For ease of notation, we often denote a weak Taylor groupoid by the underlying
groupoid $\mathcal{C}$, omitting the additional data.

######  Definition 10.4.

Let $\mathcal{C}$ be a weak Taylor groupoid, and let $f:x\rightarrow y$ be a
morphism in $\mathcal{C}$. For each $n\geq 1$, we define the $n$th truncation
of $f$, dented $T_{n}(f)$, to be the tuple $(x,y,a(f),c_{1}(f),...,c_{n}(f))$.

######  Definition 10.5.

Let $\mathcal{C}$ be a weak Taylor groupoid. Then $\mathcal{C}$ is a Taylor
groupoid if the following hold:

  1. (1)

If $g\circ f=h$ as morphisms in $\mathcal{C}$, then for each $n$, the $n$th
truncation of $h$ is determined by the $n$th truncations of $f$ and $g$.

  2. (2)

If $f$ is a morphism in $\mathcal{C}$, then $f$ is determined by its sequence
of $n$th truncations for $n\geq 1$.

  3. (3)

For fixed $n\geq 1$, every $(x,y,a,c_{1},...,c_{n})\in K^{2}\times G\times
K^{n}$ with $c_{1}\neq 0$ is the $n$th truncation of a morphism in
$\mathcal{C}$.

Taylor groupoids are very close to asserting Taylor’s Theorem about a certain
class of functions in an analytic context: each function is definably assigned
a countable sequence of ‘coefficients’ at each point, which determine the
function in a neighborhood of the point, and these coefficients carry
definable composition and inverse operations which are well-defined on finite
truncations. The main difference, and the reason for the term ‘generalized’,
is the group $G$. One should think of elements of $G$ as powers of the
Frobenius map, which are necessary to cover all the functions we need in
positive characteristic.

Since the various properties required in the two notions are quite similar, it
is not hard to see that a ‘big enough’ Taylor groupoid implies that $(K,v)$
has definable slopes:

######  Lemma 10.6.

Let $\mathcal{C}$ be a Taylor groupoid. Assume that the set of objects of
$\mathcal{C}$ is $K$, and that every basic invertible arc (in the sense of
Definition 7.5) is a morphism in $\mathcal{C}$. Then $(K,v)$ has definable
slopes.

###### Proof.

Definition 10.5(1) implies that for each $n$, multiplication and inverse in
$\mathcal{C}$ induce a groupoid structure on the $n$th truncations of all
morphisms in $\mathcal{C}$. One then defines $n$-slopes as $n$th truncations,
and the axioms of definable slopes follow easily. We point out that the
uniform definability of slopes, and the definability of composition and
inverse of $n$-slopes, follow from the $\emptyset$-ind-definability of
truncation, composition, and inverse in $\mathcal{C}$; while the fact that
$\mathcal{C}$ is the inverse limit of the truncation categories follows from
compactness and Definition 10.5 (2) and (3). ∎

Note that the groupoids and functors constructed in Lemma 10.6 are canonically
determined by $\mathcal{C}$. Thus we define:

######  Definition 10.7.

If $\mathcal{C}$ is a Taylor groupoid as in Lemma 10.6, we will call the
system of groupoids and truncation maps constructed in Lemma 10.6 the
definable slopes on $(K,v)$ induced by $\mathcal{C}$.

###  10.2. Etale Functions at 0

Our goal is to construct increasingly large Taylor groupoids until we reach
one as in Lemma 10.6. We start with étale functions at $(0,0)$.

######  Definition 10.8.

Let $f:0\rightarrow 0$ be a morphism in $\mathcal{LDH}$ (i.e. a germ of a
definable homeomorphism of neighborhoods of 0).

  1. (1)

We say that $f$ is a basic étale function at $(0,0)$ if there is a polynomial
$P(x,y)\in K[x,y]$ such that $P(0,0)=0$, both partial derivatives of $P$ are
non-zero at $(0,0)$, and $f$ is given by the restriction of $P(x,y)=0$ to a
neighborhood of $(0,0)$.

  2. (2)

We say that $f$ is an étale function at $(0,0)$ if it is a composition of
finitely many basic étale functions at $(0,0)$.

It is evident from the definition that étale functions are closed under
composition and inverse, and thus define a $\emptyset$-ind-definable groupoid
with 0 as its only object.

######  Notation 10.9.

We let $\mathcal{EFO}$ (étale functions at the origin) denote the
$\emptyset$-ind-definable groupoid of étale functions at $(0,0)$.

Our first goal is to extend $\mathcal{EFO}$ to a Taylor groupoid, with the
group $G$ interpreted as the trivial group.

######  Lemma 10.10.

Let $f$ be an étale function, and fix $n\geq 1$. Then there is exactly one
polynomial $P(x)\in K[x]$ of degree at most $n$ such that $\lim_{x\rightarrow
0}\frac{P(x)-f(x)}{x^{n}}=0$.

###### Proof.

The lemma can be expressed using infinitely many first order sentences. In
particular, it suffices to carry out the proof in a complete model. So let
$\mathbb{K}$ be a complete model of the same characteristics as $K$. It
follows from the implicit function theorem that every basic étale function at
$(0,0)$ is analytic, i.e. is given by a convergent power series in a
neighborhood of 0. Thus, so is $f$. Write
$f(x)=\sum_{i=1}^{\infty}c_{i}x^{i}$. Then the desired polynomial $P$ is
$\sum_{i=1}^{n}c_{i}x^{i}$. ∎

######  Notation 10.11.

Let $f$ be an étale function at $(0,0)$, and $n\geq 1$. We define the $n$th
coefficient of $f$, denoted $c_{n}(f)$, to be the coefficient of $x^{n}$ in
the polynomial provided in Lemma 10.10.

We now show:

######  Proposition 10.12.

With the maps $c_{n}$ defined above, the groupoid $\mathcal{EFO}$ becomes a
Taylor groupoid.

###### Proof.

It is clear from the statement of Lemma 10.10 that the maps $c_{n}$ are
$\emptyset$-ind-definable, so (interpreting the group $G$ as the trivial
group) we obtain a weak Taylor groupoid. We now verify (1)-(3) in Definition
10.5:

  1. (1)

As in the proof of Lemma 10.10, the statement is first-order, so we can work
in a complete model. Now let $g\circ f=h$ be étale functions at $(0,0)$. Then
the $n$th truncations of $f,g,h$ are just the truncations of their power
series expansions at $(0,0)$. In other words, (1) just says that composition
and inverse are well-defined on truncated polynomials. This is well-known and
easy to check.

  2. (2)

Suppose $f$ and $g$ have the same $n$th truncations for all $n$. It follows
that, if we let $h=f-g$, then $\lim_{x\rightarrow 0}\frac{h(x)}{x^{n}}=0$ for
all $n$. If $h$ is identically 0, we are done. Otherwise, in a small enough
ball, we may assume that all fibers of $h$ have size at most some integer $d$,
and that $h(x)=0$ only holds for $x=0$. Now let $\Gamma$ be the value group,
and for $r\in\Gamma$, define $s(r)$ to be the supremum of all $v(h(x))$ for
$x$ with $v(x)=r$. If $s(r)=\infty$ for some $r$, it follows by the Swiss
Cheese decomposition that the image under $h$ of the elements of value $r$
contains all non-zero points in some open ball at 0. In particular, since $h$
is at most $d$-to-one, there are only finitely many such $r$. So, passing to a
smaller neighborhood of 0, we may assume $s(r)<\infty$ for all $r$. But for
each $n$, since $\lim_{x\rightarrow 0}\frac{h(x)}{x^{n}}=0$, it follows that
for sufficiently large $r\in\Gamma$ we have $s(r)\geq nr$. In particular,
$s:\Gamma\rightarrow\Gamma$ grows faster than any linear function. This
contradicts that $\Gamma$ is a pure divisible ordered abelian group.

  3. (3)

Equivalently, we want to show that every $(c_{1},...,c_{n})\in K^{n}$ with
$c_{1}\neq 0$ is the $n$th truncation of an étale function. This is trivial,
since $(c_{1},...,c_{n})$ is the $n$th truncation of the polynomial
$\sum_{i=1}^{n}c_{i}x^{i}$, which is clearly an étale function since
$c_{1}\neq 0$.

∎

###  10.3. Generalized Etale Functions at 0

We next generalize $\mathcal{EFO}$ to include Frobenius powers.

######  Definition 10.13.

Let $G$ denote the Frobenius group of maps $K\rightarrow K$, defined as
follows: if $K$ has characteristic zero, $G=\\{\mathrm{id}\\}$. If $K$ has
characteristic $p>0$, $G$ is the group of maps $x\mapsto x^{p^{n}}$.

