# A Pila–Wilkie theorem for Hensel minimal curves

Victoria Cantoral Farfán  Victoria Cantoral-Farfán, Georg-August Universität
Göttingen, Mathematisches Institut, Bunsenstrasse 3-5, 37073 Göttingen,
Deutschland<EMAIL_ADDRESS>https://sites.google.com/view/victoriacantoral ,  Kien Huu Nguyen  Kien Huu
Nguyen, KU Leuven, Department of Mathematics, B-3001 Leuven, Belgium, and
Thang Long Institute of Mathematics and Applied Sciences, Hanoi, Vietnam
<EMAIL_ADDRESS>https://sites.google.com/site/nguyenkienmath ,
Mathias Stout  Mathias Stout, KU Leuven, Department of Mathematics, B-3001
Leuven, Belgium<EMAIL_ADDRESS>and  Floris Vermeulen  Floris
Vermeulen, KU Leuven, Department of Mathematics, B-3001 Leuven, Belgium
<EMAIL_ADDRESS>https://sites.google.com/view/floris-vermeulen

###### Abstract.

Recently, a new axiomatic framework for tameness in henselian valued fields
was developed by Cluckers, Halupczok, Rideau-Kikuchi and Vermeulen and termed
Hensel minimality. In this article we develop Diophantine applications of
Hensel minimality. We prove a Pila–Wilkie type theorem for transcendental
curves definable in Hensel minimal structures. In order to do so, we introduce
a new notion of point counting in this context related to dimension counting
over the residue field. We examine multiple classes of examples, showcasing
the need for this new dimension counting and prove that our bounds are
optimal.

###### Key words and phrases:

Pila–Wilkie theorem, non-archimedean geometry, tame geometry on henselian
valued fields, analogues to o-minimality, model theory of valued fields,
rational points of bounded height, parametrizations

###### 2010 Mathematics Subject Classification:

Primary 03C99, 14G05; Secondary 03C65, 11G50, 11D88, 03C98

The author V.C-F. was partially supported by KU Leuven IF C14/17/083. The
author K.H.N. was partially supported by Research Foundation - Flanders (FWO)
12X3519N and by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 101.04-2019.316. The author M.S. was
supported by KU Leuven. The author F.V. was partially supported by KU Leuven
IF C14/17/083, and partially by F.W.O. Flanders (Belgium) with grant number
11F1921N

##  1\. Introduction

###  1.1. The Pila-Wilkie theorem

In 1989, Bombieri and Pila developed a very fruitful method to count integral
and rational points on various types of geometric objects in
${\mathbb{R}}^{2}$ [2]. This method is now called the determinant method and
is especially well-suited for proving uniform upper bounds on points of
bounded height. For a subset $X\subseteq{\mathbb{R}}^{n}$, recall that the
counting function is defined as

$N(X;B)=\\{x\in X\cap{\mathbb{Q}}\mid H(x)\leq B\\},$

where $H(a/b)=\max(\lvert a\rvert,\lvert b\rvert)$ for coprime integers $a$
and $b$, with $b\neq 0$. For example, if $f:[0,1]\to[0,1]$ is an analytic
transcendental function and $X$ denotes its graph, Bombieri and Pila proved
that for any $\varepsilon>0$, there is a constant $c_{\varepsilon}$ such that
for all $B>0$

$N(X;B)\leq c_{\varepsilon}B^{\varepsilon}.$

A vast generalization of this result is the celebrated Pila–Wilkie theorem
[13]. It states that, if $X\subseteq{\mathbb{R}}^{n}$ is definable in an
o-minimal structure, then for any $\varepsilon>0$, there exists a constant
$c_{\varepsilon}$ such that for all $B>0$

(1.1.1) $N(X^{\mathrm{trans}};B)\leq c_{\varepsilon}B^{\varepsilon}.$

Here $X^{\mathrm{trans}}$ denotes the transcendental part of $X$, obtained
from $X$ by removing all positive-dimensional connected semialgebraic subsets
of $X$. The proof of this result is based heavily on the existence of
$C^{r}$-parametrizations, which were originally developed by Gromov and Yomdin
[16, 15, 11].

In the non-archimedean setting, such parametrization results were first proved
in [4], and a corresponding Pila–Wilkie theorem was obtained for subanalytic
sets in ${\mathbb{Q}}_{p}$. These results were further improved in [5], where
a uniform version of these bounds was proved for subanalytic sets in
${\mathbb{Q}}_{p}$ and ${\mathbb{F}}_{p}((t))$.

###  1.2. Hensel minimality

Hensel minimality, or h-minimality for short, is a recent framework for tame
non-archimedean geometry, developed by Cluckers, Halupczok and Rideau-Kikuchi
in equicharacteristic zero and extended to mixed characteristic together with
the fourth author [8, 6]. It encompasses the aforementioned analytic structure
on ${\mathbb{Q}}_{p}$ as a special case, but it applies more broadly, see e.g.
[8, Sec. 6] for several examples.

Hensel minimality bears a striking resemblance to the classical theory of
o-minimality. In an o-minimal structure $K$, each definable subset $X\subseteq
K$ is a finite union of intervals and points. In other words, there is some
finite tuple $(a_{i})_{i\in I}$ such that $X$ is a union of fibres of the map
$x\mapsto(\operatorname*{{sgn}}(x-a_{i}))_{i\in I}$. Roughly speaking,
h-minimality replaces the sign map by the _leading term_ map
$\operatorname{rv}\colon K\to K^{\times}/(1+{\mathcal{M}}_{K})\cup\\{0\\}$,
where $K$ is a valued field and ${\mathcal{M}}_{K}$ is the maximal ideal of
its valuation ring.

The goal of this article is to develop an analogue of the Pila–Wilkie theorem
in an h-minimal context. For this purpose, we need two important consequences
of Hensel minimality: a cell decomposition statement (Theorem 3.2.5) and the
Jacobian property (Theorem 3.2.1). These theorems are analogues of o-minimal
cell decomposition and the monotonicity theorem, respectively. We use them to
prove a _$T_{r}$ -parametrization_ statement for curves definable in Hensel
minimal structures (Theorem 4.1.1). These $T_{r}$-parametrizations are
analogues of the $C^{r}$-parametrizations used in the proof of the o-minimal
Pila-Wilkie theorem, and form a key technical ingredient.

###  1.3. Counting dimension

Let $k$ be a field of characteristic zero and denote by $k((t))$ the field of
Laurent series over $k$. For any natural number $s$, we let $k[t]_{s}\subseteq
k((t))$ be the set of polynomials in $t$ of degree less than $s$. For
(transcendental definable) curves $X\subseteq k((t))^{n}$ we study the number
of points on $X_{s}\coloneqq X\cap(k[t]_{s})^{n}$ as a valued field analogue
of rational points of bounded height. In particular, we are interested in
bounding the growth in terms of $s$, similar to the Pila-Wilkie theorem.

When $k={\mathbb{C}}$ and $X$ is a transcendental curve, definable in the
subanalytic structure on ${\mathbb{C}}((t))$, then $X_{s}$ is finite, for each
$s\in{\mathbb{N}}$ [1, Thm. 1]. However, this need not be the case for
definable curves in arbitrary h-minimal structures (see Section 2.4). This
issue is resolved by counting relative to the residue field. More precisely,
we introduce the notion of _counting dimension_ , or $\\#\text{-}\dim$ for
short, and consider bounds of the form

(1.3.1) $\\#\text{-}\dim(X_{s})\leq(N(s),d,e(s)),$

where $N$ and $e$ are functions ${\mathbb{N}}\to{\mathbb{N}}$ and
$d\in{\mathbb{N}}$ is constant. Intuitively, the above inequality can be
thought of as a means of expressing that

$\\#X_{s}\leq N(s)\\#(k[t]/(t^{e(s)}))^{d},$

even when $k$ is infinite. When $N=N(s)$ is constant, $d\cdot e(s)$ can be
thought of as bounding the growth of the $k$-dimension of $X_{s}$.

For $k={\mathbb{F}}_{p}$, note that the right-hand side becomes
$N(s)p^{de(s)}$. Comparing this to the known results for transcendental
definable curves in ${\mathbb{F}}_{p}((t))$ ([5, Thm. B]), leads to the
following question for transcendental curves $X\subseteq k((t))^{n}$: if
$\varepsilon>0$ is given, can we take $N=N_{\varepsilon}$ constant, $d=1$ and
$e(s)=\lceil\varepsilon\cdot s\rceil$ in (1.3.1)? We stress that the
importance of the counting dimension is that it makes this question meaningful
when $k$ is infinite.

We will return to the motivations for the counting dimension, after precisely
defining it for any henselian valued field $K$ (and not just $k((t))$).
Theorem 2.2.1 then positively answers our question above: if $K$ is an
equicharacteristic zero henselian valued field which is h-minimal, satisfying
some mild extra conditions, then for any transcendental definable curve
$X\subseteq K^{n}$ and any $\varepsilon>0$ there exists some constant
$N_{\varepsilon}>0$ such that

$\\#\text{-}\dim(X_{s})\leq(N_{\varepsilon},1,\lceil\varepsilon\cdot
s\rceil).$

We moreover show that this bound is optimal, by constructing certain
transcendental definable curves of a specific form.

Additionally, we consider the case of algebraic curves in Theorem 2.2.3, and
prove the analogue of the classical Bombieri–Pila bound here. Let us also
mention that the main obstacle in extending these results to higher dimensions
is that these parametrization results are only known for curves under Hensel
minimality. Especially the higher-dimensional geometry under Hensel minimality
has to develop further.

### Acknowledgements

The authors would like to thank Raf Cluckers for suggesting this topic and the
various discussions surrounding it. The authors also thank the anonymous
referee for the many helpful comments which improved this article.

##  2\. The counting dimension

###  2.1. Counting in valued fields

In this section we introduce the counting dimension and prove some basic
results about it. We then state our main results on the counting dimension of
transcendental and algebraic curves.

Let $K$ be a non-archimedean valued field equipped with an
${\mathcal{L}}$-structure, for some language ${\mathcal{L}}$ expanding the
language of valued fields
${\mathcal{L}}_{\mathrm{val}}=\\{0,1,+,\cdot,{\mathcal{O}}_{K}\\}$. For
$A\subset K$, a set $X\subset K^{n}$ is called $A$-definable if it is
definable in ${\mathcal{L}}$, using parameters only from $A$. We call a set
definable if it is $K$-definable. We denote by $k$ the residue field of $K$,
by ${\mathcal{O}}_{K}$ the valuation ring and by ${\mathcal{M}}_{K}$ the
maximal ideal. Let $\Gamma_{K}^{\times}$ be the value group, where the
valuation is written multiplicatively $|\cdot|:K\to\Gamma_{K}$.

Assume that $K$ is henselian of equicharacteristic zero. Then there always
exists a lift $\tilde{k}\subset K$ of the residue field $k$, i.e. a subfield
of $K$ which maps bijectively to $k$ under the reduction map
${\mathcal{O}}_{K}\to k$. Fix also a _pseudo-uniformizer_ $t\in K$, recall
that this is any non-zero element of $K$ with $|t|<1$. For a positive integer
$s$, define

$\tilde{k}[t]_{s}=\left\\{\sum_{i=0}^{s-1}a_{i}t^{i}\mid
a_{i}\in\tilde{k}\right\\}.$

If $X$ is a subset of $K^{n}$ we define $X_{s}$ to be
$X\cap\tilde{k}[t]_{s}^{n}$. We call $X_{s}$ the _set of rational points of
height at most $s$ on $X$_. Note that this set depends on the choice of
pseudo-uniformizer $t$ and on the lift $\tilde{k}$.

