1},D_{2}\subseteq{\mathscr{F}}$ are definable subsets strongly internal to
$\mathscr{K}$ then so is $D_{1}\times D_{2}$, and since $\mathscr{K}$
eliminates imaginaries, so is $(D_{1}\times D_{2})/E$ for any definable
equivalence relation on $D_{1}\times D_{2}$. In particular, as $\mathscr{K}$
eliminates imaginaries, $D_{1}+D_{2}=\\{g+h:g\in D_{1},\,h\in D_{2}\\}$ is
strongly internal to $\mathscr{K}$ (operations are taken in ${\mathscr{F}}$).

What follows is, essentially, a local version of Zil’ber’s indecomposability
theorem, not assuming (a priori) an ambient $\omega$-stable group.

We work inside $Y-Y+Y\supseteq Y$. It is $\omega$-stable (since it is strongly
internal to $\mathscr{K}$), and since $Y$ is $\mathscr{K}$-critical
$\mathrm{dp\text{-}rk}(Y-Y+Y)=\mathrm{dp\text{-}rk}(Y)$. So, in fact,
$\operatorname{RM}(Y-Y+Y)=\mathrm{dp\text{-}rk}(Y-Y+Y)=\mathrm{dp\text{-}rk}(Y)=\operatorname{RM}(Y)$.

Let $R=\\{(g,h)\in
Y^{2}:\operatorname{RM}((g+Y)\triangle(h+Y))<\operatorname{RM}(Y)\\}$. The set
$R$ is definable since $\mathscr{K}$ is strongly minimal and
$Y\subseteq\mathscr{K}^{n}$ (see [33, Lemma 6.2.20]). Moreover, it is an
equivalence relation on $Y$ (since $\mathrm{dM}(Y)=1$).

Let $k=\operatorname{DM}(Y+Y)$. We claim that there are at most $k$-many
$R$-equivalence classes in $Y$. For assume that there were at least $k+1$
classes represented by $g_{1},\dots,g_{k+1}$. Since
$(\operatorname{RM},\operatorname{DM})(g_{i}+Y)=(\operatorname{RM},\operatorname{DM})(Y)=(\operatorname{RM}(Y),1)$,
by the definition of $R$ this means that $\operatorname{RM}(g_{i}+Y\cap
g_{j}+Y)<\operatorname{RM}(Y)$. Since $g_{i}+Y\subseteq Y+Y$ for all $i$, it
follows that either $\operatorname{DM}(Y+Y)>k$ or
$\operatorname{RM}(Y+Y)>\operatorname{RM}(Y)$. Either option is impossible.

Hence there are only finitely many $R$-classes in $Y$ and since they cover $Y$
one of them has Morley rank $\operatorname{RM}(Y)$. So there is a definable
subset $Y_{1}\subseteq Y$ with $\operatorname{RM}(Y_{1})=\operatorname{RM}(Y)$
satisfying that for any $g,h\in Y_{1}$,
$\operatorname{RM}((g+Y)\cap(h+Y)=\operatorname{RM}(Y)$.

Let $p$ the unique generic type of $Y$ and observe that for $g\in Y$, $g+p$ is
the unique generic type of $g+Y$. Let
$H=\mathrm{Stab}(p)=\\{a\in{\mathscr{F}}:a+p=p\\}$. Note that, if $a\in H$
then $a+p=p$ so that $(a+Y)\cap Y$ is generic in $Y$, and in particular
$H\subseteq Y-Y\subseteq Y-Y+Y$. Hence $H$ is a type-definable group in
$\mathscr{K}$, and therefore $H$ is, in fact, definable by e.g. [33, Theorem
7.5.3]. By the definition of $Y_{1}$ above, we have $Y_{1}-Y_{1}\subseteq H$,
hence

$\operatorname{RM}(Y)=\operatorname{RM}(Y_{1}-Y_{1})\leq\operatorname{RM}(H)\leq\operatorname{RM}(Y-Y)=\operatorname{RM}(Y).$

So $\operatorname{RM}(H)=\operatorname{RM}(Y)$.

Replace $H$ by $H^{0}$, its connected component, and let $q$ be its generic
type. We claim that $H$ is invariant under ${\mathscr{F}}$-multiplication. Let
$c\in{\mathscr{F}}$. We work now in $cH+H$, that is still strongly internal to
$\mathscr{K}$. If $cH\not\subseteq H$ then $H/(cH\cap H)>1$. Since $cH$ is
connected as well it must be infinite. On the other hand $H/(cH\cap
H)\cong(cH+H)/H$, so if the latter is infinite
$\operatorname{RM}(cH+H)>\operatorname{RM}(H)=\operatorname{RM}(Y)$,
contradicting the assumption that $Y$ is $\mathscr{K}$-critical.

As a result, $H$ is a non-zero ideal of ${\mathscr{F}}$ i.e. $H={\mathscr{F}}$
and hence ${\mathscr{F}}$ is an $\omega$-stable field so algebraically closed.
If $\mathscr{K}$ is pure then by [41], $H$ is definably isomorphic to
$\mathscr{K}$. ∎

At the level of generality we are working in the statement of Proposition 4.24
is optimal:

######  Example 4.25.

Let $\mathcal{K}\models$ ACVF and k its residue field. Expand k by fusing it
(in the sense of [20]) with an algebraically closed field ${\mathscr{F}}$ of a
different characteristic. By (the proof of) [9, Propostion 5.9] the resulting
expansion of ${\mathcal{K}}$ is still dp-minimal, and k is still stably
embedded in the expanded structure. So ${\mathcal{K}}$ is an SW-uniformity.
However, the field ${\mathscr{F}}$ (with its induced structure) is not
definably isomorphic to the field k (with its expanded structure).

As a corollary we obtain another isomorphism result.

######  Corollary 4.26.

Let ${\mathcal{M}}$ be a multi-sorted structure and $K$ an infinite dp-minimal
pure field of characteristic $0$ definable in ${\mathcal{M}}$. If
${\mathscr{F}}$ is a field of finite dp-rank which is locally strongly
internal to $K$ then ${\mathscr{F}}$ is definably isomorphic to a finite
extension of $K$. In particular, any field ${\mathscr{F}}$ definable in a pure
dp-minimal field $K$ of characteristic $0$ is definably isomorphic to a finite
extension of $K$.

