# Strongly minimal reducts of ACVF

Santiago Pinzon

(October 31, 2022)

Abstract

Let $\mathbb{K}=(K,+,\cdot,v,\Gamma)$ be a valued algebraically closed field
of characteristic and $(G,\oplus)$ be a $\mathcal{K}$-interpretable group that
is either locally isomorphic to $(K,+)$ or to $(K,\cdot)$. Then if
$\mathcal{G}=(G,\oplus,\ldots)$ is a strongly minimal non locally modular
structure intepretable in $\mathbb{K}$, it interprets a field.

We also present an strategy for proving the same without the assumption of
having a definable group operation.

This document is the PhD thesis of the author and it was advised by professors
Alf Onshuus and Assaf Hasson.

###### Contents

  1. 1 Introduction and Preliminaries
    1. 1.1 Introduction
    2. 1.2 Preliminaries on Model Theory
    3. 1.3 Group Configurations
    4. 1.4 Preliminaries on valued fields
    5. 1.5 Partial Isomorphism
  2. 2 Some facts of definable groups contained in $K$
    1. 2.1 Some reductions
    2. 2.2 Good families of curves
  3. 3 Additive Case
    1. 3.1 Power Series
    2. 3.2 Getting a good family of curves on $\mathcal{G}$
  4. 4 Multiplicative Case
    1. 4.1 Finding a Group
    2. 4.2 Finding a Field
  5. 5 General Case

##  Chapter 1 Introduction and Preliminaries

###  1.1 Introduction

Given any field $k$ there are many families of curves in $k^{2}$ that one can
define. For example one may define the family of all lines on $k^{2}$. This,
together with the points of $k^{2}$, form a plane geometry in the sense of
Chapter 2.1 of [Art57]. In here Artin proved a converse for this. Namely if
one starts with a set $E$ and an incidence system of lines and points on $E$
satisfying some geometrical axioms, then one can to build a field $k$ such
that the lines of $E$ are given by linear equations on $k$.

One can wonder whether this result in some other settings where one has a good
notion of dimension and a big enough family of curve -in the sense of that
dimension-. In this way, in [Rab93], Rabinovich proved that if $k$ is an
algebraically closed valued field and $\mathcal{D}=(D,\ldots)$ is some
structure whose universe is $\mathbb{A}^{1}(k)$ and whose definable sets are
constructible sets, then if $\mathcal{D}$ has a big enough definable family of
curves contained in $D^{2}$, then $\mathcal{D}$ interprets an infinite field.

An example of such a family is the family of all lines contained in
$\mathbb{A}^{1}\times\mathbb{A}^{1}$. This is a two dimensional family of
curves, and in this case is not hard to see that one can recover an infinite
field. For example if one consideres the sub-family of lines passing trough
$(0,0)$ this can be identified with $\\{l_{m}:m\in k\\}\cup l_{\infty}$. Where
for $m\in K$ $l_{m}$ is the line of equation $y=mx$ and $l_{\infty}$ is the
line of equation $x=0$. In this case one has that $l_{m}\circ l_{n}=l_{m\cdot
n}$ so one can recover the multiplicative group of $k$, $\mathbb{G}_{m}$ from
composition between elements of that subfamily. With some more work we can
also recover the additive group $\mathbb{G}_{a}$ and also the action of
$\mathbb{G}_{m}$ on $\mathbb{G}_{a}$.

In late 1970s, Zilber abstracted this principle in the following conjecture:

######  Conjecture 1.1.1.

(Zilber’s Trichotomy Principle)

If $\mathcal{D}$ is a strongly minimal structure and there is a big enough
$\mathcal{D}$-definable family of plane curves, then $\mathcal{D}$ interprets
an infinite field.

In the conjecture, “big enough” will be read for us as “morley rank $2$”. The
family of lines in the plane is an example.

Ravinovich’ result is a very particular case of this conjecture where
$\mathcal{D}$ is some reduct of the full field structure on the affine line of
an algebraically close field.

This conjecture was proved false by Hrushovski in [Hru93]. He built a class of
examples of strongly minimal sets that are not trivial or locally modular but
does not interpret any field. However, the principle of Zilber’s trichotomy
has still an important role in modern model theory.

In [HZ96] Hrushovski and Zilber defined (1-dimensional) Zariski geometries and
proved that Zilber’s trichotomy holds for them. This covers a vast class of
examples, generalizing the algebraic case: if $k$ is an algebraically closed
field and $N$ are the $k$-points of an algebraically curve (over $k$) then if
$\mathcal{N}$ is the structure with universe $N$ and whose definable subsets
of $N^{k}$ are all the $k$-constructible sets, then $\mathcal{N}$ is a Zariski
Geometry and then it interprets a field.

There are other settings where the conjecture has been proved true without the
use of Zariski Geometries but using intersection theory. That is the case of
[HS17]. Here Hasson and Sustretov proved that if $D$ has as universe an
algebraic curve over an algebraically closed field $k$ and all the definable
sets on $D^{k}$ are definables in the field structure, then Zilber’s
trichotomy it is true for $D$. Note that it is a generalization of Ravinovich
result.

In some other settings the conjecture has proved to be true by using
intersection theory coming from continuous open functions. This is the case of
[EHP21], here is proved that if $(D,+)$ is a definable group in $k$, an
o-minimal expansion of a real closed field, and $\mathcal{D}=(D,+,\ldots)$ is
a o-minimal structure that is a reduct of the structure induced by $k$,
containing a big enough family of curves, then $\mathcal{D}$ interprets a
field.

There is also cases where the conjecture was proved to be true using
intersection theor coming from analytico functions. This is the case of
[KR16]. Here it is proved that if $D$ has as universe a valued field $K$ with
$\text{char}K=0$ and we assume that addition is definable and that all the
definable sets of $D^{k}$ are definable in the valued field structure, then
$D$ satisfies Zilber’s trichotomy.

This is the setting in which we are interested. In the introduction of [HS17]
is suggested that their methods should be suitable to be used for proving
generalizations of [KR16], for example proving the result for positive
characteristic or get rid of the assumption that $+$ is definable on $D$.

In this thesis we work on both possible generalizations:

First we prove that results on [KR16] are true even in positive
characteristic. This is Theorem 3.0.1.

Moreover we also prove that if $D$ is an expansion of the multiplicative group
$(K\setminus 0,\cdot)$ then it interprets a field if it is not locally
modular.

We expect that our results can be used for proving Zilber’s trichotomy for
$D$. Assuming that $D$ is any definable subset of an algebraically close
valued field $K$ whose Zariski closure is $1$-dimensional and whose definables
sets are definable in the valued fields structure. This can be done finding a
group interpretable in $\mathcal{D}$ as in Section 4.1 and then proving that
such a group is locally isomorphic either to $(K,+)$ or to $(K\setminus
0,\cdot)$, and then we can use our result to conclude.

Now we present the structure of the document:

In Chapter 1 we present the basic preliminaries on model theory and valued
fields that we will need.

In Chapter 2 we prove some facts about definable groups contained in $K$. In
Section 2.2 we introduce the notion of good families of curves for a definable
group $G$ and prove Theorem 2.2.2 that we will use for defining a field.

In Chapter 3 we construct a good family of curves if $G$ is the additive group
of $K$ so using results on Chapter 2 we prove Theorem 3.0.1.

In Chapter 4 we deal with the case in wich $D$ is an expansion of the
multplicative group. In here we firs build an interpretable group that is
locally isomorphic to $(K,+)$. Then we use results of Chapters 2 and 3 for
interpret a field.

###  1.2 Preliminaries on Model Theory

Let $\mathcal{L}$ be a first order language and let $\mathcal{P}=(P,\ldots)$
be an infinite $\mathcal{L}$ structure.

We adopt the usal definitios of $\mathcal{P}$-definable and
$\mathcal{P}$-intepretable sets. See for example Section 1.3 of [Mar06].
Assume moreover that $\mathcal{P}$ is $\kappa$-saturated, for some big enough
cardinal $\kappa$. In particular $\kappa>\omega$.

######  Definition 1.2.1.

Given $D$ be an interpretable set on $\mathcal{P}$.

We say that $D$ is strongly minimal if $D$ is infinite and the only definable
subsets of $D$ are the finite and the cofinite sets.

We say that $\mathcal{P}$ is strongly minimal if $P$ is strongly minimal as a
definable set on $\mathcal{P}$.

If $T$ is a theory we say that $T$ is strongly minimal if $\mathcal{P}$ is
strongly minimal for all $\mathcal{P}\models T$

We will use the following notion of dimension:

######  Definition 1.2.2.

If $X\subseteq P^{n}$ is a definable set we say that
$\operatorname{RM}_{\mathcal{P}}(X)\geq 0$ if $X$ is non empty and for an
ordinal $\alpha$, $\operatorname{RM}_{\mathcal{P}}(X)\geq\alpha$ if there are
$X_{1},X_{2}\ldots$ infinitely many definable subsets of $X$ such that:

  * •

$\bigcup X_{i}=X$,

  * •

for all $i\neq j$, $X_{i}\cap X_{j}=\emptyset$ and

  * •

for all $i$ and for all $\beta<\alpha$,
$\operatorname{RM}_{\mathcal{P}}(X_{i})\geq\beta$..

We say that $\operatorname{RM}_{\mathcal{P}}(X)=\alpha$ if
$\operatorname{RM}_{\mathcal{P}}(X)\geq\alpha$ and is not the case that
$\operatorname{RM}_{\mathcal{P}}(X)\geq\alpha+1$.

If there is some $\alpha<\kappa$ such that $\operatorname{RM}(X)=\alpha$, we
say that $X$ has bounded morley rank and that $\alpha$ is the morley rank of
$X$.

If $\operatorname{RM}(X)=n\in\omega$ then we say that $X$ has finite morley
rank.

The following is well known, a proof can be find in Lemma 6.2.7 of [Mar06].

######  Fact 1.2.3.

If $X,Y$ are subsets of $P^{n}$, then

We omit the subscript if $\mathcal{P}$ is clear from the context.

######  Definition 1.2.4.

If $p(x)\in S_{n}(A)$ we define

$\operatorname{RM}(p)=\min\\{\operatorname{RM}(X):X\in p\\}.$

For $a\in D^{n}$ let $\operatorname{RM}(a/p)=\operatorname{RM}(tp(a/A))$.

The following is well known (and easy to prove) and can be found in Lemma
6.2.7 of [Mar06].

######  Fact 1.2.5.

If $\mathcal{P}=(P,\ldots)$ is any structure and $X,Y$ are definable subsets
of $P^{n}$, then:

  1. 1.

If $X\subseteq Y$ then $\operatorname{RM}(X)\leq\operatorname{RM}(Y)$.

  2. 2.

$\operatorname{RM}(X\cup Y)=\max(\operatorname{RM}(X),\operatorname{RM}(Y))$.

  3. 3.

If $X\neq\emptyset$ then $\operatorname{RM}(X)=0$ if and only if $X$ is
finite.

######  Definition 1.2.6.

If $\operatorname{RM}(X)=\alpha$ then there is no a partition of $X$ in
infinitelly many definable subsets of morley rank greater than $\alpha$.
Therefore by compactness there is a maximal natural number $n$ such that there
are $X_{1},\ldots,X_{n}$ definable and disjoints subsets of $X$ covering $X$
with $\operatorname{RM}(X_{i})=\alpha$. We define the morley degree of $X$ as
$\operatorname{DM}(X)=n$.

We say that $X$ is sationary if $\operatorname{DM}(X)=1$.

######  Lemma 1.2.7.

A set $D$ is strongly minimal if and only if
$\operatorname{RM}(D)=\operatorname{DM}(D)=1.$

###### Proof.

Suppose $D$ is strongly minimal, then as $D$ is infinite and $D=\cup_{d\in
D}\\{d\\}$ one has that $\operatorname{RM}X\geq 1$. As each infinite definable
subset of $D$ is cofinite it is not the case that there are two infinite and
disjoints definable subsets of $D$. It shows that $\operatorname{RM}(X)=1$ and
also that $\operatorname{DM}(X)=1$.

Now assume that $\operatorname{RM}X=\operatorname{DM}X=1$, and suppose by
contradiction that there is an infinite definable set $Y\subseteq X$ such that
$X\setminus Y$ is also infinite. Then as $X=Y\cup(X\setminus Y)$ one has that
$\operatorname{DM}(X)\geq 2$, a contradiction.

∎

The following can be found in page 196 of [Mar17].

######  Fact 1.2.8.

If $\mathcal{P}=(P,\ldots)$ is strongly minimal then
$\operatorname{RM}(P^{n})=n$.

######  Corollary 1.2.9.

If $\mathcal{P}=(P,\ldots)$ is strongly minimal and $X\subseteq P^{n}$ is
$\mathcal{P}$-definable, then $X$ has finite Morley Rank.

###### Proof.

By Fact 1.2.8 one has that $\operatorname{RM}P^{n}=n$ and by clause 1 of Fact
1.2.5 one conclude that $\operatorname{RM}X\leq n$, in particular $X$ has
finite Morley Rank. ∎

######  Definition 1.2.10.

Let $\mathcal{P}=(P,\ldots)$ be a strongly minimal structure and $X$ be a
$\mathcal{P}$-interpretable set with parameters $A$. We say that a tuple
$z=(z_{1},\ldots,z_{n})\in X^{n}$ is generic independent (over $A$) if
$\operatorname{RM}(z/A)=n\operatorname{RM}(X)$

We now define curves and families of curves.

######  Definition 1.2.11.

If $\mathcal{P}=(P,\ldots)$ is a strongly minimal structure, a plane curve $C$
(of $\mathcal{P}$) is a $\mathcal{P}$-definable and one dimensional subset of
$P^{2}$.

A definable family of plane curves is a $\mathcal{M}$-definable set
$X\subseteq P^{2+n}$ such that for all $a$ in some $\mathcal{P}$-definable set
$Q\subseteq P^{n}$ one has that

$X_{a}:=\\{(x,y)\in P^{2}:(x,y,a)\in X\\}$

is a plane curve.

We usually write such a family as $(X_{a})_{a\in Q}$.

######  Definition 1.2.12.

We say that a family of curves $(X_{a})_{a\in Q}$ is almost disjoint if for
any $b\in Q$ the set $\\{a\in Q:|X_{a}\Delta X_{b}|<\infty\\}$ is finite.

From now when we say “definable family of curves” we mean “definable and
almost disjoint family of curves”.

Now we define the notion of trivial and locally modular strongly minimal
structures.

######  Definition 1.2.13.

If $\mathcal{P}$ is strongly minimal we say that $\mathcal{P}$ is trivial if
for any definable set $A\subseteq P^{n}$ one has that

$\operatorname{acl}(A)=\bigcup_{a\in A}\operatorname{acl}(a).$

######  Definition 1.2.14.

If $\mathcal{P}$ is strongly minimal we say that $\mathcal{P}$ is locally
modular if for any $X,Y$ definable subsets of $P^{n}$, if
$X=\operatorname{acl}(X)$ and $Y=\operatorname{acl}(Y)$ then one has that

$\operatorname{RM}(X\cup
Y)=\operatorname{RM}X+\operatorname{RM}Y-\operatorname{RM}(X\cap Y).$

Zilbers Trichotomy states:

Let $\mathcal{P}$ be strongly minimal, then and assume that is no trivial and
is no locally modular, then there is an infinite field $k$ interpretable in
$\mathcal{P}$.

