1 Introduction
1.1 Background
For a long time, there has been a search for adequate analogues of ominimality in settings other than for real geometry. Several notions have been put forward, each with certain strengths and weaknesses. In Part I [Reference Cluckers, Halupczok and Rideau10], together with this sequel paper, we put forward a notion for tame geometry on nonArchimedean valued fields, called Hensel minimality, both simple and strong and providing a common variant for the settings from, among others, [Reference Haskell16; Reference Macpherson and Steinhorn18; Reference Haskell and Macpherson15; Reference Hrushovski and Kazhdan17; Reference Cluckers and Loeser11].
Let us first explain the key flavor of the definition of Hensel minimality. Often, finding a correct analogue involves some reformulations. Here is a way to reformulate ominimality: a structure on the real field ${\mathbb R}$ is ominimal if and only if for each definable subset X of ${\mathbb R}$ , there exists a finite set $C\subset {\mathbb R}$ , such that, for any x in ${\mathbb R}$ , the condition $x\in X$ only depends on the signs of $xc$ for c in C, with the sign being negative, zero, or positive. In an equicharacteristic zero valued field K, we replace the sign map by the projection map $\operatorname {rv}:K\to K/(1+{\mathcal M}_K)$ , where ${\mathcal M}_K$ is the maximal ideal of the valuation ring of K, and $K/(1+{\mathcal M}_K)$ is the quotient of multiplicative semigroups. In the mixed characteristic case, one additionally uses ideals of the form $N\cdot {\mathcal M}_K$ for nonzero integers N, instead of just ${\mathcal M}_K$ itself. Apart from replacing the sign map, we keep the condition of every definable subset of K being ‘controlled’ by a finite set $C\subset K$ . In the real case, C is automatically definable over the same parameters as X; in the valued field case, this is no longer automatic and needs to be imposed. Depending on the precise kind of parameters one allows, one obtains the notion of $0$ hminimality, $\omega $ hminimality or something inbetween: $\ell $ hminimality for integers $\ell>0$ ; see Section 2 for the detailed definitions. When we do not want to be precise about the specific version, we just say ‘Hensel minimality’.
Hensel minimality is similar to ominimality [Reference van den Dries13; Reference Pillay and Steinhorn21] not only in its definition but also for having strong consequences. In Part I [Reference Cluckers, Halupczok and Rideau10], we focused on the equicharacteristic zero case. In this sequel, we focus on the mixed characteristic case. We obtain analogues of many of the results from [Reference Cluckers, Halupczok and Rideau10]. In addition, in Section 4, we give a new diophantine application, similar to the results by PilaWilkie [Reference Pila and Wilkie20] from in ominimal case; for the moment, this application is for curves only. Many proofs from [Reference Cluckers, Halupczok and Rideau10] adapt to the mixed characteristic case in a rather straightforward way; nevertheless, many proofs are repeated in this sequel, and for many others, we sketch the key ideas so that the reader has the choice between reading the short version here or the detailed version in [Reference Cluckers, Halupczok and Rideau10].
There are several ways to adapt the notions of Hensel minimality from [Reference Cluckers, Halupczok and Rideau10] to the mixed characteristic case, based on coarsenings, on compactness or on a more literal adapation of the definitions for the equicharacteristic zero case. We treat several of these variants, and in one of the most important cases (namely, $1$ hminimality), we show that they are equivalent. From this equivalence, we derive that many of the geometric results of [Reference Cluckers, Halupczok and Rideau10] also hold in the mixed characteristic case, with some strong results on Taylor approximation of definable functions. In [Reference Vermeulen23], equivalences of these notions are shown also in other cases.
The name ‘Hensel minimality’ comes from the intuition that a Hensel minimal valued field behaves as nicely as a henselian valued field in the pure valued field language. In equicharacteristic zero, this is reflected by the fact that a pure valued field K is Hensel minimal if and only if K is henselian (where any version of Hensel minimality can be used). In mixed characteristic, this is still true if K is finitely ramified (see Remark 2.2.3), but in general, Hensel minimality only captures tameness from the ramification degree on.
1.2 Overview
Let us give a short overview of the paper. In Section 2, we introduce several variants of Hensel minimality (for both the mixed and the equicharacteristic cases), and we prove that all of them are equivalent in the key case of $1$ hminimality (Theorem 2.2.8). We do this for a large part by classical strategies: coarsening of valuations of mixed characteristic to equicharacteristic zero, and model theoretic compactness. A combination of these classical strategies with new geometric and model theoretic arguments leads to our strongest results, like Theorem 3.1.2 on Taylor approximation. It is precisely this result, Theorem 3.1.2 on Taylor approximation, that plays a key role in our diophantine application Theorem 4.0.7, where we estimate the number of rational points of bounded height on transcendental curves. This is the Hensel minimal analogue of point counting on transcendental curves case as treated by Bombieri and Pila [Reference Bombieri and Pila5] and ominimally by PilaWilkie [Reference Pila and Wilkie20], and it is the axiomatic analogue of point counting on subanalytic sets as studied in [Reference Cluckers, Comte and Loeser8] and [Reference Cluckers, Forey and Loeser9], and on analytic sets in [Reference Binyamini and Kato4].