We view $G$ as a $\emptyset$-ind-definable group acting $\emptyset$-ind-
definably on $K$. The following are the key properties we need, which are
evident from the definition:

######  Fact 10.14.

For the group $G$ defined above:

  1. (1)

Every element of $G$ is a $\emptyset$-definable automorphism of $(K,v)$, so in
particular also a homeomorphism.

  2. (2)

If $a\in G$ is an étale function, then $a=\mathrm{id}$.

######  Definition 10.15.

A generalized étale function at $(0,0)$ is a morphism at $(0,0)$ in
$\mathcal{LDH}$ of the form $f\circ a$, where $a\in G$ and $f$ is an étale
function at $(0,0)$.

Since $G$ is $\emptyset$-ind-definable, the class of generalized étale
functions at $(0,0)$ is also $\emptyset$-ind-definable. We want to show that
this class forms a Taylor groupoid in a natural way. Everything will follow
from the ensuing three lemmas:

######  Lemma 10.16.

Each generalized étale function at $(0,0)$ has exactly one expression as
$f\circ a$, where $a\in G$ and $f$ is an étale function at $(0,0)$.

###### Proof.

Assume that $f\circ a=g\circ b$ are two such expressions. Simplifying yields
$g^{-1}\circ f=b\circ a^{-1}$. But $g^{-1}\circ f$ is an étale function at
$(0,0)$, so then $b\circ a^{-1}\in G$ is also an étale function at $(0,0)$. By
Fact 10.14, $b\circ a^{-1}=\mathrm{id}$. Thus $a=b$, and thus $f=g$. ∎

######  Notation 10.17.

Let $a\in G$, and let $f$ be an étale function at $(0,0)$. By $f^{a}$ we mean
the function whose graph is the image of $f$ under the automorphism $a$: that
is, the map sending $a(x)$ to $a(y)$ whenever $f(x)=y$.

######  Lemma 10.18.

Let $f$ be an étale function at $(0,0)$, and let $a\in G$. Then:

  1. (1)

$f^{a}$ is an étale function at $(0,0)$.

  2. (2)

The truncations of $f^{a}$ are the images of the truncations of $f$ under $a$.

  3. (3)

$a\circ f=f^{a}\circ a$.

###### Proof.

Both (1) and (2) follow from the fact that $a$ is an automorphism, since the
class of étale functions at $(0,0)$ and their truncation maps are
$\emptyset$-ind-definable. (3) is immediate from the definition of $f^{a}$. ∎

######  Lemma 10.19.

Let $f$ and $g$ be étale functions at $(0,0)$, and let $a,b\in G$.

  1. (1)

$(g\circ b)\circ(f\circ a)=(g\circ f^{b})\circ(b\circ a)$.

  2. (2)

$(f\circ a)^{-1}=f^{a^{-1}}\circ a^{-1}$.

  3. (3)

In particular, the set of generalized étale functions is closed under
composition and inverse, so forms a $\emptyset$-ind-definable groupoid.

###### Proof.

(1) and (2) follow from Lemma 10.18(3). (3) follows from (1) and (2). ∎

Now let $\mathcal{GEFO}$ be the $\emptyset$-ind-definable groupoid of
generalized étale functions at $(0,0)$. If $f\circ a$ is a morphism in
$\mathcal{GEFO}$, we define its $n$th truncation to be
$(0,0,a,c_{1},...,c_{n})$, where $(0,0,\mathrm{id},c_{1},...,c_{n})$ is the
$n$th truncation of $f$. By Lemma 10.16, this is well-defined.

######  Proposition 10.20.

With the Frobenius group $G$ and the truncation maps defined above,
$\mathcal{GEFO}$ is a Taylor groupoid.

###### Proof.

The $\emptyset$-ind-definability of the truncation maps follows from Lemma
10.16, the $\emptyset$-ind-definability of $G$, and the $\emptyset$-ind-
definability of truncation in $\mathcal{EFO}$.

Next, it follows from Lemma 10.19(1) that composition in $\mathcal{GEFO}$
induces composition in $G$. We have now shown that $\mathcal{GEFO}$ is a weak
Taylor groupoid. We proceed to verify (1)-(3) in the definition of Taylor
groupoids (Definition 10.5:

  1. (1)

It follows from Lemmas 10.19 and 10.18(2) that composition and inverse are
well-defined on truncations.

  2. (2)

Suppose $f\circ a$ and $g\circ b$ are morphisms in $\mathcal{GEFO}$ with the
same $n$th truncations for all $n$. By definition of the truncations in
$\mathcal{GEFO}$, this means that $a=b$ and $f$ and $g$ have the same $n$th
truncations for all $n$. Since $\mathcal{EFO}$ is a Taylor groupoid, this
implies $f=g$.

  3. (3)

For any $a\in G$ and any $c_{1},...,c_{n}\in K$ with $c_{1}\neq 0$, we want to
find a morphism in $\mathcal{GEFO}$ with $n$th truncation
$(0,0,a,c_{1},...,c_{n})$. Let $P(x)=\sum_{i=1}^{n}c_{i}x^{i}$. Then the
desired truncation is achieved by $P\circ a$.

∎

###  10.4. Generalized Étale Functions

Finally, we now extend $\mathcal{GEFO}$ to include maps at arbitrary points in
$K^{2}$, not just the origin.

######  Definition 10.21.

Let $(x_{0},y_{0})\in K^{2}$. A generalized étale function at $(x_{0},y_{0})$
is a morphism at $(x_{0},y_{0})$ in $\mathcal{LDH}$ of them form
$f(x-x_{0})+y_{0}$, where $f$ is a morphism in $\mathcal{GEFO}$. A generalized
étale function is a generalized étale function at some point.

It is clear that each generalized étale function at $(x_{0},y_{0})$ can be
expressed as $f(x-x_{0})+y_{0}$ for a unique morphism $f$ from
$\mathcal{GEFO}$.

######  Notation 10.22.

We denote the generalized étale function $f(x-x_{0})+y_{0}$ by
$f_{x_{0}}^{y_{0}}$.

The following properties are immediate:

######  Lemma 10.23.

Let $f_{x}^{y}$ and $g_{y}^{z}$ be generalized étale functions.

  1. (1)

$g_{y}^{z}\circ f_{x}^{y}=(g\circ f)_{x}^{z}$.

  2. (2)

$(f_{x}^{y})^{-1}=(f^{-1})_{y}^{x}$.

  3. (3)

In particular, the generalized étale functions form the morphisms of a
$\emptyset$-ind-definable groupoid whose set of objects is $K$.

######  Notation 10.24.

Let $\mathcal{GEF}$ be the $\emptyset$-definable groupoid of generalized étale
functions.

We now point out that $\mathcal{GEFO}$ extends canonically to a Taylor
groupoid on $\mathcal{GEF}$.

######  Definition 10.25.

If $f_{x}^{y}$ is a generalized étale function and $n\geq 1$, we define the
$n$th truncation of $f_{x}^{y}$ to be $(x,y,a,c_{1},...,c_{n})$, where
$(0,0,a,c_{1},...,c_{n})$ is the $n$th truncation of $f$.

This time, there is really nothing to check. We leave the details to the
reader, and just state:

######  Proposition 10.26.

With the group $G$ and the truncation maps given above, $\mathcal{GEF}$ forms
a Taylor groupoid.

We conclude:

######  Theorem 10.27.

$(K,v)$ has definable slopes, induced as in Lemma 10.6 by the Taylor groupoid
$\mathcal{GEF}$.

###### Proof.

By Lemma 10.6, we only need to show that every basic invertible arc is a
morphism in $\mathcal{GEF}$. So, let $X\subset K^{2}$ be $A$-definable of
dimension 1, with both projections $X\rightarrow K$ finite-to-one. Let
$(x_{0},y_{0})\in K^{2}$ be generic over $A$. By genericity, we may assume
that $X$ is defined in a neighborhood of $(x_{0},y_{0})$ by an irreducible
polynomial equation $P(x,y)=0$ with coefficients in $\operatorname{acl}(A)$.

Note that the map $x\mapsto x^{p}$ defines a generalized étale function at
every point. Indeed, for any $a$, we have $x^{p}=(x-a)^{p}+a^{p}$, which is
clearly a generalized étale function at $(a,a^{p})$. In particular, we may
freely replace $X$ with its image under applying any Frobenius power to either
coordinate. Since the Frobenius is $\emptyset$-definable, this will send
$(x_{0},y_{0})$ to a generic point of the resulting set; and since the
Frobenius is a homeomorphism, the local behavior of $X$ is preserved.