In all of its generality it was proved false by Hrushovski on [Hru93]. But we
can re state it relative to restricted setting:

Let $T$ be some complete theory

######  Conjecture 1.2.15.

(Zilber’s trichotomy principle relative to $T$)

Let $\mathcal{N}=(N,\ldots)$ be any model of $T$ and let $P$ be some
$\mathcal{N}$-interpretable set. Let $\mathcal{P}=(P,\ldots)$ be some
$\mathcal{L}^{\prime}$-structure over $P$ (for some language
$\mathcal{L}^{\prime}$) such that any $\mathcal{P}$-definable set is also
$\mathcal{N}$-definable. Then if $\mathcal{P}$ is non locally modular, there
is an infinite field interpretable in $\mathcal{P}$.

There are several instances of this conjecture that has been proved true, more
relevant for us are:

######  Fact 1.2.16.

(Theorem 4.3.3 of [HS17]) If $T$ is the theory of algebraically closed fields
of a fixed characteristic (as defined in Section 1.4) then Conjecture 1.2.15
is true if we assume that $P$ is an algebraic curve.

Recently Castle proved in [Cas22] the following:

######  Fact 1.2.17.

If $\mathcal{N}$ is an algebraically closed field of characteristic zero,
then, Conjecture 1.2.15 is true.

The following is also true:

######  Fact 1.2.18.

(Theorem 3.17 on [KR16]) If $T$ is the theory of algebraically closed valued
field of characteristic zero (as defined in Section 1.4) then Conjecture
1.2.15 is true if we assume that $\mathcal{N}=(N,+,\cdot,v,\Gamma)$, $P=N$ and
addition is definable in $\mathcal{P}$.

We will use techniques of both, [HS17] and [KR16] for almost all of our work.

###  1.3 Group Configurations

In this section we introduce the main tools in order to interpret groups and
fields in strongly minimal structures.

From now we fix $\mathcal{N}=(N,\ldots)$ an strongly minimal structure.

######  Definition 1.3.1.

A $d$-dimensional group configuration for $\mathcal{N}$ over a set of
parameters $A$ is a $6$-tuple
$\mathfrak{g}=(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3})$ where each $b_{i}$ and
$a_{i}$ are tuples of elements of $N$ such that:

  * •

$\operatorname{RM}\mathfrak{g}=3d+3$

  * •

$\operatorname{RM}(\alpha,\beta/A)=\operatorname{RM}(\alpha/A)+\operatorname{RM}(\beta/A)$
for all $\alpha\neq\beta\in\mathfrak{g}$,

  * •

$\operatorname{RM}(b_{i}/A)=d$ for $i=1,2,3$,

  * •

$\operatorname{RM}(a_{i}/A)=1$ for $i=1,2,3$,

  * •

$\operatorname{RM}(b_{1},b_{2},b_{3}/A)=3d$,

  * •

$\operatorname{RM}(b_{1},a_{2},a_{3})=\operatorname{RM}(b_{2},a_{1},a_{3})=\operatorname{RM}(b_{3},a_{1},a_{2})=d+1$.

######  Definition 1.3.2.

If $G$ is an interpretable group of dimension $d$, a group configuration of
$G$ is

$\mathfrak{g}_{G}=(a,b\cdot a,c\cdot b\cdot a,b,c,cb)$

for some choice of $a,b,c$ generic independent elements of $G$

######  Definition 1.3.3.

We say that $\mathfrak{g}=(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3})$ is a reduced
group configuration for $\mathcal{N}$ if it is a group configuration and in
addition if $a^{\prime}_{i}\in\operatorname{acl}(a_{i})$ are such
$\mathfrak{g}^{\prime}=(a^{\prime}_{1},a^{\prime}_{2},a^{\prime}_{3},b_{1},b_{2},b_{3})$
is still a group configuration, then
$a_{i}\in\operatorname{acl}(a^{\prime}_{i})$

If $\mathfrak{g}_{1}$ is another group configuration we say that
$\mathfrak{g}$ and $\mathfrak{g}_{1}$ are interalgebraic if the correspondent
coordinates satisfies $\operatorname{acl}(a)=\operatorname{acl}(a^{\prime})$.
The following is due to Hrushovski ([Hru86]). The precise statement we need
can be find in Facts 4.4 and 4.6 of [HS17].

######  Fact 1.3.4.

Let $\mathcal{N}$ be an strongly minimal structure, then if $\mathfrak{g}$ is
a group configuration for $\mathcal{N}$ (over some set of parameters), there
is a minimal group $G$, an strongly minimal set $X$ and a faithful action of
$G$ on $X$ all of the data interpretable in $N$.

In addition $\mathfrak{g}$ is reduced, then a generic group configuration of
$G$ is interalgebraic with $\mathfrak{g}$. In particular
$\operatorname{RM}G=d$.

The following is also due to Hrushovski [Hru86].

######  Fact 1.3.5.

Let $G$ be a group interpretable in $\mathcal{N}$ acting transitively and
faithfully on a strongly minimal set $X$. Assume $\operatorname{RM}(G)=2$ then
there is a field structure definable in $X$ and $G$ is isomorphic to the
semidirect product of the multiplicative and the additive group of such a
field.

In particular if $\mathfrak{g}$ is a group configuration of dimension $2$ then
$\mathcal{N}$ interprets a field.

So it makes sense to define:

######  Definition 1.3.6.

A field configuration is a $2$-dimensional group configuration.

###  1.4 Preliminaries on valued fields

We start with some basic definitions.

######  Definition 1.4.1.

Given a field $\mathbb{K}=(K,+,\cdot,0,1)$ we treat it as a first order
structure in the language of rings $\mathcal{L}_{R}=\\{+,\cdot,0,1\\}$. Let
$ACF_{p}$ be the first order theory of algebraically closed fields of
characteristic $p$. That is $ACF_{p}$ is the theory of fields of
characteristic $p$:

That is $(K,+,0)$ is an abelian group:

  * •

$\forall x(x+0=x)$,

  * •

$\forall x\exists y(x+y=0)$,

  * •

$\forall x,y,z((x+y)+z=x+(y+z))$ and

  * •

$\forall x,y(x+y=y+x)$.

The product ($\cdot$) is a binary operation defined on $K\times K$ such that
$(K\setminus 0,\cdot,1)$ is also an abelian group:

  * •

$\forall x(x\cdot 1=x)$,

  * •

$\forall x(x\neq 0\implies\exists y(x\cdot y=1))$,

  * •

$\forall x,y,z[(z\neq 0\wedge y\neq 0\wedge z\neq 0)\implies(x\cdot
y)+z=x+(y+z)]$ and

  * •

$\forall x,y(x\cdot y=y\cdot x)$.

And product distributes over addition:

$\forall x,y,z(x\cdot(y+z)=x\cdot y+x\cdot z,$

together with the scheme of axioms given by

$\forall a_{0}\ldots\forall a_{n}\exists x(a_{0}+a_{1}x+\ldots a_{n}x^{n}=0)$

for all $n\geq 1$.

And we say that $\text{char}(K)=p$, so if $p=0$ we add the scheme of axioms

$\phi_{n}:=1+1+1\cdots+1\neq 0$

where $1$ is added $n$ times.

If $\text{char}(K)=p>0$ we add the axioms $\phi_{n}$ for $n<p$ plus the axiom
$\neg\phi_{p}$.

######  Definition 1.4.2.

Given a field $(K,+,\cdot,0,1)$ we say that a set $X\subseteq K^{n}$ is
zariski closed if there is a set of polynomials $\mathfrak{a}\subseteq
K[x_{1},\ldots,x_{n}]$ such that for all $x\in K^{n}$, $x\in X$ if and only if
$f(x)=0$ for all $f\in\mathfrak{a}$.

######  Fact 1.4.3.

(It follows for example from Theorem 3.2.2 of [Mar06]) The theory $ACF_{p}$ is
complete and strongly minimal

We will need:

######  Fact 1.4.4.

(Bezout Theorem)

Let $k$ be algebraically closed field and let $F(x,y)$ and $G(x,y)$ be
polynomials with coefficients on $k$ with no common non constant divisors,
then if $V$ is the set of zeros for $F$ and $W$ is the set of zeros for $G$
then $V\cap W$ is finite and it has less than $\deg(F)\deg(G)$ points counting
multiplicities. If we considerate the closures of $V$ and $W$ in the
projective space $\mathbb{P}^{2}$ then the number of points intersection is
exactly $\deg(F)\deg(G)$ (counting multiplicities).

######  Definition 1.4.5.

If $K$ is a field, a valuation on $K$ is an ordered abelian group $\Gamma$
togheter with a valuation map $v:K\to\Gamma\cup\\{\infty\\}$ (where $\infty$
is an extra element such that $\infty>\gamma$ and
$\infty+\gamma=\gamma+\infty=\infty$ for each $\gamma\in\Gamma$) such that for
all $x,y\in K$, $v$ satisfies:

  1. 1.

$v(x)=\infty$ if and only if $x=0$.

  2. 2.

$v(x\cdot y)=v(x)+v(y)$.

  3. 3.

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

If $K$ is valued, the valuation ring is $\mathcal{O}_{K}=\\{x\in K:v(x)\geq
0\\}$. It is easy to see that is a subring of $K$ and its only maximal ideal
is: $\mathfrak{m}=\\{x\in K:v(x)>0\\}$. The residue field is
$k=\mathcal{O}_{K}/\mathfrak{m}$.

We treat a valued field as a two sorted structure $(K,\Gamma,v)$ where $K$ is
a field, $\Gamma$ is an algebraically closed ordered group and
$v:K\to\Gamma\cup\\{\infty\\}$ is the valuation. Let $\mathcal{L}_{R,v}$ be
the correspondent two sorted language.

######  Definition 1.4.6.

Let ACVF be the first order theory in the language $\mathcal{L}_{R,v}$ saying
that $K$ is an algebraically closed field and $v$ is a valuation into an
ordered abelian group $\Gamma$:

In ACVF we have the axioms that says that $\Gamma$ is an abelan group plus the
extra axioms

$(\Gamma,\leq)$ is a linear order:

  * •

$\forall x,y\in\Gamma(x\leq y\vee y\leq x)$,

  * •

$\forall x,y,z\in\Gamma(x\leq y\wedge y\leq x\implies x\leq z)$,

  * •

$\forall x,y\in\Gamma(x\leq y\wedge y\leq x\implies x=y)$ and

  * •

$\forall x\in\Gamma(x\leq x)$.

And addition on $\Gamma$ is compatible with $\leq$:

$\forall x,y,z\in\Gamma(x\leq y\implies x+z\leq y+z)$

From now we fix $\mathbb{K}=(K,+,\cdot,\Gamma,v)$ a model of ACVF.

######  Definition 1.4.7.

An open ball is a subset of $K$ of the form

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

where $a\in K$ and $\gamma\in\Gamma$. A closed ball is a subset of $K$ of the
form

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

In this setting we have two natural topologies: the Zariski and the valuation
topologies. The latter is generated by the balls. When we say ‘open’ or
‘closed’ we mean in the valuation topology. When we want to refer to the
Zariski topology we will be explicit about it.

######  Definition 1.4.8.

If $D\subseteq K^{n}$ is a $\mathbb{K}$-definable set we define $\dim D$ as
the usual algebraic dimension of the Zariski closure of $D$. Moreover if
$p(x)$ is an $n$-type over $A$, we define $\dim p=\min\\{\dim X:X\in p\\}$ and
if $a\in K^{n}$ then $\dim(a/A)$ is defined as $\dim(tp(a/A))$

######  Definition 1.4.9.

If $X\subseteq K^{n}$ is $\mathbb{K}$-definable over $A$, we say that
$x=(x_{1}\ldots,x_{m})\in X^{m}$ is a tuple of independent generics of $X$ if
$\dim(x/A)=m\dim(X)$

We have the next classic theorem that follows from Holly’s work in [Hol95]. As
stated here is Theorem 7.1 of [HHM05].

######  Fact 1.4.10.

(Quantifier Elimination) $K$ has Quantifier elimination in the language
$\mathcal{L}_{R,v}$. In particular any definable subset of $K^{n}$ is the
intersection of a (valued) open subset of $K^{n}$ with a Zariski closed set.

As a corollaries we have:

######  Fact 1.4.11.

Suppose $Y\subseteq K^{2}$ is $\mathbb{K}$-definable and infinite. Then $Y$
can be written as a finite union of subsets of $K^{2}$ that are relatively
open subsets of irreducible Zariski closed sets.

######  Fact 1.4.12.

If $a\in\operatorname{acl}_{\mathbb{K}}(B)$ for some set of parameters $B$
then there is a finite set definable just with the field sructure of
$\mathbb{K}$ with parameters $B$ containing $a$.

In other words
$\operatorname{acl}_{\mathbb{K}}=\operatorname{acl}_{\mathbb{K}^{f}}$ where
$\mathbb{K}^{f}$ is $\mathbb{K}$ seen as a structure in the language of rings.

Now we fix some concepts:

######  Definition 1.4.13.

If $U\subseteq K^{n}$ is an open set and $f:U\to K$ is a function, given $a\in
U$ we say that $f$ is analytic at $a$ if there is some power series

$g=\sum a_{I}x^{I}$

converging in a neighborhood $U^{\prime}$ of $0$ such that $f(z+a)=g(z)$ for
every $z$ in $U^{\prime}$.

We say that $f$ is analytic in $U$ if it is analytic at $a$ for every $a\in U$

######  Definition 1.4.14.

If

$F(x_{1}\ldots,x_{n})=\sum_{I=(i_{1},\ldots,i_{n})}a_{I}x^{(i_{1},\ldots,t_{n})},$

is a power series then $F_{x_{k}}(x_{1}\ldots,x_{n})$ is the (formal) partial
derivative of $F$ respect to $x_{k}$, that is

$F_{x_{k}}(x_{i},\ldots,x_{n})=\sum_{I=(i_{1},\ldots,i_{n})}i_{k}a_{I}x^{(i_{1},\ldots,i_{k}-1,\ldots,i_{n})}.$

######  Definition 1.4.15.

Let $U\subseteq K$ be an open set and let $a\in U$. Suppose $f:U\to K$
analytic at $a$ and its expansion around $a$ is

$f(x)=\sum_{n\geq 0}b_{n}x^{n}.$

We say that $a$ is a zero of $f$ with multiplicity $d$ if $f(a)=0$ and
$d=\min\\{n:b_{n}\neq 0\\}$.

######  Lemma 1.4.16.

Let $f:U\to V$ be a $\mathbb{K}$ definable function with $U$ and $V$ open
subsets of $K^{n}$ and $K^{m}$ respectively, then there is some $a\in U$ such
that $f$ is analytic at $a$.