###### Proof.

By Johnson’s theorem (Fact 2.4), $K$ is either algebraically closed, real
closed or admits a definable henselian valuation. If $K$ is algebraically
closed, the result follows from Proposition 4.24. The remaining cases follow
from Theorem 4.21 and the fact that definable functions in a pure real closed
field or a pure dp-minimal valued filed satisfy (Gen-Dif) (by o-minimality in
the former case and by 1-h-minimality in the latter case). ∎

###  4.7. From a finite-to-finite correspondence to strong internality

The results above were all proved assuming the existence of a definable
injection from an infinite subset of ${\mathscr{F}}$ into $K^{n}$. As we note
here, the injectivity assumption can often be relaxed to a finite-to-finite
correspondence. Let us, first, clarify our terminology:

######  Definition 4.27.

Let $X,Y$ be any sets. A relation $C\subseteq X\times Y$ is a _finite-to-
finite correspondence between $X$ and $Y$_ if the projections $\pi_{1}:C\to X$
and $\pi_{2}:C\to Y$ are surjective with finite fibres.

We also fix some notation, for the following standard notion:

######  A EfI.

A definable set $D$ in a (multi-sorted) structure ${\mathcal{M}}$ has
_elimination of finite imaginaries_ if whenever $\\{X_{t}\subseteq D^{n}:t\in
T\\}$ is a definable family of finite sets, uniformly bounded in size then
there exists a definable map $f:T\to D^{m}$ for some integer $m$, with the
property that $f(t_{1})=f(t_{2})$ if and only if $X_{t_{1}}=X_{t_{2}}$.

The condition (EfI) is satisfied whenever $D$ expands a definable field (using
symmetric functions) or a linear order. It allows us, under additional
assumptions, to replace definable finite-to-finite correspondences by
definable bijections:

######  Lemma 4.28.

Assume that $X$ and $Y$ are definable in some $|T|^{+}$-saturated (multi-
sorted) structure ${\mathcal{M}}$ and there is a finite-to-finite definable
correspondence between infinite subsets of $X$ and $Y$.

Assume also that

  1. (1)

$X$ has (EfI), and

  2. (2)

either $Y$ supports an SW-uniform structure or $Y$ has (EfI).

Then $X$ is locally strongly internal to $Y$.

###### Proof.

Restricting $X$ and $Y$ we may assume that there is $C\subseteq X\times Y$, a
definable finite-to-finite correspondence between $X$ and $Y$. For $x\in X$
and $y\in Y$ denote

$C_{x}=\\{y\in Y:(x,y)\in C\\},\,C^{y}=\\{x\in X:(x,y)\in C\\}$

and note that by $\aleph_{0}$-saturation, they are uniformly bounded in size
by some integer.

We first claim that there is a finite-to-one function from an infinite subset
of $Y$ into $X$. Since $X$ satisfies (EfI), each $C^{y}$ is coded by some
element of $X^{m}$, for some integer $m$, thus we obtain a definable finite-
to-one function $f$ from a $\pi_{2}(C)\subseteq Y$ into $X^{m}$. Amongst all
definable finite-to-one functions from an infinite definable subset of
$Y^{\prime}\subseteq Y$ into $X^{m}$, choose one, $h$, with $m$ minimal. We
claim that necessarily $m=1$.

Otherwise, consider the projection $\pi:X^{m}\to X^{m-1}$ onto the first $m-1$
coordinates and let $W=\pi(X^{\prime})$, for $X^{\prime}=h(Y^{\prime})$. If
there is some $w\in W$ such that $\pi^{-1}(w)$ is infinite then we obtain a
finite-to-one map from an infinite subset of $Y$ into $\pi^{-1}(w)$ so also
into $X$, contradicting the fact that $m>1$

Otherwise $\pi^{-1}(w)$ is finite for all $w\in W$. The function
$h_{1}=\pi\circ h$ is again finite-to-one from $Y^{\prime}$ into $W\subseteq
X^{m-1}$, contradicting the minimality of $m$.

We thus showed the existence of a definable finite-to-one $h:Y^{\prime}\to X$,
for some infinite definable $Y^{\prime}\subseteq Y$. Without loss of
generality, $Y^{\prime}=Y.$

Assume that $Y$ is an SW-uniformity, and all the data is definable over $A$.
Let $c\in Y$ be with $\mathrm{dp\text{-}rk}(c/A)=1=\mathrm{dp\text{-}rk}(Y)$.
Since the topology on $Y$ is Hausdorff, we can find a relatively open subset
$U\subseteq Y$ with $c\in U$ such that $h^{-1}(h(c))\cap U=\\{c\\}$. By
Proposition 3.12 there exists some $B\supseteq A$ and a $B$-definable open
neighborhood $U_{0}\subseteq U$ of $c\in U_{0}$ such that
$\mathrm{dp\text{-}rk}(c/B)=\mathrm{dp\text{-}rk}(c/A)=1$.

Now let $Y_{0}=\\{y\in U_{0}:|h^{-1}(h(y))|=1\\}$. By the above, $Y_{0}$ is
$B$-definable. As $c\in S$ and $\mathrm{dp\text{-}rk}(c/B)=1$, $Y_{0}$ must be
infinite and hence $h\restriction Y_{0}$ is injective.

Finally, assume that $Y$ has (EfI). Each of the fibers of the map $h$ can be
coded by an element of $Y^{k}$ for some fixed $k$, so we obtain a definable
injection from an infinite subset of $X$ into $Y^{k}$. ∎

######  Corollary 4.29.

Let ${\mathcal{M}}$ be a multi-sorted structure, $(K,+,\cdot)$ an SW-uniform
field satisfying (Gen-Dif) and ${\mathscr{F}}$ a definable field of finite dp-
rank. If there exists an infinite definable $S\subseteq{\mathscr{F}}$ and a
finite-to-finite definable correspondence from $S$ into $K^{n}$ then
${\mathscr{F}}$ is definably isomorphic to a finite extension of $K$.

###### Proof.

Let $f$ be the definable correspondence from $S$ into $K^{n}$. As $K$ is dp-
minimal it eliminates $\exists^{\infty}$ by [5, Lemma 2.2] and we may thus
assume $K$ is $|T|^{+}$-saturated. Both ${\mathscr{F}}$ and ${\mathcal{K}}$
are fields so satisfy (EfI). By Lemma 4.28, ${\mathscr{F}}$ is locally
strongly internal to $K$. We are now in the situation to apply Theorem 4.21
and conclude the proof. ∎

##  5\. Reduction to the distinguished sorts

Throughout this section, and until the end of the paper we set:

######  Assumption.

Let ${\mathcal{K}}=(K,v,\dots)$ denote a dp-minimal expansion of a valued
field.