The prototypical example to keep in mind is $K=k((t))$ for some characteristic
zero field $k$, with $\tilde{k}=k$ and $t$ as pseudo-uniformizer. Here
$\tilde{k}[t]_{s}$ is simply the set of polynomials over $k$ of degree at most
$s-1$.

The set $X_{s}$ is considered as a suitable analogue for the set of rational
points of bounded height on $X$, where the height is captured by $s$. We will
be interested in bounding the size of $X_{s}$, as $s$ grows, for various types
of subsets of $K^{n}$. We cannot simply use the number of points on $X_{s}$ as
a measure of size, since this set will typically be infinite. Instead we
introduce the _counting dimension_ , to measure the size of $X_{s}$ relative
to the residue field.

######  2.1.1 Definition.

Let $K$ be a henselian valued field of equicharacteristic zero equipped with
an ${\mathcal{L}}$-structure, for some language ${\mathcal{L}}$ expanding the
language of valued fields. Fix a pseudo-uniformizer $t$ of $K$ and a lift
$\tilde{k}$ of the residue field. Let $X$ be a subset of $K^{n}$, let $d$ be a
positive integer and let $N,e:{\mathbb{N}}\to{\mathbb{N}}$ be functions. Then
we say that $X$ has _counting dimension bounded by $(N,d,e)$_ if there exists
a definable function $f:X\to{\mathcal{O}}_{K}^{d}$ such that for every
positive integer $s$, the composition

$X_{s}\xrightarrow{f}{\mathcal{O}}_{K}^{d}\xrightarrow{\operatorname{proj}}\left(\frac{{\mathcal{O}}_{K}}{(t^{e(s)})}\right)^{d}$

has finite fibres of size at most $N(s)$. Here $\operatorname{proj}$ is the
componentwise reduction map modulo $t^{e(s)}$.

We will use the notation

$\\#\text{-}\dim X_{s}\leq(N(s),d,e(s))$

to mean that the counting dimension of $X$ is bounded by $(N,d,e)$.

This definition depends on the choice of pseudo-uniformizer $t$, the lift
$\tilde{k}$ and on the language ${\mathcal{L}}$. However, we suppress these in
notation and always assume a fixed choice of $t,\tilde{k}$ and
${\mathcal{L}}$.

Our definition of the counting dimension is motivated on the one hand by
counting rational points on definable subsets in local fields of mixed
characteristic as in [4, 5, 6]. In that case, one has to use a different
notion of rational points of bounded height, as there is no lift of the
residue field. Consider for example $K={\mathbb{Q}}_{p}$. Then, if $X$ is a
subset of ${\mathbb{Q}}_{p}^{n}$, define

$X_{s}=\\{x\in X\cap{\mathbb{Z}}^{n}\mid 0\leq x_{i}\leq s\text{ for all
}i\\}.$

Using this definition of rational points, if $X$ has counting dimension
bounded by $(N(s),d,e(s))$, then for every $s$, $X_{s}$ will contain no more
than $N(s)p^{de(s)}$ points. So a bound on the counting dimension gives a
corresponding bound on the number of points in $X_{s}$.

Our second motivation comes from the relation of the counting dimension to the
Zariski dimension over the residue field, as in [4, Sec. 5]. Let
$K={\mathbb{C}}((t))$ and let $X$ be a subset of $K^{n}$. Then for every $s$,
$X_{s}$ is a subset of ${\mathbb{C}}[t]_{s}$, which can be naturally
identified with ${\mathbb{C}}^{ns}$. If $X$ is algebraic, then $X_{s}$ will be
a constructible set in ${\mathbb{C}}^{ns}$, and one can wonder how the Zariski
dimension of $X_{s}$ grows with $s$. In [4, Sec. 5], a bound for this quantity
is provided. We obtain a similar bound using the counting dimension instead of
the Zariski dimension.

###  2.2. Main results

Our main results concern the counting dimension of algebraic and
transcendental curves definable in h-minimal structures.

By a _curve $C\subset K^{n}$_ we mean a set for which there exist a linear map
$p:K^{n}\to K$ such that $p(C)$ is infinite and such that $p$ has finite
fibers on $C$. If the theory of $K$ in ${\mathcal{L}}$ is 1-h-minimal, see [8,
Def. 2.3.3] or section 3.1 below, then by dimension theory [8, Thm. 5.3.4], a
definable curve in $K^{n}$ is the same as a definable set of dimension 1. We
call a curve $C\subset K^{n}$ _transcendental_ if every algebraic curve in
$K^{n}$ has finite intersection with $C$.

Let $K$ be a henselian valued field equipped with an ${\mathcal{L}}$-structure
and assume that $\operatorname{Th}_{\mathcal{L}}(K)$ is 1-h-minimal. Then we
say that $\mathrm{acl}=\mathrm{dcl}$ for $K$ if algebraic Skolem functions
exist in every model of $\operatorname{Th}_{\mathcal{L}}(K)$. By this we mean
that for any model $K^{\prime}$ of $\operatorname{Th}_{\mathcal{L}}(K)$ and
any subset $A\subset K^{\prime}$ we have that
$\mathrm{acl}_{K^{\prime}}(A)=\mathrm{dcl}_{K^{\prime}}(A)$.

Our main result is the following analogue of the Pila–Wilkie theorem on the
counting dimension of transcendental curves definable in Hensel minimal
structures.

######  2.2.1 Theorem.

Suppose that $K$ is a henselian valued field of equicharacteristic $0$
equipped with a 1-h-minimal structure. Fix a pseudo-uniformizer $t$ and a lift
of the residue field $\tilde{k}$. Suppose that $\mathrm{acl}=\mathrm{dcl}$ in
$K$ and that the subgroup of $b$-th powers in $k^{\times}$ has finite index,
for some integer $b>1$. Let $C\subset{\mathcal{O}}_{K}^{n}$ be a
transcendental definable curve. Then for each $\varepsilon>0$ there is a
constant $N$ such that for each integer $s\geq 0$

$\\#\text{-}\dim(C_{s})\leq(N,1,\lceil\varepsilon\cdot s\rceil).$

Furthermore, the constant $N$ can be taken to hold uniformly throughout all
transcendental members of a given definable family of definable curves.

The key aspect here is the slow growth of the last component of the counting
dimension, similar to the Pila–Wilkie theorem in the o-minimal setting [13,
Thm. 1.10]. The strategy of the proof is as follows. We use the notion of
$T_{r}$-parametrizations, which form a suitable analogue for
$C^{r}$-parametrizations in the non-archimedean setting, see e.g. [4, 5].

  1. (1)

We apply cell decomposition to find a $T_{1}$-parametrization of $C$. The
existence of such a cell decomposition follows from 1-h-minimality under the
extra assumption that $\mathrm{acl}=\mathrm{dcl}$ in $K$, see [8, Thm. 5.2.4
Add. 5] or section 3.1 below.

  2. (2)

Using substitutions of the form $x\mapsto x^{r}$, we may even assume that we
have a $T_{r}$-parametrization, for some suitably chosen integer $r$. For
this, we need that the subgroup of $b$-th powers of $k^{\times}$ has a finite
index in $k^{\times}$.

  3. (3)

We then use an adaptation of the Bombieri–Pila determinant method to catch all
rational points of bounded height in a small ball in a single hypersurface.

  4. (4)

Finally, the fact that $C$ is transcendental and definable in a 1-h-minimal
structure then gives the desired result. Indeed, this will follow from uniform
finiteness in definable families, see [8, Lem. 2.5.3] or section 3.1 below.

A mixed characteristic analogue of this result in ${\mathbb{Q}}_{p}$ was
proven by Cluckers–Halupczok–Rideau-Kikuchi–Vermeulen [6, Thm. 4.1.6]. Here
the notion of rational points of bounded height is defined as above. Namely,
for $X$ a subset of ${\mathbb{Q}}_{p}^{n}$ define

$X_{s}=\\{x\in X\cap{\mathbb{Z}}^{n}\mid 0\leq x_{i}\leq s\text{ for all
}i\\}.$

Then [6, Thm. 4.1.6] states that if ${\mathbb{Q}}_{p}$ carries a $1$-h-minimal
structure with $\mathrm{acl}=\mathrm{dcl}$, and if
$C\subset{\mathbb{Q}}_{p}^{n}$ is a transcendental definable curve, then for
each $\varepsilon>0$ there is a constant $c$ such that for all $H\geq 1$ we
have

$\\#C_{s}\leq cs^{\varepsilon}.$

In this article, we will restrict to equicharacteristic zero. The methods for
both proofs are quite similar, one major difference being that the residue
field is no longer finite. This is the reason for introducing the counting
dimension. Restricting to equicharacteristic zero has the added benefit that
Hensel minimality is slightly easier to work with.

We also prove that Theorem 2.2.1 is optimal, in the sense that last component
$\lceil\varepsilon\cdot s\rceil$ of our bound cannot be improved. In more
detail, one cannot replace it by a sublinear function $e(s)$, even if $N(s)$
is allowed to be completely arbitrary.

######  2.2.2 Theorem.

Let $k$ be a field of characteristic zero. There exists a $1$-h-minimal
structure on $k((t))$ with $\mathrm{acl}=\mathrm{dcl}$ satisfying the
following. Given a sublinear function $e\colon{\mathbb{N}}\to{\mathbb{N}}$ and
any $N\colon{\mathbb{N}}\to{\mathbb{N}}$, there exists a definable
transcendental curve $C\subseteq k((t))^{2}$ such that its counting dimension
is not bounded by $(N(s),1,e(s))$.

In contrast with this result, we consider in Section 5 a specific analytic
structure on ${\mathbb{Q}}_{p}((t))$ for which we are able to prove that the
counting dimension of every definable transcendental curve is bounded by
$(N,1,1)$ for some integer $N>0$. This structure already contains many
interesting examples of transcendental definable curves. For example, the
graph of the exponential function
$\exp:p{\mathbb{Z}}_{p}+t{\mathbb{Q}}_{p}[[t]]\to{\mathbb{Q}}_{p}((t))$ is
definable.

As for algebraic curves, we prove the following theorem, generalizing the
results from [4, Sec. 5].

######  2.2.3 Theorem.

Let $K$ be a henselian valued field of equicharacteristic $0$ equipped with a
1-h-minimal structure. Assume that $\mathrm{acl}=\mathrm{dcl}$ and that the
subgroup of $b$-th powers in $k^{\times}$ has finite index, for some $b>1$.
Let $C\subset K^{2}$ be an irreducible algebraic curve of degree $d$, for some
positive integer $d$. Then there exists a constant $c_{d}$, depending only on
$d$, such that

$\\#\text{-}\dim C_{s}\leq(c_{d}s,1,\lceil s/d\rceil).$

This theorem can be considered the analogue of the classical Bombieri–Pila
theorem [2]. By considering the example $y=x^{d}$, we will show that one
cannot improve the last component of the counting dimension in this result.

###  2.3. Some basic properties

Let us list some basic properties of the counting dimension. We will often use
these implicitly in our proofs.