###### Proof.

∎

We will also need an implicit function theorem, the following is well known
and can be found for example in [Abh01] (Theorem (10.8), page 84)

######  Fact 1.4.17.

(Implicit Function Theorem) Suppose $K$ is complete, $U\subseteq K^{n}$ is
open and $F:U\to K$ is an analytic function at $z\in U$. Assume $z$ is such
that $F_{x_{n}}(z)\neq 0$, then there are $U_{1}\subseteq K^{n-1}$ and $U_{2}$
open subsets of $K$, $U^{\prime}\subseteq U$ open with $z\in
U^{\prime}\cap(U_{1}\times U_{2})$ and $f:U_{1}\to U_{2}$ analytic such that:

$\\{u\in U^{\prime}:F(u)=0\\}=\\{(x,f(x)):x\in U_{1}\\}.$

As a corollary of this we have the Inverse Function Theorem:

######  Fact 1.4.18.

(Inverse Function Theorem, Theorem 10.10 in [Abh01]) Let $F(x)$ be analytic at
$0$, assume $F(0)=0$ and $F_{x}(0)\neq 0$ then there is an unique analytic
function $G$ converging in a neighborhood of $0$ such that $F(G(y))=y$ for
each $y$ in such a neighborhood.

Now we state some strong results on valued fields, first a theorem about
continuity of roots:

######  Fact 1.4.19.

(Continuity of Roots)

Assume $K$ is complete. Suppose $U_{1},U_{2}$ are open subsets of $K$ and

$F:U_{1}\times U_{2}\to K$

is analytic at $(a,b)\in U_{1}\times U_{2}$. Assume that the function $F(*,b)$
has a zero of multiplicity $d>0$ at $a$. Then there are open sets $U_{a}$ and
$U_{b}$ with $a\in U_{a}\subseteq U_{1}$ and $b\in U_{b}\subseteq U_{2}$ such
that:

  1. 1.

$a$ is the only $x\in U_{a}$ such that $F(x,b)=0$

  2. 2.

For each $y\in U_{b}$ the function $F(*,b)$ has exactly $d$ zeros in $U_{a}$
(counting multiplicities)

###### Proof.

We may assume $(a,b)=(0,0)$. By Theorem (10.3)(2) of [Abh01] there is some
$F^{*}(x,y)=f_{0}(y)+f_{1}(y)x+\ldots+x^{d}$ such that each $f_{j}$ is
analytic, $f_{i}(0)=0$ for $i=1,2,\ldots,d-1$, and there is some $\delta$ an
unit of $K[[x,y]]$ such that $F=F^{*}\delta$. Because of Theorem (11.3) of
[Abh01] there is an open set $U_{b}$ such that each $f_{i}$ is convergent in
$U_{b}$, $0\in U_{b}$ and for all $y\in K$ if $y\in U_{b}$ and $F^{*}(x,y)=0$
then $x\in U_{1}$. Moreover for all $y\in U_{b}$ the polynomial $F^{*}(*,y)$
is a polynomial of degree $d$ in the first variable so as $K$ is algebraically
closed there are exactly $d$ roots (counting multiplicities) and all of them
belongs to $U_{1}$ ∎

We will also need an identity theorem for expansion of power series:

######  Fact 1.4.20.

(Identity Theorem (10.5.2) of [Abh01])

Let

$f(x_{1},\ldots,x_{n})=\sum_{I}a_{I}x_{1}^{i_{1}}\ldots x_{x}^{i_{n}}$

be a power series converging in a neighborhood $D(f)$ of $(0,\ldots,0)$.
Assume that for all $x$ in some open set $U\subseteq D(f)$ one has that
$f(x)=0$, then $a_{I}=0$ for all $I$.

###  1.5 Partial Isomorphism

In this section we present the proof of Theorem 1.5.1, we follow lines on
[HP94].

From now let $\mathcal{H}=(H,\otimes)$ be a fixed group definable in
$\mathbb{K}$ with parameters $B$.

######  Theorem 1.5.1.

Let $(G,\oplus)$ be a group definable in the structure $\mathbb{K}$ with
parameters $B$. Assume $a,b\in G$ are such that
$\dim_{\mathbb{K}}(a/B)=\dim_{\mathbb{K}}(b/B)=\dim(G)$. And
$\dim_{\mathbb{K}}(ab/B)=2\dim(G)$. Let $c=a\oplus b$. Assume moreover that
there are $a^{\prime},b^{\prime}\in H$ with
$\dim_{\mathbb{K}}(a^{\prime}/B)=\dim_{\mathbb{K}}(b^{\prime}/B)=\dim H$ and
$\dim_{\mathbb{K}}(a^{\prime}b^{\prime}/B)=2\dim(H)$. Assume as well that
$\operatorname{dcl}_{\mathbb{K}}(aB)=\operatorname{dcl}_{\mathbb{K}}(a^{\prime}B)$,
$\operatorname{dcl}_{\mathbb{K}}(bB)=\operatorname{dcl}_{\mathbb{K}}(b^{\prime}B)$
and
$\operatorname{dcl}_{\mathbb{K}}(cB)=\operatorname{dcl}_{\mathbb{K}}(c^{\prime}B)$
where $c^{\prime}=a^{\prime}\otimes b^{\prime}$. Then there is a
$\mathbb{K}$-definable analytic, local isomorphism between a neighborhood of
the identity of $G$ and a neighborhood of the identity of $H$. That is, there
is a neighborhood $U_{1}$ of the identity of $G$, a neighborhood $U_{2}$ of
the identity of $H$ and an analytic invertible function $\phi:U_{1}\to U_{2}$
whose inverse is analytic such that if $u,v\in U_{2}$ are such that $u\oplus
v\in U_{1}$, then $\phi(u\oplus v)=\phi(u)\otimes\phi(v)$.

###### Proof.

Our assumptions on $a$ and $a^{\prime}$ implies that there is a formula
$\phi(x,y)$ (with parameters in $B$) such that $\phi(a,a^{\prime})$ holds and
there is just one $x$ such that $\phi(x,a^{\prime})$ holds and just one $y$
such that $\phi(a,y)$ holds. Our dimension assumptions on $a$ and $a^{\prime}$
implies that there is an open neighborhood $U_{1}$ of $a$ and $U_{2}$ of
$a^{\prime}$ such that for all $x\in U_{1}$ there is just one $y$ such that
$\phi(x,y)$ holds and for all $y\in U_{2}$ there is just one $x$ such that
$\phi(x,y)$ holds. Therefore $\phi(x,y)$ defines the graph of a function
$f:U_{1}\to H$ and the opposite defines a function $f^{-1}:U_{2}\to G$.

Using again our dimension assumptions on $a$ and $a^{\prime}$ we have that
$a\in U_{1}$ and $a^{\prime}\in U_{2}$. So we get open sets $U$ and
$U^{\prime}$ with $a\in U$ and $a^{\prime}\in U^{\prime}$ and an analytic
invertible function $f:U\to U^{\prime}$ with analytic inverse such that
$f(a)=a^{\prime}$.

We can do the same thing for $b,b^{\prime}$ and $c,c^{\prime}$ finding open
neighborhoods $V$, $V^{\prime}$, $W$ and $W^{\prime}$ of $b,b^{\prime},c$ and
$c^{\prime}$ respectivelly and $\mathbb{K}$-definable invertible functions
$g:V\to V^{\prime}$ and $h:W\to W^{\prime}$ such that $g(b)=b^{\prime}$ and
$h(c)=c^{\prime}$.

Now as $f(a)\otimes g(b)=h(c)$ and $\dim(ab/B)=2\dim G$ there is an open
neighborhood $Z$ of $(a,b)$ in $G^{2}$ such that for all $(x,y)\in Z$ one has
that $f(x)\otimes g(y)=h(x\oplus y)$. Moreover we may shrink $U$ and $V$ to
ensure that $U\times V\subseteq Z$. So for all $x\in U$ and $y\in V$ one has
that $f(x)\otimes g(y)=h(x\oplus y)$. We may also shrink $U$ an $V$ to ensure
that $U\oplus V\subseteq W$

Let $x,y$ and $z$ be elements of $U$ such that $a=x\oplus z^{-1}\oplus y$ we
claim that $f(a)=f(x)\otimes f(z)^{-1}\otimes f(y)$, for this let
$b_{1}=x^{-1}\oplus c$ and $c_{1}=z\oplus b_{1}$. Then as $a$ and $b$ are
independent over $B$ one has that $b_{1}\in V$ and $c_{1}\in W$. Moreover
$y\oplus b=c_{1}$. So because of previous observation one has:

$f(y)\otimes g(b)=h(c_{1})$

$f(x)\otimes g(b_{1})=h(c_{1})$

$f(x)\otimes g(b_{1})=h(c)$

Thus $f(x)\otimes f(z)^{-1}\otimes f(y)\otimes g(b)=f(x)\otimes
g(b_{1})=h(c)=f(a)\otimes g(b)$ and we get our claim.

Now define $\phi:U^{-1}\oplus a\to(U^{\prime})^{-1}\otimes a^{\prime}$ as
$\phi(x^{-1}\oplus a)=(f(x))^{-1}\otimes a^{\prime}$ we claim that it is a
local isomorphism between $U_{1}:=U^{-1}\oplus a$ and
$U_{1}^{\prime}:=(U^{\prime})^{-1}\otimes a^{\prime}$. For this let $u,v\in
U_{1}$ such that $u\oplus v\in U_{1}$ we want to prove that $\phi(u\oplus
v)=\phi(u)\otimes\phi(v)$. Suppose $u=x^{-1}\oplus a$ and $v=y^{-1}\oplus a$
with $x,y\in U$. Therefore $u\oplus v=z^{-1}\oplus a$ where
$z^{-1}=x^{-1}\oplus a\oplus y^{-1}$ with $z\in U$. So using the claim just
proved, we have that $f(z)^{-1}=f(x)^{-1}\otimes f(a)\otimes f(y)^{-1}$ and
then $\phi(z^{-1}\oplus a)=f(z)^{-1}\otimes f(a)=f(x)^{-1}\otimes f(a)\otimes
f(y)^{-1}\otimes f(a)=\phi(x^{-1}\oplus a)\otimes\phi(y^{-1}\oplus a)$ and we
get the result. ∎

##  Chapter 2 Some facts of definable groups contained in $K$

In this chapter we give some basics on $\mathbb{K}$-definable groups whose
universe is contained in $K$. We also prove the main tool we will use in order
to construct inerpretable fields, namely Theorem 2.2.2.

In the first section present some reductions that will be use in the second
one. In the second section we prove Theorem 2.2.2 that ensures the existence
of an interpretable field under the assumption of some kind of families of
curves.

For the rest of the chapter we fix $(G,\oplus)$ a $\mathbb{K}$-definable group
of dimension one and we assume $G\subseteq K$. Let $e$ be the neutral element
of $G$. We assume that the map $(x,y)\mapsto x\oplus y^{-1}$ is continuous.

######  Definition 2.0.1.

If $X\subseteq G^{2}$ is a $\mathbb{K}$-definable set we say that $X$ is
$G$-affine if it is a boolean combination of cosets of $\mathbb{K}$-definable
subgroups of $G^{2}$.

We fix $X\subseteq G^{2}$ a non affine curve and assume that
$\mathcal{G}=(G,+,X)$ is strongly minimal.

###  2.1 Some reductions

In this section we give some basic definitions and reductions that we use in
Chapter 3 and 4. The main result is Proposition 2.1.7. We would like to treat
$X$ (generically) as the graph of a function using Fact 1.4.17 but in order to
do so we need the following:

######  Lemma 2.1.1.

Let $F(x,y)\in K[x,y]$ be an irreducible polynomial and suppose that $F$ is
not constant. If there are infinitely many points $(a,b)\in K^{2}$ such that

$F(a,b)=F_{y}(a,b)=0,$ (2.1)

then $F(x,y)=G(x,y^{p})$ for some polynomial $G$.

###### Proof.

Note first that $\deg(F_{y})<\deg(F)$. So as $F$ is irreducible, if $F_{y}\neq
0$ we can use Bézout’s Theorem to conclude that the number of common zeros of
$F$ and $F_{y}$ equals $\deg F\cdot\deg F_{y}$. So if $F_{y}$ has infinitely
many common zeros with $F$ it is because $F_{y}=0$.

If char$(K)=0$ and $F_{y}=0$ using Fact 1.4.20 we can see that
$F(x,y)=kx+l=F(x,y^{p})$ with $k,l\in K$.

So assume char $(K)=p>0$. Let $V\subseteq K^{2}$ the set of zeros for $F$.

Assume $F_{y}$ has infinitely many zeros at $V$ so $F_{y}=0$. Write

$F(x,y)=f_{0}(x)+f_{1}(x)y+\ldots+f_{n}(x)y^{n}$

with $f_{i}(x)\in K[x]$ for $i=0,1,\ldots,n$.

$F_{y}(x,y)=f_{1}(x)+2f_{2}(x)y+\ldots+nf_{n}(x)y^{n-1}=0,$

so for any $a$ and any $i=1,\ldots,n$ we have that $if_{i}(a)=0$. If some
$f_{i}$ is different from zero we can take $a\in K$ such that $f_{i}(a)\neq 0$
and $if_{i}(a)=0$ implies that $i=0$ so $i=mp$ for some $m$. Therefore if
$f_{i}$ is different from zero implies $p\mid i$ so $F(x,y)$ is a polynomial
in the variable $y^{p}$. ∎

As an immediate corolary we have:

######  Lemma 2.1.2.

Assume $F(x,y)$ is an irreducible polynomial and let $V$ the set of common
zeros of $F$ then there is some finite subset of $V$, $E$ such that either for
all $a\in V\setminus E$, $F_{x}(a)\neq 0$ or for all $a\in V\setminus E$,
$F_{y}(a)\neq 0$.