In this section we aim to show that if ${\mathcal{K}}$ is either $P$-minimal,
$C$-minimal or power bounded $T$-convex then every infinite field
${\mathscr{F}}$ interpretable in ${\mathcal{K}}$ is locally strongly internal
to one of the distinguished sorts, $K$, $\Gamma$, k or $K/{\mathcal{O}}$. In
practice, at this level of generality, we show a weaker result (replacing the
definable bijection in the definition of strong internality with a finite-to-
finite correspondence), still sufficient for our needs. We start with a
general observation assuring that ${\mathscr{F}}$ is locally strongly internal
to a unary imaginary sort:

######  Lemma 5.1.

Let ${\mathcal{M}}$ be an arbitrary $\aleph_{0}$-saturated structure, and $E$
a definable equivalence relation on $X\subseteq M^{k}$ with infinitely many
classes. Then there exists an infinite definable quotient
$X^{\prime}/E^{\prime}$, with $X^{\prime}\subseteq M$ and a finite-to-finite
definable correspondence between $X^{\prime}/E^{\prime}$ and an infinite
subset of $X/E$.

###### Proof.

We fix $n$ minimal with respect to the following property: There is a finite-
to-finite definable correspondence between $X^{\prime}/E^{\prime}$ and an
infinite subset of $X/E$, such that $X^{\prime}\subseteq M^{n}$. Our goal is
to prove that that $n=1$.

Thus we may already assume that $X\subseteq M^{n}$ for this minimal $n$, and
towards contradiction we assume that $n>1$. Let $\pi:X^{\prime}\to M^{n-1}$ be
the projection onto the first $n-1$ coordinates. Let $W=\pi(X^{\prime})$. If
for some $w\in W$ the set $X^{\prime}_{w}/E^{\prime}$ is infinite, where
$X^{\prime}_{w}:=X^{\prime}\cap\pi^{-1}(w)$, we get a contradiction to the
minimality of $n$ (since we can definably identify $X^{\prime}_{w}$ with a
subset of $M$ and then obtain a one-to-one map from
$X^{\prime}_{w}/E^{\prime}$ into $X/E$).

So we assume that $|X^{\prime}_{w}/E|$ is finite for all $w\in W$. As
${\mathcal{M}}$ is $\aleph_{0}$-saturated, Without loss of generality, we may
assume that for some fixed $s\in\mathbb{N}$, for all $w\in W$, $|X_{w}/E|=s$.

We now define on $W$ the equivalence relation

$w_{1}E_{1}w_{2}\Leftrightarrow X_{w_{1}}/E=X_{w_{2}}/E.$

Because $X/E$ is infinite and each $X_{w_{1}}/E$ is finite, the quotient
$W/E_{1}$ is infinite. For $\alpha\in W/E_{1}$ we let $X_{\alpha}:=X_{w}/E$,
for any $w\in\alpha$. By our assumption, $|X_{\alpha}|=s$ for all $\alpha\in
W/E_{1}$. Notice that if $s=1$ then, by the definition of $E_{1}$, we get an
injection from $W/E_{1}$ into $X/E$, contradicting the minimality of $n$
(since $X_{1}\subseteq M^{n-1}$). We assume then that $s>1$.

Case 1. For every $\beta\in X/E$, $\beta$ belongs to at most finitely many
(finite) sets of the form $X_{\alpha}$, $\alpha\in W/E_{1}$.

In this case, the set

$C=\\{(\alpha,\beta)\in W/E_{1}\times X/E:\beta\in X_{w}/E\\}$

is a finite-to-finite definable correspondence between $W/E_{1}$ and $X/E$,
contradicting the minimality of $n$.

Case 2. There exists $\beta\in X/E$ which belongs to infinitely many sets of
the form $X_{\alpha}$, $\alpha\in W/E_{1}$.

Fix such a $\beta$ and consider the infinite definable set $X^{\beta}=\\{w\in
W:\beta\in X_{w}/E\\}$. By our assumption, $X^{\beta}/E_{1}$ is infinite. So
we may replace $W$ by $X^{\beta}$ and assume that for each $\alpha\in
W/E_{1}$, we have $\beta\in X_{\alpha}$.

We now replace $X$ by $X_{1}=X\setminus\beta$ (namely we remove one $E$-class,
$\beta$, from $X$). Since we assumed that for every $\alpha$,
$|X_{\alpha}|=s>1$ and $\beta\in X_{\alpha}$, the set
$(X_{1})_{\alpha}=(X_{1})_{w}/E$, for $w\in\alpha$ has one element less,
namely consists of $s-1$ elements. By repeating the process we can finally
reach the situation where $|X_{\alpha}|=1$, for every $\alpha$ and finish as
above.∎

From now on we focus our attention on quotients of the form $K/E$, for some
definable equivalence relation $E$ on $K$.

###  5.1. Inter-algebraicity with the distinguished sorts

We first introduce some notation. We let $\mathscr{B}^{op}_{\gamma}$ be the
set of open balls in $K$ of radius $\gamma$ and $\mathscr{B}^{cl}_{\gamma}$
the set of closed balls in $K$ of radius $\gamma$. These are clearly definable
families of sets and thus as we vary $\gamma\in\Gamma$ we obtain
$\mathscr{B}^{op}$ and $\mathscr{B}^{cl}$ the families of open and closed
balls, respectively. We let $\mathscr{B}=\mathscr{B}^{op}\cup\mathscr{B}^{cl}$
be the family of all balls. When $\Gamma$ is discrete every closed ball is
also open and thus $\mathscr{B}^{op}=\mathscr{B}^{cl}$.

Note that for any $\gamma_{1},\gamma_{2}\in\Gamma$, there is a definable
bijection (possibly using an additional parameter) between the set
$\mathscr{B}^{op}_{\gamma_{1}}$ and $\mathscr{B}^{op}_{\gamma_{2}}$ and
similarly between $\mathscr{B}^{cl}_{\gamma_{1}}$ and
$\mathscr{B}^{cl}_{\gamma_{2}}$. In particular, there is a definable bijection
between every $\mathscr{B}^{cl}_{\gamma}$ and $K/{\mathcal{O}}$, as the latter
is just $\mathscr{B}^{cl}_{0}$. Similarly, there is a definable injection of k
into $\mathscr{B}^{op}_{0}$.

Given a set $X\subseteq K$, a maximal ball inside $X$ is a ball $b\subseteq X$
such that there does not exist a ball $b^{\prime}$, $b\subsetneq
b^{\prime}\subseteq X$. Both $b$ and $b^{\prime}$ can be either closed or
open. As for any two balls $b_{1},b_{2}$, either $b_{1}\cap b_{2}=\emptyset$
or one of the balls is contained in the other, any two maximal balls in $X$
are necessarily disjoint.

By the same observation as above, if $X\subseteq K$ is definable in a
structure expanding a valued field and $x_{0}$ is an interior point of $X$,
then the family of balls $b\subseteq X$ containing $x$ is a definable chain of
balls. If the induced structure on $\Gamma$ is o-minimal (or, more generally,
definably complete) then the infimum of the radii of these balls exists and
thus there is a maximal ball $b$ containing $x_{0}$ in $X$ (which could be
closed or open, depending on whether this infimum is attained or not). The
family of all maximal balls in $X$ is thus a definable family and if $\Gamma$
is definably complete then its union covers $X$.