In Section 2.6, we develop some resplendency results analogous to the ones from [Reference Cluckers, Halupczok and Rideau10, Section 4] – namely, that Hensel minimality is preserved under certain expansions of the structure (e.g., arbitrary expansions of the value group and the residue field). Those results are used to show the equivalences of Theorem 2.2.8. It follows from the equivalence with item (4) in Theorem 2.2.8 that $1$ hminimality is preserved under coarsening of the valuation; such a coarsening result was previously known only for $\omega $ hminimality by [Reference Cluckers, Halupczok and Rideau10, Corollary 4.2.4]. Note that in [Reference Vermeulen23], several results of this paper and of part I [Reference Cluckers, Halupczok and Rideau10] are extended from $1$ hminimality to $\ell $ hminimality for each $\ell \ge 1$ (for example, their preservation under coarsenings of the valuation).
We end the paper with some open questions. One of the big challenges in the current framework is to push our diophantine application farther towards arbitrary dimension and thus get a full Hensel minimal analogue of the results by PilaWilkie [Reference Pila and Wilkie20]. Other main challenges are further developments of the geometry, like tstratifications building on and extending [Reference Cluckers, Halupczok and Rideau10, Section 5.5]. It may also be interesting to compare the notions of Hensel minimal structures with distal expansions on valued fields, as studied recently in [Reference Aschenbrenner, Chernikov, Gehret and Ziegler1].
2 Hensel minimality in characteristic zero (including mixed characteristic)
In this whole section, we fix a theory ${\mathcal T}$ of valued fields of characteristic zero in a language ${\mathcal L}$ , expanding the language ${\mathcal L}_{\mathrm {val}}= \{+,\cdot ,{\mathcal O}_K\}$ of valued fields. Note that ${\mathcal T}$ is allowed to be noncomplete and that each model K of ${\mathcal T}$ is a valued field of characteristic zero, which includes both possibilities that K has mixed characteristic or equicharacteristic zero.
In this section, we give four alternative definitions of $1$ hminimality for ${\mathcal T}$ (see Definitions 2.2.1, 2.2.4, 2.2.5, 2.2.7), and we show that they are equivalent in Theorem 2.2.8. To keep the generality of [Reference Cluckers, Halupczok and Rideau10], we will treat more generally $\ell $ hminimality for $\ell \ge 0 $ either an integer or equal to $\omega $ . The first definition is a close adaptation of the main notion of Hensel minimality of [Reference Cluckers, Halupczok and Rideau10, Definition 2.3.3]. Definitions 2.2.4 and 2.2.5 are based on coarsenings, and Definition 2.2.7 corresponds to the criterion for $1$ hminimality from [Reference Cluckers, Halupczok and Rideau10, Theorem 2.9.1]. Since for $\ell =1$ these notions coincide, we simply will call them $1$ hminimality.
2.1 Basic terminology
We use the following terminology, notation and concepts from [Reference Cluckers, Halupczok and Rideau10]. By a valued field we mean a nontrivially valued field (i.e., the field of fractions of a valuation ring which is not a field itself). Any valued field K is a structure in the language ${\mathcal L}_{\mathrm {val}}= \{+,\cdot ,{\mathcal O}_K\}$ of valued fields, where $+$ and $\cdot $ are addition and multiplication on K, and where ${\mathcal O}_K$ is (a predicate for) the valuation ring of K. The maximal ideal of ${\mathcal O}_K$ is denoted by ${\mathcal M}_K$ . We use multiplicative notation for the value group, which we denote by $\Gamma _K^{\times }$ , and we write $\Gamma _K := \Gamma _K^{\times } \cup \{0\}$ . We write $\cdot \colon K \to \Gamma _K$ for the valuation map. By an open ball, we mean a set of the form $B_{<\lambda }(a)=\{x\in K\mid xa < \lambda \}$ for some $\lambda $ in $\Gamma _K^{\times }$ and some a in K. We define $\operatorname {rad}_{\mathrm {op}} B_{<\lambda }(a)=\lambda $ to be the open radius of such a ball. Similarly, a closed ball is a set of the form $B_{\leq \lambda }(a) = \{x\in K\mid xa\leq \lambda \}$ . Its closed radius is $\operatorname {rad}_{\mathrm {cl}} B_{\leq \lambda }(a)=\lambda $ .
For any proper ideal I of the valuation ring ${\mathcal O}_K$ , we write $\mathrm {RV}_I$ for the corresponding leading term structure (i.e., the disjoint union of $\{0\}$ with the quotient group $K^{\times } / (1+I)$ , and $\operatorname {rv}_I:K\to \mathrm {RV}_I$ for the leading term map – that is, the quotient map extended by sending sending $0$ to $0$ ). When I is the open ball $\{x \in K \mid x < \lambda \}$ for some $\lambda \leq 1$ in $\Gamma _K^{\times }$ , we simply write $\mathrm {RV}_{\lambda }$ and $\operatorname {rv}_{\lambda }$ instead of $\mathrm {RV}_I$ and $\operatorname {rv}_I$ , and we write $\mathrm {RV}$ and $\operatorname {rv}$ instead of $\mathrm {RV}_1$ and $\operatorname {rv}_1$ .
We will sometimes also write $\mathrm {RV}_K$ for $\mathrm {RV}$ and similarly $\mathrm {RV}_{K,\lambda }$ for $\mathrm {RV}_{\lambda }$ if multiple fields are under consideration.
Definition 2.1.1. Let $\lambda \leq 1$ be an element of $\Gamma _K^{\times }$ .