###### Proof.

Assume it is not the case, then, both $F_{x}$ and $F_{y}$ have infinitelly
many zeros in $V$. So using Lemma 2.1.1 one can conclude that
$F(x,y)=G(x^{p},y^{p})$ for some polynomial $G$ but then
$F(x,y)=(\tilde{G}(x,y))^{p}$ for some polynomial $\tilde{G}$ and then $F$ is
not irreducible. ∎

######  Lemma 2.1.3.

There is a finite set $F\subseteq X$ such that for any $z=(z_{1},z_{2})\in
X\setminus F$ either there is an open set $U^{\prime}$ with $z_{1}\in
U^{\prime}$ and an analytic function $f:U^{\prime}\to K$ such that
$(x,f(x))\in X$ for all $x\in U^{\prime}$ or there is $U^{\prime\prime}$ open
with $z_{2}\in U^{\prime\prime}$ and an analytic function
$g:U^{\prime\prime}\to K$ such that $(g(x),x)\in X$ for all $x\in
U^{\prime\prime}.$

###### Proof.

Decompose $X=X_{0}\cup\ldots\cup X_{n}$ as in Fact 1.4.11 with $X_{0}$ finite
and each $X_{i}$ in an open set of some Zarizki closed set $C_{i}$ for $i>0$.
By adding the intersection points to $X_{0}$ we can take the union to be
disjoint. As $C_{i}$ is the set of zeros of some irreducible polynomial
$F_{i}$ we can apply Lemma 2.1.2 to each $F_{i}$, either $(F_{i})_{x}$ or
$(F_{i})_{y}$ has just finitely many zeros at $C_{i}$. Call $E_{i}$ this
finite set intersected with $X_{i}$ and put $E=X_{0}\cup E_{1}\cup\ldots\cup
E_{n}$. If $z\in X\setminus F$ there is just one $i$ such that $z\in X_{i}$,
and either $(F_{i})_{x}(z)$ or $(F_{i})_{y}(z)$ is different from zero.
Suppose $(F_{i})_{y}(z)\neq 0$ by 1.4.17 we get an open set $U$ with $z_{1}\in
U$ and an analytic function $f:U^{\prime}\to K$ such that $\\{(x,f(x)):x\in
U^{\prime}\\}\subseteq C_{i}$ as $X_{i}$ is an open subset of $C_{i}$. Replace
$U^{\prime}$ by $U^{\prime}\cap f^{-1}(X_{i})$ if necessary, we get
$\\{(x,f(x)):x\in U^{\prime}\\}\subseteq X_{i}$. This proves the lemma. ∎

######  Definition 2.1.4.

If $W\subseteq K^{2}$ is a one dimensional $\mathbb{K}$-definable set we can
decompose $W=W_{0}\cup W_{1}\cup\ldots\cup W_{n}$ where $W_{0}$ is finite and
for $i\geq 1$, $W_{i}$ is an open subset of some irreducible Zariski closed
set $\tilde{W}_{i}$. If $a\in W_{i}$ we say that $\tilde{W}_{i}$ is a branch
of $W$ at $a$.

For $i>0$ suppose $\tilde{W}_{i}$ is the set of zeros of an irreducible
polynomial $F_{i}$, then

$\begin{array}[]{ll}W^{2}:=&\\{a\in W:a\in W_{i},\ (F_{i})_{y}(a)\neq 0,\
1\leq i\leq n\\},\text{ and }\\\ W^{1}:=&\\{a\in W:a\in W_{i},\
(F_{i})_{x}(a)\neq 0,\ 1\leq i\leq n\\}.\end{array}$

######  Definition 2.1.5.

Let $Z\subseteq K\times K$ be the set of zeros of some irreducible polynomial
$f(x,y)$. If $a\in Z$ we define:

$m_{1}(a,Z)=-f_{y}(a)/f_{x}(a)$

and

$m_{2}(a,Z)=-f_{x}(a)/f_{y}(a).$

We say that $m_{2}(a,Z)$ is the slope of $Z$ at $a$ and $m_{1}(a,Z)$ is the
inverse slope of $Z$ at $a$ when they are both well defined.

######  Lemma 2.1.6.

Let $W\subseteq K\times K$ be a $\mathbb{K}$-definable set whose Zariski
closure has algebraic dimension $1$. If $a=(a_{1},a_{2})\in W^{2}$ there is
$U^{\prime}$ open neighborhood of $a_{1}$ and an analytic function
$f:U^{\prime}\to K$ such that $f(a_{1})=a_{2}$, $\\{(x,f(x)):x\in
U^{\prime}\\}\subseteq W$ and $m_{2}(a,\tilde{W}_{i})=f^{\prime}(a_{1})$.
Where $W_{i}$ is the only component such that $a\in W_{i}$.

###### Proof.

Suppose $\tilde{W}_{i}$ is the zero set of $G_{i}(x,y)$, so
$(G_{i})_{y}(a)\neq 0$ so using Fact 1.4.17 there are open sets $U_{1}$ and
$U_{2}$ with $a\in U_{1}\times U_{2}$; and there is also a function
$f:U_{1}\to U_{2}$ analytic at $a$ and an open set $U^{\prime}\subseteq
U_{1}\times U_{2}$ such that

$\Gamma:=\\{u\in U^{\prime}:G_{i}(u)=0\\}=\\{(x,f(x)):x\in U_{1}\\};$

as $W\cap\tilde{W}_{i}$ is open in $\tilde{W}_{i}$ we can assume
$\Gamma\subseteq W$. Now as $G_{i}(x,f(x))=0$ in a neighborhood of $a_{1}$, if
we derive with respect to $x$ the function $G_{i}(x,f(x))$ we get
$(G_{i})_{x}+(G_{i})_{y}f^{\prime}(x)=0$ so
$f^{\prime}(a_{1})=-(G_{i})_{x}(a)/(G_{i})_{y}(a)=m_{2}(a,\tilde{W}_{i})$. ∎

######  Proposition 2.1.7.

There is a set $Y\subseteq G\times G$ definable in $(G,\oplus,X)$ such that:
The $\dim Y=1$ and for each $(x_{0},y_{0})\in Y$ there is an analytic function
$g$ defined in a neighborhood $U_{0}$ of $x_{0}$ and some natural number $n$
such that $\text{Fr}^{-n}(g(x_{0}))=y_{0}$ and

$\left(z,\text{Fr}^{-n}\left(g(z)\right)\right)\in Y$

for each $z\in U_{0}$. Moreover, for $x_{0}=e$ we can take $n=0$; that is,
there is an analytic function $h$ defined in an open neighborhood $U$ of $e$
such that $(x,h(x))\in Y$ for all $x\in U$.

###### Proof.

Let $F\subseteq X$ be as in Lemma 2.1.3. Let $x=(x_{1},x_{2})\in X\setminus F$
so by the conclusion either $F\setminus X$ or $(F\setminus X)^{-1}$ contains
the graph of a function. Define $Y^{\prime}$ as $X\setminus F$ if it contains
the graph of an analytic function converging in a neighborhood of $x_{1}$ or
as $(X\setminus F)^{-1}$ if it doesn’t. Define $y^{\prime}=(x_{1},x_{2})$ if
$Y^{\prime}=F\setminus X$ and $y^{\prime}=(x_{2},x_{1})$ if
$Y^{\prime}=(F\setminus X)^{-1}$. So if $y^{\prime}=(y_{1},y_{2})$ there is an
analytic function converging in a neighborhood of $y_{1}$ whose graph is
contained in $Y^{\prime}$.

Now let $Y^{\prime\prime}=Y^{\prime}_{y^{\prime}}$, so there is an analytic
function $h$ converging in a neighborhood of $e$ whose graph is contained in
$Y^{\prime\prime}$.

Write $Y^{\prime\prime}=Y^{\prime\prime}_{0}\cup\cdots\cup
Y^{\prime\prime}_{n}$ and let $G_{i}$ be a polynomial whose set of zeros is
the Zariski closure of $Y^{\prime\prime}_{i}$.

If $(G_{i})_{y}(x,y)$ has just finite zeros at $Y^{\prime\prime}_{i}$ then
define $E_{i}$ as this set of zeros.

If $(G_{i})_{y}$ is zero for infinitely many points of $Y^{\prime\prime}_{i}$,
then, by Lemma 2.1.1 $G_{i}(x,y)$ is a function in the variable $y^{p^{n}}$
for some $n$, that is $G_{i}(x,y)=G(x,y^{p^{n}})$ for some polynomial $G$ and
some natural number $n$. Let $k$ be the maximum among those possible $n$ and
let $G$ be the corresponding polynomial. Therefore $G(x,y)$ is not a function
in the variable $y^{p}$ so the derivative $G_{y}$ has just finitely many zeros
in $Y_{i}$. Let $E_{i}$ this finite set. Then for each $(x_{0},y_{0})\in
Y_{i}\setminus E_{i}$ there is an analytic function $g$ defined in some
neighborhood of $x_{0}$ such that $G_{i}(z,g(z))=0$ for each $z$ in that
neighborhood. So if we define $\text{Fr}^{k}:K\to K$ as
$\text{Fr}^{k}(x)=x^{p^{k}}$ and $\text{Fr}^{-k}$ it’s inverse,
$F_{i}(z,\text{Fr}^{-k}(g(z)))=0$ so $Y^{\prime\prime}_{i}$ contains the graph
of $z\mapsto\text{Fr}^{-k}(g(z))$ in some neighborhood of $x_{0}$.

Therefore if we define $E=Y^{\prime\prime}_{0}\cup E_{1}\cup\ldots E_{n}$ then
$Y=Y^{\prime\prime}\setminus E$ satisfies conclusion of proposition.

∎

For the rest of this section we fix the following notation:

Let $Y$ be as provided by Proposition 2.1.7. Decompose $Y=Y_{1}\cup\ldots\cup
Y_{n}$ with $Y_{i}$ some open subset of a Zariski closed set $C_{i}$. And let
$F_{i}$ be an irreducible polynomial whose zero set is $C_{i}$.

######  Definition 2.1.8.

If $a\in Y^{2}$ let $Y^{\prime}(a)=m_{2}(a,C_{i})$ where $C_{i}$ is the only
branch of $Y$ at $a$.

The following is Fact 3.8 of [KR16], although the proof is just to apply usual
$\epsilon-\delta$ formula.

######  Fact 2.1.9.

Let $h$ be as in Proposition 2.1.7, then the function $z\mapsto h^{\prime}(z)$
with domain $U$ is definable in $\mathbb{K}$.

######  Definition 2.1.10.

If $V,W$ are subsets of $G^{2}$ and $a=(a_{1},a_{2})\in V$ we define:

  * •

$V\oplus W=\\{(x,y\oplus z):(x,y)\in V\text{ and }(x,z)\in W\\}$

  * •

$V\setminus W=\\{(x,y\cdot z^{-1}):(x,y)\in V\text{ and }(x,z)\in W\\}$

  * •

$V\circ W=\\{(x,y):\exists z(x,z)\in W\wedge(z,y)\in V\\}$

  * •

$V_{a}=\\{(v_{1}\oplus a_{1}^{-1},v_{2}\oplus a_{2}^{-1}):(v_{1},v_{2})\in
V\\}$

For the rest of the Chapter we will make the following assumption on $G$:

######  Assumption 2.1.11.

For each $U\subseteq K$ open set with $e\in U\subseteq G$ and each pair of
analytic functions $f,g:U\to K$ with $f(U)\cup g(U)\subseteq G$ such that
$f(e)=g(e)=e$, if we define $h(x)=f(x)\cdot g(x)$ then $h$ is analytic and
$h^{\prime}(e)=f^{\prime}(e)+g^{\prime}(e)$. And if we define
$i(x)=(f(x))^{-1}$ (the inverse in $G$) then $i^{\prime}(e)=-f^{\prime}(e)$

Note that if $G=(B,+)$ is a subgroup of $(K,+)$ or
$G=(K\setminus\\{0\\},\cdot)$, then $G$ satisfies Assumption 2.1.11.

######  Lemma 2.1.12.

If $a,b\in Y^{2}$ there is a branch $Z$ of $Y_{a}\oplus Y_{b}$ at $(e,e)$ such
that

$m_{2}((e,e),Z)=Y^{\prime}(a)+Y^{\prime}(b).$

There is also a branch $\hat{Z}$ of $Y_{a}\circ Y_{b}$ at $(e,e)$ such that

$m_{2}((e,e),\hat{Z})=Y^{\prime}(a)Y^{\prime}(b).$

###### Proof.

Write $a=(a_{1},a_{2})$ and $b=(b_{1},b_{2})$. Let $h:U\to K$ be an analytic
function whose graph is contained in $Y$ with $a_{1},b_{1}\in U$. Then the
graph of the function $h_{a}(x):=h(x\oplus a_{1})\oplus a^{-1}_{2}$ is
contained in $Y_{a}$. Similarly the graph of $h_{b}(x):=h(x\oplus b_{1})\oplus
b^{-1}_{2}$ is contained in $Y_{b}$. So the graph of $x\mapsto h_{a}(x)\oplus
h_{b}(x)$ is contained in $Y_{a}\oplus Y_{b}$ and we can take $Z$ as the
Zariski closure of such graph and use Assumption 2.1.11. In the same way, the
graph of $h_{a}\circ h_{b}$ is contained in $Y_{a}\circ Y_{b}$. ∎

We can therefore define:

######  Definition 2.1.13.

For $a,b\in Y^{2}$ let

$(Y_{a}\oplus Y_{b})^{\prime}((e,e))=m_{2}((e,e),Z)$

and

$(Y_{a}\circ Y_{b})^{\prime}((e,e))=m_{2}((e,e),\hat{Z})$

where $Z$ and $\hat{Z}$ are as in Lemma 2.1.12

We will also need a technical lemma:

######  Lemma 2.1.14.

Let $V$ and $W$ be one dimensional $\mathbb{K}$-definable subsets of $G^{2}$.
Suppose moreover that they do not have isolated points, then $V\oplus W$ does
not have isolated points. If, moreover, we assume that $W$ does not contain
infinitely many points in any horizontal line and $V$ does not contain
infinitely many points in any vertical line, then the composition $V\circ W$
does not have isolated points.

###### Proof.

If $(a,b)\in V\oplus W$ there are $z$, $w$ such that $(a,c)\in V$, $(a,d)\in
W$ and $b=c\oplus d$. Take an open sets $B_{1}$ containing $a$ and $B_{2}$
containing $b$. As $\oplus$ is continuous one can find $D$ and $E$ open sets
containing $c$ and $d$ respectively such that $D\oplus E\subseteq B_{2}$.

Given that $(a,c)\in V$ is not isolated there is a polynomial $F$ such that
$F(a,c)=0$ and for all $(a^{\prime},c^{\prime})$ in some neighborhood of
$(a,c)$ if $F(a^{\prime},c^{\prime})=0$ then $(a^{\prime},c^{\prime})\in V$.
If the polynomial $F_{a}(x):=F(a,x)$ is not zero we can use continuity of
roots and get $B$, a neighborhood of $a$ (we can assume $B\subseteq B_{1}$)
such that and for each $a^{\prime}\in B$ there is $c^{\prime}\in D$ with
$F(a^{\prime},c^{\prime})=0$ and $(a^{\prime},c^{\prime})\in V$. In the same
way (by restricting $B$ if necessary) for each $a^{\prime}\in B$ there is
$d^{\prime}\in E$ such that $(a^{\prime},d^{\prime})\in W$ so for any such
$a^{\prime}$ the point $(a^{\prime},c^{\prime}\oplus d^{\prime})\in(V\oplus
W)\cap(B_{1}\times B_{2})$ and then $(a,b)$ is not isolated in $V\oplus W$.

Note that if one of $F(a,\\_)$ is the zero polynomial then $V$ contains
infinitely many points in the vertical line $\\{a\\}\times K$, and so with
$V\oplus W$, so that any open containing $(a,b)$ contains points in such a
line. Similarly for $W$.