In order to study uniformly definable finite sets of balls, the following
additional assumption regarding the valued field $K$ will be useful.

######  A Cballs.

For every $\widehat{\mathcal{K}}\equiv\mathcal{K}$, if $X\subseteq\widehat{K}$
is a definable subset intersecting infinitely many closed $0$-balls then it
contains a closed ball of radius $<0$.

######  Remark 5.2.

  1. (1)

It is easy to verify that if ${\mathcal{K}}$ satisfies (Cballs) then it
satisfies the same statement with $0$ replaced by any $\gamma_{0}\in\Gamma$.

  2. (2)

Under our assumptions, if k is finite then $\Gamma$ is discrete (by Johnson,
[28, Lemma 4.3.1], see also [9, Lemma 2.7] for a concise overview). Assuming
(Cballs), if $\Gamma$ is discrete then k is finite. Indeed, if k were infinite
then the definable set ${\mathcal{O}}\setminus\\{0\\}$ would contain
infinitely many closed balls of radius $1$, but no closed ball of radius $0$,
contradicting (Cballs).

######  Lemma 5.3.

If ${\mathcal{K}}$ satisfies (Cballs) and $X\subseteq K$ is definable then for
every $\gamma_{0}\in\Gamma$, $X$ contains at most finitely many maximal closed
balls of radius $\gamma_{0}$.

###### Proof.

Assume towards a contradiction that $X$ contained infinitely many closed
maximal $\gamma_{0}$-balls. Let $X^{\prime}\subseteq X$ be the union of all
the closed maximal $\gamma_{0}$-balls in $X$. The set $X^{\prime}$ is
definable and each of these closed $\gamma_{0}$-balls is still maximal in
$X^{\prime}$. By (Cballs), $X^{\prime}$ must contain a closed ball of radius
$<\gamma_{0}$. But then one of the $\gamma_{0}$-balls in $X^{\prime}$ is not
maximal. ∎

Recall that $\Gamma$ has Definable Skolem Functions if given a definable
family $\\{X_{t}:t\in T\\}$ of non-empty subsets of $\Gamma$, there is a
definable function $c:T\to\Gamma$ such that $c(t)\in X_{t}$. The following
definition is based on [40].

######  Definition 5.4.

We say that a structure ${\mathcal{M}}$ is surgical if every definable
equivalence relation on $M$ has at most finitely many infinite classes.

######  Proposition 5.5.

Assume that ${\mathcal{K}}=(K,v,\ldots)$ is a sufficiently saturated expansion
of a dp-minimal valued field satisfying

  1. (1)

(Cballs).

  2. (2)

$\Gamma$ is definably complete and has definable Skolem functions.

  3. (3)

k is surgical.

For every definable $X\subseteq K^{n}$ and definable equivalence relation $E$
on $K$ with $X/E$ infinite, there exists a definable infinite $T\subseteq X/E$
and a definable finite-to-finite correspondence between $T$ and a definable
subset of $K$, $\Gamma$, k or $K/{\mathcal{O}}$.

###### Proof.

By Lemma 5.1, there exists $T\subseteq X/E$ and a finite-to-finite definable
correspondence between $T$ and $K/E^{\prime}$, for some definable equivalence
relation $E^{\prime}$ on $K$. Thus it is sufficient to prove the proposition
for $X\subseteq K$. So, let $E$ be a definable equivalence relation on $X$
with $T=X/E$ infinite and for each $t\in T$ we let $E_{t}\subseteq X$ be the
corresponding $E$-class.

If there are infinitely many $t\in T$ with $E_{t}$ finite, by passing to a
definable subset of $T$ (and using the fact that ${\mathcal{K}}$ eliminates
$\exists^{\infty}$ by [5, Lemma 2.2]), we may assume that $E_{t}$ is finite
for all $t\in T$. This gives a one-to-finite map between $T$ and $K$ as
needed.

We now assume that $E_{t}$ is infinite for all $t\in T$. By the definable
completeness of $\Gamma$, each $x\in E_{t}$ is contained in a (unique) maximal
ball inside $E_{t}$. Thus, $E_{t}$ can be written as a disjoint union of
maximal sub-balls, and the map which assigns to each $t\in T$ the set
$S_{t}\subseteq\mathscr{B}$ of maximal sub-balls (open or closed) of $E_{t}$
is definable.

Since $\Gamma$ has definable Skolem functions, we can definably choose for
each $t\in T$ a radius $r(t)\in\Gamma$ of one of the balls in $S_{t}$. By
shrinking $X$ and $E$ (but not $K/E$) we may assume that each $E_{t}$ is a
union of maximal balls, all of fixed radius $r(t)$.

Case 1 The map $t\mapsto r(t)$ is finite-to-one.

This immediately yields a finite-to-one map from $T$ into $\Gamma$. A finite-
to-one map is a finite-to-finite correspondence, so we are done.

Case 2 There exists $\gamma_{0}\in\Gamma$ such that
$T^{\prime}=r^{-1}(\gamma_{0})$ is infinite.

By replacing $T$ with $T^{\prime}$ we may assume that for all $t\in T$,
$E_{t}$ is a union of maximal balls, all of radius $\gamma_{0}$. Using a
definable bijection we may assume that $\gamma_{0}=0$, and so all maximal
balls in $E_{t}$ are of radius $0$.

Assume first that there are infinitely many $t\in T$ such that one of the
maximal balls in $E_{t}$ is closed. By restricting $T$ we may assume that for
all $t\in T$, one of the maximal balls in $E_{t}$ is closed. By Lemma 5.3,
each $E_{t}$ contains at most finitely many maximal closed $0$-balls and the
map $t\mapsto S_{t}\subseteq\mathscr{B}^{cl}_{0}=K/{\mathcal{O}}$ sending
$t\in T$ to the finite set of its closed maximal $0$-balls in $E_{t}$ is
definable. Since the $E_{t}$ are pairwise disjoint this gives rise to a one-
to-finite correspondence between $T$ and an infinite subset of
$K/{\mathcal{O}}$.

Consequently, we may now assume that for every $t\in T$, each maximal ball in
$E_{t}$ is open of radius $0$. By (Cballs), each $E_{t}$ can intersect only
finitely many closed $0$-balls (otherwise, it will contain a closed ball of
radius $<0$, contradicting the maximality of all the $0$-balls).

Case 2.a There exists a fixed closed $0$-ball $b_{0}$ intersecting infinitely
many classes $E_{t}$.

