\documentclass[JEP,XML,SOM,Unicode,ThmDefs,NoEqCountersInSection,published]{cedram} \datereceived{2022-10-21} \dateaccepted{2024-06-18} \dateepreuves{2024-06-24} \usepackage[scr=rsfs,cal=euler]{mathalfa} \newenvironment{enumeratei} {\bgroup\def\theenumi{\roman{enumi}}\def\theenumii{\arabic{enumii}}\begin{enumerate}} {\end{enumerate}\egroup} \newenvironment{enumeratea} {\bgroup\def\theenumi{\alph{enumi}}\begin{enumerate}} {\end{enumerate}\egroup} \newcommand{\bbullet}{{\scriptscriptstyle\bullet}} \newcommand{\cbbullet}{{\raisebox{1pt}{$\bbullet$}}} \newcommand{\psfrac}[2]{\sfrac{(#1)}{#2}} \newcommand{\Psfrac}[2]{(\sfrac{#1}{#2})} \let\oldimplies\implies \renewcommand\implies{\mathchoice{\oldimplies}{\Rightarrow}{\Rightarrow}{\Rightarrow}} \makeatletter \newcounter{paragraphe} \renewcommand\theparagraphe{\thesubsubsection.\arabic{paragraphe}} \makeatother \hyphenation{retrac-tion defi-na-ble defect-less Berko-vich} \usepackage{scalerel} \usepackage{tikz-cd} \DeclareUnicodeCharacter{00A0}{\relax} \OneNumberAllTheorems \NumberTheoremsAs{subsection} \SwapTheoremNumbers \newcommand{\opcit}{op.\@~cit.\@} \newcommand{\abs}[1]{\mathopen|#1\mathclose|} \newcommand{\an}{^{\mathrm{an}}} \newcommand{\cl}{\mathrm{cl}} \renewcommand{\d}{\mathrm d} \newcommand{\val}{\mathrm{val}} \renewcommand{\H}{\mathrm H} \newcommand{\gm}{\bG_{\mathrm m}} \newcommand{\tp}{\mathrm{tp\;}} \newcommand{\acvsf}{\mathrm{ACV}^2\mathrm{F}} \newcommand{\vsf}{\mathrm{V}^2\mathrm{F}} \newcommand{\acvf}{\mathrm{ACVF}} \newcommand{\DOAG}{\mathrm{DOAG}} \newcommand{\RES}{\mathrm{RES}} \newcommand{\res}{\mathrm{res}} \newcommand{\VF}{\mathrm{VF}} \newcommand{\bA}{\mathbb A} \newcommand{\bG}{\mathbb G} \renewcommand{\P}{\mathbb P} \newcommand{\F}{\mathbb F} \newcommand{\N}{\mathbb N} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\Z}{\mathbb Z} \renewcommand{\SS}{\mathbb S} \let\phi\varphi \let\epsilon\varepsilon \let\subset\subseteq \renewcommand{\labelitemi}{$\bbullet$} \renewcommand{\labelitemii}{$\diamond$} \theoremstyle{plain} \newtheorem{claim}[subsection]{Claim} \newtheorem*{mtheo}{Main Theorem} \theoremstyle{definition} \newtheorem*{Examples*}{Examples} \newcommand\stda[1]{{#1}^{\#}} \datepublished{2024-06-24} \begin{document} \frontmatter \title{Tropical functions on a skeleton} \author[\initial{A.} \lastname{Ducros}]{\firstname{Antoine} \lastname{Ducros}} \address{Sorbonne Université, Université Paris-Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche,\\ Campus Pierre et Marie Curie, case 247, 4 place Jussieu, 75252 Paris cedex 5, France} \email{antoine.ducros@imj-prg.fr} \urladdr{https://webusers.imj-prg.fr/~antoine.ducros/} \author[\initial{E.} \lastname{Hrushovski}]{\firstname{Ehud} \lastname{Hrushovski}} \address{Mathematical Institute, University of Oxford, Andrew Wiles Building,\\ Radcliffe Observatory Quarter Woodstock Road, Oxford OX2 6GG, UK} \email{Ehud.Hrushovski@maths.ox.ac.uk} \urladdr{https://www.maths.ox.ac.uk/people/ehud.hrushovski} \author[\initial{F.} \lastname{Loeser}]{\firstname{François} \lastname{Loeser}} \address{Institut universitaire de France, Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche CNRS,\\ Campus Pierre et Marie Curie, case 247, 4 place Jussieu, 75252 Paris cedex 5, France} \email{francois.loeser@imj-prg.fr} \urladdr{https://webusers.imj-prg.fr/~francois.loeser/} \author[\initial{J.} \lastname{Ye}]{\firstname{Jinhe} \lastname{Ye}} \address{Mathematical Institute, University of Oxford, Andrew Wiles Building\\ Radcliffe Observatory Quarter Woodstock Road, Oxford OX2 6GG, UK} \email{Jinhe.ye@maths.ox.ac.uk} \urladdr{https://sites.google.com/view/vincentye} \thanks{The first and third authors were partially supported by ANR-15-CE40-0008 (Défigéo). Part of this work was done during a stay of the first author at Weizmann Institute, funded by a Jakob Michael visiting professorship. The third author was partially supported by the Institut Universitaire de France. The fourth author was partially supported by the Fondation Sciences Mathématiques de Paris and GeoMod AAPG2019 (ANR-DFG), \textit{Geometric and Combinatorial Configurations in Model Theory}} \begin{abstract} We prove a general finiteness statement for the ordered abelian group of tropical functions on skeleta in Berkovich analytifications of algebraic varieties. Our approach consists in working in the framework of stable completions of algebraic varieties, a model-theoretic version of Berkovich analytifications, for which we prove a similar result, of which the former one is a consequence. \end{abstract} \subjclass{03C98, 14G22, 14T20} \keywords{Berkovich spaces, tropical geometry, skeleta, stable completion, Abhyankar valuations} \altkeywords{Espaces de Berkovich, géométrie tropicale, squelettes, complété stable, valuations d'Abhyankar} \alttitle{Fonctions tropicales sur un squelette} \begin{altabstract} Nous démontrons un résultat général de finitude pour le groupe abélien ordonné des fonctions tropicales sur un squelette dans l'analytifié de Berkovich d'une variété algébrique. Notre approche consiste à travailler dans le cadre des complétés stables de variétés algébriques, une version modèle théorique de l'analytification de Berkovich, pour lesquels nous démontrons un énoncé similaire dont notre résultat est une conséquence. \end{altabstract} \maketitle \tableofcontents \mainmatter \section{Introduction} \subsection{The general context: skeleta in Berkovich geometry} Let $F$ be a complete non-archimedean field. Among the several frameworks available for doing analytic geometry over $F$ (Tate, Raynaud, Berkovich, Huber...), Berkovich's is the one that encapsulates in the most natural way the deep links between non-archimedean and tropical (or polyhedral) geometry. Indeed, every Berkovich space $X$ over $F$ contains plenty of natural ``tropical'' subspaces, which are called \textit{skeleta}. Roughly speaking, a skeleton of~$X$ is a subset~$S$ of~$X$ on which the sheaf of functions of the form $\log \abs f$ with $f$ a section of $\mathscr O_X^\times$ induces a piecewise linear structure; \ie using such functions one can equip $S$ with a piecewise linear atlas, whose charts are modeled on (rational) polyhedra and whose transition maps are piecewise affine (with rational linear part). This definition is rather abstract, but there are plenty of concrete examples of skeleta. The prototype of such objects is the ``standard skeleton'' $S_n$ of~$(\gm^n)\an$, that consists of all Gauss norms with arbitrary real parameters; the family \[ (\log \abs{T_1},\ldots, \log\abs{T_n}) \] induces a piecewise-linear isomorphism $S_n\simeq \R^n$. Now if $X$ is an arbitrary analytic space and if $\phi_1,\ldots, \phi_m$ are quasi-finite maps from~$X$ to $(\gm^n)\an$, then $\bigcup_j \phi_j^{-1}(S_n)$ is a skeleton by \cite[Th.\,5.1]{confluentes}, (it consists only of points whose Zariski-closure is $n$\nobreakdash-dimen\-sional, so it is empty if $\dim Xn$ for all $n\in\N$. Since $\tp(1,c)=\tp(\alpha,c)$ for any $1>\alpha>0$ and $g(1,c)=c$, so $g'(\alpha,c)=c$. However $g'(0,c)=g(0,c)=0$, in contradiction with the continuity of $g'$. For a connected version of this example, replace $D$ by $D'=D\cup (\{0\}\times [1,2])$ and set $g=0$ on $\{0\}\times [1,2]$. \end{exem} This example suggests that interaction of the ambient space and the topology of~$D$ plays a role in distinguishing the two notions of combinations. To proceed towards a topological characterization for such properties, we need the following. \begin{defi}\label{def:convex} Let $T$ be an o-minimal expansion of $\DOAG$ and $\Gamma\models T$ with $D\subseteq \Gamma^n$ definable. We say that $D$ is convex if for any $u$ and $v$ in $D$, $\psfrac{u+v}{2}\in D$. \end{defi} \begin{rema}\label{rmk:rcf} When $T$ is an o-minimal expansion of the theory of real closed fields $\mathrm{RCF}$, this is equivalent to the usual definition of convexity for definable sets. For $u,v\in D$, let $L\subseteq [0,1]$ be $\{\alpha: \alpha u+(1-\alpha)v \in D\}$. By our notion of convexity, $L$~contains $\Z[1/2]\cap [0,1]$. By o-minimality, $L$ must be $[0,1]$ with at most finitely many points in $(0,1)$ removed. But removing any point from $(0,1)$ would lead to a violation of convexity. Note further that for $D$ convex, working inside the smallest affine subspace containing $D$, we may assume that $\cl(\mathrm{int}(D))=\cl(D)$. \end{rema} Lastly, recall that for any definable subset $D$ of some $\Gamma^n$, a function $f:D\to \Gamma$ is called \emph{$\Q$-affine} if $f=\sum^n_{i=1} m_ix_i+c$ where $m_i\in \Q$ and $c\in \Gamma$. Such functions are the most basic definable continuous functions on $D$. We say $f$ is \emph{$\Z$-affine} if the $m_i$ are all in $\Z$. \begin{prop}\label{prop:t-l-equiv} Let $\Gamma$ be a divisible ordered abelian group and let $f_1,\ldots,f_m$ be $\Q$-affine functions on $\Gamma^n$. Let $D\subseteq \Gamma^n$ be definable and $g:D\to \Gamma$ be a continuous definable function. Assume that $g$ is a $w$-combination of $f_1,\ldots,f_m$. Then the following are equivalent: \begin{enumerate} \item\label{prop:t-l-equiv1} $g$ is an $\ell$-combination of $f_1,\ldots,f_m$. \item\label{prop:t-l-equiv2} $g$ extends to a continuous definable function $g':\Gamma^n\to \Gamma$ that is a $w$\nobreakdash-combi\-na\-tion of $f_1,\ldots,f_m$. \item\label{prop:t-l-equiv3} $g$ extends to a continuous definable function $g':D'\to \Gamma$ on some convex definable set $D'$ containing $D$ that is a $w$-combination by $f_1,\ldots,f_m$. \item\label{prop:t-l-equiv4} For any $x,y\!\in\! D$, there is $i \!\in\! \{1,\ldots,m\}$ such that $f_i(x)\!\leq\! g(x)$ and $g(y)\!