Now, by the above paragraph, we may assume that $P(x,y)$ does not belong to
either $K[x^{p},y]$ or $K[x,y^{p}]$. Let $V$ be the variety defined by $P$. It
follows that each projection of $V$ to $K$ induces a separable extension of
function fields, which implies that each such projection is generically étale.
Since $(x_{0},y_{0})$ is generic in $X$, each projection $V\rightarrow K$ is
étale at $(x_{0},y_{0})$. In particular, the partial derivatives of $P$ at
$(x_{0},y_{0})$ are both non-zero.

Now, at this point, it is harmless to translate $X$ and assume
$x_{0}=y_{0}=0$. Then since both partial derivatives of $P$ are non-zero at
$(0,0)$, $P$ defines a basic étale function at $(0,0)$. ∎

###  10.5. TIMI in ACVF

It remains to show that with respect to the notion of definable slopes we have
constructed, ACVF satisfies TIMI. We need some preliminaries. First, we note
the following, which is clear:

######  Lemma 10.28.

Let $P(x,y,\overline{z})$ be a polynomial over $K$ in $m+2$ variables, and fix
$(x_{0},y_{0},\overline{z}_{0})$ with $P(x_{0},y_{0},\overline{z}_{0})=0$.
Assume the partial derivatives of $P$ with respect to $x$ and $y$ are non-zero
at $(x_{0},y_{0},\overline{z}_{0})$. Then the set
$\\{(x,y):P(x,y,\overline{z}_{0})=0\\}\subset K^{2}$ restricts to a
generalized étale function $f$ at $(x_{0},y_{0})$, such that $a(f)$ (the
associated element of the Frobenius group $G$) is the identity.

######  Notation 10.29.

Let $P(x,y,\overline{z})$ be a polynomial whose graph near some point
$(x_{0},y_{0},\overline{z}_{0})$ gives a generalized étale function at
$(x_{0},y_{0})$ in the sense of the above lemma. We denote this generalized
étale function by $P\restriction(x_{0},y_{0},\overline{z}_{0})$.

Note that if the partial derivatives are non-zero at
$(x_{0},y_{0},\overline{z}_{0})$ as in Lemma 10.28, then they are non-zero in
a neighborhood of $(x_{0},y_{0},\overline{z}_{0})$. Thus, the notation
$P\restriction(x_{0},y_{0},\overline{z}_{0})$ is well-defined in a
neighborhood of $(x_{0},y_{0},\overline{z}_{0})$.

Now our main tool for proving TIMI is the following:

######  Lemma 10.30.

Let $P(x,y,\overline{z})$ and $Q(x,y,\overline{w})$ be polynomials in
$m_{P}+2$ and $m_{Q}+2$ variables, respectively, and let $I\subset K^{2}\times
K^{m_{P}}\times K^{m_{Q}}$ be the set of $(x,y,\overline{z},\overline{w})$
with $P(x,y,\overline{z})=Q(x,y,\overline{w})=0$. Fix
$(x_{0},y_{0},\overline{z}_{0},\overline{w}_{0})\in I$ so that all partial
derivatives of $P$ and $Q$ in the $x$ and $y$ variables are non-zero at
$(x_{0},y_{0},\overline{z}_{0},\overline{w}_{0})$. Assume for some $n$ that
the $n$th truncations of $P\restriction(x_{0},y_{0},\overline{w}_{0})$ and
$Q\restriction(x_{0},y_{0},\overline{z}_{0})$ coincide, but every neighborhood
of $(x_{0},y_{0},\overline{z}_{0},\overline{w}_{0})$ contains some
$(x,y,\overline{z},\overline{w})$ so that the $n$th truncations of
$P\restriction(x,y,\overline{w})$ and $Q\restriction(x,y,\overline{z})$ do not
coincide. Then the projection $I\rightarrow K^{m_{P}}\times K^{m_{Q}}$ is not
injective in any neighborhood of
$(x_{0},y_{0},\overline{z}_{0},\overline{w}_{0})$.

###### Proof.

The lemma can be expressed by infinitely many first-order sentences, so it
suffices to carry out the proof in a complete model. In this case, by the
Implicit Function Theorem, the graph of $P(x,y,\overline{z})=0$ is, in a
neighborhood of $(x_{0},y_{0},\overline{z}_{0})$, the graph of an analytic
function $(x,\overline{z})\mapsto y_{P}(x,\overline{z})$. Similarly, the graph
of $Q$ is given locally by an analytic function $y_{Q}(x,\overline{w})$. Let
$g$ be the analytic function $(x,\overline{w},\overline{z})\mapsto
y_{P}(x,\overline{z})-y_{Q}(x,\overline{w})$. For given
$\overline{z},\overline{w}$, we let $g_{\overline{z}\overline{w}}$ be the
analytic function in one variable given by restricting $g$ to tuples with the
values $\overline{z}$ and $\overline{w}$ in the $K^{m_{P}}$ and $K^{m_{Q}}$
coordinates.

Now the fact that the $n$th truncations of
$P\restriction(x_{0},y_{0},\overline{w}_{0})$ and
$Q\restriction(x_{0},y_{0},\overline{z}_{0})$ coincide, equivalently stated,
gives that $x_{0}$ is a root of multiplicity $n$ of
$g_{\overline{z}_{0}\overline{w}_{0}}$. By the continuity of roots (this
follows easily, say, from [CW02]), the number of roots of $g_{zw}$ including
multiplicity, in any sufficiently small neighborhood $U$ of
$(\overline{x}_{0},\overline{y}_{0})$, is constant in a neighborhood of
$(\overline{z}_{0},\overline{w}_{0})$. But if
$(x,y,\overline{z},\overline{w})\in I$ is sufficiently close to
$(x_{0},y_{0},\overline{z}_{0},\overline{w}_{0})$ such that the $n$th
truncations do not coincide on $(x,y,\overline{z},\overline{w})$, then $(x,y)$
is a root of multiplicity less than $n$ of $g_{zw}$, implying that $g_{zw}$
has at least one other root in $U$. Such a root corresponds to another point
$(x^{\prime},y^{\prime},\overline{z},\overline{w})\in I$, violating local
injectivity of $I\rightarrow K^{m_{P}}\times K^{m_{Q}}$. ∎

We now conclude:

######  Theorem 10.31.

With the definable slopes induced by $\mathcal{GEF}$, $(K,v)$ satisfies TIMI.

###### Proof.

We are given two $A$-definable families of correspondences in $K$, $X\subset
K^{2}\times T$ and $Y\subset K^{2}\times U$, and generic points over $A$,
$(x_{0},y_{0},t_{0})\in X$ and $(x_{0},y_{0},u_{0})\in Y$. We assume that the
$n$-slopes of $X_{t_{0}}$ and $Y_{u_{0}}$ coincide at $(x_{0},y_{0})$, but
that in some neighborhood of $u_{0}$, if
$u\models\operatorname{tp}(u_{0}/x_{0}y_{0})$ with $u$ independent from
$t_{0}$ over $x_{0}y_{0}$, then the $n$th slopes of $X_{t_{0}}$ and $Y_{u}$ do
not coincide at $(x_{0},y_{0})$. We want to show that
$(x_{0},y_{0},t_{0},u_{0})$ is a topological ramification point of the
projection $I\rightarrow T\times U$, where $I$ is the set of $(x,y,t,u)$ with
$(x,y)\in X_{t}\cap Y_{u}$.

Now, $X_{t_{0}}$ and $Y_{u_{0}}$ define generalized étale functions at
$(x_{0},y_{0})$. It is harmless to apply any Frobenius power to any copy of
$K$, since the Frobenius is $\emptyset$-definable (so doesn’t change which
points are generic), is a homeomorphism (so doesn’t change the topological
ramification locus), and is a generalized étale function at every point (so
doesn’t change the relation of two curves having the same $n$-slope). In
particular, we may assume the Frobenius map associated to the generalized
étale functions $X_{t_{0}}$ and $Y_{u_{0}}$ is the identity.

It is also harmless to restrict to any $A$-definable neighborhood of
$(x_{0},y_{0},t_{0},u_{0})$, and to replace $T$ and $U$ by any sets
$A$-definably homeomorphic to $T$ and $U$. In particular, since $t_{0}\in T$
and $u_{0}\in U$ are generic, we may assume $T$ and $U$ are open subsets of
powers of $K$.