By translating, we may assume that $b_{0}={\mathcal{O}}$. By intersecting each
$E_{t}$ with ${\mathcal{O}}$ we may assume, by further shrinking $T$, that
every $E_{t}$ is contained in ${\mathcal{O}}$. Thus, each $E_{t}$ is sent by
$\mathrm{res}:{\mathcal{O}}\to\textbf{k}$ into a subset of k and for
$t_{1}\neq t_{2}$, we have
$\mathrm{res}(E_{t_{1}})\cap\mathrm{res}(E_{t_{2}})=\emptyset$. This induces a
definable equivalence relation on k so by our assumption, only finitely many
of these classes are infinite. As k is a field of finite dp-rank, by [5, Lemma
2.2] it eliminates $\exists^{\infty}$ so by reducing $T$, we obtain a
definable one-to-finite correspondence between $T$ and an infinite subset of
k.

Case 2.b Every closed $0$-ball intersects only finitely many classes $E_{t}$.

The map sending $t\in T$ to the finite set
$F_{t}\subseteq\mathscr{B}^{cl}_{0}$ of closed $0$-balls intersecting $E_{t}$
is definable. By our assumption, each ball $b\in F_{t}$ intersects at most
finitely many of the $E_{t}$, thus the function $t\to F_{t}$ is finite-to-one,
and we obtain a finite-to-finite correspondence between (infinite subsets of)
$T$ and $K/{\mathcal{O}}$. ∎

In the rest of this section we examine various settings in which the
assumptions of Proposition 5.5 hold.

####  5.1.1. The C-minimal case

Recall the definition of a C-minimal valued field from Section 2.3.2.

######  Proposition 5.6.

Let $(K,v,\dots)$ be a C-minimal valued field. Then

  1. (1)

$\Gamma$ is definably complete and has definable Skolem functions, and k is
surgical.

  2. (2)

${\mathcal{K}}$ satisfies (Cballs).

###### Proof.

(1) Follows from the o-minimality of $\Gamma$ and the strong minimality of k.

(2) Assume that $X\subseteq K$ intersects infinitely many closed $0$-balls. We
want to show that it contains a closed ball of negative radius. Since $X$ is a
finite union of Swiss cheeses one of them intersects infinitely many closed
$0$-balls. So we may assume that $X$ is a Swiss Cheese, namely
$X=b\setminus\bigcup\limits_{i=1}^{s}b_{i}$.

For every closed $0$-ball $b_{0}$ intersecting $X$, either $b\subseteq b_{0}$
or $b_{0}\subseteq b$. In the first case, $b_{0}$ is the only $0$-ball
intersecting $b$ thus $b_{0}\subsetneq b$. Note that there are only finitely
many closed $0$-balls intersecting $X$ but not contained in $X$. Indeed, if
$b_{0}\nsubseteq X$ is such a ball then there is some $1\leq i\leq s$ such
that $b_{i}\subseteq b_{0}$ (for otherwise $b_{0}\subseteq b_{i}$ and hence
$b_{0}\cap X=\emptyset$). So there are only $s$-many possible such $0$-balls

Consequently, there exists some closed $0$-ball $b_{0}=B_{\geq
0}(x_{0})\subseteq X$ with $b_{0}\subsetneq b$. If $s=0$ then $X=b$ is a ball
of negative radius, since $b_{0}$ is closed. Thus, assume that $s\geq 1$, and
for $1\leq i\leq s$ choose a center $x_{i}$ of $b_{i}$. Let
$\gamma_{1}=\max\\{v(x_{0}-x_{i}):1\leq i\leq s\\}$. Since $b_{0}$ is disjoint
from each of the $b_{i}$, necessarily $v(x_{0}-x_{i})<0$ for all $i$ and hence
$\gamma_{1}<0$. As a result, the open ball $B_{>\gamma_{1}}(x_{0})$ properly
contains $B_{\geq 0}(x_{0})$ (since $\Gamma$ is dense) and is still disjoint
from the $b_{i}$. In particular, we do not have $b\subseteq
B_{>\gamma_{1}}(x_{0})$, so we must have $B_{>\gamma_{1}}(x_{0})\subsetneq b$.

It follows that $B_{>\gamma_{1}}(x_{0})$ is contained in $X$, thus we found an
open ball of negative radius in $X$. Finally, to get a closed such ball of
negative radius, we use the fact that $\Gamma$ is dense. ∎

####  5.1.2. The P-minimal case

Recall the notion of P-minimal valued field from Section 2.3.1. Let
${\mathcal{K}}=(K,v,\dots)$ be a P-minimal valued field.

Note that since $\Gamma$ is discrete,
$\mathscr{B}^{op}=\mathscr{B}^{\operatorname{cl}}$. It follows from Hensel’s
Lemma (see Fact 5.7 below) that each $P_{n}$ is open, implying that a
definable subset $X\subseteq K$ is infinite if and only if it has non-empty
interior.

We thank D. Macpherson for sketching for us the proof the proposition below.
We first recall:

######  Fact 5.7.

[15, Lemma 2.3] Let $(K,v)$ be a $p$-adically closed field and let
$n\in\mathbb{N}$ with $n>1$ and $x,y,a\in K$. Suppose that
$v(y-x)>2v(n)+v(y-a)$. Then $x-a,y-a$ are in the same coset of $P_{n}$.

######  Proposition 5.8.

Let ${\mathcal{K}}=(K,v,\dots)$ be a P-minimal valued field. Then

  1. (1)

$\Gamma$ is definably complete and has definable Skolem functions.

  2. (2)

$K$ satisfies (Cballs).

###### Proof.

(1) Both properties follow from the fact that every definable set bounded
below has a minimum.

(2) Let $X\subseteq K$ be a definable subset intersecting infinitely many
closed $0$-balls. We need to show that $X$ contains a closed ball of radius
$-1$.

Partitioning $X$ into cells and translating by an element of $K$, we may
assume that $X$ has the form

$\\{x\in K:\gamma_{1}<v(x)<\gamma_{2}\wedge P_{n}(\lambda\cdot x)\\},$

where $\lambda\in K$, $\gamma_{1}<\gamma_{2}\in\Gamma\cup\\{\infty,-\infty\\}$
and $n\in\mathbb{N}$. Since $\Gamma$ is discrete and k is finite, the
assumption that $X$ intersects infinitely many $0$-balls implies that $X$ is
not contained in any ball $B_{\geq-m}(0)$ with $m\in{\mathbb{N}}$ (for every
such ball intersects only finitely many closed $0$-balls). So for every
$k\in\mathbb{N}$, there is some $y_{0}\in X$ such that $v(y_{0})<-k$.