\leq\! f_i(y)$. \item\label{prop:t-l-equiv5} For some collection $S$ of subsets of $\{1,\ldots,m\}$, $g=\min_{X\in S} \max_{i \in X} f_i$. \end{enumerate} \end{prop} \begin{proof} The implications $\eqref{prop:t-l-equiv5}\implies \eqref{prop:t-l-equiv1}\implies \eqref{prop:t-l-equiv2}\implies \eqref{prop:t-l-equiv3}$ are clear. For $\eqref{prop:t-l-equiv3}\implies \eqref{prop:t-l-equiv4}$, by working in an elementary extension, we may assume that $\Gamma$ is a model of the theory of real closed fields $\mathrm{RCF}$. By Remark~\ref{rmk:rcf} and after replacing~$D$ by the convex set $D'$ in \eqref{prop:t-l-equiv3}, we may assume the line segment $[x,y]$ connecting $x,y$ is in $D$. Replace $g$ by $g'$ given by \eqref{prop:t-l-equiv3} as well. Let $I_j\subseteq [x,y]$ be $\{z: g(z)=f_j(z)\}$. By~continuity of $g$ and o-minimality, we know that the sets $I_j$ are finite unions of closed intervals and $\bigcup^m_{j=1} I_j=[x,y]$. Consider the canonical parameterization $h:[0,1]\to [x,y]$, $\alpha\mapsto \alpha y+(1-\alpha)x$, and let $f_i'=f_i\circ h$, $g'=g\circ h$ and $I'_j=h^{-1}(I_j)$. Since the functions $f_i$ are $\Q$-affine, the functions $f'_i$ are of the form $a_ix+b_i$ for some $a_i,b_i\in \Gamma$. Let $k$ be the $j$ such that $a_j$ is the greatest amongst all the $j$ such that $I'_j\neq \emptyset$. If~there are multiple such $j$, pick any. By induction, for $a$ to the right of $I'_k$, we have $g'(a)\leq f'_k(a)$. Similarly, for $a$ to the left of $I'_k$, we have $f'_k(a)\leq g'(a)$. In particular we have $f'_k(0)=f_k(x)\leq g'(0)=g(x)$ and $g'(1)=g(y)\leq f_k(y)=f'_k(1)$. For $\eqref{prop:t-l-equiv4}\implies \eqref{prop:t-l-equiv5}$, consider $S$ to be the collection of subsets $X\subseteq \{1,\ldots,m\}$ such that $g\leq \max_{i\in X} f_i$ on the entire $D$. Set $f:=\min_{X\in S}\max_{i\in X} f_i$. We claim that $g= f$. Clearly $g\leq f$, so it suffices to show that $g\geq f$. For each $W\notin S$, there is some $y_W$ such that $g(y_W)>f_i(y_W)$ for every $i \in W$. By \eqref{prop:t-l-equiv4}, for each $x\in D$, there is $i^x_{W}$ such that $f_{i^x_{W}}(x)\leq g(x)$ and $f_{i^x_W}(y_W)\geq g(y_W)$. Note that $i^x_W\notin W$. Let $X=\{i^x_W:W\notin S\}$. We have that $X\in S$ because otherwise, $i^x_X\in X$. For this $x$, we have that $\max_{i\in X} f_i(x)\geq g(x)$ and $f_i(x)\leq g(x)$ for any $i\in X$, hence $f(x)\leq \max_{i\in X} f_i(x)=g(x)$. \end{proof} \begin{coro}\label{cor:cordinate} Let $D\subseteq \Gamma^n$ be a definable convex set. The set of definable continuous functions from $D$ to $\Gamma$ is $(\ell,+)$-generated by the constants and all rational multiples of coordinate functions. \end{coro} \begin{proof} By quantifier elimination, we can find $\Q$-affine functions $f_1,\ldots,f_n$ such that $g$ is a $w$-combination of $f_1,\ldots,f_n$. By Proposition~\ref{prop:t-l-equiv}, we have that $g$ is in fact an $\ell$-combination of $f_1,\ldots,f_n$. \end{proof} Proposition~\ref{prop:t-l-equiv} suggests that the agreement of $w$-combination and $\ell$-combination is related to the existence of continuous extensions to an ambient convex space. This motivates the following definition. \begin{defi}\label{def:Lipschitz} For a tuple $x\in \Gamma^n$, define $|x|=\max^n_{i=1}|x_i|$. Let $D\subseteq\Gamma^n$ and $f:D\to \Gamma$ a definable function. We say $f$ is \emph{Lipschitz} if there is some $M\in \N$ such that $|f(x)-f(y)|\leq M|x-y|$. \end{defi} Note that Lipschitz functions are automatically continuous and clearly the class of Lipschitz functions depends on the embedding of $D$ in $\Gamma^n$. Our purpose is now to investigate Lipschitz definable functions on closed definable sets; a first step will consist in reducing to the definably compact case, by using the two following lemmas. \begin{lemm}\label{lem:lipschitz-closure} Let $\Gamma$ be a model of $\DOAG$, let $D$ be a subset of $\Gamma^n$ definable over some set $A$ of parameters, and let $f\colon D\to \Gamma$ be a Lipschitz $A$-definable map. Let $(f_i)$ be a finite family of $\Q$-affine $A$-definable functions such that $f$ is a $w$-combination of the $f_i|_D$. Then $f$ admits a unique continuous extension $\overline f$ to $\cl (D)$, the set $\cl (D)$ and the function $\overline f$ are $A$-definable, and $\overline f$ is Lipschitz and if a $w$-combination of the $f_i|_{\cl (D)}$. \end{lemm} \begin{proof} The uniqueness of $\overline f$ is clear, as well as the $A$-definability of $\cl (D)$ and $\overline f$ if the latter exists, as one sees by using the definition of the closure and of the limit (with $\epsilon$ and $\delta$\ldots). The same reasoning also shows that the set of points of $\cl (D) \setminus D$ at which $f$ admits a limit is $A$-definable. Moreover if $\overline f$ exists it inherits obviously the Lipschitz property of~$f$, and it is also $w$-generated by the (restrictions of) the~$f_i$: indeed, the subset of $\cl (D)$ consisting of points $x$ such that there is some $i$ with $f(x)=f_i(x)$ is closed and contains $D$, thus is the whole of $\cl (D)$. It thus remains to show the existence of $\overline f$, and this can be done after enlarging the model $\Gamma$. We can thus suppose that it is equal to the additive group of some real closed field. Let $x$ be a point of $\cl (D) \setminus D$. There exists a half-line $L$ emanating from $x$ such that $(x,y)\subset D$ for some $y$; taking $y$ close enough to $x$ we can assume that $f=f_j$ on $(x,y)$ for some $j$. Then the limit of~$f$ at $x$ along the direction of $L$ exists and is equal to $f_j(x)$. The Lipschitz property then ensures that this limit does not depend on $L$, let us denote it by $\overline f(x)$. Since $D$ is defined by affine inequalities, there is a positive $\gamma\in \Gamma$ such that for every $y$ in $\Gamma^n$ with $\|x-y\|<\gamma$ (say for the Euclidean norm) then either $(x,y)\subset D$ or $(x,y)\cap D=\emptyset$. Thus if $y$ is a point of $D$ with $\|x-y\|<\gamma$ then $\abs{f(y)-\overline f(x)}\leq N\|x-y\|$ where $N$ is an upper bound for the slopes of the $f_i$. So $f(y)$ tends to $\overline f(x)$ when the point $y$ of $D$ tends to $x$. \end{proof} \begin{lemm} \label{lem:comp-l-gen} Let $M$ be either $\{0\}$ or a model of $\DOAG$, let $\Gamma$ be a model of $\DOAG$ containing $M$, and let $\rho$ be an element of $\Gamma$ with $\rho >M$. Let $Z\subseteq \Gamma^n$ be an $M$-definable subset. Let $x_1,\ldots,x_n:Z \to \Gamma$ denote the coordinate functions of $Z$ and let $h:Z\to \Gamma$ be an $M$-definable function. Assume that there exists a term $t$ in $\{+,-,\max,\min\}$ and $\gamma=(\gamma_1,\ldots,\gamma_l)$ in $\Gamma^\ell$ such that $h|_{Z_\rho}=t(x_1,\ldots,x_n,\gamma)|_{Z_\rho}$, where $Z_\rho=Z\cap [-\rho,\rho]^n$. Then there is a term $t'$ in $\{+,-,\max,\min\}$ and a finite tuple $\beta$ of elements of~$M$ such that $h=t'(x_1,\ldots,x_n,\beta)$. \end{lemm} \begin{proof} Assume first that $M$ is a model of $\DOAG$. By our assumption, there exists a term $t$ in $\{+,-,\max,\min\}$ and a tuple $\gamma=(\gamma_1,\ldots,\gamma_l)\in \Gamma^\ell$ such that $h|_{Z_\rho}=t(x_1,\ldots,x_n,\gamma)|_{Z_\rho}$. By model-completeness of $\DOAG$, the $\gamma_i$ can be chosen in $M\oplus \Q\cdot \rho$. Thus there is $m>0$ such that for each $i$, there exist integers $k_i$ and $\beta_i\in M$ with $\gamma_i=\Psfrac{k_i}{m}\rho+\beta_i$. Let $\nu$ denote $\rho/m$. We have \[ h|_{Z_{m\nu}}=t(x_1,\ldots,x_n,(k_i\nu+\beta_i))|_{Z_{m\nu}}. \] Viewing the above as a first-order formula with constants in the model $M$ and a variable for $\nu$, using o-minimality and model-completeness of~$M$, we have some $\nu_0\in M_{>0}$ such that for any $\nu'>\nu_0$, the following holds in $M$: \[ h|_{Z_{m\nu'}}=t(x_1,\ldots,x_n,(k_i\nu'+\beta_i))|_{Z_{m\nu'}}. \] Take $\nu(x)=\max \{|x_1|,\ldots,|x_n|,2\nu_0\}$ and \[ t'(x_1,\ldots,x_n,\beta)=t(x_1,\ldots,x_n,(k_i\nu(x)+\beta_i)). \] We then have \[ h=t'(x_1,\ldots,x_n,\beta) \] by construction, which ends the proof when $M\neq \{0\}$. If $M=\{0\}$, set $\Gamma'=\Gamma\oplus \Q\cdot \delta$ where $\delta$ is positive and infinitesimal with respect to~$\Gamma$, set $M'=\Q\cdot \delta$ and let us denote by $Z'$ and $h'$ the objects deduced from $Z$ and $h$ by base-change to $\Gamma'$. Applying the above yields a term $\theta$ in $\{+,-,\max,\min\}$ and a tuple $\beta$ of elements of $\Q\cdot \delta$ such that $h'=\theta(x_1,\ldots,x_n,\beta)$. By reducing modulo the convex subgroup $\Q\cdot \delta$ of $\Gamma'$ we see that $h=\theta(x_1,\ldots,x_n,0)$. \end{proof} We can now state the main result of this section. \begin{theo}\label{thm:lip-t-gen} Let $M\models \DOAG$ or $M=\{0\}$ and let $\Gamma$ be a model of $\DOAG$ containing $M$. Let $D\subseteq \Gamma^m$ be an $M$-definable set. Let $g:D\to \Gamma$ be a Lipschitz definable function over $M$. Let $f_1,\ldots,f_n$ be $\Q$-affine functions over $M$ such that $g$ is a $w$-combination of $f_1,\ldots,f_n$. Then $g$ is an $(\ell, +)$-combination of the $f_i$, the constant $M$-valued functions and the coordinate functions. \end{theo} Before proving this result we will need some preliminaries on cell decomposition in $\DOAG$. \subsection{Cell decomposition} Fix a model $\Gamma$ of $\DOAG$. We shall use the notion of special linear decompositions from~\cite{elef}. In~\cite{elef}, Eleftheriou defines the notion of linear decomposition, which is a cell decomposition using only graphs of $\Q$-affine functions instead of general piecewise $\Q$-affine functions. In fact we will need only to consider bounded linear cells in $\Gamma^n$. They are defined by induction on $n$. In $\Gamma^0$ the origin is a bounded linear cell. If $C$ is a bounded linear cell in $\Gamma^{n-1}$, $f$ and $g$ are $\Q$\nobreakdash-affine functions on $\Gamma^{n-1}$, with $f < g$ on $C$, the relative interval \[ (f < g)_C = \{(x', y) \in C \times \Gamma; f (x') < y < g(x')\} \] and the graph $\Gamma (f)_C = \{(x', y) \in C \times \Gamma; f (x') = y\}$ are bounded linear cells in $\Gamma^n$. If $Y$ is a bounded definable set in $\Gamma^n$, a linear decomposition of $Y$ is a partition of $Y$ into (finitely many) bounded linear cells. We denote by $\pi: \Gamma^n \to \Gamma^{n-1}$ the projection to the $n-1$ first coordinates. A~special linear decomposition of a bounded definable set $Y \subseteq \Gamma^n$ is defined recursively in~\cite{elef} as follows. When $n = 1$ any cell decomposition of $Y$ is special. If $n > 1$, a~linear decomposition $\mathcal{C}$ of $Y$ is special if the following conditions are satisfied:\enlargethispage{\baselineskip}% \begin{enumerate} \item $\pi (\mathcal{C})$ is a special linear decomposition of $\pi (Y)$. \item For every pair of cells $\Gamma (f)_S$ and $\Gamma (g)_T$ in $C$ with $S$ in the closure of $T$, $f|_S g|_S$ or $f|_S =g|_S$. \item\label{item3} For every pair of cells $(f0$, we have that $\pi (D_1) \cap \pi (D_2) = \cl (C')$ for some cell $C'$ of the projection of the decomposition. Since for $i = 1, 2$, $D_i\cap \pi^{-1} (C')$ is either of the form $(f_i \leq g_i)_{C'}$ or $\Gamma (f_i)_{C'}$, it follows from condition \eqref{item3} of being a special linear decomposition that either $D_1 \cap D_2 \cap \pi^{-1} (C') = (f_1\leq g_1)_{C'} = (f_ 2\leq g_2)_{C'}$ or $D_1 \cap D_2 \cap \pi^{-1} (C') = \Gamma (f_1)_{C'} = \Gamma (f_2)_{C'}$, from which the statement follows. \end{proof} We shall also need the following statement. \begin{lemm}\label{lemm:ineg}Let $D$ be a closed bounded definable subset of $\Gamma^n$. Assume $D$ is convex. Let $h$ be a $\Q$-affine function on $\Gamma^n$ such $h \geq 0$ on $D$. Let $D_0$ be the zero locus of $h$ in $D$. We assume that $D_0$ is non-empty and $D_0 \neq D$. Let $f$ be a $\Q$-affine function on $\Gamma^n$ which vanishes on $D_0$. Then there exists a positive integer $M$ such that, for every $x \in D$, $\vert f (x) \vert \leq M h (x)$. \end{lemm} \begin{proof}Let $\mathcal{D}$ be a linear decomposition of $D$. We consider the set $\mathcal{F}$ of all sets $F$ of the form $F = \cl (C)$, with $C$ a 1-dimensional cell in $\mathcal{D}$, that intersect the hyperplane $h = 0$ and are not contained in $h=0$. For such an $F$ we denote by $p_F$ its intersection point with $h = 0$. There exists a positive integer $M_F$ such that $\vert f (x) \vert \leq M_F h (x)$ on~$F$. Indeed, the restrictions of both $h$ and $g$ to the line segment $F$ are linear functions on~$F$ vanishing at the endpoint $p_F$ of $F$ and the restriction of $h$ is not identically zero, which yields the existence of some $M_F$. In fact the inequality $\vert f (x) \vert \leq M_F h (x)$ holds on the whole half-line $L_F$ containing $F$ with origin $p_F$. Take $M = \max_{\mathcal{F}} (M_F)$. Now consider $R$ a $\mathrm{RCF}$-expansion of $\Gamma$. Let $Y$ be the convex hull of the half-lines $L_F$ in that expansion. We have $\vert f (x) \vert \leq M h (x)$ on $Y$. But $Y$ contains $D (R)$, since if $P$ is a convex definably compact polyhedron of $R^n$, and if $F$ is a face of $P$ of any dimension, then the convex hull of all half-lines directed by $1$-faces intersecting $F$ contains $P$, hence the result, taking $P = D$ and $F = D_0$. \end{proof} The following statement about separation by hyperplanes will play a key role in our proof of Theorem \ref{thm:lip-t-gen}. \begin{prop}\label{prop:cell}Fix a special linear cell decomposition of a closed bounded definable subset of $\Gamma^n$ and let $C_1$ and $C_2$ be two cells. Assume $C_1 \neq C_2$. Set $D_1 = \cl (C_1)$ and $D_2 = \cl (C_2)$. Then there exists a $\Z$-affine function $h$ such that $h \geq 0$ on $D_1$, $h \leq 0$ on $D_2$, and the hyperplane $H =h^{-1}(0)$ satisfies $D_1\cap D_2 =D_1 \cap H=D_2\cap H$. \end{prop} \begin{proof}We shall proceed by induction on $n$, the case $n=1$ being clear. If $\pi (C_1) = \pi (C_2)$, then the statement is clear. Indeed, for each $i = 1, 2$, we have $C_i = (f_i < g_i)_S$ or $C_i = \Gamma(f_i)_S$. In the second case we set $g_i = f_i$. We may assume that $C_1$ is above~$C_2$. The graph of the average of $f_1$ and $g_2$ provides the required hyperplane. Thus we will assume from now on that $\pi (C_1) \neq \pi (C_2)$. We set $C'_i = \pi (C_i)$ for $i = 1, 2$. By Lemma \ref{lemm:cell}, if $D_1 \cap D_2$ is non-empty, there exists a cell $C$ such that $D_1 \cap D_2 = \cl (C)$. \subsubsection*{Case 1: $D_1 \cap D_2$ is non-empty and $C$ is of the form $(f < g)_S$} In this case, for $i = 1, 2$, $C_i$ is necessarily of the form $(f_i < g_i)_{C'_i}$ where $f_i$ and $g_i$ are $\Q$-affine functions coinciding with $f$ and $g$ on $S$, since we are working with a special linear cell decomposition. Furthermore $D_i = (f_i \leq g_i)_{\cl (C'_i)}$ and we have $f_1 = f_2$ and $g_1 = g_2$ on $\cl (C'_1) \cap \cl (C'_2)$. It follows that $D_1 \cap D_2 = (f_i \leq g_i)_{\cl (C'_1) \cap \cl (C'_2)}$, for $i = 1,2$. By the induction hypothesis, there exists an hyperplane $h'$ in $\Gamma^{n-1}$ given by a $\Z$-affine equation satisfying the conditions of Proposition \ref {prop:cell} relatively to $\cl (C'_1)$ and $\cl (C'_2)$. Consider the vertical hyperplane $H$ above $h'$ (the hyperplane defined by the same equation in $\Gamma^n$). It follows from our description of $D_1 \cap D_2$ that $H$ satisfies the required conditions. \subsubsection*{Case 2: $D_1 \cap D_2$ is non-empty and $C$ is of the form $\Gamma (f)_S$} By the induction hypothesis, there exists an hyperplane $h'$ given by an equation $h' (x')=0$ in $\Gamma^{n-1}$, with $h'$ a $\Z$-affine function, $x'=(x_1,\ldots, x_{n-1})$ fulfilling the conditions of Proposition \ref {prop:cell} relatively to $\cl (C'_1)$ and $\cl (C'_2)$. In particular $h' \geq 0 $ on $\pi (D_1)$ and $h' \leq 0$ on $\pi (D_2)$. We denote by $H$ the hyperplane with equation $h'(x')=0$ in $\Gamma^{n}$.\enlargethispage{\baselineskip}% The set $C_1$ is of the form $(f_1 < g_1)_{C'_1}$ or $\Gamma (f_1)_{C'_1}$. In the second case we set $g_1 = f_1$. Similarly $C_2$ is of the form $(f_2 g_1$ over $\pi (D_1) \cap \pi (D_2)$. Note that if we intersect the cells of a special linear cell decomposition of some bounded set $W$ with $H$ we get a special linear cell decomposition of $W \cap H$. Thus we can apply the induction hypothesis to~$D'_1$ and~$D'_2$, and there exists a $\Z$-affine function $f$ on $\Gamma^n$ such that $f >0$ on $D'_1$ and $f <0$ on~$D'_2$. We claim that for $M$ a large enough integer the hyperplane $M h' +f = 0$ will separate~$D_1$ and~$D_2$. To prove this we proceed similarly as in the proof of Lemma \ref{lemm:ineg}. We consider the set $\mathcal{F}$ of all sets $F$ of the form $F = \cl (C)$, with $C$ a 1-dimensional cell contained in~$D_1$, that intersect $H$ and are not contained in $H$. For such an $F$ denote by $p_F$ the intersection point of $F$ with $H$. The restriction of~$f$ to $F$ can be written as $f (p_F) + \ell_F$ with $\ell_F$ a $\Q$-linear function on $F$. Since $h'$ is strictly positive on $F$ outside $p_F$, there exists a positive integer $M_F$ such that $M_F h' + \ell_F \geq 0 $ on $F$. This still holds on the whole half-line $L_F$ containing $F$ with origin $p_F$. Since $f (p_F) > 0$ by assumption, we~get that $M_F h' + f > 0$ on $L_F$. Take $M_1 = \max_{\mathcal{F}} (M_F)$. Proceeding as in the proof of Lemma \ref{lemm:ineg} we deduce that $M_1 h' + f >0$ on $D_1$. One proves similarly the existence of $M_2$ such that $M_2 h' + f <0$ on $D_2$. Thus we can take $M = \max (M_1, M_2)$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:lip-t-gen}] By Lemma \ref{lem:lipschitz-closure} we can assume that $D$ is closed. We may then enlarge the model $\Gamma$ and assume that it contains some $\rho$ with $\rho>M$. Let $D_\rho$ denote the intersection of $D$ with $[-\rho,\rho]^m$. This is a definably compact subset of $\Gamma^m$ which is definable over $M_\rho:=M\oplus \Q\cdot \rho$. If we prove that $g|_{D_\rho}$ is an $(\ell,+)$-combination of the $f_i$, the coordinate functions and some constant functions with values in $M_\rho$, Lemma \ref{lem:comp-l-gen} above will allow us to conclude that $g$ is an $(\ell,+)$-combination of the $f_i$, the coordinate functions and some constant functions with values in $M$. We thus may and do assume that $D$ is definably compact. By considering a submodel of~$M$ over which everything is defined, we reduce to the case where $M$ has exactly $r$ non-trivial convex subgroups, and we proceed by induction on $r$. The case $r=0$ is obvious since the definably compact set $D$ is then either empty or equal to $\{0\}$. Assume now that $r>0$ and that the result holds true for smaller values of $r$.\enlargethispage{.5\baselineskip}% Let $M_0$ be the smallest non-trivial convex subgroup of~$M$, and $\overline{M}=M/M_0$ be the quotient. We are first going to explain why we can assume that $g(D(M)) \subseteq M_0$; this is tautological if $M_0=M$, so we assume (just for this reduction step) that $M_0\neq M$. In this case $\overline M$ is a model of $\DOAG$ with $r-1$ non-trivial convex subgroups, and the natural map carrying $M$ to $\overline{M}$ induces a map that carries $D(M)$ to a definably compact definable subset $\overline{D}$ of $\overline{M}^m$ (see~\cite[Th.