We will now prove that $V\circ W$ doesn’t contain any isolated point. Let
$(a,b)\in V\circ W$ so there is $c$ with $(a,c)\in V$ and $(c,b)\in W$. If
$B_{1}$ and $B_{2}$ are open sets containing $a$ and $b$ respectively, as in
the addition one can find $B$ containing $c$ such that for each $c^{\prime}\in
B$ there is $a^{\prime}\in B_{1}$ such that $(a^{\prime},c^{\prime})\in Y_{b}$
(using that the closure of $W$ doesn’t contain any horizontal line) and some
$b^{\prime}\in B_{2}$ such that $(c^{\prime},b^{\prime})\in V$ (using that the
closure of $V$ doesn’t contain any vertical line) so that for each such
$c^{\prime}$ the point $(a^{\prime},b^{\prime})\in(V\circ W)\cap(B_{1}\times
B_{2})$ and $(a,b)$ is not an isolated point of $V\circ W$. ∎

###  2.2 Good families of curves

In this section we define the notion of a good family of curves for
$\mathcal{G}$ and we show how it can be used to interpret a field in
$\mathcal{G}$. This is done in Theorem 2.2.2.

######  Definition 2.2.1.

A good family of curves definable in $\mathcal{G}$ is given by the following
data:

  1. 1.

A $\mathcal{G}$-definable set $Y\subseteq G\times G$ and a
$\mathcal{G}$-definable family of curves $(X_{a})_{a\in Y}$.

  2. 2.

$U\subseteq G$, an open neighborhood of $e$, and an analytic function
$H(x,s):U\times U\to G$.

  3. 3.

An analytic function $h:U\to G$.

Such that:

  1. 1.

For all $s\in U$ if we define $\bar{s}:=(s,h(s))$ then $\bar{s}\in Y$.

  2. 2.

Each $X_{a}$ is a one dimensional subset of $K^{2}$ with no isolated points.

  3. 3.

For all $s,x\in U$, $(x,H(x,s))\in X_{\bar{s}}$.

  4. 4.

$H(e,s)=e$ for all $s\in U$.

  5. 5.

The set of derivatives $\displaystyle\\{H_{x}(e,s):s\in U\\}$ is infinite.

  6. 6.

For each $s\in U$ and $(x,y)\in X_{\bar{s}}$ there is a neighborhood $V_{x}$
of $x$ and a neighborhood $U_{s}\subseteq U$ of $s$, an analytic function
$\Phi(z,\delta)$ defined in $V_{x}\times U_{s}$ and a natural number $n$ such
that for all $(w,\delta)\in V_{x}\times U_{s}$ one has that
$(w,\text{Fr}^{-n}(\Phi(w,\delta)))\in X_{\bar{\delta}}$.

We will dedicate the rest of the section to proving:

######  Theorem 2.2.2.

If there is a good family of curves definable in $\mathcal{G}$ then
$\mathcal{G}$ interprets a field.

So for the rest of this section we fix $Y$, $(X_{a})_{a\in Y}$, $U$, $H$ and
$h$ as in the data of a good family of curves.

######  Lemma 2.2.3.

There exists a ball $B\subseteq K$ with center $t$ contained in
$\\{H_{x}(e,s):s\in U\\}$ and $\mathfrak{a}=(a_{1},a_{2},b_{1},b_{2},b)\in
K^{5}$ with $\dim\mathfrak{a}=5$ such that

$\\{ta_{1},\ t+a_{2},\ tb_{1},\ t+b_{2},\ ta_{1}a_{2},\ t+a_{1}b_{2}+a_{2},\
t+b,\ t+a_{1}b+a_{2},\ t+a_{1}b_{1}b+b_{1}a_{2}+b_{2}\\}\subseteq B$

.

###### Proof.

As $\\{H_{x}(e,s):s\in U\\}$ is infinite and $\mathbb{K}$-definible, it
contains an open ball say $B\subseteq G$ with $0\neq t=H_{x}(e,d)\in B$ for
some $d\in U$. Then $\mathfrak{a}\in K^{5}$ exists because $B$ is a non empty
open and both addition and multiplication are continuous functions.

∎

For $s\in U$ define $Y^{\prime}(s)=H_{x}(e,s)$. So we have:

######  Lemma 2.2.4.

Let $\mathfrak{a}$ as in Lemma 2.2.3, then there are tuples:
$\alpha=(\alpha_{1},\alpha_{2}),\ \beta=(\beta_{1},\beta_{2})$ and
$\gamma=(\gamma_{1},\gamma_{2})$ contained in $U^{2}$ such that:

  * •

$Y^{\prime}(\alpha_{1})=ta_{1}$, $Y^{\prime}(\alpha_{2})=t+a_{2}$,

  * •

$Y^{\prime}(\beta_{1})=tb_{1}$, $Y^{\prime}(\beta_{2})=t+b_{2}$, and

  * •

$Y^{\prime}(\gamma_{1})=ta_{1}b_{1}$,
$Y^{\prime}(\gamma_{2})=t+a_{1}b_{2}+a_{2}$.

There is also a triple $(p,q,r)\in U^{3}$ such that:

  * •

$Y^{\prime}(p)=t+b$,

  * •

$Y^{\prime}(q)=t+a_{1}b+a_{2}$, and

  * •

$Y^{\prime}(r)=t+a_{1}b_{1}b+b_{1}a_{2}+b_{2}$.

###### Proof.

It just follows from the fact that

$\\{ta_{1},\ t+a_{2},\ tb_{1},\ t+b_{2},\ ta_{1}a_{2},\ t+a_{1}b_{2}+a_{2},\
t+b,\ t+a_{1}b+a_{2},\ t+a_{1}b_{1}b+b_{1}a_{2}+b_{2}\\}\subseteq B$

and

$B\subseteq\\{H_{x}(e,s):s\in U\\}.$

∎

For the rest of the section fix $B$,
$\mathfrak{a}=(a_{1},a_{2},b_{1},b_{2},b)$ and
$G_{1}=(\alpha,\beta,\gamma,p,q,r)$ as before.

######  Proposition 2.2.5.

$G_{1}$ is a field configuration for $\mathcal{M}$.

###### Proof.

Using Fact 2.1.9 the $\mathcal{G}$-independence relations follows, so we only
need to prove $\mathcal{G}$-dependence relations.

Let us prove for example that $q\in\text{acl}_{\mathcal{G}}(\alpha,p)$.

We have

$Y^{\prime}(q)=t+a_{1}b+a_{2}=t+t^{-1}Y^{\prime}(\alpha_{1})(Y^{\prime}(p)-t)+Y^{\prime}(\alpha_{2})-t.$

Multiplying by $t$ we get:

$tY^{\prime}(q)=Y^{\prime}(\alpha_{1})Y^{\prime}(p)-tY^{\prime}(\alpha_{1})+tY^{\prime}(\alpha_{2}).$
(2.2)

Let $d\in U$ such that $H_{x}(e,d)=t$. Call $X=X_{(d,h(d))}$ and define the
family of curves

$Z_{\delta}:=(X_{\alpha_{1}}\circ X_{p})\ominus(X\circ
X_{\alpha_{1}})\oplus(X\circ X_{\alpha_{2}})\ominus(X_{\delta}\circ X)$ (2.3)

for $\delta\in Y$.

Here the $-$ is taken as in Definition 2.1.10.

For $\delta\in Y$ define

$Z_{\delta}^{e}=\\{x\in G:(x,e)\in Z_{\delta}\\}$

Since the family $(Z_{\delta})_{\delta\in Y}$ is $\mathcal{G}$-definable with
parameters $\alpha$ and $p$ it is enough to show that

$\\{\delta\in Y:|Z_{\delta}^{e}|\leq|Z_{q}^{e}|\\}$

is finite. As we may assume that Morley degree of $Y$ computed in
$\mathcal{G}$ is $1$, it is enough to show:

######  Claim 2.2.6.

There is an open neighborhood $W$ containing $q$ such that for all $\delta\in
W\setminus\\{q\\}$, $|Z_{\delta}^{e}|>|Z_{q}^{e}|$

For this, list $Z_{q}^{e}=\\{e=x_{1},x_{2},\ldots,x_{n}\\}$

Define the function

$F:U\times U_{2}\to K$ $F(x,s)=H(H(x,p),\alpha_{1})\oplus
H(H(x,\alpha_{1}),d)^{-1}\oplus H(H(x,\alpha_{2}),d)\oplus H(H(x,d),s)^{-1}$

Where $U_{2}\subseteq U$ is some neighborhood of $q_{1}$ contained in $U$ such
that everything is well defined.

If $\delta\in U_{2}$ then the graph of the function $F(\\_,\delta)$ is
contained in $Z_{\delta}$ in a neighborhood of $0$.

Using Equation (2.2) it is easy to see that $F_{x}(e,q_{1})=0.$ Therefore the
function $F(\\_,q_{1})$ has a zero of multiplicity $d\geq 2$ at $x=e$. Apply
Fact 1.4.19 to the function $F$ and get open sets $V$ and $W$ with
$(e,q_{1})\in V\times W\subseteq U\times U_{2}$ such that for all $y\in W$ the
function $F(\\_,y)$ has exactly $d$ roots (counting multiplicities) at $V$.

Note that $F_{x}(e,s)=0$ just for finitely many $s\in U$. That is because

$F(x,s)=T(x)\oplus H(H(x,d),s)^{-1}$

where $T(x)$ is a function depending just on $x$. Therefore

$F_{x}(e,s)=T^{\prime}(e)+H_{x}(H(e,d),s)H_{x}(e,d)=T^{\prime}(e)+H_{x}(e,s)t,$

here we are using that $H(e,d)=e$. Thus if $F_{x}(e,s)$ is zero for infinitely
many $s$ then $H_{x}(e,s)=-T^{\prime}(e)/t$ for all such $s$ and then as the
function $H_{x}(e,s)$ is $\mathbb{K}$ definable, Fact 1.4.10 tells us that
$\\{s:H_{x}(e,s)=0\\}$ contains an open ball and Fact 1.4.20 implies that
$s\mapsto H_{x}(e,s)$ is a constant function on $U$ but this contradicts
clause 5 of the definition of a good family of curves.

Therefore we may assume that for each $s\in W\setminus\\{q_{1}\\}$ the
function $F(\\_,s)$ has a simple zero at $x=e$.

Because of our choice of $V$ and $W$, for each $s\in W$ the function
$F(\\_,s)$ has at least two zeros at $V$ (counting multiplicities) and if
$s\neq q_{1}$, as $F(\\_,s)$ has only simple roots, there are two different
points $c_{1},c_{2}\in V$ such that $F(c_{i},s)=0$ for $i=1,2$. Thus, for any
$\delta\in W\setminus\\{q_{1}\\}$ there are at least two different $x\in V$
such that $x\in Z^{0}_{\bar{\delta}}$.

######  Claim 2.2.7.

For $i>1$ take pairwise disjoint open sets $V_{i}$ containing $x_{i}$.
Remember that $x_{i}$ are the elements of $Z_{q}^{e}$. Then we can find open
sets $W_{i}\subseteq U$ with $(x_{i},q_{1})\in V_{i}\times W_{i}$ such that
for each $s\in W_{i}$ there is at least one $x\in Z_{s}^{0}\cap V_{i}$.

###### Proof.

(Proof of Claim 2.2.7) Call

$P:=(X_{\alpha_{1}}\circ X_{p})\ominus(X\circ X_{\alpha_{1}})\oplus(X\circ
X_{\alpha_{2}}).$

So there is some $y_{i}$ such that

$(x_{i},y_{i})\in P\cap X_{q}\circ X.$

Then there is some $z_{i}$ such that $(x_{i},z_{i})\in X$ and
$(z_{i},y_{i})\in X_{q}$.

We can find am analytic function $\Phi_{1}(x,s)$ and a natural number $n_{1}$
such that

$\left(x,\text{Fr}^{-n_{1}}(\Phi_{1}(x,s))\right)\in X_{s}$

for all $x$ in some neighborhood of $x_{i}$ and all $s$ in a neighborhood of
$e$. In the same way we can find $\Phi_{2}$ and $n_{2}$ such that

$\left(z,\text{Fr}^{-n_{2}}(\Phi_{2}(z,s))\right)\in X_{s}$

for all $z$ in a neighborhood of $z_{i}$ and all $s$ in a neighborhood of $q$.

For $j=1,2$ let $\Psi_{j}=\text{Fr}^{-n_{j}}\circ\Phi_{j}$.

Therefore

$(x,\Psi_{2}(\Psi_{1}(x,d),q))\in X_{q}\circ X$

for all $x$ in some neighborhood of $x_{i}$. Moreover if $\delta$ is close
enough to $q_{1}$ then the graph of

$x\mapsto\Psi_{2}(\Psi_{1}(x,d),\delta)$

is contained in $X_{\delta}\circ X$.

Call

$\Psi(x,s)=\Psi_{2}(\Psi_{1}(x,d),s)$

.

As $(x_{i},y_{i})\in P$ and $P$ has no isolated points there is a polynomial
$G(x,y)$ such that for $(x,y)$ close enough to $(x_{i},y_{i})$ if $G(x,y)=0$
then $(x,y)\in P$.

So we would be done if we are able to prove that for all $\delta$ close enough
to $q_{1}$ there is some $x\in V_{i}$ such that

$G(x,\Psi(x,\delta))=0.$

For this note that

$\Psi(x,z)=\text{Fr}^{-n}(\tilde{\Phi}_{2}(\Phi_{1}(x,d),z)$

where $n=n_{1}+n_{1}$ and $\tilde{\Phi}_{2}$ is a function such that
$\text{Fr}^{-n_{1}}(\Phi_{2}(x,s))=\tilde{\Phi}_{2}(\text{Fr}^{-n_{1}}(x),s)$
for all $(x,s)$ so $\tilde{\Phi}_{2}$ is the function obtained by changing all
the coefficients of $\phi_{2}$ for its $p^{n_{1}}$-power. That is, if

$\Phi_{2}(x,s)=\sum_{I}a_{I}x^{i_{1}}s^{i_{2}},$

then

$\tilde{\Phi}_{2}(x,s)=\sum_{I}a_{I}^{p^{n_{1}}}x^{i_{1}}s^{i_{1}p^{n_{1}}}.$

Let

$\tilde{\Psi}(x,s):=\tilde{\Phi}_{2}(\Phi_{1}(x,e),z)$

an analytic function.

So we want to prove that for $\delta$ close enough to $q_{1}$ the function
$G(x,\text{Fr}^{-n}(\tilde{\Psi}(x,\delta)))$ has a zero at $V_{i}$ but again

$G\left(x,\text{Fr}^{-n}(\tilde{\Psi}(x,z))\right)=\text{Fr}^{-n}\left(\tilde{G}(x,\tilde{\Psi}(x,z))\right)$

where $\tilde{G}$ is obtained obtained from $G$ changing all the coefficients
for it’s $p^{n}$-power. That is, if

$G(x,y)=\sum_{I}a_{I}x^{i_{1}}y^{i_{2}},$

then

$\tilde{G}(x,y)=\sum_{I}a_{I}^{p^{n}}x^{i_{1}p^{n}}y^{i_{2}.}$

Now as $\tilde{G}(x,\tilde{\Psi}(x,z))$ is an analytic function that is zero
at $(x_{i},q_{1})$ we can apply Fact 1.4.19 and find some neighborhood $W_{i}$
of $q_{1}$ such that for all $\delta\in W_{i}$ there is $x\in V_{i}$ with
$\tilde{G}(x,\Psi(x,\delta))=0$ but then

$G\left(x,\text{Fr}^{-n}(\tilde{\Psi}(x,\delta))\right)=\text{Fr}^{-n}\left(\tilde{G}(x,\tilde{\Psi}(x,\delta))\right)=\text{Fr}^{-n}(0)=0.$

End of proof of Claim 2.2.7 ∎

Then if $\delta\in\bigcap W_{i}\setminus\\{q_{1}\\}$, $Z_{\delta}^{0}$
contains at least two zeros at $V$ and at least one at $V_{i}$ for $i>1$ so it
has at least $n+1$ points and since there are infinitely many such $z$’s, we
are done.