(1) Given an arbitrary set $X \subset K$ and a finite nonempty set $C \subset K$ , we say that X is $\lambda $ prepared by C if the condition whether some $x\in K$ lies in X depends only on the tuple $(\operatorname {rv}_{\lambda }(xc))_{c\in C}$ .

(2) We say that a ball $B \subset K$ is $\lambda $ next to an element $c \in K$ if
$$\begin{align*}B= \{x\in K\mid \operatorname{rv}_{\lambda}(xc) = \xi \} \end{align*}$$for some (nonzero) element $\xi $ of $\mathrm {RV}_{\lambda }$ . 
(3) We say that a ball $B \subset K$ is $\lambda $ next to a finite nonempty set $C \subset K$ if B equals $\bigcap _{c\in C} B_c$ with $B_c$ a ball $\lambda $ next to c for each $c\in C$ .
One easily verifies that for fixed C and $\lambda $ , the set of all balls $\lambda $ next to C forms a partition of $K \setminus C$ . Moreover, a set $X \subset K$ is $\lambda $ prepared by C if and only if $X \setminus C$ is a union of parts of this partition (without any condition on $X \cap C$ ). Decreasing $\lambda $ refines this partition (i.e., if $\lambda ' < \lambda $ and $C \ \lambda $ prepares X, then it also $\lambda '$ prepares X). Similarly, if $C' \supset C$ and $C \ \lambda $ prepares X, then $C'$ also $\lambda $ prepares X.
Example 2.1.2. Fix any $\lambda \le 1$ in $\Gamma _K^{\times }$ .