\,4.1.1]{BP} for example). Furthermore, since $g$ is Lipschitz, it descends to a definable function $\overline{g}:\overline{D}(\overline{M})\to \overline{M}$, which is Lipshitz as well and is a $(w,+)$-combinations of the $\overline{f_i}$. By the induction hypothesis, we then know that $\overline{g}$ is of the form $\tau(\overline{f_1},\ldots,\overline{f_n})$, where $\tau$ is a term involving constants, projections and $+,-,\min,\max$ only. Replacing $g$ by $g-\tau(f_1,\ldots,f_n)$, we may assume that $g(D)\subseteq M_0$, as announced. By~\cite[Fact 2.2]{elef} there exists a special linear decomposition $\mathcal{D}$ of $D$ such that~$g$ is $\Q$-affine on each cell. Clearly $D$ is covered by the closed sets $D_i = \cl (C_i)$, for~$C_i$ in~$\mathcal{D}$. In fact if one considers the set $\mathcal{D}'$ of all $C\in \mathcal{D}$ such that, for any $C'\neq C$, $C$~is not contained in the closure of $C'$, it follows from \cite[Fact 2.3]{elef} that the closed sets $D_i = \cl (C_i)$ for $C_i$ in $\mathcal{D}'$ already cover $D$, but we will not use this. Sets of the form $\cl (C)$ with $C \in \mathcal{D}$ will be referred to as closed cells. We will now use the separating hyperplanes provided by Proposition \ref{prop:cell} to build affine functions that will appear in the $(\ell,+)$-combination we are seeking for describing~$g$. For this purpose, the inclusion~$g(D(M))\subseteq M_0$ will be crucial. \begin{claim}\label{claim:separation} Let $C'$ and $C''$ be any two distinct cells in $\mathcal{D}$. Set $D' = \cl (C')$ and $D'' = \cl (C'')$. There exists a function $f_{D',D''}$ in the group generated by $f_1,\ldots,f_n$, the constant functions and the coordinate functions such that\vspace*{-3pt} \begin{equation}\label{eqn:separation}\tag{$*$} g|_{D'}\leq f_{D',D''}|_{D'} \text{ and } f_{D',D''}|_{D''}\leq g|_{D''}. \end{equation} \end{claim} \begin{proof}[Proof of the claim] By Proposition~\ref{prop:cell} there exists a $\Z$-affine function $h$ such that the hyperplane $H =h^{-1}(0)$ satisfies $D' \cap D'' =D' \cap H=D''\cap H$, $h \geq 0$ on $D'$ and $h \leq 0$ on $D''$. If $D'\cap D''=\emptyset$, using definable compactness of $D'$ and $D''$, we get that there exists $a \in M_0$ such that $h|_{D''}<-a<0 ma - b$ on $D'$ and $mh - g < -ma + b$ on $D''$. Since $M_0$ is archimedean, for $m$ large enough, we have $ma > b$, hence condition~(\ref{eqn:separation}) is satisfied for $f_{D',D''} = mh$. If $D'\cap D''\neq \emptyset$, take $c\in D'\cap D''$ and let $G$ be the $\Q$-affine function such that $g=G$ on $D'$. Replacing $g$ by $g-G$, we may assume that $g=0$ on $D'$. Translating our entire set by $c$, we may assume that $c$ is the origin. Thus $g(0)=0$ and $g$ is actually the restriction of a $\Q$-linear function on $D''$. On $D'$, for any positive integer $m$, we have $mh\geq0=g$. For any $b\in D''$, if $h(b)=0$, then $b\in D''\cap H=D'\cap H$, hence $g(b)=0$. Thus, by Lemma \ref{lemm:ineg}, there exists a positive integer $m$ such that $-g \leq -mh$ on $D''$. For such an integer $m$, we have $g \leq mh$ on $D'$ and $g \geq mh$ on $D''$. \end{proof} We can now conclude the proof of Theorem \ref{thm:lip-t-gen}. Note that $g$ is a $w$-combination of the functions $f_i$; it is thus a fortiori a $w$-combination of the set of functions obtained by adding all the functions $f_{D',D''}$ from Claim \ref{claim:separation} to the functions $f_i$. Take~$x$ and~ $y$ in $D$. If they belong to the same closed cell $D' = \cl(C')$, then $g (x) = f_i (x)$ and $g (y) = f_i (y)$ for some $i$ and condition \eqref{prop:t-l-equiv4} in Proposition~\ref{prop:t-l-equiv} is satisfied. If they belong to two distinct closed cells $D'$ and $D''$, then $g (x) \leq f_{D',D''} (x)$ and $ f_{D',D''} (y) \leq g (y)$ by Claim \ref{claim:separation}. Thus, by the implication $\eqref{prop:t-l-equiv4}\implies\eqref{prop:t-l-equiv1}$ in Proposition~\ref{prop:t-l-equiv}, we obtain that~$g$ is an $\ell$-combination of the functions $f_i$ and $f_{D',D''}$, which concludes the proof. \end{proof} The proof of Claim~\ref{claim:separation} actually yields the following convenient way to check if a given function is Lipschitz on a definably compact set. \begin{coro} Let $D'$ and $D''$ be two definably compact convex sets such that $D'\cap D''=D'\cap H_a=D''\cap H_a\neq \emptyset$ with $H_a$ an hyperplane defined by a $\Z$-affine function. Assume further that $g$ a continuous function that is affine on $D'$ and $D''$ respectively, then $g$ is Lipschitz on $D'\cup D''$. \end{coro} \begin{rema} Note that one can have definably compact versions of Example~\ref{eg:dl} by replacing $[0,\infty)$ with $[0,c]$ for some $c>n\cdot 1$ for all $n\in \N$. However, the function~$g$ there is not Lipschitz because $|(0,c)-(1,c)|=1$ and $|g(0,c)-g(1,c)|=c$. \end{rema} \begin{rema}In the case of homogeneous linear equations, with no parameters, equivalence of $\ell$-combination and $w$-combination goes back to work of Beynon \cite{beynon}, see also \cite[\S5.2]{glass} and \cite{Ovchinnikov} for related results. In 2011, as a student, Daniel Lowengrub rediscovered and partially generalized Beynon's results. He also gave Exam\-ple \ref{eg:dl} showing that they do not hold over non-archimedean parameters. Here we fully generalized them, after replacing continuity by a Lipschitz condition. Our proofs in this section make use of his ideas. \end{rema} \section{Complements about \texorpdfstring{$\Gamma$}{G}-internal sets} \subsection{Preimages of the standard $\Gamma$-internal subset of $\widehat{\gm^n}$}\label{ss-preimsn} Let $k$ be an algebraically closed valued field and let $X$ be an irreducible $n$\nobreakdash-dimen\-sional $k$-scheme of finite type. Let $\Sigma_n$ be the image of the definable topological embedding from $\Gamma^n$ into $\widehat{\gm^n}$ that sends a $n$-tuple $\gamma$ to the generic point $r_\gamma$ of the closed $n$-ball with valuative radius~$\gamma$ and centered at the origin. This set $\Sigma_n$ is the archetypal example of a $\Gamma$-internal subset, and it is contained in $(\gm^n)^{\#}_{\mathrm{gen}}$. Let $\phi$ be any morphism from $X$ to $\gm^n$. Set $\Upsilon=\phi^{-1}(\Sigma_n)$. If $\dim\phi(X)\min_I \val (a_I)+\val (f^I(x)).\] Let $V$ be the subset of~$X$ defined by the inequality \[\val\left(\sum a_I f^I\right)>\min_I \val (a_I)+\val (f^I).\] It is a definable subset of~$X$, and its stable completion is an open neighborhood of $x$ in $\widehat X$. Moreover by the very definition of $V$, for every $y\in V$ there exists two distinct multi-indices $I$ and $J$ with $\val (a_I)+\val (f^I(y))=\val (a_J)+\val (f^J(y))$, which shows that $\val (f) (V)$ is contained in a finite union of $(n-1)$-dimensional subspaces of $\Gamma^n$. As a consequence, the tropical dimension of~$f$ at $x$ is at most $n-1$. Conversely, let $x\in \Upsilon$ and let $V$ be a definable open subset of~$X$ such that $\widehat V$ is a neighborhood of $x$ in $\widehat X$. Since $x\in \Upsilon$, it is contained in $X^\#_{\mathrm{gen}}$. There is a dense open subset $U$ of~$X$ such that $f$ induces a finite flat map from $U$ to a dense open subscheme of $\gm^n$; then the induced map $\widehat U\to \widehat {\gm^n}$ is open by \cite[Cor.\,9.7.2]{HL}, and since $x\in X^\#_{\mathrm{gen}}$, it belongs to $\widehat U$; as a consequence, $f$ is open around $x$. In particular, $f(\widehat V)$ contains a neighborhood $\Omega$ of $f(x)$. Since $x\in \Upsilon$, the image $f(x)$ is equal to $r_\gamma$ for some $\gamma \in \Gamma^n$. The intersection $\Omega \cap \Sigma_n$ then contains $\{r_\delta\}_{\delta \in B}$ for $B$ some product of~$n$ open intervals containing $\gamma$, so $\val (f)(\widehat V)$ contains $B$, and is in particular $n$\nobreakdash-dimen\-sional. The tropical dimension of~$f$ at $x$ is thus equal to $n$. \end{proof} We are now ready to establish the announced description of a $\Gamma$-internal subset $\Upsilon$ of $X^{\#}_{\mathrm{gen}}$. The case where $\Upsilon$ is purely $n$\nobreakdash-dimen\-sional will rely of the description of the maximal tropical dimension locus given by the above proposition. The general case will then be handled by embedding $\Upsilon$ into a purely $n$\nobreakdash-dimen\-sional $\Gamma$-internal subset of $X^{\#}_{\mathrm{gen}}$ -- the basic idea for doing this is to increase the dimension of $\Upsilon$ (until $n$ is achieved) by ``following'' it along a deformation retraction as built in \cite{HL}. \begin{theo}\label{structure-pure-gammainternal} Let $X$ be an $n$\nobreakdash-dimen\-sional integral scheme of finite type over $k$, and let $\Upsilon\subseteq X^\#_{\mathrm{gen}}$ be a $\Gamma$-internal subset defined over $k$. There exists a dense open subset $U$ of~$X$ and finitely many morphisms $\phi_1,\ldots, \phi_m$ from $U$ to $\gm^n$ such that $\Upsilon\subseteq\bigcup_j \phi_j^{-1}(\Sigma_n)$. \end{theo} \begin{proof} Let us first assume that $\Upsilon$ is purely $n$\nobreakdash-dimen\-sional. Since $k$ is algebraically closed, after shrinking $X$ we might assume that there exist finitely many invertible functions $f_1,\ldots, f_r$ on $X$ such that $\val (f)$ induces a $k$-definable homeomorphism between $\Upsilon$ and a definable subset of $\Gamma^r$ (\ref{gamma-internal-generic}). For every subset $I$ of $\{1,\ldots, r\}$ of cardinality $n$, let $f_I$ be the map from $X$ to $\gm^I$ given by the $f_i$ with $i\in I$. Since $\Upsilon$ is of pure dimension $n$, for every $x\in \Upsilon$ there is at least one subset $I$ of $\{1,\ldots, r\}$ of cardinality $n$ such that the tropical dimension of $f_I$ at $x$ is $n$. By Proposition \ref{prop-dimtrop}, this means that \[\Upsilon \subseteq\bigcup_{\substack{I\subseteq\{1,\ldots, r\}\\ |I|=n}} f_I^{-1}(\Sigma_n),\] which ends the proof in this particular case. As a by-product, we get in view of Theorem \ref{theo-union-skeleta} that a finite union of purely $n$\nobreakdash-dimen\-sional $\Gamma$-internal subset of $X^\#_{\mathrm{gen}}$ is still $\Gamma$-internal (and of course purely $n$\nobreakdash-dimen\-sional). Let us now go back to an arbitrary $\Upsilon$. In order to prove the theorem, it suffices by the above to show that $\Upsilon$ is contained in some purely $n$\nobreakdash-dimen\-sional $\Gamma$-internal subset of $X^\#_{\mathrm{gen}}$. By shrinking $X$ we can assume that it is quasi-projective. We have already noticed that a finite union of purely $n$\nobreakdash-dimen\-sional $\Gamma$-internal subset of $X^\#_{\mathrm{gen}}$ is still $\Gamma$-internal and purely $n$\nobreakdash-dimen\-sional, which allows ourselves to cut $\Upsilon$ into finitely many $k$-definable pieces and to argue piecewise. We thus can assume that $\Upsilon$ is purely $d$-dimensional for some $d$, and we argue by descending induction on $d$, so we assume that our statement holds if the $\Gamma$-internal subset involved is equidimensional of dimension $>d$. Let $\alpha$ be a $k$-definable embedding from $\Upsilon$ into some $\Gamma^m$ given by finitely many non-zero rational functions. By~\cite[Th.