Now by the genericity of $(x_{0},y_{0},t_{0})\in X$ and
$(x_{0},y_{0},u_{0})\in Y$, after restricting to small enough neighborhoods we
can assume $X$ and $Y$ are defined by polynomial equations $P(x,y,t)=0$ and
$P(x,y,u)=0$. Since the generalized étale functions of $X_{t_{0}}$ and
$Y_{u_{0}}$ have no Frobenius power, it follows that the partial derivatives
of $P$ in $Q$ in the $x$ and $y$ variables are non-zero at
$(x_{0},y_{0},t_{0},u_{0})$. It follows that we are in the situation of Lemma
10.30: indeed, by compactness, any neighborhood of $u_{0}$ contains some
$u\models\operatorname{tp}(u/x_{0}y_{0})$ with $u$ independent from $t_{0}$
over $x_{0}y_{0}$ – thus, by assumption, $(x_{0},y_{0},t_{0},u_{0})$ belongs
to the closure of points $(x,y,t,u)\in I$ where the $n$th slopes of $X_{t}$
and $Y_{u}$ do not coincide. ∎

##  11\. Proofs of the Main Theorems

We now prove our main theorems for definable relics of ACVF. For this, we need
to recall the main result of the recent preprint [HOP24] (Fact 11.3 below). We
use some ad hoc terminology to recall their setting most easily.

Recall the notion of almost embeddability (Definition 2.1).

######  Definition 11.1.

Let $(K,v)$ be an $\aleph_{1}$-saturated algebraically closed valued field.
Let $(G,\oplus_{G})$ and $(H,\oplus_{H})$ be interpretable groups in $(K,v)$
over some parameter set $A$. Assume that each of $G$ and $H$ is almost
embeddable into $K$. We say that $G$ and $H$ are locally equivalent if there
are $g_{12},g_{23},g_{13}\in G$ and $h_{12},h_{23},h_{23}\in H$ such that:

  1. (1)

$\dim(g_{12}g_{23}/A)=\dim(h_{12}h_{23}/A)=2$.

  2. (2)

$g_{12}\oplus_{G}g_{23}=g_{13}$ and $h_{12}\oplus_{H}h_{23}=h_{13}$.

  3. (3)

For each $1\leq i<j\leq 3$, $g_{ij}$ and $h_{ij}$ are interalgebraic over $A$.

######  Remark 11.2.

The definition of local equivalence in [HOP24] is more general. It does not
assume almost embeddability into $K$, and uses the more general $dp$-rank
(discussed in the next section) instead of dimension. However, we will only
encounter groups almost embeddable into $K$; and for these groups, the two
notions coincide (see [Sim14, Theorem 0.3], and use [DGL11] to apply it).

######  Fact 11.3.

[HOP24] Let $(K,v)$ be an $\aleph_{1}$-saturated algebraically closed valued
field. Let $(G,\oplus_{G})$ be a $(K,v)$-interpretable group which is almost
embeddable into $K$ and locally equivalent to either $(K,+)$ or
$(K^{\times},\times)$. Let $\mathcal{G}=(G,\oplus_{G},...)$ be a non-locally
modular strongly minimal reduct of the full $(K,v)$-induced structure on $G$.
Then $\mathcal{G}$ interprets a field $F$, $(K,v)$-definably isomorphic to
$K$. Furthermore, the $\mathcal{G}$-induced structure on $F$ is a pure
algebraically closed field.

Let us proceed with our main results. First, Theorem 11.4 is a composite of
our results on ACVF thus far, which is independent of Fact 11.3:

######  Theorem 11.4.

Let $\mathcal{K}=(K,v)$ be an $\aleph_{1}$-saturated algebraically closed
valued field. Let $\mathcal{M}$ be a non-locally modular strongly minimal
definable $\mathcal{K}$-relic. Then $\mathcal{M}$ interprets a strongly
minimal group that is almost embeddable into $K$ and locally equivalent to
either $(K,+)$ or $(K^{\times},\times)$.

###### Proof.

By Corollary 9.60, $(K,v)$ is a Hausdorff geometric structure with the open
mapping property and ramification purity. Moreover, by Theorems 10.27 and
10.31, $(K,v)$ has definable slopes induced by $\mathcal{GEF}$ and satisfying
TIMI. Now apply Theorem 8.1. The result is that $\dim(M)=1$ and $\mathcal{M}$
interprets a strongly minimal group satisfying all of the conclusions of
Proposition 7.33. To recall, this gives us the following (where $n_{0}$ is the
smallest $n$ such that there is a non-algebraic coherent $n$-slope):

  * •

A parameter set $B$,

  * •

An $\mathcal{M}(B)$-interpretable group $(G,\cdot)$ which is strongly minimal
as an interpretable set in $\mathcal{M}$,

  * •

A generic element $y\in K$ over $\emptyset$,

  * •

For each $i,j\in\\{1,2,3\\}$ with $i<j$, an element $g_{ij}\in G$, and

  * •

For each $i,j\in\\{1,2,3\\}$ with $i<j$, a morphism $f_{ij}:y\rightarrow y$ in
$\mathcal{IA}_{n_{0}}$,

such that:

  1. (1)

$y$ is $\mathcal{K}(B)$-definable (that is,
$y\in\operatorname{dcl}_{\mathcal{K}}(B)$).

  2. (2)

The $(n_{0}-1)$-th truncation of each $f_{ij}$ is the identity at $y$.

  3. (3)

$g_{12}$ and $g_{23}$ are independent generics in $G$ over $B$.

  4. (4)

$g_{13}=g_{23}\cdot g_{12}$ and $f_{13}=f_{23}\circ f_{12}$.

  5. (5)

For each $i,j\in\\{1,2,3\\}$ with $i<j$, $g_{ij}$ and $f_{ij}$ are
interalgebraic over $B$.

Our goal is to show that $(G,\cdot)$ is almost embeddable into $K$ and locally
equivalent to either the additive or multiplicative group of $K$. The first
clause is easy: By strong minimality, $G$ is in finite correspondence with
$M$; but $M$ admits a finite-to-one map to $K$ (by using that $\dim(M)=1$ and
taking piecewise projections) – so composing gives a finite correspondence
between $G$ and a subset of $K$. Thus $G$ is almost embeddable into $K$.

Now we move toward local equivalence with the additive or multiplicative
group. Recalling that our definable slopes in ACVF are induced by the groupoid
$\mathcal{GEF}$ from the previous section, we can express each $f_{ij}$ as an
$n_{0}$-truncated morphism $y\rightarrow y$ in $\mathcal{GEF}$ with trivial
$(n_{0}-1)$-th truncation. Recall that such truncations are notated
$(y,y,a,c_{1},...,c_{n_{0}})$, where $a$ is a Frobenius power and
$c_{1},...,c_{n_{0}}$ are the coefficients of an $n_{0}$-truncated polynomial
sending 0 to 0. Since our $(n_{0}-1)$-th truncations are trivial, all but the
last entry in such an expression are determined. Precisely, we have the
following options for $f_{ij}$:

  * •

Case 1: $n_{0}=1$. Then we have

$f_{ij}=(y,y,\textrm{id},h_{ij})$

for some $h_{ij}\in K$.

  * •

Case 2: $n_{0}>1$. Then we have

$f_{ij}=(y,y,\textrm{id},1,0,0,...,0,h_{ij})$

for some $h_{ij}\in K$.

The first case corresponds to scaling maps $x\mapsto h_{ij}x$, and the second
corresponds to maps of the form $x+h_{ij}x^{n_{0}}$. In any case, we can fix
$h_{ij}\in K$ as above for each $i<j$. Since
$y\in\operatorname{dcl}_{\mathcal{K}}(B)$, each $h_{ij}$ is interdefinable
with $f_{ij}$ over $B$, and so is also interalgebraic with $g_{ij}$ over $B$.
One thus easily concludes (1) and (3) from Definition 11.1, interpreting $H$
as either $(K,+)$ or $(K^{\times},\times)$ (and replacing $A$ from Definition
11.1 with our $B$). So to complete the proof, we must show that either
$h_{13}=h_{12}+h_{23}$ or $h_{13}=h_{12}h_{23}$. But $f_{13}=f_{12}\circ
f_{23}$, and composition of truncations in $\mathcal{GEF}$ corresponds to
composition of truncated polynomials – so we just need to check the
composition operation on the truncated polynomials in the two cases above:

  * •

Case 1: $n_{0}=1$. Then $h_{13}x$ is the composition of $h_{12}x$ and
$h_{23}x$, which is $h_{12}h_{23}x$, and thus $h_{13}=h_{12}h_{23}$.

  * •

Case 1: $n_{0}=2$. Then $x+h_{13}x^{n_{0}}$ is the $n_{0}$-truncated
composition of $x+h_{12}x^{n_{0}}$ and $x+h_{23}x^{n_{0}}$. One easily checks
this truncated composition to be $x+(h_{12}+h_{23})^{n_{0}}$, and thus
$h_{13}=h_{12}+h_{23}$.