######  2.3.1 Proposition.

Let $K$ be a non-archimedean valued field in some language ${\mathcal{L}}$
expanding the language of valued fields and let $X,X^{\prime}$ be definable
subsets of $K^{n}$. Assume that

$\\#\text{-}\dim(X_{s})\leq(N(s),d,e(s)),\quad\\#\text{-}\dim(X^{\prime}_{s})\leq(N^{\prime}(s),d^{\prime},e^{\prime}(s))$

for some integers $d,d^{\prime}$ and some functions
$N,N^{\prime},e,e^{\prime}:{\mathbb{N}}\to{\mathbb{N}}$. Then

  1. (1)

$\\#\text{-}\dim((X\cup
X^{\prime})_{s})\leq(N(s)+N^{\prime}(s),\max\\{d,d^{\prime}\\},\max\\{e(s),e^{\prime}(s)\\})$,

  2. (2)

$\\#\text{-}\dim((X\times
X^{\prime})_{s})\leq(N(s)N^{\prime}(s),d+d^{\prime},\max\\{e(s),e^{\prime}(s)\\})$,

  3. (3)

if $f:X^{\prime}\to X$ is a definable map with finite fibres of size at most
$N^{\prime\prime}$, for some integer $N^{\prime\prime}$, then the counting
dimension of $X^{\prime}$ is bounded by $(N^{\prime\prime}N,d,e)$. If moreover
$f$ is surjective and $\mathrm{acl}=\mathrm{dcl}$ in $K$, then the counting
dimension of $X$ is bounded by $(N^{\prime},d^{\prime},e^{\prime})$.

###### Proof.

The proof of the first two properties is straightforward, so let us prove the
last property. Let $g:X\to{\mathcal{O}}_{K}^{d}$ be a definable map such that
for every positive integer $s$, the composition

$X_{s}\xrightarrow{g}{\mathcal{O}}_{K}^{d}\xrightarrow{\operatorname{proj}}\left(\frac{{\mathcal{O}}_{K}}{(t^{e(s)})}\right)^{d}$

has finite fibres of size at most $N(s)$ and take a similar such definable map
$g^{\prime}:X^{\prime}\to{\mathcal{O}}_{K}^{d^{\prime}}$ for $X^{\prime}$.
Then for every positive integer $s$, the composition

$\operatorname{proj}\circ g\circ
f:X^{\prime}_{s}\to\left(\frac{{\mathcal{O}}_{K}}{(t^{e(s)})}\right)^{d}$

has finite fibres of size at most $N^{\prime\prime}N(s)$, so the counting
dimension of $X^{\prime}$ is bounded by $(N^{\prime\prime}N,d,e)$.

Conversely, using $\mathrm{acl}=\mathrm{dcl}$ there exists a section
$f^{\prime}:X\to X^{\prime}$ for $f$. Then for every positive integer $s$, the
composition

$\operatorname{proj}\circ g^{\prime}\circ
f^{\prime}:X_{s}\to\left(\frac{{\mathcal{O}}_{K}}{(t^{e^{\prime}(s)})}\right)^{d^{\prime}}$

has finite fibres of size at most $N^{\prime}(s)$. ∎

###  2.4. Some examples

We give some examples of bounding the counting dimension of transcendental
curves, which the reader can keep in mind throughout this article. In each of
these, we consider a curve $C$ in a valued field $K$ which is definable in
some 1-h-minimal structure on $K$. We then give an upper bound for the
counting dimension of $C$ by simply computing the sets $C_{s}$.

######  2.4.1 Example.

Consider the valued field $K={\mathbb{Q}}_{p}((t))$ with valuation ring
${\mathbb{Q}}_{p}[[t]]$. In Section 5 we argue that there is a 1-h-minimal
structure on $K$ in which the exponential map

$\exp:p{\mathbb{Z}}_{p}+t{\mathbb{Q}}_{p}[[t]]\to{\mathbb{Q}}_{p}((t)):z\mapsto\sum_{i\geq
0}\frac{z^{i}}{i!}$

is definable. Let us write $U=p{\mathbb{Z}}_{p}+t{\mathbb{Q}}_{p}[[t]]$.

Let $C$ be the graph of this exponential function. We claim that this is a
transcendental set. Indeed, suppose that $f(x,y)\in K[x,y]$ is a non-zero
polynomial such that $f(x,\exp x)=0$ for infinitely many $x\in U$. We take
such an $f$ of minimal degree. By h-minimality [8, Lem. 2.5.2], there is then
an open ball $B\subset U$ on which this holds. Moreover, $\exp$ is
differentiable on $U$ with derivative $\exp$. Write
$f(x,y)=\sum_{i=0}^{d}f_{i}(x)y^{i}$. Define the polynomial

$g(x,y)=\sum_{i=0}^{d}f_{i}^{\prime}(x)y^{i}+\sum_{i=0}^{d-1}(i-d)f_{i}(x)y^{i}.$

This polynomial is non-zero and has degree strictly smaller than $f$ and for
$x\in B$ we have

$g(x,\exp x)=\dfrac{\partial}{\partial x}(f(x,\exp x))-df(x,\exp x)=0.$

This is the desired contradiction, showing that $C$ is a transcendental set.

To compute the counting dimension, we use $t$ as a uniformizer and
${\mathbb{Q}}_{p}\subset{\mathbb{Q}}_{p}((t))$ as a lift of the residue field.
Now, if $x$ is in $p{\mathbb{Z}}_{p}$ then $\exp x$ is in ${\mathbb{Q}}_{p}$.
Hence

$C_{1}=\\{(x,\exp x)\mid x\in p{\mathbb{Z}}_{p}\\}.$

In particular, $C_{1}$ is infinite. We claim that the counting dimension of
$C$ is bounded by $(1,1,1)$. For this purpose, consider the map
$C\to{\mathcal{O}}_{K}:(x,\exp x)\mapsto x$ followed by projection
${\mathcal{O}}_{K}\to{\mathcal{O}}_{K}/{\mathcal{M}}_{K}={\mathbb{Q}}_{p}$.
This clearly has finite fibres on $C_{1}$. Even more, if $x$ is an element of
$U\cap{\mathbb{Q}}_{p}[t]$ which is not in $p{\mathbb{Z}}_{p}$ then
automatically $\exp x$ is not in ${\mathbb{Q}}_{p}[t]$. Thus for any positive
integer $s$, $C_{1}=C_{s}$ and the counting dimension of $C$ is bounded by by
$(1,1,1)$.

######  2.4.2 Example.

Consider the field ${\mathbb{R}}((t))$ in the language of ordered valued
fields. We expand the language by the full Weierstrass system ${\mathcal{B}}$
as in [7, Sec. 3.1]. In more detail, let

$A_{n,\alpha}(({\mathbb{Z}}))=\\{\sum_{i\in I}f_{i}t^{i}\mid f_{i}\in
A_{n,\alpha},I\subset{\mathbb{Z}}\textnormal{ well ordered}\\},$

where $A_{n,\alpha}$ is the ring of real power series in
${\mathbb{R}}[[\xi_{1},...,\xi_{n}]]$ with radius of convergence $>\alpha$,
and define $B_{n,\alpha}=A_{n,\alpha}({\mathbb{Z}})$. This is a real
Weierstrass system in the sense of [7, Def. 3.1.1], and we equip
${\mathbb{R}}((t))$ with real analytic ${\mathcal{B}}$-structure. In work in
progress, we prove that the theory of ${\mathbb{R}}((t))$ in this language is
$1$-h-minimal. In particular, we have function symbols for all elements of the
Weierstrass system ${\mathcal{B}}$. Now, by our choice of ${\mathcal{B}}$ the
exponential

$\exp:(-1,1)+t{\mathbb{R}}[[t]]\to{\mathbb{R}}((t)):z\mapsto\sum_{i\geq
0}\frac{z^{i}}{i!}$

is definable. Denote by $C$ the graph of $\exp$, which as above is a
transcendental set. We use ${\mathbb{R}}\subset{\mathbb{R}}((t))$ as a lift of
the residue field and $t$ as our choice of uniformizer. Then $C_{1}$ is
infinite, since if $x$ is in $(-1,1)$ then $\exp(x)$ is again real. Consider
the reduction map

$C\to{\mathcal{O}}_{K}/(t):(x,\exp x)\mapsto x\bmod t.$

Then this has finite fibres of cardinality at most $1$ above $C_{1}$. In fact,
if $x$ is in ${\mathbb{R}}[t]_{s}$ but not in ${\mathbb{R}}$ then $\exp(x)$ is
never in ${\mathbb{R}}[t]$. Thus for any $s\geq 1$ we have $C_{s}=C_{1}$ and
so the counting dimension of $C$ is bounded by $(1,1,1)$.

######  2.4.3 Example.

Denote by ${\mathcal{L}}_{\mathrm{omin}}$ the language of ordered rings. For
any real number $r>0$, let $f_{r}$ denote the function
${\mathbb{R}}_{>0}\to{\mathbb{R}}_{>0}:x\mapsto x^{r}$ and consider the
expansion ${\mathcal{L}}$ of ${\mathcal{L}}_{\mathrm{omin}}$ where we have a
function symbol for every $f_{r}$. The theory of ${\mathbb{R}}$ in
${\mathcal{L}}$ is o-minimal, since it is a reduct of ${\mathbb{R}}_{\exp}$.
Let $K$ be a proper elementary extension of ${\mathbb{R}}$. Then we may turn
$K$ into a valued field by taking for ${\mathcal{O}}_{K}$ the convex closure
of ${\mathbb{R}}$. Note that the theory ${\mathcal{T}}$ is power-bounded, so
by [8, Thm. 7.2.4] the theory of $K$ in the language
${\mathcal{L}}\cup\\{{\mathcal{O}}_{K}\\}$ is 1-h-minimal. Let $t$ be a
pseudo-uniformizer of $K$ and denote by $\tilde{k}$ a lift of the residue
field. Then $\tilde{k}$ is again a real closed field. For example, $K$ might
be the field of Hahn series ${\mathbb{R}}((t^{\mathbb{R}}))$ with pseudo-
uniformizer $t$ and $\tilde{k}={\mathbb{R}}$.

Now let $C$ be the graph of the function $K_{>0}\to K_{>0}:x\mapsto x^{\pi}$.
This is a definable set by our choice of language. Clearly, this is also a
transcendental set. But $C_{1}$ is simply the graph of $x\mapsto x^{\pi}$ on
$\tilde{k}$, which certainly contains this graph on ${\mathbb{R}}$. In
particular, this set is infinite. Also note that $C_{s}=C_{1}$ for any $s\geq
1$ so that the counting dimension of $C$ is bounded by $(1,1,1)$.

##  3\. Notation and background

###  3.1. Hensel minimality

In this section, we record some background material on Hensel minimality. We
refer to [8] and [6] for further details.

Let ${\mathcal{L}}$ be a language containing the language of valued fields
${\mathcal{L}}_{\mathrm{val}}=\\{0,1,+,\cdot,{\mathcal{O}}_{K}\\}$. Let
${\mathcal{T}}$ be a complete ${\mathcal{L}}$-theory whose models are non-
trivially valued fields of equi-characteristic zero. Let $K$ be a model of
${\mathcal{T}}$. We denote by ${\mathcal{O}}_{K}$ the valuation ring of $K$
and by $\Gamma_{K}^{\times}$ the valuation group. The valuation will be
denoted by $|\cdot|:K\to\Gamma_{K}=\Gamma_{K}^{\times}\cup\\{0\\}$. By an open
ball we will mean a set of the form $B_{<\lambda}(a)=\\{x\in
K\mid|x-a|<\lambda\\}$, where $a\in K$, $\lambda\in\Gamma_{K}^{\times}$.
Similarly, a closed ball is a set of the form $B_{\leq\lambda}(a)=\\{x\in
K\mid|x-a|\leq\lambda\\}$. If $B$ is an open ball as above, we denote by
$\operatorname{rad}_{\mathrm{op}}B$ its radius $\lambda$, and similarly we use
$\operatorname{rad}_{\mathrm{cl}}B$ for $\lambda$ if $B$ is closed ball.