(1) A finite set $X \subset K$ is $\lambda $ prepared by C if and only if C contains X.

(2) A subset $X \subset K$ is $\lambda $ prepared by the set $C = \{0\}$ if and only if X is a (possibly infinite) union of fibers of the map $\operatorname {rv}_{\lambda }$ – that is, if it is of the form $X = \operatorname {rv}_{\lambda }^{1}(\Xi )$ for an arbitrary subset $\Xi \subset \mathrm {RV}_{\lambda }$ .

(3) Every open ball $B = B_{< \lambda }(a)$ of radius $\lambda $ contained in $\mathcal {O}_K$ is $\lambda $ prepared by $C = \{0\}$ , though B might not be $\lambda $ next to C. (It is $\lambda $ next to C if and only if $a = 1$ .)
The following notation specific to mixed characteristic already appears in [Reference Cluckers, Halupczok and Rideau10, Section 6].
Definition 2.1.3 (Equicharacteristic zero coarsening).
Given a model $K \models {\mathcal T}$ , we write ${\mathcal O}_{K,{\mathrm {ecc}}}$ for the smallest subring of K containing ${\mathcal O}_K$ and ${\mathbb Q}$ , and we let $\cdot _{{\mathrm {ecc}}}\colon K \to \Gamma _{K,{\mathrm {ecc}}}$ be the corresponding valuation. (Thus, $\cdot _{{\mathrm {ecc}}}$ is the finest coarsening of $\cdot $ which has equicharacteristic zero; note that $\cdot _{{\mathrm {ecc}}}$ can be a trivial valuation on K.) If $\cdot _{{\mathrm {ecc}}}$ is a nontrivial valuation (i.e., ${\mathcal O}_{K,{\mathrm {ecc}}} \ne K$ ), then we also use the following notation: $\operatorname {rv}_{{\mathrm {ecc}}}\colon K \to \mathrm {RV}_{\mathrm {ecc}}$ is the leading term map with respect to $\cdot _{{\mathrm {ecc}}}$ . Given $\lambda \in \Gamma _{K,{\mathrm {ecc}}}$ , $\operatorname {rv}_{\lambda }\colon K \to \mathrm {RV}_{\lambda }$ is the leading term map with respect to $\lambda $ , and ${\mathcal L}_{{\mathrm {ecc}}}$ is the extension of ${\mathcal L}$ by a predicate for ${\mathcal O}_{K,{\mathrm {ecc}}}$ . More generally, for any nontrivial coarsening $\cdot _c$ of the valuation on K, write ${\mathcal L}_c$ for the extension of ${\mathcal L}$ by a predicate for the valuation ring for $\cdot _c$ .
2.2 Equivalent definitions
There are several natural notions of Hensel minimality in mixed characteristic. We give four possible definitions. Theorem 2.2.8 states that these are all equivalent, in the case of $1$ hminimality.
We first adapt the main definition of Hensel minimality from [Reference Cluckers, Halupczok and Rideau10, Definition 2.3.3] to include the mixed characteristic case. Recall that ${\mathcal T}$ is a theory of valued fields of characteristic zero in a language ${\mathcal L}\supset {\mathcal L}_{\mathrm {val}}$ .
Definition 2.2.1. Let $\ell \geq 0$ be either an integer or $\omega $ . Say that ${\mathcal T}$ is $\ell $ h $^{\mathrm {mix}}$ minimal if for each model K of ${\mathcal T}$ , for each integer $n \ge 1$ , each $\lambda \le 1$ in $\Gamma _K^{\times }$ , each $A\subset K$ , each finite $A'\subset \mathrm {RV}_{\lambda }$ of cardinality $\# A'\le \ell $ and each $(A\cup \mathrm {RV}_{n} \cup A')$ definable set $X\subset K$ , there exists an integer $m \ge 1$ such that X is $m\lambda $ prepared by a finite Adefinable set $C\subset K$ . If all models of ${\mathcal T}$ are of equicharacteristic zero, we also call ${\mathcal T}$ simply $\ell $ hminimal.
As one sees from the definition, bigger values of $\ell $ yield stronger conditions (i.e., for $\ell < \ell '$ , $\ell '$ h $^{\mathrm {mix}}$ minimality implies $\ell $ h $^{\mathrm {mix}}$ minimality).
Clearly, if all models K of ${\mathcal T}$ are of equicharacteristic zero, then one can take $n=m=1$ , so the above definition of $\ell $ hminimality agrees with [Reference Cluckers, Halupczok and Rideau10, Definition 2.3.3]. In the mixed characteristic case, one is obliged to take the valuation of integers into account in Definition 2.2.1. The general philosophy is that $\lambda $ preparation in equicharacteristic zero becomes $\lambda \cdot m$ preparation in mixed characteristic, for some integer $m \ge 1$ , as the following example illustrates.
Example 2.2.2. The set X of cubes in the $3$ adic numbers ${\mathbb Q}_3$ cannot be $1$ prepared by any finite set C, since each of the infinitely many disjoint balls $27^r(1+3{\mathbb Z}_3)$ , $r \in {\mathbb Z}$ , contains both cubes and noncubes. However, X is a union of (infinitely many) fibers of the map $\operatorname {rv}_{3}\colon {\mathbb Q}_3 \to \mathrm {RV}_{3}$ , so it is $3$ prepared by the set $\{0\}$ .