We can thus fix $y_{0}\in X$, such that $v(y_{0})+2v(n)<-2$. We claim that the
closed ball $B_{\geq-1}(y_{0})$ is contained in $X$.

Indeed, assume that $v(x-y_{0})\geq-1$. Then, since $v(y_{0})<-2$, we have
$v(x)=v(y_{0})$ and therefore $\gamma_{1}<v(x)=v(y_{0})<\gamma_{2}$. Thus it
is sufficient to see that $x$ and $y_{0}$ are in the same $P_{n}$-coset.

By our choice of $y_{0}$ we have $v(x-y_{0})\geq-1>v(y_{0})+2v(n)$, hence by
Fact 5.7 (with $a=0$ there), $x$ and $y_{0}$ are in the same $P_{n}$-coset, so
$x\in X$. Thus $B_{\geq-1}(y_{0})\subseteq X$. We have thus shown that $X$
contains at least one closed ball of radius $-1$. ∎

####  5.1.3. The weakly o-minimal case

By [2], the theory of real closed convexly valued fields, known as _Real
Closed Valued Field_ , RCVF for short, is weakly o-minimal.

######  Lemma 5.9.

Let $(K,v)$ be a real closed valued field. If $X\subseteq K$ is a convex set
and every open ball $b\subseteq X$ has nonnegative radius then $X$ is
contained in a single closed ball of radius $0$.

###### Proof.

We first claim that for each $x_{1},x_{2}\in X$, we must have
$v(x_{1}-x_{2})\geq 0$. Indeed, assume that $v(x_{1}-x_{2})=\gamma_{0}<0$ and
let $x_{0}=\frac{x_{1}+x_{2}}{2}$. Then

$v(x_{0}-x_{1})=v(x_{0}-x_{2})=v((x_{1}-x_{2})/2)=\gamma_{0}.$

Thus the open ball $B_{>\gamma_{0}}(x_{0})$ does not contain $x_{1},x_{2}$.
Since the ball is convex and $X$ is convex we have
$B_{>\gamma_{0}}(x_{0})\subseteq X$, contradicting the assumption on $X$. It
follows that for every $x_{0}\in X$, the ball $B_{\geq 0}(x_{0})$ contains
$X$. ∎

######  Lemma 5.10.

Assume that ${\mathcal{K}}=(K,<,v,\ldots)$ is a weakly o-minimal expansion of
a real closed valued field such that $\Gamma$ and k are o-minimal. Then

  1. (1)

$\Gamma$ is definably complete and has definable Skolem functions, and k is
surgical.

  2. (2)

${\mathcal{K}}$ satisfies (Cballs).

###### Proof.

(1) Follows from the o-minimality of $\Gamma$ and of k. (2) Let $X\subseteq K$
be a definable set intersecting infinitely many closed $0$-balls. By weak
o-minimality, $X$ is a finite union of convex sets, so one of these intersects
infinitely many closed $0$-balls. By Lemma 5.9, this component necessarily
contains a ball of negative radius.∎

######  Remark 5.11.

By Section 2.3.4, T-convex power-bounded valued fields satisfy the assumptions
of Lemma 5.10.

###  5.2. Strong internality to the distinguished sorts

We can finally show that, under suitable assumptions, any interpretable field
${\mathscr{F}}$ is locally strongly internal to one of the distinguished
sorts. We focus on the case where $\Gamma$ is dense as we do not know the
results in the discrete case (and we do not need them for the proof of our
main theorem).

We first show that $K/{\mathcal{O}}$ is an SW-uniform structure with respect
to a natural topology:

######  Definition 5.12.

Let $(K,v)$ be a valued field. The _thick ball topology_ on $K/{\mathcal{O}}$
is the topology generated by $\\{\pi(B):B\in\mathscr{B},r(B)<0\\}$. Where
$\pi:K\to K/{\mathcal{O}}$ is the natural projection, and $r(B)$ is the
valuative radius of $B$.

######  Lemma 5.13.

Let ${\mathcal{K}}=(K,v,\dots)$ be a dp-minimal valued field with a dense
value group. If $K$ satisfies (Cballs) then $K/{\mathcal{O}}$ is an SW-
uniformity with respect to the thick ball topology.

###### Proof.

By [47, Example 3, page 3] to show that the thick ball topology gives rise to
a uniform structure on $K/{\mathcal{O}}$ we only have check that it is a group
topology. If we denote $\tau_{a}$ the collection of basic neighbourhoods of
$a$ then $\tau_{a}=\tau_{0}+a$. Therefore the map $x\mapsto-x$ is a
homeomorphism, and to check that addition is continuous it suffices to check
continuity at $(0,0)$. This follows from the fact that the pre-image under
addition (in $K^{2}$) of a ball $B\subseteq K$ containing $0$ is $B\times B$,
and since $\pi(B\times B)=\pi(B)\times\pi(B)$ continuity of addition (in
$K/{\mathcal{O}}$) at $(0,0)$ follows.

So it remains to check that $K/{\mathcal{O}}$ has no isolated points and that
every infinite $S\subseteq K/{\mathcal{O}}$ has non-empty interior. The first
property follows from the fact that $\Gamma$ is dense (and therefore k is
infinite): if $a\in K/{\mathcal{O}}$ and $a_{0}\in\pi^{-1}(a)$ is any point
then a basic open neighbourhood of $a$ is the image under $\pi$ of a ball
$B\ni a_{0}$ with radius $\gamma>0$. Since k is infinite, $B$ contains
infinitely many $0$-balls, and so $\pi(B)$ is infinite. The latter property is
automatic from (Cballs). ∎

With this last observation in hand, we can finally show:

######  Proposition 5.14.

Let $\mathcal{K}=(K,v,\dots)$ be either a C-minimal valued field or a weakly
o-minimal convexely valued field whose value group and residue field are
o-minimal. Any interpretable infinite field is locally strongly internal to
either $K$, $K/{\mathcal{O}}$, $\Gamma$ or k.

###### Proof.

We may assume that ${\mathcal{K}}$ is sufficiently saturated. Let
${\mathscr{F}}=X/E$, $X\subseteq K^{n}$, be an interpretable infinite field.
By Proposition 5.6 (for the C-minimal case) and Lemma 5.10 (for the weakly
o-minimal case), the assumptions of Proposition 5.5 hold.

Consequently there exists an infinite definable subset
$S\subseteq{\mathscr{F}}$ and a finite-to-finite definable correspondence
between $S$ and either $K$, $K/{\mathcal{O}}$, $\Gamma$ or k.