For $\lambda\leq 1$ an element of $\Gamma_{K}^{\times}$ let $I_{\lambda}$ be
the ideal $\\{x\in K\mid|x|<\lambda\\}$. We define
$\mathrm{RV}_{\lambda}^{\times}$ to be $K^{\times}/(1+I_{\lambda})$, with
quotient map

$\operatorname{rv}_{\lambda}:K^{\times}\to\mathrm{RV}_{\lambda}^{\times}.$

We also consider
$\mathrm{RV}_{\lambda}=\mathrm{RV}_{\lambda}^{\times}\cup\\{0\\}$. The map
$\operatorname{rv}_{\lambda}$ extends to $K\to\mathrm{RV}_{\lambda}$ via
$\operatorname{rv}_{\lambda}(0)=0$. We will write
$\mathrm{RV}=\mathrm{RV}_{1}$ and $\operatorname{rv}=\operatorname{rv}_{1}$.
The set $\mathrm{RV}$ combines information from the residue field and the
value group. Indeed, there is a short exact sequence

$1\to\left({\mathcal{O}}_{K}/{\mathcal{M}}_{K}\right)^{\times}\to\mathrm{RV}^{\times}\to\Gamma_{K}^{\times}\to
1.$

Now let $\lambda\leq 1$ be in $\Gamma_{K}^{\times}$ and let $X$ be a subset of
$K$. We say that a finite set $C$ _$\lambda$ -prepares $X$_ if the following
holds: for any $x,y\in K$, if

$\operatorname{rv}_{\lambda}(x-c)=\operatorname{rv}_{\lambda}(y-c),\text{ for
all }c\in C$

then either $x$ and $y$ are both in $X$, or they are both not in $X$. If
$(\xi_{c})_{c}\in\mathrm{RV}_{\lambda}^{\\#C}$ then the set

$\\{x\in K\mid\operatorname{rv}_{\lambda}(x-c)=\xi_{c}\text{ for all }c\in
C\\}$

is said to be a ball _$\lambda$ -next to $C$_ (if it is disjoint from $C$).
Note that if such a set is disjoint from $C$, then it is indeed an open ball.
We can rephrase preparing as follows. A finite set $C$ $\lambda$-prepares $X$
if for any ball $B$ $\lambda$-next to $C$, either $B\subseteq X$ or $B\cap
X=\emptyset$. Note also that the balls $1$-next to a finite set $C$ are
precisely the maximal open balls disjoint from $C$.

###  3.2. Consequences of Hensel minimality

For this section, we fix a field $K$ of equicharacteristic zero equipped with
an ${\mathcal{L}}$-structure which is $1$-h-minimal. By [8], we may freely add
constants from $K$ to the language ${\mathcal{L}}$ and preserve
$1$-h-minimality. Many of the results below are formulated only for
$\emptyset$-definable objects, but therefore hold just as well for
$A$-definable objects, for $A\subset K$.

Hensel minimality implies tameness results on various definable objects. For
functions there is the Jacobian property and Taylor approximation.

######  3.2.1 Theorem ([8, Cor. 3.2.6], Jacobian property).

Let $f:K\to K$ be a $\emptyset$-definable function. Then there exists a finite
$\emptyset$-definable set $C$ such that for every $\lambda\leq 1$ in
$\Gamma_{K}^{\times}$, every ball $B$ $\lambda$-next to $C$ and every
$x_{0},x\in B$, $x\neq x_{0}$, we have that

  1. (1)

the derivative $f^{\prime}$ (as defined in the usual way) exists on $B$ and
$\operatorname{rv}_{\lambda}\circ f^{\prime}$ is constant on $B$,

  2. (2)

$\operatorname{rv}_{\lambda}((f(x)-f(x_{0}))/(x-x_{0}))=\operatorname{rv}_{\lambda}(f^{\prime})$,

  3. (3)

for any open ball $B^{\prime}\subset B$, $f(B^{\prime})$ is either a point or
an open ball.

Note in particular that (1) implies that $|f^{\prime}|$ is constant on balls
1-next to $C$, since $\Gamma_{K}$ is a quotient of $\mathrm{RV}_{\lambda}$. We
will also use the following corollary. Recall that $\tilde{k}\subset K$ is a
lift of the residue field.

######  3.2.2 Corollary.

Let $\tilde{k}\subset K$ be a lift of the residue field of $K$. Let
$f\colon{\mathcal{O}}_{K}\to{\mathcal{O}}_{K}$ be a $\emptyset$-definable
function. Then, for all but finitely many $a\in\tilde{k}$ the following
property holds: for all $x,x_{0}\in a+{\mathcal{M}}_{K}$

$\lvert f(x)-f(x_{0})\rvert\leq\lvert x-x_{0}\rvert.$

###### Proof.

By the Jacobian property there exists a finite $\emptyset$-definable set
$C\subseteq{\mathcal{O}}_{K}$ such that for every ball $B$ which is 1-next to
$C$ there is a $\mu_{B}\in\Gamma_{K}$ such that for all $x,x_{0}\in B$

$\lvert f(x)-f(x_{0})\rvert=\mu_{B}\lvert x-x_{0}\rvert.$

Moreover, by [8, Cor. 3.1.6] we may additionally assume that if such a $B$ is
open, then $f(B)$ is an open ball of radius
$\mu_{B}\operatorname{rad}_{\mathrm{op}}(B)$. As $C$ is finite and the balls
$a+{\mathcal{M}}_{K}$ with $a\in\tilde{k}$ are pairwise disjoint, it follows
that only finitely many $a+{\mathcal{M}}_{K}$ contain a point of $C$. Thus all
but finitely many $a+{\mathcal{M}}_{K}$ are $1$-next to $C$. Now suppose
$\mu_{B}>1$ for some $B=a+{\mathcal{M}}_{K}$ which is disjoint from $C$. This
implies that $f(a+{\mathcal{M}}_{K})={\mathcal{O}}_{K}$ (and this is only
possible if the value group is discrete). In particular we can only have
$\mu_{B}>1$ for finitely many such $B$. Indeed, else $f^{-1}(y)$ would be
infinite for all $y\in{\mathcal{O}}_{K}$, contradicting [8, Lem. 2.8.1].
Hence, the desired property holds for cofinitely many $a\in\tilde{k}$. ∎

The second result we need is about Taylor approximation. For a function
$f:X\subset K\to K$ on an open set $X$ which is $r$-fold differentiable and
$x_{0}\in X$ we define the _$r$ -th order Taylor polynomial of $f$ at $x_{0}$_
to be as usual

$T_{f,x_{0}}^{\leq
r}(x)=T_{f,x_{0}}^{<r+1}(x)=\sum_{i=0}^{r}\frac{f^{(i)}(x_{0})}{i!}(x-x_{0})^{i}.$

The following result basically states that any definable function can be well
approximated by its Taylor polynomial up to some fixed order, at least away
from finitely many points.

######  3.2.3 Theorem ([8, Thm. 3.2.2], Taylor approximation of order $r$).

Let $f:K\to K$ be a $\emptyset$-definable function and fix a positive integer
$r$. Then there exists a finite $\emptyset$-definable set $C$ such that for
every ball $B$ 1-next to $C$, $f$ is $(r+1)$-fold differentiable on $B$,
$|f^{(r+1)}|$ is constant on $B$ and for $x,x_{0}\in B$ we have

$|f(x)-T_{f,x_{0}}^{\leq r}(x)|\leq|f^{(r+1)}(x_{0})(x-x_{0})^{r+1}|.$

We will also need results on cell decomposition. If ${\mathcal{T}}$ is a
1-h-minimal theory, then by [8, Prop. 4.3.3] there is an expansion of the
language by predicates on cartesian powers of $\mathrm{RV}$ such that the
resulting structure is still 1-h-minimal and we have
$\mathrm{acl}=\mathrm{dcl}$. In particular, we can typically assume that
$\mathrm{acl}=\mathrm{dcl}$ without any problems.

######  3.2.4 Definition.

Let $A\subset K$ be a parameter set. For $n\geq m$, denote by $\pi_{\leq
m}:K^{n}\to K^{m}$ the projection on the first $m$ coordinates. Let $X\subset
K^{n}$ be an $A$-definable set. Consider, for $i=1,...,n$, values
$j_{i}\in\\{0,1\\}$ and $A$-definable functions $c_{i}:\pi_{<i}(X)\to K$. Fix
also an $A$-definable set

$R\subseteq\prod_{i=1}^{n}(j_{i}\cdot\mathrm{RV}^{\times}),$

where $0\cdot\mathrm{RV}^{\times}=\\{0\\}$. We say that $X$ is an
$A$-definable cell if

$X=\\{x\in K^{n}\mid\operatorname{rv}(x_{i}-c_{i}(\pi_{<i}(x)))_{i=1,...,n}\in
R\\}.$

We call $X$ a cell of type $(j_{1},...,j_{n})$. The functions $c_{i}$ are
called the _cell centers_. A _twisted box_ of the cell $X$ is a set of the
form

$\\{x\in
K^{n}\mid\operatorname{rv}(x_{i}-c_{i}(\pi_{<i}(x)))_{i=1,...,n}=r\\},$

for $r\in R$.

By [8, Thm. 5.2.4], for a $\emptyset$-definable set $X\subseteq K^{n}$ there
always exists a $\emptyset$-definable _cell decomposition_ , i.e. a partition
of $X$ into finitely many $\emptyset$-definable cells $A_{\ell}$. We will need
the following variant. Recall that a function $f:X\subset K^{n}\to K^{m}$ is
said to be _1-Lipschitz_ if for all $x,x^{\prime}\in X$ we have

$|f(x)-f(x^{\prime})|\leq|x-x^{\prime}|,$

where we use the maximum norm on $K^{n}$.

######  3.2.5 Theorem ([8, Thm. 5.2.4, Add. 5], Cell decomposition).

Assume that $K$ carries a 1-h-minimal structure with
$\mathrm{acl}=\mathrm{dcl}$. Let $X\subset K^{n}$ be $\emptyset$-definable.
Then there exist a partition of $X$ into finitely many $\emptyset$-definable
sets $A_{\ell}$ such that for every $\ell$ there is some coordinate
permutation $\sigma_{\ell}:K^{n}\to K^{n}$ such that $\sigma_{\ell}(A_{\ell})$
is a cell of type $(1,...,1,0,...,0)$ and such that each component of each
center is 1-Lipschitz.

###  3.3. $T_{r}$-approximation

We recall some useful definitions and results from [5, §4.2] about
$T_{r}$-approximation. Let $K$ be a henselian valued field of
equicharacteristic zero which is 1-h-minimal in some language ${\mathcal{L}}$
expanding the language of valued fields.