In [Reference Dolich and Goodrick12], Dolich and Goodrick introduced the notion of a visceral structure (a structure equipped with a uniformly definable topological base satisfying certain simple axioms) and showed several tameness results in this abstract setting. It is easy to see that $0$ h $^{\mathrm {mix}}$ minimality implies viscerality in this sense.
Remark 2.2.3. In the equicharacteristic zero case, $0$ hminimality implies definable spherical completeness (meaning that every definable family of nested balls has nonempty intersection), and this in turn implies henselianity (see [Reference Cluckers, Halupczok and Rideau10], Lemma 2.7.1 and Theorem 2.7.2). A similar implication holds in mixed characteristic if we restrict to finitely ramified fields K. If such a K is $0$ h $^{\mathrm {mix}}$ minimal, then it is definable spherical complete by [Reference BradleyWilliams and Halupczok6, Proposition 1.4], and this in turn implies that K is henselian using the usual argument via Newton approximation (see the proof of [Reference Cluckers, Halupczok and Rideau10, Theorem 2.7.2] for details). In contrast, note that if K is not finitely ramified, then $0$ h $^{\mathrm {mix}}$ minimality does not imply definable spherical completeness; see Example 1.5 in [Reference BradleyWilliams and Halupczok6].
The following two definitions use equicharacteristic zero coarsenings of the valuation to define mixed characteristic Hensel minimality in terms of the equicharacteristic notions from [Reference Cluckers, Halupczok and Rideau10, Definition 2.3.3] (which we just recalled in Definition 2.2.1). We use notation from Definition 2.1.3.
Definition 2.2.4. Let $\ell \geq 0$ be either an integer or $\omega $ . We say that ${\mathcal T}$ is $\ell $ h $^{\mathrm {ecc}}$ minimal if for every model $K \models {\mathcal T}$ , the following holds. If the valuation $\cdot _{{\mathrm {ecc}}}$ on K is nontrivial, then the ${\mathcal L}_{{\mathrm {ecc}}}$ theory of K, when considered as a valued field with the valuation $\cdot _{{\mathrm {ecc}}}$ , is $\ell $ hminimal.
One can also require every equicharacteristic zero coarsening to be $\ell $ hminimal, leading to the following definition.
Definition 2.2.5. Let $\ell \geq 0$ be either an integer or $\omega $ . We say that ${\mathcal T}$ is $\ell $ h $^{\mathrm {{coars}}}$ minimal if for every model $K \models {\mathcal T}$ and each nontrivial equicharacteristic coarsening $\cdot _c$ of the valuation on K, the ${\mathcal L}_{c}$ theory of K, when considered as a valued field with the valuation $\cdot _{c}$ , is $\ell $ hminimal.
Remark 2.2.6. Since in an equicharacteristic zero valued field K, we have ${\mathcal O}_{K,{\mathrm {ecc}}} = {\mathcal O}_K$ , the theory of such a field is $\ell $ h $^{\mathrm {ecc}}$ minimal if and only if it is $\ell $ hminimal, for each $\ell \ge 0$ . For more subtle reasons, $\omega $ h $^{\mathrm {{coars}}}$ minimality is equivalent to $\omega $ h $^{\mathrm {ecc}}$ minimality by [Reference Cluckers, Halupczok and Rideau10, Corollary 4.2.4]. Even more, in [Reference Vermeulen23], it is shown that in equicharacteristic zero, $\ell $ h $^{\mathrm {{coars}}}$ minimality is equivalent to $\ell $ h $^{\mathrm {ecc}}$ minimality for any $\ell \geq 1$ . In mixed characteristic, we will show the analogue of this (and more) for $\ell =1$ in Theorem 2.2.8.
By the usual play of compactness, preparation results that hold in each model of ${\mathcal T}$ also hold uniformly for all models of ${\mathcal T}$ , and results in equicharacteristic zero can be transfered to mixed characteristic. We will give examples in the proofs for Corollary 2.3.5 and Corollary 2.5.5 to illustrate how compactness is used for these purposes.
In [Reference Cluckers, Halupczok and Rideau10, Theorem 2.9.1], a geometric criterion for $1$ hminimality is given in equicharacteristic zero. The following is a mixed characteristic version of that criterion.
Definition 2.2.7. Let $f: K\to K$ be Adefinable for some set $A\subset K\cup \mathrm {RV}_{n}$ , for some positive integer n. We define the following two properties:

(T1^{mix}) There exists a finite Adefinable $C\subset K$ and a positive integer m such that for every ball $B \ m$ next to C, there exists $\mu _B\in \Gamma _K$ such that for $x,y\in B$ , we have
$$\begin{align*}\mu_B\cdotm\cdot xy\leq f(x)f(y) \leq \mu_B\cdot \left\frac{1}{m}\right \cdot xy. \end{align*}$$ 
(T2) The set $\{y\in K\mid f^{1}(y) \text { is infinite}\}$ is finite.
We say that ${\mathcal T}$ satisfies (T1,T2) if for all f, A and n as above, the two conditions (T1 $^{\mathrm {mix}}$ ) and (T2) hold.
We now have several variants of 1hminimality in mixed characteristic. The main result of this section is the following, stating that all of the above definitions agree.
Theorem 2.2.8. The following are equivalent, for a theory ${\mathcal T}$ of valued fields of characteristic zero (possibly of mixed characteristic) in a language ${\mathcal L}$ containing ${\mathcal L}_{\mathrm {val}}$ .

(1) ${\mathcal T}$ is 1h $^{\mathrm {ecc}}$ minimal.

(2) ${\mathcal T}$ is 1h $^{\mathrm {mix}}$ minimal.

(3) ${\mathcal T}$ satisfies (T1,T2).

(4) ${\mathcal T}$ is 1h $^{\mathrm {{coars}}}$ minimal.
Therefore, we will call this common notion simply 1hminimality.
We will prove Theorem 2.2.8 in Section 2.7. Until then, we continue distinguishing between the four notions.
2.3 Basic results under 1h $^{\mathrm {mix}}$ minimality
Many basic results from [Reference Cluckers, Halupczok and Rideau10] also hold in mixed characteristic with only minor changes. In particular, this includes all the results of Sections 2.4–2.6 from [Reference Cluckers, Halupczok and Rideau10], and those of Section 2.8 up to Lemma 2.8.5. In the following, we state the precise version of the mixed characteristic results. When a proof is (almost) identical to the corresponding one from [Reference Cluckers, Halupczok and Rideau10], we only give a sketch so that the reader has the choice between reading the short version here or the long version in [Reference Cluckers, Halupczok and Rideau10].
Lemma 2.3.1. Assume that ${\mathcal T}$ is $0$ h $^{\mathrm {mix}}$ minimal. The following results are true for any model K of ${\mathcal T}$ .