To obtain strong internality from this finite-to-finite correspondence we aim
to apply Lemma 4.28. To do so, we need to show that the source of the
correspondence, ${\mathscr{F}}$, satisfies (EfI) (which is clear) and the
target (each of the distinguished sorts) is either an SW-uniformity, or,
itself satisfies (EfI). Since $K$ and k are fields they both satisfy (EfI),
and so does $\Gamma$ by virtue of being linearly ordered. This leaves the case
of $K/{\mathcal{O}}$. As ${\mathcal{K}}$ satisfies (Cballs), $K/{\mathcal{O}}$
is an SW-uniformity by Lemma 5.13. ∎

##  6\. Eliminating the sorts $K/{\mathcal{O}}$ and $\Gamma$

The results collected up until this point allow us, given an infinite
inpterpretable field ${\mathscr{F}}$ to construct a finite-to-finite
correspondence between a dp-minimal subset of ${\mathscr{F}}$, strongly
internal to $K/E$ (for some definable equivalence relation $E$) into one of
$K$, $K/{\mathcal{O}}$, $\Gamma$ or k. In the present section we develop the
tools allowing (under suitable assumptions) to eliminate $K/{\mathcal{O}}$ and
$\Gamma$ from the list.

###  6.1. Opaque Equivalence Relations

Towards studying $K/{\mathcal{O}}$, we prove a general domination result for
generic types in $(K/E)^{n}$ when $E$ is an _opaque_ equivalence relation.
Recall;

######  Definition 6.1.

[12, Definition 11.2] Let $D$ be a definable set in some structure and $E$ a
definable equivalence relation on $D$.

  1. (1)

A definable set $X\subseteq D$ _crosses_ an $E$-class, $a/E$, if both $X$ and
its complement in $D$ intersect $a/E$.

  2. (2)

The equivalence relation $E$ is opaque if every definable $X\subseteq D$
crosses at most finitely many classes.

We will also refer to $D/E$ being opaque, meaning that $E$ is.

######  Example 6.2.

  1. (1)

If $(K,v,\dots)$ is $C$-minimal then $K/{\mathcal{O}}$ and $K/\textbf{m}$ are
opaque, see [12, Lemma 11.13(i)].

  2. (2)

If $(M,<,\dots)$ is weakly o-minimal then every definable convex equivalence
relation on $K$ is opaque. Indeed, if $X$ consists of $r$ convex sets then
since each $E$-class is convex, $X$ crosses at most $2r$-many classes.

  3. (3)

If $(K,v,\dots)$ is P-minimal then $K/{\mathcal{O}}$ is opaque. This follows,
essentially, from Fact 5.7 (and quantifier elimination).

The next lemma can be viewed as a statement on domination by opaque imaginary
sorts.

Below, to simplify notation we use $\pi$ to denote each of the projections
$D_{i}\to D_{i}/E_{i}$ and let
$\pi_{k}:\prod_{i=1}^{k}D_{i}\to\prod_{i=1}^{k}(D_{i}/E_{i})$ be the natural
projection.

######  Proposition 6.3.

Let ${\mathcal{M}}$ be a structure of finite dp-rank. Let $D_{1},\ldots,D_{n}$
be any definable sets, and for each $i=1,\ldots,n$, let $E_{i}$ be a definable
opaque relation on $D_{i}$, such that the structure on $D_{i}/E_{i}$ is dp-
minimal. Let $E=\prod_{i}E_{i}$ be the equivalence relation on
$D=\prod_{i}D_{i}$.

Assume that $X\subseteq P\subseteq D$ are definable sets such that
$\pi_{n}(X)=\pi_{n}(P)$ (i.e. $X$ intersects every $E$-class which $P$ does).
Then $\mathrm{dp\text{-}rk}(\pi_{n}(P\setminus X))<n$.

###### Proof.

We proceed by induction on $n$. For $n=1$, we may assume that $P=D_{1}$ and
$X\subseteq D_{1}$ is definable and intersects every $E$-class. It follows
that $D_{1}\setminus X$ intersects only those classes which $X$ crosses, and
by opacity there are only finitely many such $E$-classes. Thus
$\pi_{1}(D_{1}\setminus X)$ is finite so has
$\mathrm{dp\text{-}rk}(\pi_{1}(D_{1}\setminus X))=0$, as required.

Assume that the proposition is proved for all $n^{\prime}<n$, where $n>1$, and
now let $P$ and $X$ be as in the statement. We aim to prove that
$\mathrm{dp\text{-}rk}(\pi_{n}(P\setminus X))<n$.

We begin with some notation and elementary observations. We let
$D^{\prime}=\prod_{i=1}^{n-1}D_{i}$. For $i=1,\ldots,n$ we write
$\overline{D}_{i}=D_{i}/E_{i}$ and $\overline{D}=D/E$.

For $a\in D^{\prime}$, $b\in D_{n}$ and $Y\subseteq D$, we let

$Y_{a}=\\{b^{\prime}\in D_{n}:(a,b^{\prime})\in
Y\\}\,\,,\,\,Y^{b}=\\{a^{\prime}\in D^{\prime}:(a^{\prime},b)\in Y\\}.$

We shall sometimes identify elements in $\overline{D}$ with the corresponding
$E$-class inside $D$. In particular, an element $\beta\in\overline{D}_{n}$ is
also a subset of $D_{n}$, and for $Y\subseteq D$ we define

$Y^{\beta}:=\bigcup_{b\in\beta}Y^{b}=\\{a\in D^{\prime}:\exists
b\in\beta\,(a,b)\in Y\\}.$

It is a subset of $D^{\prime}$ and we have
$\pi_{n-1}(Y^{\beta})=(\pi_{n}(Y))^{\beta}$ (where on the right, $\beta$ is
taken as an element of $\overline{D}_{n}$), and for $a\in D^{\prime}$ and
$\beta\in D_{n}$, we have $a\in Y^{\beta}\Leftrightarrow\beta\cap
X_{a}\neq\emptyset$.

Since $\pi_{n}(P\setminus X)\subseteq\pi_{n}(P)$, we may assume that
$\mathrm{dp\text{-}rk}(\pi_{n}(P))=n$. Let

$X^{*}=\\{(a,b)\in(D^{\prime}\times D_{n})\cap X:X_{a}\mbox{ crosses
}b/E_{n}\\}.$

Clearly $X^{*}\subseteq X$ is definable.

($\clubsuit$) A special case: $X^{*}=\emptyset$ (i.e for all $a\in
D^{\prime}$, $X_{a}$ does not cross any $E_{n}$-class).

Since $\pi_{n}(X)=\pi_{n}(P)$ it immediately follows that for every
$\beta\in\overline{D}_{n}$ we get $\pi_{n-1}(X^{\beta})=\pi_{n-1}(P^{\beta})$.
For each $\beta\in\overline{D}_{n}$ we may apply induction to
$X^{\beta}\subseteq P^{\beta}\subseteq D^{\prime}$ to conclude that
$\mathrm{dp\text{-}rk}(\pi_{n-1}(P^{\beta}\setminus X^{\beta}))<n-1$.