######  3.3.1 Definition.

Let $U\subseteq K^{m}$ be an open set, let
$\psi=(\psi_{1},...,\psi_{n}):U\to\mathcal{O}_{K}^{n}$ be a function, and let
$r>0$ be an integer. We say that $\psi$ satisfies _$T_{r}$ -approximation_ if
for each $y\in U$ there is an $n$-tuple $T_{y}^{<r}$ of polynomials with
coefficients in $\mathcal{O}_{K}$ and of degree less than $r$ that satisfies

(3.3.1) $|\psi(x)-T_{y}^{<r}(x)|\leq|x-y|^{r}\qquad\text{for all }\;x\in U.$

Let $X$ be a definable subset of ${\mathcal{O}}_{K}^{n}$ of dimension $m$. We
say that a definable family $(\varphi_{i})_{i\in I}$ of functions
$\varphi_{i}:U_{i}\to X_{i}\subseteq\mathcal{O}_{K}^{n}$ is a _$T_{r}$
-parametrization of $X$_ if $X=\bigcup_{i\in I}X_{i}$ and each $\varphi_{i}$
is surjective and satisfies $T_{r}$-approximation.

######  3.3.2 Definition.

Let $\alpha=(\alpha_{1},...,\alpha_{m})\in{\mathbb{N}}^{m}$ and define
$|\alpha|=\alpha_{1}+\cdots+\alpha_{m}$. We define the following sets and
numbers:

  * •

$\Lambda_{m}(k):=\\{\alpha\in{\mathbb{N}}^{m};\;|\alpha|=k\\}$,

  * •

$\Delta_{m}(k):=\\{\alpha\in{\mathbb{N}}^{m};\;|\alpha|\leq k\\}$,

  * •

$L_{m}(k):=\\#\Lambda_{m}(k)$ and $D_{m}(k):=\\#\Delta_{m}(k)$

Note that $L_{m}(k)$ (resp. $D_{m}(k)$) is the number of monomials of degree
exactly (resp. at most) $k$ in $m$ variables.

Fix an integer $d$ and define, for all integers $n$ and $m$ such that $m<n$
the following integers.

(3.3.2) $\displaystyle\mu(n,d)$ $\displaystyle=D_{n}(d)\qquad$ $\displaystyle
r(m,d)$
$\displaystyle=\min\\{x\in{\mathbb{Z}};\;D_{m}(x-1)\leq\mu<D_{m}(x)\\}$
$\displaystyle V(n,d)$ $\displaystyle=\sum_{k=0}^{d}kL_{n}(k)\qquad$
$\displaystyle e(n,m,d)$ $\displaystyle=\sum_{k=1}^{r-1}kL_{m}(k)+r(\mu-
D_{m}(r-1)).$

To apply the determinant method, we need the following lemma.

######  3.3.3 Lemma.

Let $K$ be a henselian field of equicharacteristic $0$. Let $t$ be a pseudo-
uniformizer of $K$. Fix integers $\mu,r$ and $U$ an open subset of $K^{m}$
which is contained in a product of $m$ closed balls of radius $|t|^{\rho}$,
where $\rho\geq 0$ is an integer. Fix $x_{1},...,x_{\mu}\in U$ and functions
$\psi_{1},...,\psi_{\mu}:U\to K$. Assume that the $\psi_{i}$ satisfy
$T_{r}$-approximation on $U$ for some integer $r$ with

$D_{m}(r-1)\leq\mu<D_{m}(r).$

Then

$|\det(\psi_{i}(x_{j}))_{i,j})|\leq|t|^{\rho e}.$

###### Proof.

Straightforward adapation from [4, Lem. 3.3.1]. ∎

##  4\. Counting rational points on transcendental and algebraic curves

In this section we prove Theorems 2.2.1, 2.2.2, and 2.2.3. Throughout, let $K$
be an equicharacteristic zero henselian valued field, equipped with a
1-h-minimal ${\mathcal{L}}$-structure, for some language ${\mathcal{L}}$
expanding the language of valued fields. We assume that
$\mathrm{acl}=\mathrm{dcl}$ in $K$ and that the subgroup of $b$-th powers in
the residue field $k^{\times}$ has finite index, for some $b>1$. Fix a lift
$\tilde{k}$ of the residue field and fix a pseudo-uniformizer $t$.

###  4.1. $T_{r}$-parametrizations

Crucial to our approach is the following theorem, which asserts the existence
of $T_{r}$-parametrizations for definable planar curves.

######  4.1.1 Theorem.

Let $K$ be an equicharacteristic zero valued field, equipped with a
1-h-minimal ${\mathcal{L}}$-structure. Assume that $\mathrm{acl}=\mathrm{dcl}$
in $K$ and that the subgroup of $b$-th powers in $k^{\times}$ has finite
index, for some $b>1$. Let $Y\subset K^{n}$ be a definable set. Fix a positive
integer $r$ and let $C\subset Y\times{\mathcal{O}}_{K}^{2}$ be a definable set
such that for every $y\in Y$, $C_{y}$ is a curve. Then there exist finitely
many maps $\phi_{1},\ldots,\phi_{N}:Y\times{\mathcal{O}}_{K}\to C$ such that
for every $y\in Y$, $\phi_{1,y},\ldots,\phi_{N,y}$ form a
$T_{r}$-parametrization for $C_{y}$.

To prove this theorem, we start with a $T_{1}$-parametrization of our curve,
which exists because of Theorem 3.2.5. To move from a $T_{1}$-parametrization
to a $T_{r}$-parametrization for curves we will use power substitutions.

######  4.1.2 Lemma.

Let $X\subseteq K$ and let $f:X\to{\mathcal{O}}_{K}$ be a
$\emptyset$-definable $1$-Lipschitz function. Fix a positive integer $r$. Then
there exists a finite $\emptyset$-definable set $C$ such that the following
holds. Let $B$ be a ball $1$-next to $C$ contained in ${\mathcal{O}}_{K}$, say
$1$-next to $c\in C$. For $a,b\in{\mathcal{O}}_{K}$ consider the map
$p_{r}:K\to K:x\mapsto a(x-c)^{r}+b$. If $D$ is any open ball not containing
$0$ with $p_{r}(D)\subseteq B$ then $f\circ p_{r}$ satisfies
$T_{r}$-approximation on $D$ with respect to its Taylor polynomial. Moreover,
for $y\in D$ and $j=1,...,r$ there is the bound

$|\partial^{j}(f\circ p_{r})(y)|\leq|y|^{r-j}.$

###### Proof.

Use Theorem 3.2.1 and Theorem 3.2.3 to find a finite $\emptyset$-definable set
$C$ such that $f$ satisfies Taylor approximation up to order $r$ on balls
$1$-next to $C$ and such that the first $r$ derivatives of $f$ satisfy the
Jacobian property on balls $1$-next to $C$. Without loss of generality, let
$c=0$, $a=1$ and $b=0$ and fix $x_{0}\in D$. Then $x_{0}^{r}\in B$ and
$\operatorname{rad}_{\mathrm{op}}B=|x_{0}|^{r}$ since $B$ is $1$-next to $0$.
The fact that $f$ is $1$-Lipschitz gives that

$|f^{\prime}(x_{0}^{r})|\leq 1.$

By the Jacobian property, the first $r$ derivatives of $f$ all have a constant
norm on $B$. Thus for $i\leq r$,

$f^{(i)}(B)\subseteq\\{y\in K\mid|y|=|f^{(i)}(x_{0}^{r})|\\},$

and hence
$\operatorname{rad}_{\mathrm{op}}f^{(i)}(B)\leq|f^{(i)}(x_{0}^{r})|$. On the
other hand, the Jacobian property yields that

$|f^{(i)}(x_{0}^{r})|=\frac{\operatorname{rad}_{\mathrm{op}}f^{(i-1)}(B)}{\operatorname{rad}_{\mathrm{op}}B}\leq\frac{|f^{(i-1)}(x_{0}^{r})|}{|x_{0}^{r}|}.$

Using induction and the fact that $f$ is $1$-Lipschitz this gives for $1\leq
i\leq r$ that

(4.1.1) $|f^{(i)}(x_{0}^{r})x_{0}^{r(i-1)}|\leq 1.$

Let $x\in D$. Then $x^{r}$ is in the same ball $1$-next to $0$ as $x_{0}^{r}$
and so $|x^{r}|=|x_{0}^{r}|$. Since both $f$ and $p_{r}$ have Taylor
approximation up to order $r$ on their respective domains $B$ and $D$ we can
conclude by [4, Lem. 3.2.7]. ∎

###### Proof of Theorem 4.1.1.

By enlarging $r$ if necessary, we may assume that $r$ is a power of $b$. Using
the cell decomposition theorem 3.2.5 uniformly in $y\in Y$ we obtain for every
$y\in Y$ finitely many sets $P_{i,y}$ whose union is $C_{y}$ such that every
$P_{i,y}$ is, after a coordinate permutation, a $(1,0)$-cell or a $(0,0)$-cell
with $1$-Lipschitz centers. Since everything below works uniformly in $y$, we
drop the subscript $y$ from now on.

The $(0,0)$-cells are just singletons, so let us focus on one of the
$(1,0)$-cells, say $P_{\ell}$. After a coordinate transformation and a
translation we may assume that $P_{\ell}$ is the graph of a $1$-Lipschitz map

$\phi:P\subset{\mathcal{O}}_{K}\to{\mathcal{O}}_{K},$

where $P$ is a cell with center $0$. The group of $r$-th powers in
$k^{\times}$ has finite index in $k^{\times}$, since $r$ is a power of $b$.
Let $a_{1},...,a_{m}\in{\mathcal{O}}_{K}^{\times}$ reduce to representatives
for the cosets of $(k^{\times})^{r}$ in $k^{\times}$. For $i=1,...,m$ and
$j=0,...,r-1$ let

$D_{i,j}=\\{y\in K\mid a_{i}t^{j}y^{r}\in P\\}.$

Then the finitely many maps

$p_{i,j}:D_{i,j}\to P:y\mapsto a_{i}t^{j}y^{r}$

cover $P$. Now by Lemma 4.1.2, we can find a further subdivision of $P$ such
that the $\phi\circ p_{i,j}$ are all $T_{r}$ on open balls contained in
$D_{i,j}$. We prove that actually $\phi\circ p_{i,j}$ even has
$T_{r}$-approximation on all of $D_{i,j}$. So let $x,y\in D_{i,j}$. Since
$D_{i,j}$ is a cell with center $0$, if
$\operatorname{rv}(x)=\operatorname{rv}(y)$ then $x$ and $y$ are in the same
ball contained in $D_{i,j}$ and we are done. So assume that
$\operatorname{rv}(x)\neq\operatorname{rv}(y)$. Then

$\displaystyle\qquad|\phi(a_{i}t^{j}x^{r})-T_{\phi\circ
p_{i,j},y}^{<r}(x)|\leq\max\\{|\phi(a_{i}t^{j}x^{r})-\phi(a_{i}t^{j}y^{r})|,|\phi(a_{i}t^{j}y^{r})-T_{\phi\circ
p_{i,j},y}^{<r}(x)|\\}.$

For the first term, use that $\phi$ is $1$-Lipschitz to obtain

$|\phi(a_{i}t^{j}x^{r})-\phi(a_{i}t^{j}y^{r})|\leq|x^{r}-y^{r}|\leq\max\\{|x|^{r},|y|^{r}\\}=|x-y|^{r},$

since $\operatorname{rv}(x)\neq\operatorname{rv}(y)$. For the second term, we
use the bound provided by Lemma 4.1.2 to get that

$\displaystyle|\phi(a_{i}t^{j}y^{r})-T_{\phi\circ p_{i,j},y}^{<r}(x)|$
$\displaystyle\leq\max_{\ell=1,...,r-1}\left|\frac{\partial^{\ell}(\phi\circ
p_{i,j})(y)}{\ell!}(x-y)^{\ell}\right|$
$\displaystyle\leq\max_{\ell=1,...,r-1}|y|^{r-\ell}|x-y|^{\ell}\leq|x-y|^{r}.$

So $\phi\circ p_{i,j}$ satisfies $T_{r}$-approximation on all of $D_{i,j}$. In
conclusion, the maps

$\psi_{ij}:D_{ij}\to C:y\mapsto(p_{ij}(y),\phi(p_{ij}(y))$

all have $T_{r}$-approximation and their images cover $P_{\ell}$. ∎

###  4.2. Transcendental curves

The following lemma is an adapted version of [5, Lem. 5.1.3]. We use it to
capture rational points of bounded height in a small ball in a single
hypersurface.