∎

##  Chapter 3 Additive Case

In this chapter we prove:

######  Theorem 3.0.1.

Let $(G,+)$ be an infinite $\mathbb{K}$-definable subroup of $(K,+)$ group and
let $X\subseteq G^{2}$ be a non affine $\mathbb{K}$-definable set of dimension
one. Assume that the structure $\mathcal{G}=(G,\oplus,X)$ is strongly minimal,
then $\mathcal{G}$ interprets an algebraically closed field that is definable
isomorphic (in $\mathbb{K}$) to $(K,+,\cdot,0,1)$.

The fact that the field is isomorphic to $K$ follows from Theorem 7.1 of
[HHP22a]. So we will devote the entire chapter to find an interpretable field.

Throughout this chapter we fix $G$ and $X\subseteq G^{2}$ satisfying the
assumptions of Theorem 3.0.1. Without loss of generality we may assume that
$(0,0)\in X$.

Note that in this case Assumption 2.1.11 just says that
$(f+g)^{\prime}(0)=f^{\prime}(0)+g(0)$ and $(-f)^{\prime}(0)=-(f^{\prime}(0))$
for each pair of functions $f$ and $g$ analytic at $0$. Which is clearly true.

Therefore it is enough to provide a good family of curves definable in
$\mathcal{G}$.

Toward that end we need some preliminaries on power series.

###  3.1 Power Series

######  Definition 3.1.1.

Let

$f(x)=\sum_{n\geq 1}b_{n}x^{n}$

be a power series converging in a neighborhood $D(f)$ of $0$. For $a\in D(f)$
define $f_{a}(x)=f(x+a)-f(a)$. Suppose

$f_{a}(x)=\sum_{n\geq 1}b_{n,a}x^{n}$

and define

$\begin{split}s_{n}(f)&:=\\{b_{n,a}:a\in D(f)\\}\\\
N(f)&:=\min\\{n:s_{n}(f)\text{ is infinite}\\}\end{split}$

######  Lemma 3.1.2.

Let $f(x)$ be a function analytic at $0$ such that $f(0)=0$ and
$f^{\prime}(0)\neq 0$. Let $g(x)$ be an analytic inverse for $f$ converging in
some neighborhood of $0$. Then if $N(f)$ is finite, $N(g)\geq N(f)$.

###### Proof.

The existence of $g$ it is just Fact 1.4.18.

The inequality $N(g)\geq N(f)$ is a consequence of the usual Lagrange
inversion formula but we present a proof anyway. We will prove that $c_{n}$
depends only on $b_{1},b_{2},\ldots,b_{n}$ by induction on $n$. For $n=1$ it
is easy to see that $c_{1}=1/b_{1}$. Assume therefore that $c_{n}$ depends
only on $b_{1},\ldots,b_{n}$ and we will prove it for $n+1$. As we have that

$x=f(g(x))=\sum_{i}b_{i}\left(\sum_{j}c_{j}x^{j}\right)^{i}$

the coefficient of $x^{n+1}$ in the right side of equality has to be $0$. But
this coefficient is

$b_{1}c_{n+1}+b_{2}d_{2}+\ldots+b_{n}d_{n}+b_{n+1}c_{1}^{n+1},$

where $d_{2},\ldots,d_{n}$ are some polynomials quantities depending only on
$c_{1},\ldots,c_{n}$ that on it’s turn, by induction hypothesis, depend only
on $b_{1},\ldots,b_{n}$. Therefore

$c_{n+1}=\frac{-b_{2}d_{2}-\ldots-b_{n}d_{n}-b_{n+1}c_{1}^{n}}{b_{1}}$

depends only on $b_{1},\ldots,b_{n},b_{n+1}$.

Thus $c_{n,a}$ depends only on $b_{1,a},\ldots,b_{n,a}$ and if $k\leq N$
$b_{k,a}$ is constant as $a$ varies. Therefore if $n\leq N$ $c_{n,a}$, is
constant as $a$ varies and then $N(g)\geq N$.

∎

######  Corollary 3.1.3.

Assuming notation of Lemma 3.1.2. If $N(f)$ and $N(g)$ are both finite then
$N(f)=N(g)$.

###### Proof.

By Lemma 3.1.2 one has that $N(f)\leq N(g)$ apply same lemma for $g$ and get
that $N(g)\leq N(h)$ where $h$ is the inverse of $g$ but then $h=f$ so
$N(f)=N(g)$. ∎

######  Lemma 3.1.4.

Assume

$f=\sum_{n\geq 1}b_{n}x^{n}$

is an analytic function converging in a neighborhood of $0$. Assume moreover
that $N(f)$ is finite. Let

$l=\min\\{e\in\mathbb{N}:\exists n>1\text{ such that }p\nmid n\wedge
b_{np^{e}}\neq 0\\}.$

Then for all $k<l$ and all $m\geq 1$ such that $p\nmid m$ the coefficient
$b_{mp^{k},a}$ is constant as $a$ varies. Moreover $N(f)=p^{l}$.

###### Proof.

Assume notation of Definition 3.1.1 and let $l$ be as in the statement of the
lemma. It is well defined because of our assumption on $N(f)$: If

$\\{e\in\mathbb{N}:\exists n>1\text{ such that }p\nmid n\wedge b_{np^{e}}\neq
0\\}=\emptyset$

then $f$ is a function in which the degree of all the non zero monomials are
powers of $p$ so $f(x+a)=f(x)+f(a)$ and therefore $s_{n}(f)$ is finite for all
$n$.

We will show that for all $k<l$ and all $m\geq 1$ with $p\nmid m$ the
coefficient $b_{mp^{k},a}$ is constant as $a$ varies. And we will also show
that the coefficient $b_{p^{l},a}$ assumes infinitely many values as $a$
varies.

Let $n>1$ such that $b_{np^{l}}\neq 0$ and $p\nmid n$. Therefore

$f(x)=b_{1}x+b_{p^{2}}x^{p^{2}}+\ldots+b_{p^{l}}x^{p^{l}}+\sum_{i>p^{l}}b_{i}x^{i}.$

We will show first that if $k<l$ and $m\geq 1$ with $p\nmid m$, then the
coefficient $b_{mp^{k},a}$ is constant as $a$ varies. Remember that
$b_{mp^{k},a}$ is the coefficient of $x^{mp^{k}}$ in the expansion of
$f_{a}(x)=f(x+a)-f(a)$. In this expansion $x^{mp^{k}}$ just appears in the
terms of the form $b_{t}(x+a)^{t}$ with $t\geq mp^{k}$. Write $t=up^{s}$ with
$p\nmid u$. Suppose first $u>1$ then if $s<l$, $b_{t}=0$. If $s\geq l>k$ then
$(x+a)^{t}=(x^{p^{s}}+a^{p^{s}})^{u}$. And therefore in the expansion of
$(x+a)^{t}$ all the exponents of $x$ are multiples of $p^{s}$ so non of those
equals $mp^{k}$ (as $s>k$ and $p\nmid m$). We can assume then that $u=1$ but
in this case $(x+a)^{t}=x^{p^{s}}+a^{p^{s}}$ and the only way the term
$x^{mp^{k}}$ appears is $k=s$ and $m=1$. In this case the coefficient of
$x^{mp^{k}}$ in $f_{a}(x)$ is $b_{mp^{k}}$ so it is constant as $a$ varies.

We show now that the coefficient $b_{p^{l},a}$ is not constant as $a$ varies.
As

$f(x)=b_{1}x+b_{p^{2}}x^{p^{2}}+\ldots+b_{p^{l}}x^{p^{l}}+\sum_{i>p^{l}}b_{i}x^{i}$

and $b_{np^{k}}=0$ for all $n>0$ and $k<l$ the function
$\displaystyle\sum_{i>p^{l}}b_{i}x^{i}$ is indeed a function in the variable
$x^{p^{l}}$. So

$f(x)=b_{1}x+\ldots+b_{p^{l}}x^{p^{l}}+\sum_{i>1}b_{ip^{l}}x^{ip^{l}}$

therefore the coefficient of $x^{p^{l}}$ in $f_{a}$ is

$b_{p^{l},a}=b_{p^{l}}+\sum_{i>1}b_{ip^{l}}(a^{p^{l}})^{i}$

but this is a series in the variable $a$ with some non zero coefficients,
using Fact 1.4.20 we conclude that it is not constant as $a$ varies and we
conclude. ∎

###  3.2 Getting a good family of curves on $\mathcal{G}$

In this section we prove:

######  Proposition 3.2.1.

There is a good family of curves (as in Definition 2.2.1) definable in
$\mathcal{G}$.

###### Proof.

Take $Y$, $U$ and $h$ as provided by Proposition 2.1.7 applied to the
structure $\mathcal{K}$. If $s_{1}(h)=\\{h^{\prime}(z):z\in U\\}$ is infinite,
then for $a\in Y$ define $X_{a}:=Y_{a}$ (as in Definition 2.1.10). So we can
take

$H(x,s)=h(x-s)+h(s)$

and if $s\in U$ and $(x,y)\in Y_{\bar{s}}$ by the conclusion of Proposition
2.1.7 there is an analytic function $g$ defined in a neighborhood of $x$ and a
natural number $n$ such that $(w,\text{Fr}^{-n}(g(w)))\in Y$ for all $w$ in
that neighborhood. Now using Lemma 2.1.12, if $\delta$ is close enough to $0$
then

$(w-\delta,\text{Fr}^{-n}(g(w))-h(\delta))\in Y_{\bar{\delta}}$

so we can take

$\Phi(w,\delta)=g(w+\delta)-\text{Fr}^{n}(h(\delta)).$

Now assume that $s_{1}(h)$ is finite. Therefore there is an open neighborhood
$U$ containing $0$ such that $h^{\prime}(w)$ is constant in $U$. Using Fact
1.4.20 we have that $h(w)=Kw+f(w)$ where $K$ is some constant and $f(w)$ is a
function in the variable $w^{p^{n}}$ for some $n>0$. Take $n$ maximal with
that property. So $h(w)=Kw+F(w^{p^{n}})$ for some analytic function $F$. For
each $a\in Y$ define

$X_{a}:=(Y-Y_{a})\circ(Y-Y_{c})^{-1}$ (3.1)

where $c=(c_{1},h(c_{1}))$ for some fixed $c_{1}\in U$.

######  Claim 3.2.2.

The family $(X_{a})_{a\in Y}$ defined by Equation 3.1 is a good family of
curves for $\mathcal{G}$.

Suppose that

$h(x)=Kx+\sum_{i\geq 1}b_{ip^{n}}x^{ip^{n}}$

with $b_{p^{n}}\neq 0$. Moreover let $m$ such that $N(f)=p^{m}$ (using Lemma
3.1.4). So there is an open neighborhood $U_{0}$ of $0$ such that for all
$a\in U_{0}$, if we define

$h_{a}(x)=h(x+a)-h(a)$

and put

$h_{a}(x)=\sum_{n\geq 1}b_{n,a}x^{n}$

then

$b_{n,a}=b_{n}$

for all $n$ such that $s_{n}(h)$ is finite. Therefore $Y-Y_{a}$ contains the
graph of

$h-h_{a}=\sum_{i\geq 1}d_{i,a}x^{ip^{m}}$

where $d_{i,a}=b_{ip^{m}}-b_{ip^{m},a}$. Note that $\\{d_{1,a}:a\in U\\}$ is
infinite. In the same fashion for all $b_{1}\in U$ if we call
$b=(b_{1},h(b_{1}))$, then $Y-Y_{b}$ contains the graph of

$h-h_{b}=\sum_{i\geq 1}d_{i,b}x^{ip^{m}}$

we can pick $b_{1}$ such that $d_{1,b}\neq 0$.

Now define

$G_{a}(x)=\sum_{i\geq 1}d_{i,a}x^{i}$

so

$(h-h_{a})(x)=G_{a}(x^{p^{m}}).$

As $d_{1,b}\neq 0$ then $G^{\prime}_{b}(0)\neq 0$ so using Fact 1.4.18, exists
an analytic function $G^{-1}_{b}$ defined in some neighborhood of $0$ such
that $G_{b}(G_{b}^{-1}(z))=z$ for all $z$ in that neighborhood.

######  Claim 3.2.3.

The graph of $G_{a}\circ G_{b}^{-1}$ is contained in
$(Y-Y_{\bar{a}})\circ(Y-Y_{b})^{-1}=X_{\bar{a}}$.

###### Proof.

(Proof of Claim 3.2.3)

Let $y=G_{a}(G_{b}^{-1}(x))$ and we will prove that
$(x,y)\in(Y-Y_{\bar{a}})\circ(Y-Y_{b})^{-1}=X_{\bar{a}}$. So we have to show
that there is some $z$ with $(z,x)\in Y-Y_{b}$ and $(z,y)\in Y-Y_{\bar{a}}$.
Let $z$ such that $z^{p^{m}}=G_{b}^{-1}(x)$. Then,

$(h-h_{b})(z)=G_{b}(z^{p^{m}})=G_{b}(G_{b}^{-1}(x))=x$

and as the graph of $h-h_{b}$ is contained in $Y-Y_{b}$ then $(z,x)\in
Y-Y_{b}$. In a similar way

$(h-h_{a})(z)=G_{a}(z^{p^{m}})=G_{a}(G_{b}^{-1}(x))=y$

and as the graph of $h-h_{a}$ is contained in $Y-Y_{\bar{a}}$ then $(z,y)\in
Y-Y_{\bar{a}}$ so $(x,y)\in(Y-Y_{\bar{a}})\circ(Y-Y_{b})^{-1}$

(End of proof of Claim) ∎

Now note that

$G_{b}^{-1}(x)=\frac{1}{d_{1,b}}x+L(x)$

for some analytic function $L$ such that every non zero monomial of $L$ has
degree bigger than $1$.

Therefore

$G_{a}\circ G_{b}^{-1}(x)=\frac{d_{1,a}}{d_{1,b}}x+\sum_{i>1}e_{i,a}x^{i}$

For some coefficients $e_{i,a}$. It is easy to see that each $e_{i,a}$ is a
power series in the variable $a$, and the same is true for $d_{1,a}$.
Therefore there is a power series $H(x,a)$ such that for all $a$ in a
neighborhood of $0$ one has that $H(x,a)=(G_{a}\circ G_{b}^{-1})(x)$ so

$H_{x}(0,a)=(G_{a}\circ G_{b}^{-1})^{\prime}(0)=\frac{d_{1,a}}{d_{1,b}}$

that takes infinitely many values as $a$ varies.