∎

Now combining our work with Fact 11.3, we obtain the full restricted
trichotomy for definable relics of ACVF:

######  Theorem 11.5.

Let $\mathcal{K}=(K,v)$ be an algebraically closed valued field. Let
$\mathcal{M}$ be a non-locally modular strongly minimal definable
$\mathcal{K}$-relic. Then $\mathcal{M}$ interprets a field $F$,
$(K,v)$-definably isomorphic to $K$. Moreover, the structure induced on $F$
from $\mathcal{M}$ is a pure algebraically closed field.

###### Proof.

First, we check that we can assume $(K,v)$ is $\aleph_{1}$-saturated. This is
a straightforward reduction, and similar arguments are given in [Cas23,
Theorem 9.6] and [HS24, Lemma 4.8]. We give a sketch. Let $(K_{1},v_{1})$ be
an $\aleph_{1}$-saturated elementary extension of $(K,v)$. Then the same
formulas defining $\mathcal{M}$ in $(K,v)$ also define a definable
$(K_{1},v_{1})$-relic, say $\mathcal{M}_{1}$. It is easy to see that
$\mathcal{M}_{1}$ is an elementary extension of $\mathcal{M}$, so is strongly
minimal and not locally modular.

Now suppose $\mathcal{M}_{1}$ interprets a field definably isomorphic to
$K_{1}$, say $F$. Then there are formulas (with parameters from $K_{1}$)
defining the set $F$, the field structure on $F$, the interpretation of $F$ in
$\mathcal{M}_{1}$, and a field isomorphism $F\leftrightarrow K_{1}$. But all
of these properties are first-order, so they can be pulled down to $(K,v)$ and
$\mathcal{M}$. Thus, using different instances of the same formulas, one can
find a definably isomorphic copy of $K$ in $\mathcal{M}$.

So we assume $(K,v)$ is $\aleph_{1}$-saturated. By Theorem 11.4, $\mathcal{M}$
interprets a strongly minimal group $G$ locally equivalent to $(K,+)$ or
$(K^{\times},\times)$. Now apply Fact 11.3 to the structure induced on $G$
from $\mathcal{M}$. ∎

Finally, we deduce a full solution of the restricted trichotomy for pure
algebraically closed fields:

######  Corollary 11.6 (Restricted Trichotomy Conjecture).

Let $K$ be an algebraically closed field, and let $\mathcal{M}$ be a strongly
minimal $K$-relic. If $\mathcal{M}$ is not locally modular, then $\mathcal{M}$
interprets the field $K$.

###### Proof.

By elimination of imaginaries, we may assume $\mathcal{M}$ is a definable
$K$-relic (that is, its universe is a subset of some $K^{n}$). By an identical
argument to Theorem 11.5 above, we may assume $K$ is $\aleph_{1}$-saturated
(really, $\aleph_{0}$ is enough here), and thus there is a non-trivial
valuation on $K$. Let $v$ be such a valuation. Then $\mathcal{M}$ is still
interpreted in $(K,v)$. One can now finish in two ways: either apply Theorem
11.5 directly (thus using [HOP24]); or apply Theorem 8.1 to show $\dim(M)=1$,
and then conclude with the main result of [HS24]. ∎

##  12\. Imaginaries in ACVF

In the final section, we discuss strongly minimal structures interpreted,
rather than defined, in ACVF. That is, we allow the universe of the structure
to be in an imaginary sort. Our main result is a reduction to the real sorts
in residue characteristic zero, which works in residue characteristic $p$
assuming a conjectural condition about the distinguished sort $K/\mathcal{O}$
which we expect to be true.

We follow the strategy used in [HHP22] for classifying interpretable fields in
various valued fields by reducing them to four distinguished sorts (the valued
field, the residue field, the value group, and the set of closed balls of
radius zero). The guiding principle is that because of the ‘richness’ and
‘uniformity’ provided by families of plane curves in non-locally modular
strongly minimal structures, such a structure should be (1) only able to
‘live’ in one of these four sorts, and (2) not able to live in the two ‘non-
field’ sorts. We thus hope to reduce to structures genuinely coming from
either the valued field or the residue field.

###  12.1. The Setting

Throughout this section, $\mathcal{K}=(K,v)$ is an $\aleph_{1}$-saturated
model of ACVF. We let $\Gamma$ denote the value group, $\mathcal{O}$ the
valuation ring, and k the residue field. $\mathcal{M}=(M,...)$ is a non-
locally modular strongly minimal $\mathcal{K}$-relic, whose universe $M$ need
not be definable. Absorbing parameters into $(K,v)$ and $\mathcal{M}$, we
assume that $\operatorname{acl}_{\mathcal{M}}(\emptyset)$ is infinite, and
that every $\emptyset$-definable set in $\mathcal{M}$ is $\emptyset$-definable
in $(K,v)$.

######  Convention 12.1.

Recall (see Definition 2.1) that a finite correspondence between two
interpretable sets $X$ and $Y$ is an interpretable set $C\subset X\times Y$
such that the projections of $C$ to both factors are finite-to-one and
surjective.

###  12.2. Ranks

It was shown in [DGL11] that ACVF is dp-minimal, and it thus follows that any
ACVF relic has finite dp-rank (if the universe of the relic is $X/E$ for some
definable set $X\subseteq K^{n}$ and definable equivalence relation $E$ then
$\operatorname{dprk}(X^{n}/E)<n$). See [HHP22, §2.1] for more relevant details
on dp-rank. We remind the readers that dp-rank (as opposed to dimension) is
sub-additive, but need not be additive:

$\operatorname{dprk}(ab/A)\leq\operatorname{dprk}(a/A)+\operatorname{dprk}(b/Aa).$

We use $\operatorname{dprk}$ for dp-rank in $(K,v)$, and $\operatorname{MR}$
for Morley rank in $\mathcal{M}$. Since $M$ is infinite,
$\operatorname{dprk}(M)\geq 1$.

We will repeatedly use the following:

######  Lemma 12.2.

Let $X\subset M$ be infinite and interpretable over $A$. Let
$a=(a_{1},...,a_{n})\in X^{n}$ with
$\operatorname{dprk}(a/A)=n\cdot\operatorname{dprk}(X)$. Then $a$ is
$\mathcal{M}$-generic in $M^{n}$ over $A$.

###### Proof.

If not, then some
$a_{m}\in\operatorname{acl}_{\mathcal{M}}(Aa_{1}...a_{m-1})$. But then for
that $m$ we have

$\operatorname{dprk}(a_{m}/Aa_{1}...a_{m-1})=0,$

and so sub-additivity implies that

$\operatorname{dprk}(a/A)\leq\sum_{j=1}^{n}\operatorname{dprk}(a_{j}/Aa_{1}...a_{j-1})\leq
0+(n-1)\cdot\operatorname{dprk}(X)<n\cdot\operatorname{dprk}(X),$

a contradiction. ∎

It follows from the main result of [Joh22] that finiteness is definable in
ACVFeq. In such situations, dp-rank is also definable in definable families,
but this is not needed in the sequel.

###  12.3. Local Interalgebraicity

In [HHP22], Halevi, Hasson, and Peterzil classify the interpretable fields in
various valued fields of interest. Their main tool is the notion of an
interpretable set $Y$ being locally almost strongly internal to an
interpretable set $D$: this means that some infinite interpretable subset of
$Y$ admits an interpretable finite-to-one map to $D$. In particular, the
authors showed that every interpretable set is locally almost strongly
internal to one of four distinguished sets (the valued field, the value group,
the residue field, and the set of closed balls of radius zero). They then
classify the fields locally almost strongly internal to each of these four
sorts.

We will use a very similar strategy. However, for our purposes, it is more
convenient to work with finite correspondences instead of finite-to-one maps
(roughly, because without assuming a field, one cannot always generate enough
definable functions). Thus, we will introduce a nearly identical, but slightly
weaker, notion. Note that the reduction to the four distinguished sorts will
remain valid.

######  Definition 12.3.

Given interpretable sets $Y$ and $D$, we define the $D$-critical number of
$Y$, $\operatorname{Crit}_{D}(Y)$, to be the largest dp rank of an
interpretable subset $X\subset Y$ which is in interpretable finite
correspondence with a subset of some $D^{n}$. We moreover call any such
$X\subset Y$ $D$-critical in $Y$. Finally, we say that $Y$ is locally
interalgebraic with $D$ if $\operatorname{Crit}_{D}(Y)\geq 1$.

By compactness, we have:

######  Fact 12.4.

Let $Y$ and $D$ be interpretable, and let $X$ be $D$-critical in $Y$. Let $A$
be a parameter set, let $y\in Y$, and let $d_{1},...,d_{n}\in D$. If $y$ is
interalgebraic with $d_{1},...,d_{n}$ over $A$, then
$\operatorname{dprk}(y/A)\leq\operatorname{dprk}(X)$.