\,11.1.1]{HL}, there exists a pro-definable deformation retraction $h:I\times \widehat{X}\to\widehat{X}$ preserving $\alpha$ with a $\Gamma$-internal purely $n$\nobreakdash-dimen\-sional image $\Upsilon_{\mathrm{targ}}$ contained in $X^\#.$ Let $\Upsilon_s =\{p\in \Upsilon: h(t,p)=p \text{ for any } t\}$. By its very definition, $\Upsilon_s$ is contained in the set $\Upsilon'$ of Zariski-dense points of $\Upsilon_{\mathrm{targ}}$, which is a purely $n$\nobreakdash-dimen\-sional $\Gamma$-internal subset of $X^{\#}_{\mathrm{gen}}$. It therefore suffices to prove the proposition for the open complement of $\Upsilon_s$ in $\Upsilon$, which is still purely $d$-dimensional. In other words, we can assume that $\Upsilon_s=\emptyset$. Let $\Upsilon''=h(I,\Upsilon)$. We claim that it is iso-definable, and thus $\Gamma$-internal. By~\cite[Lem.\,2.2.8]{HL}, $\Upsilon''$ is strict pro-definable. Since $\Upsilon\subset X^\#$, the set $\Upsilon''$ is contained in~$X^\#$ as well by~\cite[Th.\,11.1.1]{HL} and the latter is strict ind-definable. Hence by compactness, we see that $h(I,\Upsilon)$ is a strict pro-definable subset of a definable set, thus is iso-definable. Note also that the homotopy built in \cite{HL} is Zariski-generalizing, so~$\Upsilon''\subseteq\nobreak X^\#_{\mathrm{gen}}$. Since $\Upsilon_s=\emptyset$, for every $p\in \Upsilon$ there are some $a_p, b_p$ in $I$ with $a_p0\;\text{for all}\;i.\] Now $L^{\mathrm h}$ is a compositum of $L$ and $K^{\mathrm h}$, so it is generated by $b$ over $K^{\mathrm h}$. Hence there exists a sub-tuple $\beta$ of $b$ of size $d$ such that $b$ is contained in the $K^{\mathrm h}$-vector space generated by $\beta$. As $K^{\mathrm h}$ is the definable closure of $K$, the latter property can be rephrased as $\Psi(a,b)$ for some quantifier-free formula $\Psi$ in the language of valued fields, with parameters in $F$. Now let $L':=F(a',b')$ be a valued extension of $F$, and set $K'=F(a')$. Assume that $\mathrm{res}(K')$ is a regular extension of $\mathrm{res}(F)$, and that \[L'\models \Phi(a',b')\;\text{and}\;\Psi(a',b').\] Then $\Psi(a',b')$ ensures that $(L')^{\mathrm h}$ is at most $d$-dimensional over $(K')^{\mathrm h}$, while $\Phi(a',b')$ ensures that $\mathrm{res}(L')$ contains a field isomorphic to $\mathrm{res}(F)(c_1,\ldots, c_r)=\mathrm{res}(L)$. Since $\mathrm{res}(K')$ is regular over $\mathrm{res}(K)=\mathrm{res}(F)$, the residue field $\mathrm{res}(L')$ contains a field isomorphic to $\mathrm{res}(L)\otimes_{\mathrm{res}(F)}\mathrm{res}(K')$, which is of degree $d$ over $\mathrm{res}(K')$. As a consequence, \[[L':K']=[\mathrm{res}(L'):\mathrm{res}(K')]=d\] and thus \[\val(L')=\val(K').\qedhere\] \end{proof} \subsection{Generic types of closed balls}\label{gentypeball} In practice, the above proposition will be applied for $a$ realizing the generic type of a ball over $F$. Let us collect here some basic facts about such types. If $\gamma$ is an element of $\Gamma$, we denote by $r_\gamma$ the type of the closed ball of (valuative) radius $\gamma$, which belongs to $\widehat{\bA^1}$ and even to $\stda{\bA^{1}}$. More generally if $\gamma=(\gamma_1,\ldots, \gamma_n)$ we shall denote by $r_\gamma$ the type $r_{\gamma_1}\otimes \ldots\otimes r_{\gamma_n}$, which is the generic type of the $n$\nobreakdash-dimen\-sional ball of polyradius $(\gamma_1,\ldots, \gamma_n)$ and belongs to $\stda{\bA^{n}}$. Now let $F$ be a valued field, let $K$ be a valued extension of $F$ and let $a_1,\ldots, a_r, a_{r+1},\ldots, a_n$ be elements of $K^\times$. Assume the following: \begin{enumerate} \item the group elements $\val(a_1), \ldots, \val(a_r)$ are $\Z$-linearly independent over $\val (F)$; \item one has $\val(a_i)=0$ for $i=r+1,\ldots, n$ and the residue classes of the $a_i$ for $i=r+1,\ldots, n$ are algebraically independent over $\mathrm{res}(F)$. \end{enumerate} Set $\gamma_i=\val(a_i)$ for $i=1,\ldots, n$. Then under these assumptions one has $a\models r_\gamma|_{F(\gamma)}$. Conversely, assume that $a\models r_\gamma|_{F(\gamma)}$. Then $\val(a_i)=0$ for $i=r+1,\ldots, n$, the residue classes of the $a_i$ for $i=r+1,\ldots, n$ are algebraically independent over $\mathrm{res}(F)$ and $\mathrm{res}(F(a_{r+1},\ldots, a_n)$ is generated by the residue classes of the $a_i$, so is purely transcendental of degree $n-r$ over $\mathrm{res}(F)$. In particular, this is a regular extension of $\mathrm{res}(F)$. Now the $\val(a_i)$ for $i=1,\ldots, r$ are $\Z$-linearly independent, the group $\val(F(a_1,\ldots, a_n))$ is generated over the group $\val(F(a_{r+1},\ldots a_n))=\val(F)$ by the $\val(a_i) $ for $i=1,\ldots, r$, so it is free of rank $r$ modulo $\val(F)$; and the residue field of $F(a_1,\ldots, a_n)$ is equal to that of $F(a_{r+1},\ldots, a_n)$, so it is purely transcendental of degree $n-r$ over $\mathrm{res}(F)$; in particular, this is a regular extension of the latter. \begin{lemm}\label{lemm-abhyankar} Let $F$ be a valued field and let $p$ be a strongly stably dominated (global) type with canonical parameter of definition $\gamma\in \Gamma^n$ over $F$. Let $b\models p|_{F(\gamma)}$ and set $K=F(b)$. Then: \begin{enumerate} \item\label{lemm-abhyankar1} $\gamma$ is definable over $F(b)$; \item\label{lemm-abhyankar2} $F(b)$ is an Abhyankar extension of $F$. \end{enumerate} \end{lemm} \begin{proof} Let us start with \eqref{lemm-abhyankar1}. Let $\Phi$ be an automorphism of the monster model fixing $F(b)$ pointwise. One has to show that $\Phi$ fixes $\gamma$, or $p$ -- this amounts to the same. Set $\delta=\Phi(\gamma)$ and $q=\Phi(p)$. Let $A$ be a $\Phi$-invariant subset of $\Gamma$ containing~$\gamma$. Since $p$ is orthogonal to $\Gamma$, the restriction $p|_{F(\gamma)}$ implies a complete type $r$ over $F(A)$, which coincides necessarily with the type of $b$ over $F(A)$. Thus $p$ contains the type of~$b$ over $F(A)$, and so does $\Phi(p)$ since both $b$ and $A$ are $\Phi$-invariant. So~$p$ and~$\Phi(p)$ are two global generically stable $F(A)$-definable types that coincide over $F(A)$; it follows that they are equal, \cf \cite[Prop.\,2.35]{simon}. Now we prove \eqref{lemm-abhyankar2}. By replacing $\gamma$ by a suitable subtuple if necessary, we may and do assume that $\gamma=(\gamma_1,\ldots, \gamma_n)$ where the $\gamma_i$ are $\Z$-linearly independent over $\val (F)$. Now choose $c=(c_1,\ldots, c_n)$ realizing $r_\gamma$ over $F(b)$. Then by stable domination, $b$ realizes~$p$ over $F(\gamma, c)$ and in particular over $F(c)$. The type $p$ is strongly stably dominated, and it is definable over $F(c)$ by construction. So $\mathrm{res}(F(b))$ is of transcendence degree $\dim X$ over $\mathrm{res}(F(c))$, and $F(b,c)$ is thus Abhyankar over $F(c)$, hence over $F$ since $F(c)$ is Abhyankar over $F$. Then $F(b)$ is Abhyankar over $F$. \end{proof} \begin{theo}\label{theo-tfinite-pure}Let $F$ be a defectless valued field with divisible value group. Let $X$ be an $n$\nobreakdash-dimen\-sional integral $F$-scheme of finite type, and let $\Upsilon$ be a $\Gamma$-internal subset of $X^{\#}_{\mathrm{gen}}\subseteq \widehat X$. Then $\SS(\Upsilon)$ is finitely $(w,+)$-generated up to constant functions. \end{theo} \begin{proof} We shall prove the following: for every $p\in \Upsilon$, there exists a $F$-definable subset $W$ of $\Upsilon$ containing $p$ and finitely many functions $a_1,\ldots, a_n$ in $F(X)^\times$ such that for every $x\in W$ and every $f\in \SS(\Upsilon)$, the element $\val(f(x))$ of $\Gamma$ belongs to the group generated by $\val(F)$ and the $\val(a_i(x))$. This will allow us to conclude. Indeed, assume that this statement has been proved. Then by compactness there is a finite cover $\mathscr W$ of $\Upsilon$ with finitely many sets $W$ as above. Hence $\SS(\Upsilon)$ is $(w,+)$-generated by the $a_i$ up to constant functions. Let $p\in \Upsilon$. This is a strongly stably dominated global type. Let $\gamma \in \Gamma^r$ be a canonical parameter of definition of $p$ and let $b$ be a realization of $p$ over $F(\gamma)$. By~Lemma \ref{lemm-abhyankar}, $\gamma$ is definable over $F(b)$ and $F(b)$ is Abhyankar over $F$. As $\gamma$ is definable over $F(b)$ and as it is defined only up to inter-definability, we can assume that $\gamma=(\gamma_1,\ldots, \gamma_r)$ where the $\gamma_i$ are $\Z$-linearly independent over $\val(F)$, and where each $\gamma_i$ is equal to $\val(a_i)$ for some $a_i\in F(b)$. Since $p$ is stably dominated every element of $\val(F(b))$ belongs to the $\Q$-vector space generated by $\val (F)$ and the $\gamma_i$. Moreover the group $\val (F(b))$ is finitely generated over $\val(F)$ because $F(b)$ is Abhyankar over~$F$ and as $\val(F)$ is divisible, $\val( F(b))$ is torsion-free modulo $\val (F)$; as a consequence, $\val(F(b))/\val(F)$ is free of finite rank. We can thus even assume that $(\gamma_1,\ldots, \gamma_r)$ is a $\Z$-basis of $\val(F(b))/\val(F)$. The valued field $F(b)$ being Abhyankar over $F$, the family $(a_1,\ldots, a_r)$ can be completed into an Abhyankar basis $(a_1,\ldots, a_r, a_{r+1},\ldots, a_n)$ of~$F(b)$ over $F$ such that $\val(a_i)=0$ for every $i\geq r+1$ and the residue classes of $a_{r+1},\dots,a_n$ are algebraically independent over the residue field of $F$. The field $F(b)$ is then algebraic over $F(a_1,\ldots, a_n)$. We set $a=(a_1,\ldots,a_n)$ and we now denote by $\gamma$ the $n$-uple $(\gamma_1,\ldots, \gamma_n)$ with $\gamma_i=0$ if $i\geq r+1$, so that $\gamma_i=\val(a_i)$ for all $i$. Since $p$ is Zariski-generic, $a=(a_1,\ldots, a_n)$ can be interpreted as an $n$-uple of rational functions on $X$, giving rise to a map $\pi$ from a dense open subset of~$X$ to $\bA^n_F$. In particular, $\pi$ induces a map (which we still denote by $\pi$) from $\Upsilon$ to $\widehat{\bA^n_F}$, and the fact that $a_1,\ldots, a_n$ is an Abhyankar basis of $F(b)$ means that $\pi(p)|_{F(\gamma)}=r_{\gamma}|_{F(\gamma)}$; as both $\pi(p)$ and $r_\gamma$ are generically stable types defined over $F(\gamma)$, it follows that $\pi(p)=r_\gamma$. Moreover, the tower $F(a_{r+1},\ldots, a_n)\subseteq F(a)\subseteq F(a,b)$ fulfills the conditions of Proposition \ref{prop-formula-valuegroup}; hence the latter provides a formula $\phi(y, x_1,\ldots, x_r)$ with coefficients in $F(a_{r+1},\ldots, a_n)$, which we can see as the evaluation at $(a_{r+1},\ldots, a_n)$ of a formula $\psi(y,x_1,\ldots, x_n)$. Now let $W$ be the subset of $\Upsilon$ defined as the set of types $q$ satisfying the following conditions, with $\delta_i:= \val(a_i(q))$:\enlargethispage{.5\baselineskip}% \begin{enumeratea} \item\label{enum:a} $\pi(q)=r_\delta$; \item\label{enum:b} $\delta_i=0$ for $r+1\leq i\leq n$; \item\label{enum:c} $\psi(b(q), a_1(q), \ldots, a_n(q))$ holds. \end{enumeratea} Then $W$ is an $F$-definable subset of $\Upsilon$ -- as far as condition \eqref{enum:a} is concerned this is by \cite[Lem.