######  4.2.1 Lemma.

Fix integers $d,m,n$ with $m<n$ and consider $r,V,e$ as defined in equation
3.3.2. Let $s$ be a positive integer, let $U\subseteq{\mathcal{O}}_{K}^{m}$
and suppose $\psi=(\psi_{1},...,\psi_{n}):U\to{\mathcal{O}}_{K}^{n}$ satisfies
$T_{r}$-approximation. For $\alpha>\frac{sV}{e}$ a positive integer, denote by
$p:{\mathcal{O}}_{K}^{m}\to({\mathcal{O}}_{K}/(t^{\alpha}))^{m}$ the
projection map. Then for any fibre $B$ of $p$, the image $\psi(B\cap U)_{s}$
is contained in an algebraic hypersurface of degree at most $d$. Moreover,
$V/e$ goes to $0$ as $d$ goes to infinity.

###### Proof.

Let $B\subseteq{\mathcal{O}}_{K}^{m}$ be a product of closed balls of radius
$|t|^{\alpha}$, i.e. a fibre of the map $p$, and take points
$P_{1},...,P_{\mu}$ in $\psi(B\cap U)_{s}$. Take $x_{i}\in B\cap U$ such that
$\psi(x_{i})=P_{i}$. Consider the determinant

$\Delta=\det((\psi(x_{i}))^{j})_{1\leq i\leq\mu,j\in\Delta_{n}(d)}.$

For $j\in\Delta_{n}(d)$, the notation $(y_{1},...,y_{n})^{j}$ is to be
interpreted as $\prod_{i}y_{i}^{j_{i}}$. Since $\psi$ satisfies
$T_{r}$-approximation, Lemma 3.3.3 gives that ${\rm ord}_{t}(\Delta)\geq\alpha
e$. Since the $P_{i}$ are in $\tilde{k}[t]^{n}_{s}$, if $\Delta$ were non-zero
then ${\rm ord}_{t}(\Delta)\leq sV$. But $\alpha>\frac{sV}{e}$ so that
$\Delta=0$.

Now we use the determinant method. Since $\Delta=0$, the $\mu$ vectors
$((\psi(x_{i}))^{j})_{j}$ are linearly dependent. This implies that there
exists some algebraic hypersurface of degree at most $d$ passing through all
of the points $\psi(x_{i})$. Since this holds for any $\mu$ points in
$\psi(B\cap U)_{s}$ we can find such a hypersurface containing all of
$\psi(B\cap U)_{s}$. The last fact follows from an easy explicit calculation,
see e.g. [12, p. 212]. ∎

We need one more projection lemma to reduce to the planar case. Let us call a
set in $K^{2}$ _non-algebraic up to degree $d$_ if it has a finite
intersection with every algebraic curve of degree at most $d$.

######  4.2.2 Lemma.

Let $C\subset K^{n}$ be a definable transcendental curve and fix a positive
integer $d$. Then there exists a finite definable partition of $C$ into sets
$C_{i}$, together with coordinate projections $\pi_{i}:C_{i}\to K^{2}$ such
that $\pi_{i}|_{C_{i}}$ is a bijection onto its image, and $\pi_{i}(C_{i})$ is
non-algebraic up to degree $d$.

###### Proof.

By the cell decomposition theorem 3.2.5, we may partition $C$ into finitely
many definable sets $C_{i}$ such that after a coordinate permutation, $C_{i}$
is a $(1,0,...,0)$ cell with $1$-Lipschitz centers. (We may disregard the
finitely many $(0,...,0)$-cells since there are just points.) In other words,
$C_{i}$ is the graph of a map

$\phi:P\to K^{n-1}:x\mapsto(\phi_{1}(x),...,\phi_{n-1}(x)),$

where all $\phi_{j}$ are $1$-Lipschitz, and $P\subset K$ is a $1$-cell. Denote
by $A_{j}\subset P$ the set of $x\in P$ such that in some neighbourhood of
$x$, the graph of $\phi_{j}$ is non-algebraic up to degree $d$. Then the
$A_{j}$ are definable sets, and they cover $P$, since else $\phi$ would not be
transcendental. Thus we can further partition $C_{i}$ into the graphs of
$\phi$ over every $A_{j}$. Over $A_{j}$, projection onto the first and $j$-th
coordinate gives the desired conclusion. ∎

######  4.2.3 Remark.

The use of this lemma can be avoided by working with cylinders over $X$ for
each possible projection $K^{n}\to K^{2}$, see [4, p. 45].

With this, we can prove our main result. Recall that the strategy of the proof
is as follows. Using $T_{r}$-parametrizations, we represent $C$ as a finite
union of graphs of functions satisfying $T_{r}$-approximation. Then Lemma
4.2.1 gives a hypersurface catching all rational points on $C$ of height at
most $s$. The fact that $C$ is transcendental then gives the desired
conclusion.

###### Proof of Theorem 2.2.1.

Recall the definition of $V,e$ and $r$ from equation 3.3.2 and recall that
$V/e$ goes to zero as $d$ goes to infinity, by Lemma 4.2.1. Take $d$ such that

$\frac{V}{e}<\varepsilon.$

Up to enlarging $r$ if necessary, we may assume that $r$ is a power of $b$. By
applying Lemma 4.2.2 we may assume without loss of generality that $C$ is a
planar curve in ${\mathcal{O}}_{K}^{2}$ which is non-algebraic up to degree
$d$.

By Theorem 4.1.1 there exist finitely many maps
$\psi_{i}:U_{i}\subset{\mathcal{O}}_{K}\to C$ which together form a
$T_{r}$-parametrization of $C$. Let us focus on one such $\psi_{i}$. By our
construction, we may apply Lemma 4.2.1 with $\alpha=\lceil
s\varepsilon\rceil$. This yields that $\psi_{i}(U_{i}\cap B)_{s}$ is contained
in an algebraic hypersurface $X$ of degree at most $d$, for any closed ball
$B\subset{\mathcal{O}}_{K}$ of radius $|t|^{\alpha}$. Since $\psi_{i}(U_{i})$
is contained in $C$, and since $C$ is non-algebraic up to degree $d$, this
intersection $X\cap C$ is finite. Even more, by uniform finiteness in
definable families [8, Lem. 2.5.2], the intersection of $X$ with $C$ is
uniformly bounded (over all such $X$) by some integer $N$. Thus the counting
dimension of $C$ is bounded by

$(NN^{\prime},1,\lceil\varepsilon s\rceil),$

where $N^{\prime}$ is the total number of sets $U_{i}$ required in the cell
decomposition for $C$.

Finally, to prove a uniform upper bound on counting dimension in definable
families, assume that $(C_{y})_{y}$ is a definable family of transcendental
curves, for $y\in Y\subseteq K^{m}$ with $Y$ definable. Then the above proof
can easily be made uniform in $C_{y}$. Indeed, by Lemma 4.2.2 we can assume
that every $C_{y}$ is a planar curve which is non-algebraic up to degree $d$.
The number of maps for a $T_{r}$-parametrization can be uniformly bounded in
$y$ since Theorem 4.1.1 is uniform in families. Similarly, the intersection of
an algebraic curve of degree at most $d$ with any $C_{y}$ is finite, and thus
uniformly bounded by [8, Lem. 2.5.2]. These two facts give the desired
conclusion. ∎

###  4.3. Linear upper bounds are optimal

We now prove Theorem 2.2.2. This shows that the bound in Theorem 2.2.1 is
optimal, in the sense that one cannot replace the last component
$\lceil\varepsilon\cdot s\rceil$ by a sublinear function $e(s)$, even if we
allow $N(s)$ to be completely arbitrary.

We first recall the notion of rings of strictly convergent power series
${\mathcal{O}}_{K}\langle x_{1},\dots,x_{n}\rangle$. By definition, their
elements are those power series $\sum_{i\in{\mathbb{N}}}a_{i}x^{i}$ with
coefficients in ${\mathcal{O}}_{K}$ such that $a_{i}\to 0$, when $\lvert
i\rvert\to\infty$. Each $f\in{\mathcal{O}}_{K}\langle
x_{1},\dots,x_{n}\rangle$ can be naturally considered as a function
${\mathcal{O}}_{K}^{n}\to K$ and then also as a function $f\colon K^{n}\to K$,
after extending by zero. By [8, Thm. 6.2.1] there exists a $1$-h-minimal
structure on $K$ in which these functions are definable. Moreover, by [8,
Prop. 4.3.3], there exists such a structure in which
$\mathrm{acl}=\mathrm{dcl}$.

###### Proof of Theorem 2.2.2.

We work in the structure on $K\coloneqq k((t))$ as outlined above, in which
all functions ${\mathcal{O}}_{K}^{n}\to{\mathcal{O}}_{K}$ defined by a
strictly convergent power series are definable.

First fix any strictly increasing continuous function
$\delta\colon{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}_{>0}$ such that
$e(s)\leq\delta(s)$ for all $s\in{\mathbb{N}}$, with
$\lim_{s\to\infty}\delta(s)=+\infty$ and $\lim_{s\to+\infty}\delta(s)/s=0$.
Then choose any strictly increasing function $F\colon{\mathbb{R}}_{\geq
0}\to{\mathbb{R}}_{>0}$ with $F({\mathbb{N}})\subseteq{\mathbb{N}}$ and such
that for all $s\in{\mathbb{N}}$

$F(\delta(s))>N(s).$

Next, take any strictly increasing sequence of natural numbers $(N_{n})_{n}$,
with the property that for all $n\in{\mathbb{N}}$

$3N_{n-1}^{2}F(N_{n-1})<\frac{\delta^{-1}(N_{n})-1}{N_{n}}.$

Such a sequence exists since $\lim_{u\to\infty}\delta^{-1}(u)/u=\infty$ as
$\delta$ is sublinear. From these data, we construct
$f\in{\mathcal{O}}_{K}\langle x\rangle$ as

$f(x)=\sum_{n=0}^{\infty}t^{N_{n}}x^{N_{n}}\prod_{i,\ell=1}^{N_{n}}\prod_{j=1}^{F(N_{n})}(x-i-
jt^{\ell}).$

Letting the sequence $(N_{n})_{n}$ grow even faster, if necessary, we may
assume that for each $d\in{\mathbb{N}}$ there is some $n_{d}$ such that for
all $n\geq n_{d}$ we have $N_{n}>d(N_{n-1}+N_{n-1}^{2}F(N_{n-1}))$. Hence, for
$M=N_{n-1}+N_{n-1}^{2}F(N_{n-1})$, the order of contact between $f$ and its
$M$-th order Taylor approximation exceeds $dM$. Bézout’s theorem thus implies
that $f$ cannot be algebraic of any degree $d\in{\mathbb{N}}$ (as in [1, Prop.
1]). Hence the graph $C$ of $f\colon{\mathcal{O}}_{K}\to K$ is a definable
transcendental curve in $K^{2}$.