The following is also clear:

######  Fact 12.5.

Suppose $Y$, $Z$, and $D$ are interpretable sets, and $Y$ is in finite
correspondence with a subset of $Z$. Then
$\operatorname{Crit}_{D}(Y)\leq\operatorname{Crit}_{D}(Z)$.

The following key observation is [HHP22, Proposition 5.5], and forms the basis
for our approach:

######  Fact 12.6.

Every infinite interpretable set is locally interalgebraic with at least one
of $K$, k, $\Gamma$, and $K/\mathcal{O}$.

We will need the following stronger result (see [HOP24, Lemma 2.8]):

######  Fact 12.7.

Let $Y$ be an infinite interpretable set which is not locally interalgebraic
with any of k, $\Gamma$, or $K/\mathcal{O}$. Then $Y$ definably embeds into
$K^{n}$ for some $n$.

###  12.4. Richness

Our main goal is to ‘rule out’ the two non-field sorts $\Gamma$ and
$K/\mathcal{O}$, by showing in some sense, that they are not complex enough to
admit a non-locally modular strongly minimal structure. Our tool for showing
this is richness. Roughly, we want to say that a sort $D$ is ‘rich’ if it
admits arbitrarily large families of infinite subsets of a fixed $D^{n}$,
parametrized by tuples from $D$, satisfying an appropriate ‘non-redundancy’
condition.

######  Definition 12.8.

Let $A$ be a parameter set, let $k$ be a non-negative integer, and $a$ and $b$
finite tuples. We call $(a,b,A)$ a $k$-rich configuration if the following
hold:

  1. (1)

$a\notin\operatorname{acl}(Ab)$.

  2. (2)

Whenever $\overline{b}=\\{b_{1},b_{2},...\\}$ is an infinite sequence of
distinct realizations of $\operatorname{tp}_{(K,v)}(b/Aa)$, then
$a\in\operatorname{acl}(A\overline{b})$.

  3. (3)

$\operatorname{dprk}(b/A)\geq k$.

######  Definition 12.9.

An interpretable set $D$ is rich if there is $n$ such that for arbitrarily
large $k$, there are $m$, elements $a\in D^{n}$ and $b\in D^{m}$, and a
parameter set $A$, such that $(a,b,A)$ is a $k$-rich configuration.

######  Example 12.10.

The motivating example of the above notions is given by a family of plane
curves in our strongly minimal relic $\mathcal{M}$. Namely, let $\\{C_{t}:t\in
T\\}$ be an almost faithful $\mathcal{M}(A)$-definable family of plane curves
in $\mathcal{M}$, where $T\subset M^{k}$ has Morley rank $k$. Let $t\in T$
with $\operatorname{dprk}(t/A)=\operatorname{dprk}(T)$, and let $x\in C_{t}$
with $\operatorname{dprk}(x/At)=\operatorname{dprk}(C_{t})$. Then one shows
easily that $(x,t,A)$ is a $k$-rich configuration. In particular, since
$\mathcal{M}$ is not locally modular (and thus $k$ can be arbitrarily large),
it follows that the universe $M$ is rich (where in the notation of Definition
12.9, we use $n=2$).

The notion of a $k$-rich configuration may seem a bit obscure. Its main
benefit to us is that it is preserved under finite correspondences – a useful
feature given the definition of local interalgebraicity:

######  Lemma 12.11.

Assume that $(a,b,A)$ is a $k$-rich configuration, and let $c$ and $d$ be
finite tuples with $\operatorname{acl}(Aa)=\operatorname{acl}(Ac)$ and
$\operatorname{acl}(Ab)=\operatorname{acl}(Ad)$. Then $(c,d,A)$ is a $k$-rich
configuration.

###### Proof.

Interalgebraicity gives automatically that $\operatorname{dprk}(d/A)\geq k$
and $c\notin\operatorname{acl}(Ad)$. Now suppose
$\overline{d}=(d_{1},d_{2},...)$ are infinitely many realizations of
$\operatorname{tp}_{(K,v)}(d/Ac)$. For each $i$, let $(a_{i},b_{i})$ be such
that $\operatorname{tp}(a_{i}b_{i}cd_{i}/A)=\operatorname{tp}(abcd/A)$. Since
$a\in\operatorname{acl}(Ac)$, we may assume after passing to a subsequence
that each $a_{i}=a_{1}$. Since each $d_{i}\in\operatorname{acl}(Ab_{i})$, and
the $d_{i}$ are distinct, we may assume after passing to a further subsequence
that the $b_{i}$ are all distinct (here we are using that all of the tuples
involved are finite). Now since
$\operatorname{tp}(a_{1}b_{1}/A)=\operatorname{tp}(ab/A)$, $(a_{1},b_{1},A)$
is a $k$-rich configuration. It follows that
$a_{1}\in\operatorname{acl}(Ab_{2}b_{3}...)$. But each $b_{i}$ is
interalgebraic with $d_{i}$ over $A$, and $a_{1}$ is interalgebraic with $c$
over $A$. Thus $c\in\operatorname{acl}(Ad_{2},d_{3},...)$. ∎

######  Corollary 12.12.

Let $X$ and $Y$ be interpretable sets. Assume that $X$ is in definable finite
correspondence with a subset of some $Y^{n}$. If $X$ is rich, then so is $Y$.

###  12.5. Main Proposition

We now give the main technical result of this section. This result should be
thought of as saying that if $M$ has any interaction at all with some sort
$D$, then it has a large amount of interaction with $D$.

######  Proposition 12.13.

Suppose $M$ is locally interalgebraic with an interpretable set $D$. Assume
that $X$ is $D$-critical, and that $X$, as well as some finite correspondence
witnessing it, are $A$-definable. Let $\mathcal{C}=\\{C_{t}:t\in T\\}$ be an
$\mathcal{M}(A$)-definable almost faithful family of plane curves with
$T\subset M^{2}$ of Morley rank 2. Let $(u_{0},v_{0})\in M^{2}$ be
$\mathcal{M}$-generic over $A$, and let $(x_{0},y_{0})\in X^{2}$ be such that
$\operatorname{dprk}(x_{0}y_{0}/Au_{0}v_{0})=2\operatorname{dprk}(X)$. Then:

  1. (1)

There is at least one, but only finitely many, $t\in T$ with
$(u_{0},v_{0}),(x_{0},y_{0})\in C_{t}$.

  2. (2)

If $(u_{0},v_{0}),(x_{0},y_{0})\in C_{t}$ for some $t$ then $C_{t}\cap X^{2}$
is infinite.

###### Proof.

It follows from Lemma 12.2 that $(x_{0},y_{0})$ is $\mathcal{M}$-generic in
$M^{2}$ over $Au_{0}v_{0}$, so that $(u_{0},v_{0})$ and $(x_{0},v_{0})$ are
$\mathcal{M}$-independent $\mathcal{M}$-generics in $M^{2}$ over $A$. Since
$M^{2}$ is stationary, this easily implies (1).

Now we show (2). Absorbing parameters, we assume $A=\emptyset$. Let $t$ be
such that $(u_{0},v_{0}),(x_{0},y_{0})\in C_{t}$. By (1),
$t\in\operatorname{acl}(u_{0}v_{0}x_{0}y_{0})$. Now assume toward a
contradiction that $C_{t}\cap X^{2}$ is finite. Then
$(x_{0},y_{0})\in\operatorname{acl}(t)$, so $(x_{0},y_{0})$ is interalgebraic
with $t$ over $(u_{0},v_{0})$. Now $t$ belongs to the set of $t^{\prime}$ with
$(u_{0},v_{0})\in C_{t^{\prime}}$, so it follows that
$\operatorname{MR}(t/u_{0}v_{0})\leq 1$. In particular, by strong minimality,
$t$ is interalgebraic with an element of $M$ over $(u_{0},v_{0})$. But since
$x_{0},y_{0}\in X$, $(x_{0},y_{0})$ is interalgebraic with a tuple $\bar{d}$
of elements of $D$. Putting everything together, the tuples $t$,
$(x_{0},y_{0})$, and $\bar{d}$ are interalgebraic over $(u_{0},v_{0})$. Since
$X$ is $D$-critical, Lemma 12.4 now implies that
$\operatorname{dprk}(x_{0}y_{0}/u_{0}v_{0})\leq\operatorname{dprk}(X)$. But by
assumption
$\operatorname{dprk}(x_{0}y_{0}/u_{0}v_{0})=2\cdot\operatorname{dprk}(X)$,
forcing $\operatorname{dprk}(X)=0$, and thus contradicting that $M$ is locally
interalgebraic with $D$. ∎

###  12.6. Corollaries of the Main Proposition

We now deduce two useful corollaries from Proposition 12.13.