######  Claim 6.3.1.

$P^{\beta}\setminus X^{\beta}=(P\setminus X)^{\beta}$.

###### Proof.

The inclusion $\subseteq$ is true without any assumptions on $X$. For the
converse, assume that $a\in(P\setminus X)^{\beta}$, namely there exists
$b\in\beta$ such that $(a,b)\in P\setminus X$. It follows that $a\in
P^{\beta}$ so we want to show that $a\notin X^{\beta}$. Indeed, since $a\notin
X^{b}$ then $b\notin X_{a}$. But since $X_{a}$ does not cross any
$E_{n}$-class then $\beta\cap X_{a}=\emptyset$, and therefore $a\notin
X^{\beta}$, so $a\in P^{\beta}\setminus X^{\beta}$. ∎

Using the claim and the induction hypothesis we conclude that for every
$\beta\in\overline{D}_{n}$,

$\mathrm{dp\text{-}rk}(\pi_{n}(P\setminus
X)^{\beta})=\mathrm{dp\text{-}rk}\left(\pi_{n-1}((P\setminus
X)^{\beta})\right)=\mathrm{dp\text{-}rk}\left(\pi_{n-1}(P^{\beta}\setminus
X^{\beta})\right)<n-1.$

Since this is true for every $\beta\in\overline{D}_{n}$, it follows from the
sub-additivity of rank that $\mathrm{dp\text{-}rk}(\pi_{n}(P\setminus X)<n.$
This concludes the proof of the special case. ∎♣

We return to the general case and first note that
$\mathrm{dp\text{-}rk}(\pi_{n}(X^{*}))<n$. Indeed, let
$\widehat{X^{*}}=\\{(a,\pi(b))\in D^{\prime}\times\overline{D_{n}}:(a,b)\in
X^{*}\\}$. By the definition of $X^{*}$ and opacity, the projection of
$\widehat{X^{*}}$ to $D^{\prime}$ is finite-to-one and hence
$\mathrm{dp\text{-}rk}(\widehat{X^{*}})<n$. It follows that
$\mathrm{dp\text{-}rk}(\pi_{n}(X^{*}))<n$ as well.

To finish the proof in the general case, let $R=\pi_{n}(X^{*})$,
$P_{1}=P\setminus\pi_{n}^{-1}(R)$ and $X_{1}=X\cap P_{1}$. Now,
$X_{1}^{*}=\emptyset$ (because $X_{1}^{*}\subseteq X^{*}$ and $P_{1}\cap
X^{*}=\emptyset$) and $\pi_{n}(X_{1})=\pi_{n}(P_{1})$ so by the special case
$\clubsuit$, we have $\mathrm{dp\text{-}rk}(\pi_{n}(P_{1}\setminus X_{1}))<n$.
Finally, since $P\setminus X\subseteq(P_{1}\setminus
X_{1})\cup\pi_{n}^{-1}(R)$

$\pi_{n}(P\setminus X)\subseteq\pi_{n}(P_{1}\setminus X_{1})\cup R.$

We have seen in the previous paragraph that $\mathrm{dp\text{-}rk}(R)<n$ so
$\mathrm{dp\text{-}rk}(P\setminus X)<n$. ∎

We now prove the domination result referred to above:

######  Lemma 6.4.

Let ${\mathcal{M}}$ be a dp-minimal structure and $E$ an $A$-definable opaque
equivalence relation on $M$. For any complete type $p\in S(A)$ concentrated on
${\mathcal{M}}^{n}/E^{n}$ with $\mathrm{dp\text{-}rk}(p)=n$, there exists a
unique complete type $q$ over $A$ , concentrated on $M^{n}$ with $\pi_{*}q=p$.
Furthermore, $\mathrm{dp\text{-}rk}(q)=n$ as well.

###### Proof.

For simplicity, we denote $E^{n}$ by $E$ and let $\pi:M^{n}\to M^{n}/E$ be the
quotient map. Note that since $\pi$ is surjective, there exists at least one
complete type, $q$ over $A$ such that $\pi_{*}(q)=p$.

To show uniqueness assume towards a contradiction that there exists an
$A$-definable subset $Z\subseteq M^{n}$ such that both $p\vdash\pi(Z)$ and
$p\vdash\pi(Z^{c})$ (where $Z^{c}:=M^{n}\setminus Z$). Then
$p\vdash\pi(Z)\cap\pi(Z^{c})$, hence
$\mathrm{dp\text{-}rk}(\pi(Z)\cap\pi(Z^{c}))=n$. Let
$P:=\pi^{-1}(\pi(Z)\cap\pi(Z^{c}))$ and $X=Z\cap P$. By definition of $P$,
$\pi(Z\cap P)=\pi(P)=\pi(Z^{c}\cap P)$. But since $P\setminus X=P\cap Z^{c}$,
$\pi(P\setminus X)=\pi(P)=\pi(X)$. By Proposition 6.3,
$\mathrm{dp\text{-}rk}(\pi(P\setminus X))<n$ and
$\mathrm{dp\text{-}rk}(\pi(X))<n$, so $\mathrm{dp\text{-}rk}(\pi(P))<n$
contradicting the above. ∎

######  Remark 6.5.

In the notation of the previous lemma, it is not hard to see that
$q=\\{Z\subseteq M^{n}:\text{$Z$ is $A$-definable, }p\vdash\pi(Z)\\}$.

###  6.2. Definable functions in $K/{\mathcal{O}}$.

Our goal in the present section is to show that, under certain assumptions,
definable functions on $(K/{\mathcal{O}})^{n}$ are locally affine (with
respect to the group structure on $K/{\mathcal{O}}$). This fact, of possible
interest on its own right, will allow us to show that, under the same
assumptions, no infinite field interpretable in $K$ is locally strongly
internal to $K/{\mathcal{O}}$. Note that the analogous statement fails if one
replaces $K/{\mathcal{O}}$ with $K/\textbf{m}$, as k itself is strongly
internal to $K/\textbf{m}$.

######  Assumption.

Until the end of this section we let ${\mathcal{K}}=(K,v,\dots)$ be a
sufficiently saturated dp-minimal valued field of characteristic $0$
satisfying (Cballs), such that $\Gamma$ is dense (equivalently k is infinite).
Throughout the topology on $K/{\mathcal{O}}$ is the thick ball topology and
the topology on $(K/{\mathcal{O}})^{n}$ is the product topology.

Let $\pi:K\to K/{\mathcal{O}}$ be the quotient map. For simplicity, for every