We now show that the counting dimension of $C$ is not bounded by $(N,1,e)$. So
let $g:K^{2}\to{\mathcal{O}}_{K}$ be any definable map. We will show that the
composition

$C_{s}\xrightarrow{g}{\mathcal{O}}_{K}\xrightarrow{\operatorname{proj}}\frac{{\mathcal{O}}_{K}}{(t^{e(s)})}$

has a fibre of size strictly larger than $N(s)$ for some sufficiently large
$s$.

Take any $n$ and let $s$ be such that

$\delta(s)\leq N_{n}<\delta(s+1).$

For each $i,j,\ell\in{\mathbb{N}}$, we have by construction that
$f(i+jt^{\ell})\in k[t]$. By our choice of $s$, we moreover have that
$t$-degree of $f(N_{n}+jt^{N_{n}})$ is strictly smaller than $s$, when $1\leq
j\leq F(N_{n})$. Indeed, it follows from the construction of $(N_{n})_{n}$
that

$\displaystyle{\rm deg}_{t}(f(N_{n}+jt^{N_{n}}))$ $\displaystyle\leq
N_{n-1}+N_{n}N_{n-1}+N_{n}N_{n-1}^{2}F(N_{n-1})$ $\displaystyle\leq
3N_{n}N_{n-1}^{2}F(N_{n-1})$ $\displaystyle<\delta^{-1}(N_{n})-1$
$\displaystyle<s.$

Now define

$S\coloneqq\\{(N_{n}+jt^{N_{n}},f(N_{n}+jt^{N_{n}}))\mid 1\leq j\leq
F(N_{n})\\}$

and note that $S\subseteq C_{s}$ by the above computation.

Define $h\colon{\mathcal{O}}_{K}\to{\mathcal{O}}_{K}\colon x\mapsto
g(x,f(x))$. By Corollary 3.2.2 we may assume that, possibly after increasing
$n$ (and $s$),

$\lvert h(N_{n}+jt^{N_{n}})-h(N_{n}))\rvert\leq\lvert t^{N_{n}}\rvert$

for all $j\in{\mathbb{N}}$. As $N_{n}\geq\delta(s)\geq e(s)$, this implies
that all points of $S$ belong to the fibre at $N_{n}$ of the composition

$C_{s}\xrightarrow{g}{\mathcal{O}}_{K}\xrightarrow{\operatorname{proj}}\frac{{\mathcal{O}}_{K}}{(t^{e(s)})}.$

But by construction of $F$, the inequality $N_{n}\geq\delta(s)$ also implies
$F(N_{n})>N(s)$. As $\\#S=F(N_{n})$, we have found a fibre containing more
than $N(s)$ elements of $C_{s}$. ∎

###  4.4. Algebraic curves

In this section we prove Theorem 2.2.3, following along the lines of [4, Sec.
5]. We need some results on Hilbert functions.

For $r$ a positive integer, denote by $K[x_{0},...,x_{n}]_{r}$ the space of
homogeneous degree $r$ polynomials. For $I$ a homogeneous ideal in
$K[x_{0},...,x_{n}]$, let $I_{r}=K[x_{0},...,x_{n}]_{r}\cap I$ and denote by
$H_{I}(r)=\dim K[x_{0},...,x_{n}]_{r}/I_{r}$ the Hilbert function of $I$. Let
$<$ be the monomial order on $K[x_{0},...,x_{n}]$ defined by
$x^{\alpha}<x^{\beta}$ if $|\alpha|<|\beta|$ or $|\alpha|=|\beta|$ and
$\alpha_{i}>\beta_{i}$ for some $i$ and $\alpha_{j}=\beta_{j}$ for $j<i$.
After reordering the variables, this is the graded reverse lexicographic order
on monomials. Denote by $\operatorname*{LT}(I)$ the ideal of leading terms of
$I$, where the leading term of a homogeneous element $p(x)$ of
$K[x_{0},...,x_{n}]$ is the monomial in $p(x)$ which is maximal for $<$. Then
$I$ and $\operatorname*{LT}(I)$ have the same Hilbert functions, by [9, Ch. 9
Prop. 3.9]. For $i\in\\{0,...,n\\}$ define

$\sigma_{I,i}(r)=\sum_{|\alpha|=r,x^{\alpha}\notin\operatorname*{LT}(I)}\alpha_{i}$

and note that $rH_{I}(r)=\sum_{i}\sigma_{I,i}(r)$. Let $X$ be an irreducible
variety in ${\mathbb{P}}^{n}_{K}$ of degree $d$ and dimension $m$, with
homogeneous ideal $I$. The Hilbert function $H_{I}(r)$ of $I$ agrees with the
Hilbert polynomial $P_{X}(r)$ of $X$, for $r$ sufficiently large. Recall that
this is a degree $m$ polynomial whose leading coefficient is $d/m!$. By [3],
for $i=0,...,n$ there are real numbers $a_{I,i}\geq 0$ such that

$\frac{\sigma_{I,i}(r)}{rH_{I}(r)}=a_{I,i}+O_{n,d}(1/r),\quad\text{ for
}r\to\infty.$

Note also that $a_{I,0}+...+a_{I,n}=1$. We can now prove Theorem 2.2.3.

###### Proof of Theorem 2.2.3.

We have an irreducible algebraic curve $C$ in ${\mathbb{A}}_{K}^{2}$ of degree
$d$. Put $C^{\prime}=C(K)\cap{\mathcal{O}}_{K}^{2}$. Consider the embedding

$\iota:{\mathbb{A}}_{K}^{2}\to{\mathbb{P}}_{K}^{2}:(x,y)\mapsto(1:x:y)$

and let $I$ be the homogeneous ideal of the closure of $\iota(C)$ in
${\mathbb{P}}_{K}^{2}$. Let $\delta$ be a positive integer, which we will
choose later depending on $s$, and define

$M(\delta)=\\{j\in{\mathbb{N}}^{3}\mid|j|=\delta,x^{j}\notin\operatorname*{LT}(I)\\}.$

Let $\mu=\\#M(\delta)=H_{I}(\delta),\sigma_{i}=\sigma_{I,i}(\delta)$ for
$i=0,1,2$ and put $e=\mu(\mu-1)/2$. By Theorem 4.1.1 there exist finitely many
maps $\phi_{1},...,\phi_{N}:Y_{i}\subset{\mathcal{O}}_{K}\to C^{\prime}$
forming a $T_{r}$-parametrization of $C^{\prime}$. Let $B_{\alpha}$ be a
closed ball in ${\mathcal{O}}_{K}$ of radius $|t|^{\alpha}$ for some integer
$\alpha$. Fix a positive integer $s$, take points $y_{1},...,y_{\mu}$ in
$(\phi_{i}(B_{\alpha}\cap Y_{i}))_{s}$ and consider the determinant

$\Delta=\det\left(\iota(y_{i})^{j}\right)_{j\in M(\delta),1\leq i\leq\mu}.$

By Lemma 3.3.3 we have that $|\Delta|\leq|t|^{\alpha e}$. On the other hand,
since $\iota(y_{i})$ has coordinates which are polynomials of degree $<s$, we
find that $\Delta$ is in $\tilde{k}[t]$ of degree

${\rm deg}_{t}\Delta\leq(s-1)(\sigma_{1}+\sigma_{2}).$

Therefore, if we take $\alpha>(s-1)(\sigma_{1}+\sigma_{2})/e$, then
$\Delta=0$. As in the proof of Lemma 4.2.1, using the determinant method, we
can find a polynomial $H$ in two variables with coefficients in
$\tilde{k}[t]$, and exponents in $M(\delta)$ which vanishes on
$(\phi_{i}(B\cap Y_{i}))_{s}$. Since the exponents of $H$ lie in $M(\delta)$,
we also see that $H$ does not vanish identically on $C$. By Bézout’s theorem,
the intersection of $H=0$ and $C$ consists of at most $\delta d$ points.

We want to conclude by taking $\alpha=\lceil s/d\rceil$, so we look for a
suitable $\delta$ now. Similarly to the proof of [4, Thm. 5.1.3], one obtains
that

$\sigma_{i}/e=2\alpha_{i}/d+O_{d}(\delta^{-1}).$

By a result of Salberger [14, Lem. 1.12], it follows that

$\frac{\sigma_{1}+\sigma_{2}}{e}\leq\frac{1}{d}+O_{d}(\delta^{-1}).$

Hence we may take $\delta=sO_{d}(1)$ so that

$\frac{(s-1)(\sigma_{1}+\sigma_{2})}{e}<\lceil s/d\rceil=\alpha.$

By Proposition 2.3.1 we can find a definable map
$f:C^{\prime}\to{\mathcal{O}}_{K}$ such that the composition

$C_{s}\to{\mathcal{O}}_{K}\to\frac{{\mathcal{O}}_{K}}{(t^{\lceil s/d\rceil})}$

has finite fibres of size at most $Nd\delta=NsO_{d}(1)$, for every $s$. Now,
the number of cells $N$ required in the $T_{r}$-parametrization of $C$ can be
made uniform in definable families. Since the set of degree $d$ curves in
$K^{2}$ is a definable family, we may assume that $N=O_{d}(1)$. So we conclude
that the counting dimension of $C$ is bounded by

$(sO_{d}(1),1,\lceil s/d\rceil).\qed$

######  4.4.1 Example.

We show that one cannot improve the last component in the counting dimension
for algebraic curves. Let $K$ be any equicharacteristic zero valued field
equipped with a 1-h-minimal structure. Fix a pseudo-uniformizer $t$ and a lift
$\tilde{k}$ of the residue field. Denote by $C$ the curve in
${\mathcal{O}}_{K}^{2}$ defined by $y=x^{d}$. We claim that the counting
dimension of $C$ is not bounded by $(N(s),1,e(s))$ for any functions
$N,e:{\mathbb{N}}\to{\mathbb{N}}$ for which $e(s)<\lceil s/d\rceil$ when $s$
is sufficiently large. Let $f:C\to{\mathcal{O}}_{K}$ be any definable map.

Take $s^{\prime}=sd+1$ sufficiently large such that $e(s^{\prime})<\lceil
s^{\prime}/d\rceil=s+1$. In particular note that $e(s^{\prime})\leq s$. Define
the map $g:{\mathcal{O}}_{K}\to C:x\mapsto(x,x^{d})$ and put $h=f\circ
g:{\mathcal{O}}_{K}\to{\mathcal{O}}_{K}$. By Corollary 3.2.2, there exists an
$a\in\tilde{k}$, such that for any $b\in\tilde{k}$ we have that

$|h(a)-h(a+bt^{s})|\leq|t^{s}|.$

This implies that $h(a)\equiv h(a+bt^{s})$ in
${\mathcal{O}}_{K}/(t^{e(s^{\prime})})$, for all $b\in\tilde{k}$. Finally, by
noting that $g(a+bt^{s})$ lies in $C_{s^{\prime}}$, one sees that the map

$C_{s^{\prime}}\xrightarrow{f}{\mathcal{O}}_{K}\xrightarrow{\operatorname{proj}}\frac{{\mathcal{O}}_{K}}{(t^{e(s^{\prime})})}$

has an infinite fibre.