So we have already proved clauses 1-5 in the definition of good family of
curves and only 6 is missing. For this let $s\in U$ and let
$(x,y)\in(Y-Y_{\bar{s}})\circ(Y-Y_{b})^{-1}$ so there is some $z$ such that
$(z,x)\in Y-Y_{b}$ and $(z,y)\in Y-Y_{\bar{s}}$. There are $y_{1},y_{2}$ such
that $(z,y_{1})\in Y$, $(z,y_{2})\in Y_{\bar{s}}$ and $y=y_{1}-y_{2}$. Again
Proposition 2.1.7 there are analytic functions $g_{1}$, $g_{2}$ defined in
some neighborhoods $U_{1}$ and $U_{2}$ of $0$, and natural numbers $n_{1}$,
$n_{2}$

$\text{Fr}^{-n_{1}}(g_{1}(z))=y_{1}$, $\text{Fr}^{-n_{2}}(g_{2}(z))=y_{2}$,

and the graph of $\text{Fr}^{-n_{1}}\circ g_{1}$ and $\text{Fr}^{-n_{2}}\circ
g_{2}$ are contained in $Y$ and $Y_{\bar{s}}$ respectively.

Moreover for $\delta$ close enough to $s$, the graph of
$w\mapsto\text{Fr}^{-n_{2}}(g_{2}(w-s+\delta))+h(s)-h(\delta)$ for $w\in
U_{2}$ is contained in $Y_{\bar{\delta}}$. For showing that we have to prove
that if

$y=\text{Fr}^{-n_{2}}(g_{2}(w-s+\delta))+h(s)-h(\delta)$

then $(w,y)\in Y_{\bar{\delta}}$ but this is equivalent to show that
$(w+\delta,y+h(\delta))\in Y$ and again this is the same to show that
$(w+\delta-s,y+h(\delta)-h(s))\in Y_{\bar{s}}$. And as the graph of
$\text{Fr}^{-n_{2}}\circ g_{2}$ is contained in $Y_{\bar{s}}$ it is enough to
prove that

$\text{Fr}^{-n_{2}}(g(w+\delta-s))=y+h(\delta)-h(s)$

but this is precisely the definition of $y$.

Note that
$\text{Fr}^{-n_{2}}(g_{2}(w-s+\delta))+h(s)-h(\delta)=\text{Fr}^{-n_{2}}(G(w,\delta))$
where

$G(w,\delta)=g_{2}(w-s+\delta)+h(s)^{p^{n_{2}}}-h(\delta)^{p^{n_{2}}},$

an analytic function.

For all $(z,x)\in Y-Y_{b}$ there is an analytic function $g$ converging in
some neighborhood of $z$ an a natural number $k$ such that
$\text{Fr}^{-k}(g(z))=x$ and the graph of $\text{Fr}^{-k}\circ g$ is contained
in $Y-Y_{b}$ in some neighborhood of $z$. Moreover we can assume that
$g^{\prime}$ is never zero. So there exists an inverse for $g$, say $f$,
converging in a neighborhood of $x^{p^{k}}$ therefore the graph of
$f\circ\text{Fr}^{k}$ is contained in $(Y-Y_{b})^{-1}$ so, for $\delta$ close
enough to $s$, the graph of

$w\mapsto\text{Fr}^{-n_{1}}(g_{1}(\hat{w}))-\text{Fr}^{-n_{2}}(G(\hat{w},\delta))$

where $\hat{w}=f(\text{Fr}^{k}(w))$ is contained in
$(Y-Y_{\bar{\delta}})\circ(Y-Y_{b})^{-1}$.

Assume $n_{1}\geq n_{2}$ so

$\text{Fr}^{-n_{1}}(g_{1}(\hat{w}))-\text{Fr}^{-n_{2}}(G(\hat{w},\delta))=\text{Fr}^{-n_{1}}\left(g_{1}(\hat{w})-\text{Fr}^{n_{1}-n_{2}}(G(\hat{w},\delta))\right)$

We can take $n=n_{1}$ and

$\Phi(w,\delta)=g_{1}(\hat{w})-\text{Fr}^{n_{1}-n_{2}}(G(\hat{w},\delta))$

in order to get clause 6 of the definition. ∎

By Theorem 2.2.2 and Proposition 3.2.1 we complete the proof of Theorem 3.0.1

##  Chapter 4 Multiplicative Case

In this chapter we call $M=K\setminus\\{0\\}$ the universe of the
multiplicative group of $K$ and $(\cdot)$ denotes the multiplication on $M$.

######  Definition 4.0.1.

Let $(H,\otimes)$ be a one dimensional group interpretable in $\mathbb{K}$. We
say that $H$ is locally isomorphic to $(M,\cdot)$ if there is $U\subseteq M$,
some open set containing $1$, and $i:U\to H$, a $\mathbb{K}$-definable
injection such that $i(1)$ is the neutral element of $H$ and for all $x,y\in
U$ if $x\cdot y\in U$ then $i(x\cdot y)=i(x)\otimes i(y)$.

The main result of the chapter is:

######  Theorem 4.0.2.

Let $(H,\otimes)$ be a $\mathbb{K}$-interpretable group of dimension one.
Suppose that $H$ is locally isomorphic to $(M,\cdot)$. Assume moreover that
$X\subseteq H^{2}$ is a $\mathbb{K}$-inerpretable set that is not a boolean
combination of subgroups of $(H,\otimes)\times(H,\otimes)$, and assume
$\mathcal{H}=(H,\otimes,X)$ is strongly minimal, then $\mathcal{M}$ interprets
a field, moreover such a field is definable isomorphic (on $\mathbb{K}$) to
$K$.

Again the fact that the field is isomorphic to $K$ follows from Theorem 7.1 of
[HHP22a] so we just have to find an interpretable field.

Throughout this chapter fix $H$ and $X\subseteq H^{2}$ as in the hypothesis of
Theorem 4.0.2 and call $\mathcal{H}:=(H,\otimes,X)$. We also fix $U\subseteq
M$ an open set containing $1$ and $i:U\to H$ given by the definition of
locally isomorphic to $(M,\cdot)$

Let $1_{H}$ be the neutral element of $H$ so without loss of generality we
shall assume that $(1_{H},1_{H})\in X$.

For $Y\subseteq H^{n}$ let $Y^{*}\subseteq B^{n}$ be the inverse image of $Y$
via $i$, that is:

$Y^{*}=\\{(y_{1},\ldots,y_{n})\in B^{n}:(i(y_{1}),\ldots,i(y_{n}))\in Y\\}$

and for $y\in H^{n}$ let $y^{*}\in B^{n}$ such that $i(y^{*})=y$.

Therefore $X^{*}$ is a $\mathbb{K}$-definable subset of $K$ of dimension one

By using same definition as in proof of Proposition 2.1.7 there is a
$\mathcal{H}$-definable set $Y\subseteq H^{2}$, an open set $V$ contained in
$U$, $V\ni 1$ and analytic function $h:V\to M$ such that $h(1)=1$ and
$(x,h(x))\in Y^{*}$ for all $x\in V$.

Changing $U$ by $U\cap V$ (and $i$ by $i|_{U\cap V}$) we may assume that
$U=V$.

Assume for a moment that $s_{1}(h):=\\{h_{a}^{\prime}(1):a\in U\\}$ is
infinite. In this case we can get a good family of curves definable in
$\mathcal{M}$ just as in the additive case taking $H(x,s)=h(xs)/h(s)$.
Therefore we can use the same definition as in Theorem 2.2.2 and get a field
interpretable in $\mathcal{H}$. So we may assume that $s_{1}(X)$ is finite and
by shrinking $B$ we may assume:

######  Assumption 4.0.3.

$h^{\prime}_{a}(1)$ is constant as $a$ varies in $B$.

Under Assumption 4.0.3 we will find a group $G$ interpretable in $\mathcal{H}$
that is (locally) isomorphic to $(K,+)$ therefore we use the results of
Chapter 3 to find a field interpretable in $G$ and therefore in $\mathcal{H}$.

In the first section of this chapter we find such a group. In the second
section we use that group for finding a field interpretable in $\mathcal{H}$.

###  4.1 Finding a Group

######  Proposition 4.1.1.

Under Assumption 4.0.3, in any $V$, open neighborhood of $1$, there is a group
tuple of elements of $U$ such that it’s image via $i$ is a configuration
$\mathcal{H}$ that is inter algebraic some group configuration of $(K,+)$.

###### Proof.

Remember that we fixed $Y\subseteq H^{2}$ a one dimensional
$\mathcal{H}$-definable subset, and $h:U\to M$ an analytic function such that
for all $x\in U$, $(x,h(x))\in Y^{*}$. Replacing $B$ by $B\cap U$ we assume
that $B=U$ and replacing $V$ for $V\cap B$ may assume that $V\subseteq B$.

Note that if $a\in U$ then the graph of $h_{a}(x):=h(xa)/h(a)$ is contained in
$(Y^{*})_{(a,h(a))}$ for $x$ in some neighborhood of $1$. For $a\in U$ we will
write $Y^{*}_{a}$ instead of $(Y^{*})_{(a,h(a))}$.

As $h_{a}$ is analytic there is some power series converging in a neighborhood
of $1$ that represents $h_{a}$, say

$h_{a}(x)=1+\sum_{n\geq 1}d_{n}(a)(x-1)^{n}.$ (4.1)

By Assumption 4.0.3 we know that $d_{1}(a)$ is finite as $a$ varies in $U$. So
we may assume it is constant. Let $N$ be the minimum natural number such that
$\\{d_{N}(a):a\in U\\}$ is infinite.

We may assume that for $n<N$ the set $\\{d_{n}(a):a\in U\\}$ is a singleton.

Exchanging $Y$ by $Y_{c}\circ Y_{b}^{-1}$ for $(b,c)$ generic enough we may
assume that for all $a\in U$, $d_{1}(1)=d_{1}(a)=1$ and for $1<n<N$,
$d_{n}(1)=d_{n}(a)=0$. That follows from a straightforward computation of the
coefficients of $h_{a}\circ h_{b}^{-1}$.

Note that if $a$ and $b$ are close enough of $1$ then the function $h_{a}\circ
h_{b}$ is analytic in some neighborhood of $1$. So if the power expansion of
$h_{a}\circ h_{b}$ is given by:

$h_{a}(h_{b}(x))=1+\sum_{n\geq 1}d_{n}(a,b)(x-1)^{n},$ (4.2)

Then we have:

######  Claim 4.1.2.

$d_{N}(a,b)=d_{N}(a)+d_{N}(b)$.

###### Proof.

(Proof of Claim 4.1.2) Note that

$\begin{array}[]{ll}h_{a}(h_{b}(x))&=1+\sum_{n\geq
1}d_{n}(a)\left(1+\sum_{k}b_{k}(b)(x-1)^{k}-1\right)^{n}\\\ &=1+\sum_{n\geq
1}d_{n}(a)\left(\sum_{k}b_{k}(b)(x-1)^{k}\right)^{n}\end{array}$

So the coefficient that multiplies $(x-1)^{N}$ is

$d_{N}(a,b)=d_{1}(a)b_{N}(b)+d_{2}(a)p_{1}(b)+\cdots+d_{N}(a)(b_{1}(b))^{N}$

Where for $1<i<N$ $p_{i}$ is some polynomial on $b$. Given that for each such
$i$ $d_{i}(a)=0$ and $d_{1}(a)=d_{1}(b)=1$ then

$d_{N}(a,b)=d_{N}(a)+d_{N}(b)$

as desired. ∎

Note that for each $n\in\mathbb{N}$ the map from $U$ to $K$ given by $a\mapsto
d_{n}(a)$ is $\mathbb{K}$-definable. So, as $\\{d_{N}(a):a\in U\\}$ is an
infinite $\mathbb{K}$-definable subset of $K$, it contains an open ball, say
$B$ with center $t$. Assume $t=d_{N}(1)$ and call $B_{0}=\\{x-t:x\in B\\}$ a
ball around $0$.

Find $a,b,e\in U$ generic independent (in the sense of $\mathbb{K}$). Then
$d_{N}(a)-t$ and $d_{N}(b)-t$ are in $B_{0}$, their sum is in $B_{0}$ so
$t+(d_{N}(a)-t)+(d_{N}(b)-t)=d_{N}(a)+d_{N}(b)-t\in B$ so there is some $c\in
U$ such that $d_{N}(c)=d_{N}(a)+d_{N}(b)-t$. Fix such $c$.

In the same way there is $f\in U$ such that $d_{N}(f)=d_{N}(a)+d_{N}(e)-t$.
And there is also $g$ such that
$d_{N}(g)=t+(d_{N}(a)-t)+(d_{N}(b)-t)+(d_{N}(c)-t)$.

Let
$G_{2}=(d_{N}(a)-t,d_{N}(b)-t,d_{N}(c)-t,d_{N}(e)-t,d_{N}(f)-t,d_{N}(g)-t)$.
It is a group configuration for $(K,+)$ in $\mathbb{K}$. Define
$G_{1}=(i(a),i(b),i(c),i(e),i(f),i(g))$. Note that $G_{1}$ is interdefinable
with $G_{2}$ (in $\mathbb{K}$) just because the map $z\mapsto
d_{N}(i^{-1}(z))-t$ is $\mathbb{K}$ definable.

######  Claim 4.1.3.

$G_{1}=(i(a),i(b),i(c),i(e),i(f),i(g))$ is a group configuration for
$\mathcal{H}$.

###### Proof.

(Proof of Claim 4.1.3)

Let us see for example that
$i(c)\in\operatorname{acl}_{\mathcal{H}}(i(a),i(b))$. Let $Z=Y_{i(a)}\circ
Y_{i(b)}\circ Y^{-1}$. This is a $\mathcal{H}$-definable set with parameters
$(i(a),i(b))$.

We will prove that if we define

$E_{Z}:=\\{e\in Y:|Y_{e}\cap Z|>|Y_{i(c)}\cap Z|\\},$

then it is an infinite set.

For this we will prove first that

$F_{Z}:=\\{e\in Y^{*}:|Y^{*}_{e}\cap Z^{*}|>|Y^{*}_{c}\cap Z^{*}|\\}$

is an infinite set.