\,8.2.9]{HL}, and it contains $p$. Now let $q$ be a point of $W$. Set $b'=b(q)$ and $a'=a(q)$, and $\gamma'_i=\val (a'_i)$ for all $i$. Conditions \eqref{enum:a} and \eqref{enum:b} ensure that $F(a')$ has a residue field which is regular over $\mathrm{res}(F(a'_{r+1},\ldots, a'_n))$. Indeed, up to applying an invertible monomial transformation to $(a'_1,\ldots, a'_r)$ and renormalizing, we can assume that there is some $s$ such that $\val(a'_1), \ldots, \val (a'_s)$ are free modulo $\val(F)$ and that $\val(a'_t)=0$ for $s+1\leq t\leq r$, in which case the result is obvious since the residue field we consider is then purely transcendental of degree $r-s$ over that of $F(a'_{r+1}, \ldots, a'_n)$. Using the fact that $F(a'_{r+1},\ldots, a'_n)\simeq F(a_{r+1}, \ldots, a_n)$ as valued extensions of $F$ (with $a'_i$ corresponding to $a_i$) and the definition of $\psi$, we see that $\val(F(b',a'))=\val(F(a'))$. In other words, the value group of $q$ is generated by the $a_i(q)$ and $\val(F)$, which ends the proof. \end{proof} Our purpose is now to show how the results of section \ref{t-finite-pure} extend quite straightforwardly, at least on affine charts, when $\Upsilon$ is not assumed to consist only of Zariski-generic points. \subsection{A more general setting} We still denote by $k$ a defectless valued field with divisible value group. Let $X$ be an \textit{affine} $k$-scheme of finite type, and let $\Upsilon$ be a $\Gamma$\nobreakdash-inter\-nal subset of $X^{\#} \subseteq \widehat X$. Let $X_1,\ldots, X_m$ be the irreducible Zariski-closed subsets of~$X$ whose generic point supports an element of $\Upsilon$ (it follows from \cite[Cor.\,10.4.6]{HL} and finiteness of the Zariski topology of $\Gamma_\infty^w$ that there is only a finite number of such irreducible subsets); for each $i$, set \[X'_i=X_i\setminus \bigcup_{j,\, X_j\subsetneq X_i}X_j\] and $\Upsilon_i=\Upsilon\cap \widehat{X'_i}$. By construction, $\Upsilon=\coprod \Upsilon_i$ and for all $i$, $\Upsilon_i$ consists only in Zariski-generic points in in $\widehat{X'_i}$. We denote by $\SS(\Upsilon)$ the set of $k$-definable functions of the form $\val (f)$ with $f$ a \emph{regular} function on $X$ (and not merely a rational function as above). \begin{prop} There exists a finite set $E$ of regular functions on $X$ such that for every $f\in \SS(\Upsilon)$, there exists a finite covering $(D_a)_a$ of $\Upsilon$ by closed definable subsets and, for each $a$, an element $\lambda$ of $k$, a finite family $(e_1,\ldots, e_\ell)$ of elements of $E$, and a finite family $(\epsilon_1,\ldots, \epsilon_\ell)$ of elements of $\{-1,1\}$ such that: \begin{itemize} \item $\epsilon_j=1$ if $e_j$ vanishes on $D_a$; \item $f=\val(\lambda e_1^{\epsilon_1}\ldots e_\ell^{\epsilon_\ell})$ identically on $D_a$. \end{itemize} \end{prop} \begin{proof} For all $i$, we can apply Theorem \ref{theo-tfinite-pure} to the integral scheme $X'_i$ and the $\Gamma$-internal set $\Upsilon_i$; let $E_i$ be the finite set of rational functions on $X'_i$ provided by this theorem. Write $E_i=\{g_{ij}/h_{ij}\}_j$ where $g_{ij}$ and $h_{ij}$ are non-zero regular functions on the integral affine scheme $X_i$. For all $(i,j)$, let $g'_{ij}$ and $h'_{ij}$ denote lifts of $g_{ij}$ and $h_{ij}$ to the ring $\mathscr O_X(X)$. We then might take for $E$ the set of all $g'_{ij}$ and $h'_{ij}$. \end{proof} \section{Specializations and Lipschitz embeddings}\label{section5} As before, $\Upsilon$ is a $\Gamma$-internal subset of $X^{\#}_{\mathrm{gen}}$ for $X$ a separated integral scheme of finite type over a valued field $K$. The goal of this section is to show the existence of regular embeddings of $\Upsilon$ in some $\Gamma^n$ such that $\SS(\Upsilon)$ becomes exactly the set of Lipschitz definable functions under certain assumptions. We begin with some definitions. \begin{defi}\label{def:embed} Let $\alpha:\Upsilon\to \Gamma^n$ be a definable and continuous map and set $W=\alpha(\Upsilon)$. \begin{enumerate} \item We say $\alpha$ is \emph{regular} if $\alpha$ is given by a tuple of regular functions $\Upsilon \to \Gamma$, \ie functions of the form $\val (f)$ with $f$ a non-zero rational function. \item If $\alpha$ is an embedding, then we say $\alpha^{-1}:W\to \Upsilon$ is an \emph{integral parameterization} if for any rational function $f$ defined on $\Upsilon$, $\val(f)\circ \alpha^{-1}$ is piecewise $\Z$-affine. We will also call $\alpha$ integral in this case. \item If $\alpha$ is an embedding, then we say $\alpha^{-1}:W\to \Upsilon$ is \emph{Lipschitz} if for any non zero rational function $f$ on $X$, $\val (f)\circ \alpha^{-1}$ is a Lipschitz function. We will also say $\alpha$ is Lipschitz. \item We say $\alpha$ is a \emph{good embedding} if it is integral and Lipschitz. \end{enumerate} \end{defi} It is immediate from the definition that if $\alpha: \Upsilon\to \Gamma^n$ is a regular embedding (\resp regular integral embedding, \resp regular Lipschitz embedding) and $f$ is another regular function $\Upsilon \to \Gamma$, then $(\alpha, \val (f))\colon\Upsilon\to\Gamma^{n+1}$ is also a regular embedding (\resp regular integral embedding, \resp regular Lipschitz embedding). \begin{lemm}\label{lemm-tfinite} Assume that $K$ is algebraically closed and let $\Upsilon$ be a $\Gamma$-internal subset of $X^{\#}_{\mathrm{gen}}$. Then there exists a regular integral embedding $\alpha\colon \Upsilon\hookrightarrow \Gamma^n$. \end{lemm} \begin{proof}By Theorem \ref{theo-tfinite-pure} there exists a finite family $\alpha = (\alpha_1, \ldots, \alpha_n)$ which $(w,+)$-generates $\SS(\Upsilon)$ modulo the constant functions. Since $K$ is algebraically closed, it~follows from \cite[Prop.\,6.2.7]{HL} that there exists a regular embedding of $\Upsilon$. We may thus enlarge $\alpha$ so that it becomes a regular embedding; it is integral by $(w,+)$-generation. \end{proof} We will now recall some basic facts about $\acvsf$ and specializations, that will provide an important criterion for the existence of good embeddings. \subsection{$\acvsf$-specializations}\label{sec:specialization} We consider a triple $(K_2,K_1,K_0)$ of fields with surjective places $r_{ij}: K_i \to K_j$ for $i > j$, with $r_{20} = r_{10} \circ r_{21}$, such structures are also called $\vsf$. The places $r_{21}$ and $r_{20}$ give rise to two valuations on $K_2$, which we denote by $\val_{21}$ and $\val_{20}$ respectively. We denote by $\Gamma_{ij}$ and $\RES_{ij}$ the corresponding value groups and residue fields. We consider $(K_2,K_1,K_0)$ as a substructure of a model of the theory $\acvsf$ introduced in~\cite[Ch.\,9.3]{HL}. We will use $K_{210}$ to denote the structure $(K_2,K_1,K_0)$. It is clearly an expansion of $(K_2, \val_{21})$ via an expansion of the residue field and an expansion of $(K_2,\val_{20})$ by a convex subgroup in the value group. We will focus on the latter expansion. Let $X$ be an affine integral scheme of finite type over $K_2$, we will use $X_{20}$ when we view $X$ as a definable set in an ambient model of $\acvf$ extending $(K_2,\val_{20})$ and $X_{21}$ is defined analogously. There is a natural map $s: X^{\#}_{20}\to X^{\#}_{21}$ which can be described as follows. Let $p \in X^{\#}_{20}$. By~\cite[Lem.\,9.3.8]{HL}, we have that $p$ generates a complete type $p_{210}$ in $\acvsf$. Furthermore, by~\cite[Lem.\,9.3.10]{HL}, $p_{210}$ as an $\acvsf$-type is stably dominated. Let $\dim(p)$ denote the dimension of the Zariski closure of $p$. Let $L\models\acvsf$ extending $K_{210}$ and $c\models p|L$. Since $p$ corresponds to an Abhyankar point in the space of valuations, we see that the residual transcendence degree of $\tp_{21}(c/L)$ is still $\dim(p)$, so $\tp_{21}(c/L)$ extends to a type $s(p)$ in $X^{\#}_{21}$. (Note that here we work in the restricted language where the only valuation is $\val_{21}$.) \begin{lemm} Let $Y\subseteq X^{\#}_{20}$ be an $\acvsf_{K_{210}}$-definable set, then $s|_Y$ is a definable function. \end{lemm} \begin{proof} By the way $s$ is defined, it is a pro-definable function by considering the $\varphi$-definitions. Note that a pro-definable function between two definable sets is definable by compactness. \end{proof} We need one last lemma before stating our criterion with respect to specializations. \begin{lemm}\label{lem:imaginary} Let $(K_2,K_1,K_0)\models \acvsf$ and $Y$ be a definable set of imaginaries in $\acvf_{K_{21}}$. If $Y$ is $\Gamma_{20}$-internal as a definable set in $\acvsf_{K_{210}}$, then $Y$ is $\Gamma$-internal in $K_{21}$. \end{lemm} \begin{proof} By the classification of imaginaries in $\acvf$~\cite[Th.\,1.01]{haskell-h-m2006}, if $Y$ is not $\Gamma$-internal in $K_{21}$, there is an $\acvf_{K_{21}}$-definable map (possibly after expanding the language by some constants) that is generically surjective onto the residue field. By~assump\-tion, $Y$ is $\Gamma_{20}$-internal as an $\acvsf_{K_{210}}$ set. This yields a generically surjective map $\Gamma_{20}\to \RES_{21}$. Composing with the dominant place $\RES_{21}\to \RES_{20}$, we obtain an $\acvsf$-definable map $\Gamma_{20}\to \RES_{20}$ that is generically surjective. By~\cite[Lem.