##  5\. Curves with uniformly bounded counting dimension

We now consider an analytic structure on ${\mathbb{Q}}_{p}((t))$ where each
$\emptyset$-definable curve $C$ has an associated constant
$N_{C}\in{\mathbb{N}}$ such that $\\#\text{-}\dim(C_{s})\leq(N_{C},1,1)$.
Contrast this with Theorem 2.2.2, where we produced curves, definable in some
analytic structure, whose counting dimension can not be bounded by any
constant triple $(N,1,1)$. The stronger upper bounds in this section result
from working in a more restricted analytic structure. The essential difference
with the setting of Theorem 2.2.2 is that we now only add function symbols for
power series whose coefficients do not involve the uniformizer $t$.

Throughout this section we will take $t\in{\mathbb{Q}}_{p}((t))$ as our chosen
pseudo-uniformizer and
$\tilde{k}={\mathbb{Q}}_{p}\subseteq{\mathbb{Q}}_{p}((t))$ as our lift of the
residue field.

######  5.0.1 Definition.

Let ${\mathcal{L}}_{{\mathbb{Z}}_{p}\langle x\rangle}$ be the language
expanding the language of valued fields
${\mathcal{L}}_{\mathrm{val}}=\\{0,1,+,\cdot,{\mathcal{O}}_{K}\\}$ by

  1. (1)

a binary function symbol “$-$” and a unary function symbol $(\cdot)^{-1}$,

  2. (2)

a unary relation symbol ${\mathcal{O}}_{K,{\mathrm{fine}}}$,

  3. (3)

$n$-ary function symbols for the elements of the rings of strictly convergent
power series ${\mathbb{Z}}_{p}\langle x_{1},\dots,x_{n}\rangle$, for
$n\in{\mathbb{N}}$.

The field $K={\mathbb{Q}}_{p}((t))$ admits a natural
${\mathcal{L}}_{{\mathbb{Z}}_{p}\langle x\rangle}$-structure. We interpret
${\mathcal{O}}_{K,{\mathrm{fine}}}$ as the valuation ring
${\mathbb{Z}}_{p}+t{\mathbb{Q}}_{p}[[t]]$ and ${\mathcal{O}}_{K}$ as its
equicharacteristic zero coarsening ${\mathbb{Q}}_{p}[[t]]$. Each function
symbol $f\in{\mathbb{Z}}_{p}\langle x_{1},\dots,x_{n}\rangle$ is interpreted
naturally as a function ${\mathcal{O}}_{K,{\mathrm{fine}}}^{n}\to K$. Finally,
“$-$” and $(\cdot)^{-1}$ are just subtraction and inversion on $K$, where the
latter is extended by by $0^{-1}=0$. Note that ${\mathbb{Q}}_{p}((t))$,
equipped with the valuation ring ${\mathcal{O}}_{K}$, is $1$-h-minimal for
this structure by [8, Thm. 6.2.1].

######  5.0.2 Theorem.

For each transcendental curve $C\subseteq{\mathbb{Q}}_{p}((t))^{n}$,
$\emptyset$-definable in ${\mathcal{L}}_{{\mathbb{Z}}_{p}\langle x\rangle}$,
there is some constant $N_{C}\in{\mathbb{N}}$ such that

$\\#\text{-}\dim(C_{s})\leq(N_{C},1,1).$

######  5.0.3 Remark.

As before, the constant $N_{C}$ can be made uniform in definable families.

######  5.0.4 Lemma.

For all $\lambda\in{\mathbb{Z}}_{p}^{\times}$ the map

$\tau_{\lambda}\colon{\mathbb{Q}}_{p}((t))\to{\mathbb{Q}}_{p}((t))\colon\sum_{j=k}^{+\infty}a_{j}t^{j}\mapsto\sum_{j=k}^{+\infty}a_{j}(\lambda
t)^{j}$

is an ${\mathcal{L}}_{{\mathbb{Z}}_{p}\langle x\rangle}$-automorphism.

###### Proof.

Fix some $\lambda\in{\mathbb{Z}}_{p}^{\times}$. It is clear that
$\tau_{\lambda}$ fixes the constants $0,1$ and that the relations
$x\in{\mathbb{Q}}_{p}[[t]]$ and $x\in{\mathbb{Z}}_{p}+t{\mathbb{Q}}_{p}[[t]]$
are invariant under $\tau_{\lambda}$. Similarly, it is straightforward to
verify that $\tau_{\lambda}$ commutes with addition, multiplication and
inversion.

It thus remains to check that $\tau_{\lambda}$ commutes with all function
symbols $f$ in ${\mathbb{Z}}_{p}\langle x_{1},\dots,x_{n}\rangle$. Take
$(x_{i})_{i=1}^{n}=(\sum_{j=k_{i}}^{+\infty}a_{ij}t^{j})_{i=1}^{n}\in{\mathbb{Q}}_{p}((t))^{n}$.
Then $f(x)$ is computed as follows.

$f(x)=\begin{cases}0&\text{ if
}x_{i}\notin{\mathbb{Z}}_{p}+t{\mathbb{Q}}_{p}[[t]]\text{ for some }i,\\\
\sum_{s=0}^{+\infty}f_{s}((a_{ij})_{i,j})t^{s}&\text{ else},\end{cases}$

where each $f_{s}$ is some quasi-homogeneous polynomial of degree $s$ in
variables $a_{ij}$, each of weight $j$, for $1\leq i\leq n$ and $0\leq j\leq
s$. This precisely means that
$f_{s}((\lambda^{j}a_{ij})_{i,j})=f_{s}((a_{i,j})_{i,j})\lambda^{s}$. It
follows that $f$ commutes with $\tau_{\lambda}$. ∎

We will need that transcendental curves only have finite intersection with any
semialgebraic curve: one-dimensional subsets of $K^{n}$ definable in
${\mathcal{L}}_{\mathrm{val}}\cup K$. This follows from the fact that
semialgebraic curves are locally algebraic, as made precise by the lemma
below.

######  5.0.5 Lemma.

Let $K$ be a valued field with valuation ring ${\mathcal{O}}_{K}$. Let $C$ be
a transcendental curve in $K^{n}$. Then, for any $K$-definable curve $X$ in
${\mathcal{L}}_{\mathrm{val}}$, the intersection $C\cap X$ is finite.

###### Proof.

By valued field-quantifier elimination for henselian valued fields [10, Prop.
4.3], $C$ is a finite union of sets $V\cap S$, where $V$ is the vanishing
locus of some polynomials with coefficients in $K$ and $S$ is of the form
$(\operatorname{rv}(f_{i}(x)))_{i=0}^{n}\in R$ for certain $f_{i}\in K[x]$ and
$R\subseteq(\mathrm{RV}^{\times})^{n}$. As $X$ is of dimension one and $S$ is
open, only the zero- and one-dimensional irreducible components of $V$ can
have non-empty intersection with $S$. In particular, if $V\cap S$ met $C$ at
infinitely many points, then there would be an algebraic curve
$X^{\prime}\subseteq V$ containing infinitely many points of $C$. ∎

We now continue along the same lines as in the proof of [1, Thm. 1], using
Lemma 5.0.4 instead of the quantifier elimination statement used there.

######  5.0.6 Proposition.

Let $C\subseteq{\mathbb{Q}}_{p}((t))^{n}$ be a transcendental curve which is
$\emptyset$-definable in ${\mathcal{L}}_{{\mathbb{Z}}_{p}\langle x\rangle}$.
Then $C_{s}\subseteq C_{1}$ for all integers $s>0$. In particular,
$\\#\text{-}\dim(C_{s})\leq(1,n,1)$.

###### Proof.

Suppose $(\sum_{j=0}^{s-1}a_{ij}t^{j})_{i}\in C_{s}\setminus C_{1}$. Consider
$C_{s}$ as a subset $A\subseteq{\mathbb{Q}}_{p}^{ns}$ via the identification

$\left(\sum_{j=0}^{s-1}b_{ij}t^{j}\right)_{i}\mapsto(b_{ij})_{i,j}.$

Let $X$ be an algebraic curve in ${\mathbb{Q}}_{p}^{ns}$containing all points
$(\lambda^{j}a_{ij})_{i,j}$ for $\lambda\in{\mathbb{Q}}_{p}$. By the above
Lemma 5.0.4, it follows that $X\cap A$ contains all points
$(\lambda^{j}a_{ij})$ for $\lambda\in{\mathbb{Z}}_{p}^{\times}$. In
particular, it is infinite.

Let $Y$ be the image of $X({\mathbb{Q}}_{p}[[t]])$ under
$(x_{ij})_{i,j}\mapsto(\sum_{j=0}^{s-1}x_{ij}t^{j})_{i}$. As this map and the
curve $X({\mathbb{Q}}_{p}[[t]])$ are definable in the $1$-h-minimal
${\mathcal{L}}_{{\mathrm{val}}}$-structure on ${\mathbb{Q}}_{p}((t))$ (for the
valuation ring ${\mathbb{Q}}_{p}[[t]]$), it follows from [8, Prop. 5.2.4
(3,4)] that $Y$ is an
${\mathcal{L}}_{\mathrm{val}}\cup{\mathbb{Q}}_{p}((t))$-definable set of
dimension at most $1$. Since the infinite set $X$ injects into $Y$, it then
follows by [8, Prop. 5.2.4 (1)] that $Y$ has dimension exactly $1$. Now use
that $X\cap A$ is infinite, whence so is $Y\cap C_{s}$. By Lemma 5.0.5, this
contradicts the assumption that $C$ is transcendental. ∎

###### Proof of Theorem 5.0.2.

By [8, Thm. 5.7.3] we may assume that $C$ is a single reparametrized cell
$(A,\sigma)$. As $C$ is one-dimensional, it follows that either $A$ is a
finite collection of points (in which case we are done) or $A$ is of type
$(1,0,\dots,0)$, up to a coordinate permutation of $K^{n}$. Thus $A$ is the
coordinate projection onto $K^{n}$ of the graph of some
${\mathcal{L}}_{{\mathbb{Z}}_{p}\langle x\rangle}$-definable function $c\colon
P\subseteq K\times\mathrm{RV}^{\ell}\to K^{n-1}$ (for some
$\ell\in{\mathbb{N}}$). By [8, Cor. 2.6.7] and [8, Lem. 2.5.2] it holds that
$\\#{c(x,\mathrm{RV}^{\ell})}\leq N$ for some $N\in{\mathbb{N}}$, independent
of $x$.

Now consider the map $g\colon K^{n}\to K$, wich is the projection onto the
first coordinate on ${\mathcal{O}}_{K}^{n}$ and is identically zero on
$K^{n}\setminus{\mathcal{O}}_{K}^{n}$. For any
$x\in{\mathcal{O}}_{K}/(t)\cong\tilde{k}\subseteq{\mathcal{O}}_{K}$, the fibre
at $x$ of the composition

$A_{1}\xrightarrow{g}{\mathcal{O}}_{K}\xrightarrow{\operatorname{proj}}\frac{{\mathcal{O}}_{K}}{(t)}$

is precisely $(\\{x\\}\times c(x,\mathrm{RV}^{\ell}))\cap A_{1}$. In
particular, it has size at most $N$. As $A_{s}=A_{1}$ for all
$s\in{\mathbb{N}}$, by proposition 5.0.6, it follows that
$\\#\text{-}\dim(A)\leq(N,1,1)$. ∎

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