For proving that list

$Y^{*}_{c}\cap Z^{*}=\\{x_{1}=(1,1),x_{2},\ldots,x_{n}\\},$

and for $i=1,\ldots,n$ take $W_{i}=W^{\prime}_{i}\times
W^{\prime\prime}_{i}\subseteq U\times U$ pairwise disjoint open sets with
$x^{*}_{i}\in W^{*}_{i}$. We will prove that there is an open subset
$V\subseteq U$ containing $c$ such that for all $e\in V\setminus\\{c\\}$ the
curve $Y^{*}_{e}$ intersects $Z^{*}$ at two distinct points in $W_{1}$ and at
least one inside of $W_{i}$ for $i>1.$

First note that there are just finitely many $z\in U$ such that
$d_{n}(z)=d_{N}(a)+d_{N}(b)-t$. Otherwise there will be an open set where the
equality holds an then by Fact 1.4.20 $d_{N}(z)$ is constant as $z$ varies in
$U$ and we are assuming that this is not the case. So we may assume that
$d_{N}(z)\neq d_{N}(a)+d_{N}(b)-t$ for $z\in U\setminus\\{c\\}$

Now define the function $H(x,e)=h_{a}\circ h_{b}\circ h^{-1}-h^{-1}_{e}(x)$
that is analytic in some neighborhood of $(1,c)$. As the function $H(\\_,c)$
has a zero of multiplicity $N$ at $x=1$. Using Fact 1.4.19 we find some open
sets $V_{1}\subseteq W^{\prime}_{1}$ and $U_{1}\subseteq U$ with $(1,c)\in
V_{1}\times U_{1}$ such that for each $e\in U_{1}$ the function $x\mapsto
H(x,e)$ has $N$ zeros (counting multiplicities) in $V_{1}$. Note that the
function $x\mapsto H(x,e)$ has a zero of order $N$ at $x=1$ just for finitely
many $e$, otherwise by 1.4.20 $d_{N}(a)$ should be constant in some open and
we are assuming that it is not the case. So, by shrinking $U$ we may assume
that if $e\neq c$ there is no zero of multiplicity $N$ at $U$. Therefore there
are at least two different zeros at $V_{1}$. That is, there is two different
points $x,x^{\prime}\in W^{\prime}_{1}$ such that $y:=h_{a}\circ h_{b}\circ
h^{-1}(x)=h_{c}(x)$. By shrinking $W^{\prime}_{1}$ one may assume that $y\in
W^{\prime\prime}_{1}$. So as the graph of $h_{a}\circ h_{b}\circ h^{-1}$ is
contained in $Z^{*}$ and the graph of $h_{c}$ is contained in $Y^{*}_{c}$ one
has that $(x,y)\in Z^{*}\cap Y^{*}_{e}\cap W_{1}$. The same is true for
$x^{\prime}$ so for each $e\in U_{1}$ we can find two different points in
$Z^{*}\cap Y^{*}_{e}\cap W_{1}$.

The same argument can be applied for finding each $i>1$, exists an open
neighborhood of $c$, say $U_{i}$, such that for all $e\in U_{i}$, $|Z^{*}\cap
Y^{*}_{e}\cap W_{i}|\geq 1$. Then for each $e\in V:=U_{1}\cap
U_{2}\cap\ldots\cap U_{n}$ there is at least $n+1$ different points at
$Z^{*}\cap Y^{*}_{e}$.

Now if

$Y^{*}_{c}\cap Z^{*}=\\{x_{1}=(1,1),x_{2},\ldots,x_{n}\\},$

then $i(x_{i})$ are disctint elements of $Y_{c}\cap Z$ So we can list

$Y_{i(c)}\cap
Z=\\{y_{1}=(1,1),y_{2}=i(x_{2}),\ldots,y_{n}=i(x_{n}),z_{1},\ldots z_{k}\\}.$

Where each $z_{j}\in H\times H\setminus i(U)\times i(U)$. Let
$z_{j}=(s_{j},t_{j})$ then if we define $U_{j}=s_{j}i(U)$ it is an infinite
set contining $s_{j}$ and $m_{j}\circ i:U\to U_{j}$ is injective, where
$m_{j}:G\to G$ is multiplication by $s_{j}$. By shrinking $U$ we may assume
that the $V_{j}$ are pairwise disjoint.

We can apply the same argument as before but changing the function $i$ by
$m_{j}\circ i$ for finding an open $V_{j}$ set around $c$ such that for each
$e$ in such a neighborhood there is at least one point at $Y_{e}\cap
Z\cap(U_{j}\times U_{j})$. Therefore for each $e\in V\cap V_{1}\cap\ldots\cap
V_{n}$ the curve $Y_{(}i(e))\cap Z$ has at least $n+1$ points at $i(U)\times
i(U)$ (just the image of the elements in $Z^{*}\cap Y_{e}^{*}$) and at least
one at $U_{i}$ for each $U_{i}$ so in total has at least $n+k+1$ intersection
points.

Therefore $E_{Z}$ is infinite and as $X$ is strongly minimal in $\mathcal{H}$
we are done.

(End of the proof of Claim 4.1.3) ∎

Therefore we use Theorem 1.3.4 and conclude that there is a group definable in
$\mathcal{H}$ and there is a group configuration for $\mathcal{H}$ that it is
inter definable with a configuration for de additive group $(K,+)$.

∎

So there is a group interpretable in $\mathcal{M}$ whose group configuration
in interalgebraic (in $\mathbb{K}$) with the group configuration of $(K,+)$.

###  4.2 Finding a Field

In this section we prove Theorem 4.0.2.

First we need a Fact from [HHP22b] (Remark 7.8)

######  Fact 4.2.1.

Let $H=F/\sim$ be a $\mathbb{K}$-interpretable group with $F\subseteq K^{m}$
$\mathbb{K}$-definable. Assume that each $\sim$-equivalence class is finite.
Then, after quotient $H$ by a finite normal subgroup, there is a
$\mathbb{K}$-definable injection $f:H^{\prime}\to K^{d}$ for some $d$, where
$H^{\prime}$ is an infinite and definable subset of $H$.

###### Proof.

(Proof of Theorem 4.0.2)

As we said, we may assume 4.0.3. Let $(G,\oplus)$ an interpretable group as
provided by Proposition 4.1.1. Let $e$ denote its neutral element. We may
assume that $G$ is strongly minimal as a interpretable set in $\mathcal{H}$.

Let $L\subseteq G$ and $f:L\to K^{d}$ as provided by Fact 4.2.1, by taking
translations one may assume that $e\in L$ an then group operation on $G$ can
be used for defining some (partial) operation on $f(L)$ moreover we can find
$a,b,c$ generic independent (in the sense of $\operatorname{acl}$) elements of
$G$ inside of $L$ and then $f(a),f(b),f(c)$ are generic independent elements
of $f(L)$.

Thus one can use Theorem 1.5.1 and find $U_{1}$, an open subset of $f(L)$
containing $f(e)$, $U_{2}$ a neighborhood of $0$ in $K$ and $\psi:U_{2}\to
U_{1}$ a $\mathbb{K}$-definable bijection with analyitic inverse. Let
$\phi=f^{-1}\circ\psi$ a $\mathbb{K}$-definable function, note that for
$x,y\in U_{2}$ if $x+y\in U_{2}$ then $\phi(x+y)=\phi(z)\oplus\phi(y)$. As
$U_{2}$ is open it contains an open ball around $0$. Replace $U_{2}$ for such
a ball and then one can assume $U_{2}$ is a subgroup of $(K,+)$, so

$H:=f^{-1}(U_{1})$

is a subgroup of $G$ isomorphic to $(U_{2},+)$.

So $\phi:U_{2}\to G$ is a $\mathbb{K}$ definable injective group homomorphism.

It is enough then if we prove the next general statement:

######  Proposition 4.2.2.

If $G$ is a $\mathbb{K}$-interpretable group and $\phi:U_{2}\to G$ is a
$\mathbb{K}$-interpretable map such $\phi(x+y)=\phi(x)\oplus\phi(y)$ for each
$x,y\in U_{2}$, then if $X\subseteq G\times G$ is a $\mathbb{K}$-definable set
of dimension one that is not $G$-affine, then the structure
$\mathcal{G}=(G,\oplus,X)$ interprets a field

###### Proof.

(Proof of Proposition 4.2.2)

For $Y\subseteq G\times G$ define

$Y^{*}:=\phi^{-1}(Y\cap H\times H)\subseteq U_{2}\times U_{2}$

and for $a\in H\times H$ define

$a^{*}:=\phi^{-1}(a)$

As $G$ with the induced structure is not locally modular there is some
$Y\subseteq G^{2}$ $\mathcal{M}$-definable that is not $G$-affine.

In this case $Y^{*}$ is a curve on $U_{2}\times U_{2}$ that is not
$(U_{2},+)$-affine.

Considerate the $\mathcal{G}$-definable family of curves

$X_{a}:=(Y\ominus Y_{a})\circ(Y\ominus Y_{c})^{-1},$

for $c\in H\times H\cap Y$ fixed and $a\in Y$.

Therefore if for $a\in Y^{*}$ we define,

$X^{*}_{a}=(Y^{*}-Y^{*}_{a})\circ(Y^{*}-Y^{*}_{c^{*}})^{-1}.$

Where $c^{*}$ is some fixed element of $Y$, we can apply Claim 3.2.2 to the
structure $\mathcal{U}$ and conclude that $(X^{*}_{a})_{a\in Y^{*}}$ is a good
family of curves for $\mathcal{U}$.

Let $U\subseteq U_{2}$ and $h:U\to U_{2}$ as given by the data of a good
family of curves. Let be

$\mathfrak{a}=(a_{1},a_{2},b_{1},b_{2},b)\in K^{5}$

as given by Lemma 2.2.3 and let

$G_{1}=(\alpha,\beta,\gamma,p,q,r)$

be as given by Lemma 2.2.4.

So if $\alpha=(\alpha_{1},\alpha_{2})$ then $\alpha_{1}\in U$ so there is some
$\hat{\alpha}_{1}\in H\times H$ such that
$(\hat{\alpha}_{1})^{*}=(\alpha_{1},h(\alpha_{1}))$ define
$\hat{\alpha}=(\hat{\alpha}_{1},\hat{\alpha}_{2})$ and the same for $\beta$,
$\gamma$ and also for $p$, $q$ and $r$.

We claim that

$\hat{G}_{1}:=(\hat{\alpha},\hat{\beta},\hat{\gamma},\hat{p},\hat{q},\hat{r})$

is a field configuration for $\mathcal{G}$.

Again we only have to prove the $\mathcal{G}$-dependence relations. So let’s
prove for example that
$\hat{q}\in\operatorname{acl}_{\mathcal{G}}(\hat{\alpha},\hat{p})$

For $\delta\in Y$ define

$Z_{\delta}:=(X_{\hat{\alpha}_{1}}\circ X_{\hat{p}})\ominus(X\circ
X_{\hat{\alpha}_{1}})\oplus(X\circ
X_{\hat{\alpha}_{2}})\ominus(X_{\delta}\circ X)$ (4.3)

It is a $\mathcal{G}$ definable family of subsets of $G\times G$.

For $\delta\in Y$ define

$Z_{\delta}^{e}=\\{x\in G:(x,e)\in Z_{\delta}\\}$

Since the family $(Z_{\delta})_{\delta\in Y}$ is $\mathcal{G}$-definable with
parameters $\alpha$ and $p$ it is enough to show that

$\\{\delta\in Y:|Z_{\delta}^{e}|\leq|Z_{q}^{e}|\\}$

is finite. As we may assume that Morley degree of $Y$ computed in
$\mathcal{G}$ is $1$, it is enough to show that

$\\{\delta\in Y:|Z_{\delta}^{0}|>|Z_{q}^{0}|\\}$

is infinite.

Note that for $\delta\in Y^{*}$,

$Z^{*}_{\delta}=(X^{*}_{\alpha_{1}}\circ X^{*}_{p})-(X^{*}\circ
X^{*}_{\alpha_{1}})+(X^{*}\circ X^{*}_{\alpha_{2}})-(X^{*}_{\delta}\circ
X^{*})$

We proceed to prove that there is an open subset of $q$ such that for all
$\delta$ in that open one has that

$|Z_{\hat{\delta}}^{e}|>|Z_{\hat{q}}^{e}|$

For this list
$Z^{e}_{\hat{q}}=\\{(e,e)=x_{1},\ldots,x_{l},x_{l+1},\ldots,x_{n}\\}$ in such
a way that $x_{1},\ldots,x_{l}\in H\times H$ and $x_{l+1},\ldots,x_{n}\in
G\times G\setminus H\times H$.

By Claim 2.2.6 there is an open subset $W\ni q$ such that for all $\delta\in
W\setminus\\{q\\}$ one has that

$|(Z^{*}_{\delta})^{e}|>|(Z^{*}_{q})^{e}|.$

Moreover

$|(Z^{*}_{\delta})^{e}|=|Z_{\hat{\delta}}^{e}\cap H\times H|.$

and the same it is true for $Z_{q}$.

So for all $\delta\in W$

$|Z_{\hat{\delta}}^{e}\cap H\times H|>|Z_{\hat{q}}^{e}\cap H\times H|$

Moreover just as in Claim 2.2.7 we have:

For all $i$ with $l<i\leq n$ if $V_{i}$ is an open set containing $x_{i}$,
there is $W_{i}$ a neighborhood of $q$ such that if $\delta\in
W_{i}\setminus\\{q\\}$ then $|Z_{\hat{\delta}}^{e}\cap V_{i}|\geq 1$

So if we take $V_{i}$ disjoint not intersecting $H$ and define

$\bar{W}=W\cap W_{l+1}\cap\ldots\cap W_{n},$

then for all $\delta\in\bar{W}\setminus q$ $Z_{\hat{\delta}}^{e}$ has more
than $l$ points on $H\times H$ and at least $1$ point in $V_{i}$ for $i>l$ so
it has more than $n$ points in total.

So $\hat{G}_{1}$ is a field configuration for $\mathcal{G}$ therefore
$\mathcal{G}$ interprets a field and then.

(End o proof of Proposition 4.2.2) ∎

Therefore as $\mathcal{G}$ is intepretable in $\mathcal{H}$, one has that
$\mathcal{H}$ interprets a field.

∎

##  Chapter 5 General Case

In this chapter we present a strategy for proving the general definable one
dimensional case, that is we present an outline of the prove of:

######  Conjecture 5.0.1.

Let $N$ be a $\mathbb{K}$-definable set of dimension one. Let
$\mathcal{N}=(N,\ldots)$ a strongly minimal, non locally modular structure
over $N$ that is definable in $\mathcal{K}$, then $\mathcal{N}$ interprets a
field.

For this the first step is finding a group, for doing so a procedure very
similar as the one presented in Section 4.1 should works. In this the group
structure of $G$ is just used for finding a family of curves that has no
analytic points and where one can use continuity of roots. One should be able
to find such a family without the group structure just as is done in the proof
of Theorem 4.25 of [HS17].

Moreover as the group configuration is constructed using the coeficients of
the Taylor expansion of the curves, there are two cases:

Either the first derivative is infinite (as one varies in the family) and in
this case the group configuration is interalgebraic with some group
configuration of $(K,+)$. Or the first derivative is finite and there is some
coefficient of the taylor expansion that is infinite.

Then by Theorem 1.5.1 either $G$ is locally isomorphic to $(K,+)$ or it is
locally isomorphic to $(M,\cdot)$. In the first case we use Proposition 4.2.2
and in the second we use Theorem 4.0.2.

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