\,9.3.1(4)]{HL}, one checks immediately that the two sorts $\Gamma_{20}$ and $\RES_{20}$ are orthogonal in $\acvsf$, hence a contradiction. \end{proof} \subsection{Specializable maps and Lipschitz condition} Now we introduce the notion of specializations of maps. Let $(K,v)$ be a valued field, we denote by $\rho(K)$ the set of convex subgroups of $\Gamma(K)$. Clearly, if $K$ is of transcendence degree $m$ over the prime field, then $|\rho(K)|\leq m+1$. For each $\Delta\in \rho(K)$, we have a valuation $\val_{21}:K\to \Gamma(K)/\Delta$ given by quotienting out by $\Delta$, which gives rise to a $\vsf$ structure we shall denote by $K[\Delta]$. Each choice of $\Delta$ specifies an expansion of $\acvf_K$ to $\acvsf_{K}$ by interpreting the convex subgroup to be the convex hull of $\Delta$. Moreover, by varying~$\Delta$ one exhausts all the possible expansions of $\acvf_K$ to $\acvsf_{K}$. Let~$X$ be an integral separated scheme of finite type over $K$ as before. We write $X_\Delta$ to denote $X$ as a definable set in $\acvf_{K[\Delta]}$. We use $s_\Delta$ to denote the map $s$ defined in Section~\ref{sec:specialization} when we expand $\acvf_K$ to $\acvsf_{K[\Delta]}$. We use $\Upsilon_\Delta$ to denote $s_\Delta(\Upsilon)$. Similarly, if $\alpha:\Upsilon\to \Gamma^n$ is some regular embedding, we use $\alpha_\Delta:\Upsilon_\Delta\to \Gamma_{21}^n$ to denote the corresponding map. \begin{defi} Let $\alpha:\Upsilon \hookrightarrow \Gamma^n$ be a regular embedding and let $K$ be a field over which $\alpha$ is defined. We say $\alpha$ is \emph{specializable} if for every convex subgroup $\Delta$ of $\Gamma(K)$ the map $\alpha_\Delta$ is still an embedding. \end{defi} \begin{rema}\label{rem:parameter} Note that the specializability of $\alpha$ does not depend on the choice of~$K$. Namely, let $L\supseteq K$ be an extension of valued fields, it suffices to show that if $\alpha$ is specializable with respect to $K$, it is so with respect to $L$. Let $\Delta_L$ be a convex subgroup of $\Gamma(L)$. Note that this gives a convex subgroup $\Delta_K$ of $\Gamma(K)$ by taking intersection. Note that whether $\alpha_{\Delta_L}$ is an embedding only depends on $\acvf_{L[\Delta_L]}$, which is an expansion of $\acvf_{K[\Delta_K]}$. Hence the specializability of $\alpha$ over $K$ guarantees that $\alpha_{\Delta_L}$ is an embedding. \end{rema} \begin{rema} If $\alpha\colon \Upsilon \hookrightarrow \Gamma^n$ is a specializable regular embedding and $\beta\colon \Upsilon \to \Gamma^m$ is any regular map, the regular embedding $(\alpha,\beta)\colon \Upsilon\to \Gamma^{n+m}$ is specializable as well. \end{rema} \begin{rema}\label{rem:specializable} Assume $\alpha:\Upsilon\to \Gamma^n$ is specializable and defined over $K$, and $\Upsilon'\subseteq \Upsilon$ is definable but not necessarily over $K$. If $\alpha$ is specializable, so is $\alpha|_{\Upsilon'}$. This follows from a similar argument as in Remark~\ref{rem:parameter}. More precisely, let $L\supseteq K$ be such that $\Upsilon'$ is defined over $L$, any expansion of $\acvf_L$ to $\acvsf_L$ by some $\Delta'$ gives an expansion of $\acvf_K$ to $\acvsf_K$ by some $\Delta$. As $\alpha$ is specializable, $\alpha_\Delta$ is an embedding for any $\Delta$, thus $\alpha|_{\Upsilon'}$ is specializable. \end{rema} \begin{theo}\label{thm:existence} Let $X$ be an affine integral scheme of finite type over a valued field $K$ and let $\Upsilon\subseteq X^{\#}$ be a $\Gamma$-internal subset. Let $F$ be a finitely generated field over which all the above is defined. Then there exists a $F^\mathrm{alg}$-definable integral regular embedding of $\Upsilon$ into $\Gamma^n$ that is specializable. \end{theo} \begin{proof} For each $\Delta\in \rho(F)$, by Lemma~\ref{lem:imaginary}, we have that $\Upsilon_\Delta\subseteq X^{\#}_\Delta$ is $\Gamma$-internal in $\acvf_{F[\Delta]}$. Consider $X$ as embedded in some affine space. By~\cite[Cor.\,6.2.5]{HL}, for each $\Delta$, there are finitely many polynomial functions $h^\Delta_i $ such that \[ h^\Delta=(\val(h_1^\Delta),\ldots,\val(h_s^\Delta)) \] is injective on $\Upsilon$. Moreover the $h^\Delta_i$'s can be found to be defined over $F^\mathrm{alg}$ by the proof of~\cite[Cor.\,6.2.5]{HL} (or more precisely,~\cite[Lem.\,6.2.2]{HL}). Since there are only finitely many such $\Delta$'s to consider, putting them as the coordinates, we get some specializable embedding as desired, which can be made integral by concatenation with an arbitrary integral regular embedding, whose existence follows from Lemma \ref{lemm-tfinite}. \end{proof} \begin{rema} In the situation of interest for classical non-archimedean geometry, the ground field $K$ will be algebraically closed and equipped with a valuation whose group embeds into $\R$ and has therefore no non-trivial proper convex subgroup. The reasoning above then shows that \textit{any} $K$-definable regular embedding from $\Upsilon$ into $\Gamma^s$ is specializable. \end{rema} \begin{prop}\label{speandlip} Let $X$ be an affine integral scheme of finite type over a valued field $K$ and let $\Upsilon\subseteq X^{\#}$ be a $\Gamma$-internal subset. If $\alpha:\Upsilon\hookrightarrow \Gamma^n$ is a specializable embedding, then the image of $\SS(\Upsilon)$ is contained in the group of Lipschitz functions. In other words, all the log-rational functions are Lipschitz and $\alpha$ is Lipschitz. \end{prop} \begin{proof} We let $W=\alpha(\Upsilon)$ and use $p_w$ to denote $\alpha^{-1}(w)$ for $w\in W$. Assume there is some $f\in K(X)$ such that $w\mapsto p_w(f)$ is not Lipschitz. Going to an elementary extension, we may assume there is $w_1,w_2 \in W$ such that $|p_{w_2}(f)-p_{w_1}(f)|>n|w_1-w_2|$ for all $n\in \N$. Take $C$ to be the convex subgroup generated by $|w_1-w_2|$. Consider~$L$ to be the same field with the valuation given by quotienting out by $C$. By our assumption on specializability, we have that $\alpha_L$ is an embedding. However, we have $\overline{w_1}=\overline{w_2}$, while $\overline{p_{w_1}(f)}=p_{w_1}(f)+C\neq p_{w_2}(f)+C=\overline{p_{w_2}(f)}$, a contradiction. \end{proof} \begin{coro}\label{goodexistence} Let $X$ be an affine integral scheme of finite type over an algebraically closed valued field $K$ and let $\Upsilon\subseteq X^{\#}$ be a $\Gamma$-internal set. Then there exists a good embedding $\Upsilon \hookrightarrow \Gamma^n$. \end{coro} \begin{proof} The embedding provided by Theorem \ref{thm:existence} is $K$-definable, and it is good in view of Proposition \ref{speandlip}. \end{proof} \section{The main theorem} In this section, we prove the theorem stated in Section~\ref{sec:about-proof} and we transfer it into the Berkovich setting. \begin{lemm}\label{lem-stable-min} Let $k$ be a valued field with infinite residue field, let $X$ be a geometrically integral $k$-scheme and let $\Upsilon\subset X^\# _{\mathrm{gen}}$ be a $k$-definable $\Gamma$-internal subset defined over $k$. The group $\SS(\Upsilon)$ is stable under $\min$ and $\max$. \end{lemm} \begin{proof} It is enough to prove stability under $\min$. Let $p$ be a point of $\Upsilon$. If there exists a scalar $a$ of valuation zero such that $\val(f(p)+ag(p))>\min(\val (f)(p), \val (g(p)))$ then $\mathrm{res}(a)$ is a well-defined element of the residue field which we call $\theta(p)$; otherwise we set (say) $\theta(p)=0$. Then $\theta$ is a $k$-definable map from the $\Gamma$-internal set~$\Upsilon$ to the residue field. By orthogonality between the value group and the residue sorts, $\theta$ has finite image. Since $k$ has infinite residue field, there exists an element $a\in\mathscr O_k^\times$ whose residue class does not belong to the image of $\theta$. Then $f+ag\neq 0$ and $\val(f(p)+ag(p))=\min (\val (f(p)), \val (g(p)))$ for all $p\in \Upsilon$. \end{proof} In the situation of the lemma above, it thus makes sense to speak about an $(\ell,+)$-generating system of elements of $\SS(\Upsilon)$. As for $(w,+)$-generation, we shall say for short that $\SS(\Upsilon)$ is finitely $(\ell,+)$-generated up to the constant functions if there exists a finite subset $E$ of $\SS(\Upsilon)$ such that $E$ and the $k$-definable constant functions (\ie the constant functions taking values in $\Q\otimes \val(k^\times)$) $(\ell,+)$-generate $\SS(\Upsilon)$. \begin{theo}\label{themaintheorem} Let $k$ be an algebraically closed valued field. Let $X$ be an integral scheme of finite type over $k$ and let $\Upsilon\subseteq X^\#_{\mathrm{gen}}$ be a $\Gamma$-internal subset defined over $k$. The group $\SS(\Upsilon)$ is stable under $\min$ and $\max$ and is finitely $(\ell,+)$-generated up to constant functions. \end{theo} \begin{proof} By Theorem~\ref{thm:existence}, there is a $k$-definable good embedding $\alpha: \Upsilon\to \Gamma^n$ for some $n$. By Theorem~\ref{theo-tfinite-pure}, $\SS(\Upsilon)$ is $(w,+)$-finitely generated up to constant functions. Let $f_1,\ldots,f_m$ be finitely many $k$-rational functions whose valuations $(w,+)$-generate $\SS(\Upsilon)$ up to constant functions, adjoining the $\val (f_i)$ as new coordinates of $\alpha$, we~may furthermore assume that $\SS(\Upsilon)$ is $(w,+)$-generated by the components of $\alpha$ and the constant functions. By possibly enlarging once again $\alpha$ and replacing $X$ with a suitable dense Zariski-open subset we can also assume that $\alpha=\val (f)$ for some closed immersion $f\colon X\to \gm^n$; in particular, $\alpha$ is definably proper and induces a definable homeomorphism $\Upsilon\simeq \alpha(\Upsilon)$. Let $f$ in $\SS(\Upsilon)$. Since $\alpha$ is a specializable embedding whose coordinates $(w,+)$-generate $\SS(\Upsilon)$ up to the constant functions, the composition $f\circ \alpha^{-1}$ viewed as a $\Gamma$\nobreakdash-valued function on $\alpha(\Upsilon)$ is piecewise $\Z$-affine and Lipschitz. In view of Theorem~\ref{thm:lip-t-gen}, this implies that $f\circ \alpha^{-1}$ is an $\ell$-combination of finitely many $\Z$-affine functions, so~that~$f$ itself is an $(\ell,+)$-combination of the components of $\alpha$ and of constant functions. \end{proof} \begin{rema}\label{wgeneration-scalarextension} Assume that $k$ is algebraically closed and let $(f_1,\ldots,f_n)$ be a family of rational functions on $X$ such that $\SS(\Upsilon)$ is $(w,+)$-generated (resp. $(\ell,+)$-generated) by the $\val (f_i)$ and the constant ($k$-definable) functions. Then for every algebraically closed extension $L$ of $k$, the $\val (f_i)$ and the $L$-definable constant functions $(w,+)$-generate (resp. $(\ell,+)$-generate) $\SS_L(\Upsilon)$ (work with a bounded family of rational functions and use compactness). \end{rema} Our purpose is now to state and prove the Berkovich avatar of our main theorem. We fix a non-archimedean complete field $F$. For all $n$, we denote by $S_{F,n}$ the closed subset $\{\eta_r\}_{r\in (\R^\times_+)^n}$ of $\bG_{\mathrm m,F}^{n,\mathrm{an}}$, where $\eta_r$ is the semi-norm $\sum a_I T \mapsto \max \abs{a_I}r^I$. In \cite[4.6]{confluentes}, a general notion of a skeleton is defined for an $F$-analytic space; the subset $S_{n,F}$ of $\bG_{\mathrm m,F}^{n,\mathrm{an}}$ is the archetypal example of such an object. But this notion is however slightly too analytic for our purposes here: indeed, if~$X$ is an algebraic variety over $F$ then $X\an$ might have plenty of skeleta in the sense of \cite{confluentes} that cannot be handled by our methods, since they would not correspond to any $\Gamma$-internal subset of $\widehat X$, by lack of algebraic definability. For instance, assume that~$F$ is algebraically closed and non-trivially valued, and let $f$ be any non-zero analytic function of $\bA^{1,\mathrm{an}}_F$ with countably many zeroes. Let $U$ be the non-vanishing locus of~$f$, and let $\Sigma$ be the preimage of $S_{1,F}$ under $f \colon U\to \gm$. Then $\Sigma$ is a skeleton in the sense of \cite{confluentes}, but topologically this is only a locally finite graph, with countably many branch points. We thus shall need to focus on ``algebraic'' skeleta. \begin{theo}\label{mtberk} Let us assume that $F$ is algebraically closed. Let $X$ be an integral $F$-scheme of finite type, and let $n$ be its dimension. Let $\phi_1,\ldots,\phi_r$ be maps $U_i\to \bG_{\mathrm m, F}^n$ where the $U_i$ are dense open subsets of~$X$, and let $S\subseteq X\an$ be a subset of $\bigcup_i \phi_i^{-1}(S_n)$ defined by a Boolean combination of norm inequalities between non-zero rational functions. There exist finitely many non-zero rational functions $f_1,\ldots, f_m$ on $X$ such that the following hold. \begin{enumerate} \item The functions $\log \abs{f_1},\ldots, \log \abs{f_m}$ identify $S$ with a piecewise-linear subset of~$\R^m$ (\ie a subset defined by a Boolean combination of inequalities between $\Q$-affine functions). \item The group of real-valued functions on $S$ of the form $\log \abs g$ for $g$ a non-zero rational function on $X$ is stable under $\min$ and $\max$ and is $(\ell,+)$-generated by the $\log \abs{f_i}$ and the constant functions of the form $\log \abs{\lambda}$ with $\lambda \in F^\times$. \end{enumerate} \end{theo} \begin{proof} The subset $\Sigma$ of $\widehat X$ given by the same definition as $S$ mutatis mutandis is a $\Gamma$-internal set contained in $X^\#_{\mathrm{gen}}$ to which we can thus apply Theorem \ref{themaintheorem}. The theorem above then follows by noticing that if $L$ denotes a non-archimedean maximally complete extension of $F$ with value group the whole of $\R_+^\times$, then $S$ is naturally homeomorphic to $\Sigma (L)$. \end{proof} \begin{rema}Note that by Theorem \ref{structure-pure-gammainternal} the condition that $S$ is a subset of some $\bigcup_i \phi_i^{-1}(S_n)$ holds as soon as $S$ is the image of $\Upsilon (L)$ under the projection $\widehat{X} (L) \to X\an$ with $\Upsilon$ some $F$-definable $\Gamma$-internal subset of $X^{\#}_{\mathrm{gen}}$ and $L$ as in the above proof. \end{rema} \begin{rema}\label{contrex} We insist that we require that the ground field be algebraically closed. Indeed, our theorems (for stable completions as well as for Berkovich spaces) definitely do not hold over an arbitrary non-Archimedean field, \textit{even in a weaker version with $(w,+)$-generation instead of $(\ell,+)$-generation}, as witnessed by a counter-example that was communicated to the authors by Michael Temkin (this counterexample involves a field with defect, we do not know if our theorem holds for defectless fields with divisible value group as in Theorem \ref{theo-tfinite-pure}). For the reader's convenience we will first detail the original counter-example which is written in the Berkovich's language, and then a model-theoretic variant thereof in the Hrushovski-Loeser's language. \subsubsection{The Berkovich version} Let $F$ be a non-archimedean field and let $\F$ be the completion of an algebraic closure of $F$; assume that the residue field of $F$ is of positive characteristic $p$ and that $F$ admits an immediate extension $L$ of degree $p$, say $L:=F(\alpha)$ with $\alpha\in \F$. By general valuation theory, the distance $r$ between $\alpha$ and $F$ is not achieved. For every $s\geq r$ let $\xi_s$ be the image on $\P^{1,\mathrm{an}}_F$ of the Shilov point of the closed $\F$-disc with center $\alpha$ and radius $s$. If $s>r$ there exists $\beta_s$ in $F$ with $\abs{\alpha-\beta_s}\leq s$, and $\xi_s$ is the Shilov point of the closed $F$-disc with center $\beta_s$ and radius $s$; but as far as $\xi_r$ is concerned, it is the Shilov point of an affinoid domain $V$ of $\P^{1, \mathrm{an}}_F$ without rational point. Let $v$ be a rigid point of $V$. It corresponds to an element $\omega$ of $\F$ algebraic over~$F$ and whose distance to $F$ is equal to $r$ and not achieved. Therefore the extension $F(\omega)$ has defect over $F$, which forces its degree to be divisible by $p$. In other words, $[\mathscr H(v):F]$ is divisible by $p$. In particular if $f$ is any non-zero element of $F(T)$, the divisor of $f|_V$ has degree divisible by $p$, so that there exists some $s>r$ such that the slope of $\log \abs f$ on $(\xi_r,\xi_s)$ is divisible by $p$. Now assume that there exists a finite set $E$ of non-zero rational functions such that on the skeleton $[\xi_r,\infty)$, every function of the form $\log \abs g$ with $g$ in $F(T)^\times$ belongs piecewise to the group generated by the $\log \abs h$ for $h\in E\cup F^\times$. Then there would exist some $s>r$ such that for every $g$ as above, all slopes of $(\log \abs g)|_{[\xi_r,\xi_s]}$ are divisible by~$p$. Taking $g=T-\beta_s$ leads to a contradiction. \subsubsection{The model-theoretic version} Let $F$ be a perfect valued field of positive residue characteristic $p$ such that there exists an irreducible separable polynomial $P\in F[T]$ with the following property: \textit{the smallest closed ball containing all roots of~$P$ has no $F$-rational points, but any bigger $F$-definable closed ball has one $F$-point} (it is not difficult to exhibit such pairs $(F,P)$; the easiest case is that of pure characteristic $p$, where one can take any perfect field $F$ with an height 1 valuation having an element $s$ with $\val(s)<0$ such that $T^p-T-s$ has no root in $F$, and take $P=:T^p-T-s$; for instance, the perfect closure of $\F_p(s)$ equipped with the $(1/s)$-adic valuation will do the job). Let $b$ be the smallest closed ball containing the roots of $P$, and let $B$ be a bigger $F$-definable closed ball. Let $I$ be the interval between their generic points; this is a $\Gamma$-internal subset of $\widehat{\P^1}$ contained in $\P^{1,\#}_{\mathrm{gen}}$. This interval is naturally parameterized by the interval $[V,v]$ where $V$ is the valuative radius of $B$ and $v$ is that of $b$, and we will identify them. In particular a linear function from $I$ to $\Gamma$ has a well-defined slope. We~will be interested in the germ of functions on $I$ towards the endpoint $v$. The number of roots in $b$ of every irreducible polynomial of $F[T]$ is divisible by $p$, for otherwise averaging the roots would produce an $F$-rational point in $b$. Hence the valuation of every polynomial, and indeed every rational function in $F(T)$, has slope divisible by $p$ on some interval $(i,v)$ inside $I$. If the group of functions $\val(f) | I$ were finitely $(w,+)$-generated up to constants, there would be a fixed $i0$ such that there exists a non-zero $L$-rational function on $X$ whose valuation has slope $1/N$ along the germ of branch emanating rightward from $\omega$ and induced by $I$. But since the group of non-zero rational functions on $X$ is universally generated by quotients of non-zero global sections of $\mathscr L$, this integer $e(I,\omega)$ can be read off from the finite set of slopes $s(I,\omega)$ (it is nothing but the lcm of their denominators). Now $F_L$ is a defectless extension of $(L(T), w_{h^+})$ if and only if the sum of all the ramification indexes of $v(I,\omega)$ for $\omega$ above $w_h$ and $I\in \mathscr I(\omega)$ is equal to $[F:K(T)]$. Thus whether $F_L$ is a defectless extension of $(L(T),w_{h^+})$ or not can be read off from the sets of slopes $s(I,\omega)$ for $\omega$ above $w_h$ and $I\in \mathscr I(\omega)$; the existence of the required $K$-definable set $D$ follows immediately. \subsubsection{Conclusion} Our purpose is to prove that $F$ is a defectless extension of $(K(T),v_g)$ or, in other words, that $g\in D(K)$, and we are in fact going to prove that~$D$ is the whole of $\Gamma$. For this, it suffices to show that $D$ is both definably open and definably closed and non-empty. \refstepcounter{paragraphe} \subsubsection*{\textup{\theparagraphe}. The set $D$ is both definably open and definably closed}\label{d-openandclosed} Let $h\in v(K^\times)$, and let $(L,w)$ be a model of $\acvf$ containing $K$ and such that $w(L^\times)$ contains two elements $h^+$ and $h^-$ infinitely close to $h$ with respect to $v(K^\times)$ and with $h^-