######  Corollary 12.14.

Suppose $M$ is locally interalgebraic with an interpretable set $D$. Then $D$
is rich.

###### Proof.

Let $X\subset M$ be $D$-critical. We show that $X$ is rich, which implies by
Corollary 12.12 that $D$ is rich.

Fix $k$. We will find a $k$-rich configuration $(a,b,A)$ where $a\in X^{2}$
and $b$ is a tuple from $X$. First let $j=k+2$. Then let
$\mathcal{C}=\\{C_{t}:t\in T\\}$ be an almost faithful family of plane curves
in $\mathcal{M}$, where $T\subset M^{j+2}$ is generic (see [Cas23], Fact 2.27
and Lemma 2.42). Adding parameters, we assume $\mathcal{C}$ is
$\mathcal{M}$-definable over $\emptyset$. Given $z\in M^{j}$, we let
$\mathcal{C}(z)$ be the family with graph $\\{(x,y,s)\in M\times M\times
M^{2}:(x,y)\in C_{(s,z)}\\}$. If $z$ is $\mathcal{M}$-generic in $M^{j}$, then
$\mathcal{C}(z)$ is an almost faithful family indexed by a generic subset of
$M^{2}$.

Let us fix $z\in X^{j}$ with
$\operatorname{dprk}(z)=j\cdot\operatorname{dprk}(X)$. It follows from Lemma
12.2 that $z$ is $\mathcal{M}$-generic in $M^{j}$.

Now let $(u,v)\in X^{2}$ with
$\operatorname{dprk}(uv/z)=2\operatorname{dprk}(X)$, and let $(x,y)\in X^{2}$
with $\operatorname{dprk}(xy/uvz)=2\operatorname{dprk}(X)$. It follows from
Lemma 12.2 that $(u,v)$ is $\mathcal{M}$-generic in $M^{2}$ over $z$.

We are in the situation of Proposition 12.13. We conclude that there is $s\in
M^{2}$ such that for $t=(s,z)$, we have $(u,v),(x,y)\in C_{t}$ and $C_{t}\cap
X^{2}$ is infinite. Now since $C_{t}\cap X^{2}$ is infinite, there is
$a=(a_{1},a_{2})\in C_{t}\cap X^{2}$ with $\operatorname{dprk}(a/uvxyt)\geq
1$. We conclude:

######  Claim 12.15.

$(a,t,ux)$ is a $k$-rich configuration.

###### Proof.

Since $\operatorname{dprk}(a/uvxyt)\geq 1$, $a\notin\operatorname{acl}(uxt)$.
Now
$\operatorname{dprk}(t)\geq\operatorname{dprk}(z)=j\cdot\operatorname{dprk}(X)$.
Since $(u,x)\in X^{2}$, $\operatorname{dprk}(ux)\leq 2\operatorname{dprk}(X)$.
So by subadditivity (remember that $j=k+2$), $\operatorname{dprk}(t/ux)\geq
k\cdot\operatorname{dprk}(X)\geq k$.

Finally, let $\bar{t}=t_{1},t_{2},...$ be distinct realizations of
$\operatorname{tp}_{(K,v)}(t/uxa)$. Since $\mathcal{C}$ is almost faithful,
there is $N$ such that $C_{t_{1}}\cap C_{t_{N}}$ is finite, which shows that
$a\in\operatorname{acl}(ux\bar{t})$. ∎

To prove the corollary, we need to modify $(a,t,ux)$ by replacing $t$ by a
tuple from $X$. The key to doing this is:

######  Claim 12.16.

$t$ is interalgebraic with $zvy$ over $ux$. In particular, $(a,zvy,ux)$ is a
$k$-rich configuration.

###### Proof.

Clearly $z\in\operatorname{acl}(t)$ because $z$ is a subtuple of $t$. Now it
follows from the non-triviality of $C_{t}$ that $v\in\operatorname{acl}(tu)$
and $y\in\operatorname{acl}(tx)$. On the other hand, by Proposition 12.13,
there are only finitely many $C_{t^{\prime}}$ containing both $(u,v)$ and
$(x,y)$, which shows that $t\in\operatorname{acl}(uvxy)$. The claim now
follows by Lemma 12.11. ∎

Finally, note that $a\in X^{2}$ and $zvy\in X^{j+2}$. So since $k$ was
arbitrary, we conclude that $X$ is rich, and thus so is $D$. ∎

######  Remark 12.17.

The above proof of Corollary 12.14 would have been easier if, in the
definition of richness of $X$, the ‘parameter’ tuple $b$ was not required to
be a sequence of elements of $X$. Indeed, in this case we could have taken
$(a,t,\emptyset)$ as our $k$-rich configuration. The problem is that our
ultimate goal is to show that $\Gamma$ and $K/\mathcal{O}$ are not rich; and
if we adopted this weaker version of richness, then $K/\mathcal{O}$ would be
rich in positive characteristic (indeed, taking $\mathcal{O}$-linear
combinations of Frobenius powers gives arbitrarily large families of subsets
of $(K/\mathcal{O})^{2}$ indexed by powers of $\mathcal{O}$). Recall, however,
that $K/\mathcal{O}$ is not stably embedded – and we expect that it is
impossible to find such large families in $(K/\mathcal{O})^{n}$ without using
external parameters.

######  Corollary 12.18.

Suppose $M$ is locally interalgebraic with an interpretable set $D$. Then $M$
is almost internal to $D$.

###### Proof.

Let $X\subset M$ be $D$-critical. Fix a rank 2 family
$\mathcal{C}=\\{C_{t}:t\in T\\}$ of non-trivial plane curves as in Proposition
12.13, and assume it is $\emptyset$-definable in $\mathcal{M}$. Now let $v$ be
any element of $M-\operatorname{acl}(0)$. Let $u\in X$ be such that
$\operatorname{dprk}(u/v)=\operatorname{dprk}(X)$, and let $(x,y)\in X^{2}$ be
such that $\operatorname{dprk}(xy/uv)=2\operatorname{dprk}(X)$. So $(u,v)$ is
$\mathcal{M}$-generic in $M^{2}$. Then by Proposition 12.13, there is $t\in T$
with $(u,v),(x,y)\in C_{t}$ and $C_{t}\cap X^{2}$ infinite. Let
$(x^{\prime},y^{\prime})\in C_{t}\cap X^{2}$ with
$\operatorname{dprk}(x^{\prime}y^{\prime}/txy)\geq 1$. So
$(x^{\prime},y^{\prime})$ is $\mathcal{M}$-generic in $C_{t}$ over $txy$, and
thus there are only finitely many curves from $\mathcal{C}$ containing both
$(x^{\prime},y^{\prime})$ and $(x,y)$ (see [Cas23], Lemma 2.38). Since $C_{t}$
is one such curve, $t\in\operatorname{acl}(xyx^{\prime}x^{\prime})$. But since
$C_{t}$ is non-trivial, we also have $v\in\operatorname{acl}(tu)$. Thus in
total we have $v\in\operatorname{acl}(uxyx^{\prime}y^{\prime})$. Now
$u,x,y,x^{\prime},y^{\prime}\in X$. By compactness, all but finitely many
elements of $M$ are in $\operatorname{acl}(X)$. Adding finitely many
parameters to cover the remaining elements, we get that $M$ is almost internal
to $X$. But by definition $X$ is almost internal to $D$, so it follows that
$M$ is almost internal to $D$. ∎

###  12.7. Ruling out the Non-field Sorts

Our next goal is to show that $\Gamma$ and $K/\mathcal{O}$ are not rich. For
$K/\mathcal{O}$, our approach will only work in residue characteristic 0.

Let $(D,+,...)$ be either $\Gamma$ or $K/\mathcal{O}$. We can equip $D$ with a
non-discrete group topology: for $\Gamma$, we use the order topology, and for
$K/\mathcal{O}$, we use the topology generated by balls of negative radius
(equivalently, the topology induced by valuation on $K/\mathcal{O}$,
identifying $v(\mathcal{O})$ with 0). It was noted in [HHP22] that, with this
topology, $D$ forms a uniform structure in the sense of Simon and Walsberg
[SW19] (the o-minimal case appears already in the work of Simon and Walsberg;
for $K/\mathcal{O}$ see [HHP22, Lemma 5.13]) We do not define this notion
here. Instead, we give the following facts that we will use:

######  Fact 12.19.

Let $X\subset D^{n}$ be $A$-definable with $\operatorname{dprk}(X)=k$. Let
$a\in X$ with $\operatorname{dprk}(a/A)=k$. Then there are a projection
$\pi:X\rightarrow D^{k}$, a set $B\supset A$, and a $B$-neighborhood $U$ of
$a$ in $X$, such that $\operatorname{dprk}(a/B)=k$ and $\pi$ restricts on $U$
to a homeomorphism with an open subset of $D^{k}$.

######  Fact 12.20.

Let $X\subset D^{n}$ be $A$-definable, and let $a\in X$ be an isolated point.
Then $a\in\operatorname{acl}(A)$.

For the first of the above facts see [SW19, Proposition 4.6] (and [HHP22,
Corollary 3.13] to ensure $\operatorname{dprk}(a/B)=k$), and for the second
see [SW19, Lemma 3.6].

The following notion was studied in [HHP22]:

######  Definition 12.21.

$D$ is locally linear if whenever $f:U\rightarrow D$ is an $A$-definable
function on an open subset of $D^{n}$, and $a\in U$ with
$\operatorname{dprk}(a/A)=n$, there is a neighborhood of $a$ on which $f$
agrees with a map of the form $x\mapsto g(x)+b$, where $g:D^{n}\rightarrow D$
is a definable group homomorphism and $b\in D$ is a constant.

One can quickly improve on this definition:

######  Lemma 12.22.

Assume $D$ is locally linear. Let $X\subset D^{n}$ be $A$-definable, and let
$a\in X$ with $\operatorname{dprk}(a/A)=\operatorname{dprk}(X)$. Then there
are $B\supset A$ and a $B$-definable neighborhood $U$ of $a$ in $X$, such that
$\operatorname{dprk}(a/B)=\operatorname{dprk}(X)$ and $X$ agrees with a coset
of a definable subgroup of $D^{n}$.

###### Proof.

Let $k=\operatorname{dprk}(X)$. By Fact 12.19, we can assume there is a
projection $\pi:X\rightarrow D^{k}$ which is a homeomorphism onto an open set.
We thus view $X$ locally as the graph of a function $D^{k}\rightarrow
D^{n-k}$. Now apply the definition of local linearity to each coordinate
component of this function. ∎

The main fact we need is:

######  Fact 12.23.

$\Gamma$ is locally linear. $K/\mathcal{O}$ is locally linear if $K$ has
residue characteristic zero.

For $\Gamma$ this is well known, since $\Gamma$ is an o-minimal pure divisible
ordered abelian group. For $K/\mathcal{O}$ this is [HHP22, Proposition 6.24].

We do not know if $K/\mathcal{O}$ is locally linear in residue characteristic
$p$, but we suspect that it is.

Continuing, we now show:

######  Proposition 12.24.

Assume $D$ is locally linear. Then $D$ is not rich.

###### Proof.

Let $a\in D^{n}$, $b\in D^{m}$, $A$, and $k$ be such that $(a,b,A)$ is a
$k$-rich configuration (for now, we do not assume anything about $k$). Let
$r=\operatorname{dprk}(ab/A)$, and let $X\subset D^{n}\times D^{m}$ be
$A$-definable with $(a,b)\in X$ and $\operatorname{dprk}(ab/A)=r$. Since
$(a,b,A)$ is a $k$-rich configuration, and using compactness, we can assume
that the intersection of any infinitely many distinct fibers
$X_{b^{\prime}}\subset D^{n}$ is finite.

By Lemma 12.22, there are $B\supset A$, a $B$-definable neighborhood $U$ of
$a$ in $X$, and a subgroup $H\leq D^{n+m}$, such that
$\operatorname{dprk}(ab/B)=r$ and $X$ agrees with the coset $H+(a,b)$ on $U$.
Note, then, that whenever $(a^{\prime},b^{\prime})\in U\cap X$, we have
$H+(a,b)=H+(a^{\prime},b^{\prime})$, and thus $X$ also agrees with
$H+(a^{\prime},b^{\prime})$ on $U$.

Now let $G\times\\{0\\}$ be the kernel of the projection $H\rightarrow D^{m}$
– so $G$ is a subgroup of $D^{n}$. By the above, whenever
$(a^{\prime},b^{\prime})\in U\cap X$, the fiber $X_{b^{\prime}}$ agrees with
$G+a^{\prime}$ in a neighborhood of $a^{\prime}$. Indeed, in some small enough
neighborhood of $a^{\prime}$, we have

$x\in X_{b^{\prime}}\iff(x,b^{\prime})\in X\iff(x,b^{\prime})\in
H+(a^{\prime},b^{\prime})\iff(x,b^{\prime})-(a^{\prime},b^{\prime})\in H$
$\iff(x-a^{\prime},0)\in H\iff x-a^{\prime}\in G\iff x\in G+a^{\prime}.$

Now in particular, this implies that for any $b,b^{\prime}$ with
$(a,b),(a,b^{\prime})\in X\cap U$, the fibers $X_{b}$ and $X_{b^{\prime}}$
agree in a neighborhood of $a$ (because they both agree with $G+a$).

Now assume toward a contradiction that $D$ is rich. This says that, in the
situation constructed above, we can make
$\operatorname{dprk}(ab/A)=\operatorname{dprk}(ab/B)$ arbitrarily large, while
leaving $\operatorname{dprk}(a/A)$ (and thus also $\operatorname{dprk}(a/B)$)
bounded. In particular, we may fix all of the data in such a way that
$b\notin\operatorname{acl}(Ba)$. Thus, we can find $\bar{b}=b_{1},b_{2},...$,
an infinite sequence of distinct realizations of
$\operatorname{tp}_{(K,v)}(b/Ba)$. So each $(a,b_{i})\in U$, and thus (by the
previous paragraph) any finitely many of the fibers $X_{b_{i}}$ coincide on a
neighborhood of $a$.

Finally, since $(a,b,A)$ is a $k$-rich configuration, we have
$a\notin\operatorname{acl}(Ab)$. So by Fact 12.20, $a$ is not isolated in
$X_{b}$, and thus $a$ is not isolated in any $X_{b_{i}}$. Thus, any finitely
many $X_{b_{i}}$ have infinite intersection. By compactness, the intersection
of all of the $X_{b_{i}}$ is infinite, a contradiction. ∎

######  Corollary 12.25.

$M$ is not locally interalgebraic with $\Gamma$. If k has characteristic zero,
$M$ is not locally interalgebraic with $K/\mathcal{O}$. If k has
characteristic $p>0$ and $K/\mathcal{O}$ is locally linear, $M$ is not locally
interalgebraic with $K/\mathcal{O}$.

###  12.8. Concluding

We now show our main result:

######  Theorem 12.26.

Assume that $K$ has residue characteristic 0, or that k has characteristic
$p>0$ and $K/\mathcal{O}$ is locally linear. Then $M$ definably embeds into a
power of $K$ or k.

###### Proof.

By Corollary 12.25, $M$ is not locally interalgebraic with $\Gamma$ or
$K/\mathcal{O}$.

First, suppose $M$ is locally interalgebraic with $k$. Then by Corollary
12.18, $M$ is almost internal to k. It follows by an easy type-counting
argument that $M$ is stable and stably embedded. But all such sets are
internal to k ([HHM06], Proposition 3.4.11). Combining with stable
embeddedness and elimination of imaginaries in k, we then embed $M$ into some
$k^{n}$.

Now assume $M$ is not locally interalgebraic with k. Then by Fact 12.7, $M$
definably embeds into a power of $K$. ∎

We now drop the ambient assumptions and end the paper by collecting all
results on ACVF:

######  Theorem 12.27.

Let $\mathcal{K}=(K,v)$ be an algebraically closed valued field, and let
$\mathcal{M}$ be a non-locally modular strongly minimal $\mathcal{K}$-relic.
Assume the residue field k has characteristic zero, or $K/\mathcal{O}$ is
locally linear. Then $\mathcal{M}$ interprets a field $(F,\oplus,\otimes)$,
$(K,v)$-definably isomorphic to either $K$ or k. Moreover, the induced
structure on $F$ from $\mathcal{M}$ is a pure algebraically closed field.

###### Proof.

By Theorem 12.26, we may assume that $M\subset K^{n}$ or
$M\subset\textbf{k}^{n}$ for some $n$. If $M\subset\textbf{k}^{n}$, we reduce
to Corollary 11.6. If $M\subset K^{n}$, we reduce to Theorem 11.5. ∎

Acknowledgements: This project started while the first and third authors
attended the 2021 Thematic Program on Trends in Pure and Applied Model Theory
at the Fields Institute. We thank the Fields Institute for their hospitality.
We would also like to thank Yatir Halevi for discussions around the model
theory of valued fields, Will Johnson for his help with visceral theories,
Amnon Yekutily and Ilya Tyomkin for their help with algebro-geometric aspects
of the paper, and Martin Hils for pointing out a mistake in an earlier version
of the paper.

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