(1) (Adding constants [Reference Cluckers, Halupczok and Rideau10, Lemma 2.4.1]). If $A\subset K\cup \mathrm {RV}_{n}^{\mathrm {eq}}$ , then $\operatorname {Th}_{{\mathcal L}(A)}(K)$ is $0$ h $^{\mathrm {mix}}$ minimal. (And similarly, if ${\mathcal T}$ is $\ell $ h $^{\mathrm {mix}}$ minimal, then $\operatorname {Th}_{{\mathcal L}(A)}(K)$ is $\ell $ h $^{\mathrm {mix}}$ minimal for any $\ell $ and any such A.)

(2) (Preparation is first order [Reference Cluckers, Halupczok and Rideau10, Lemma 2.4.2]). If $X_q, C_q$ are $\emptyset $ definable families of subsets of K with q running over an $\emptyset $ definable Q in an arbitrary imaginary sort, and $C_q$ is finite for all q, then the set of $(q, \lambda )\in Q\times \Gamma _K^{\times }$ with $\lambda \leq 1$ such that $C_q \ \lambda $ prepares $X_q$ is $\emptyset $ definable.

(3) ( $\exists ^{\infty }$ elimination [Reference Cluckers, Halupczok and Rideau10, Lemma 2.5.2]). Every infinite $\emptyset $ definable $X\subset K$ contains an open ball.

(4) (Finite sets are $\mathrm {RV}$ parametrized [Reference Cluckers, Halupczok and Rideau10, Lemma 2.5.3]). If $C_q\subset K$ is a $\emptyset $ definable family of finite sets, for q in some arbitrary $\emptyset $ definable imaginary sort Q, then there is a $\emptyset $ definable family of injections $f_q: C_q\to \mathrm {RV}_{n}^k$ (for some $k \ge 0$ and $n \ge 1$ ).

(5) ( $\lambda $ next balls as unions of fibres [Reference Cluckers, Halupczok and Rideau10, Lemma 2.5.4]). Let $C\subset K$ be a finite $\emptyset $ definable set. Then for any $\lambda \le 1$ , there is a $\{\lambda \}$ definable map $f: K\to \mathrm {RV}_{n}^k\times \mathrm {RV}_{m\lambda }$ (for some k, n and m) such that every ball $\lambda $ next to C is a union of fibres of f. If, moreover, $\lambda = n'$ for some integer $n'>0$ , then we can ensure that for some integer $p>0$ , every ball $p$ next to C is contained in a fibre of f.
In (1) of Lemma 2.3.1, by $\mathrm {RV}_{n}^{\mathrm {eq}}$ , we mean imaginary sorts of the form $(\mathrm {RV}_{n})^m/\mathord {\sim }$ , for some m and some $\emptyset $ definable equivalence relations $\sim $ .
Proof of Lemma 2.3.1.
(1) and (2) are straightforward from the definition. (3) also follows directly by preparing X.
(4) Using (3), we can assume that $\#C_q$ does not depend on q. We define $a_q := \frac 1{\#C_q}\sum _{x \in C_q} x$ and $\hat f_q\colon C_q\to \mathrm {RV}_{n}, x \mapsto \operatorname {rv}_{n}(x  a_q)$ for some n which is a multiple of $\#C_q$ . This implies that $\hat f_q$ is not constant on $C_q$ . We then apply induction to the family consisting of all fibers of $f_q$ , for all q. (We take the final n to be a multiple of all cardinalities $\#C_q$ appearing during this process.)
(5) For $x\in K$ , define
The map $a: K\to K$ has finite image, so by (4), we can find an injection $\alpha : \operatorname {im} a\to \mathrm {RV}_{n}^k$ . Let $m = \max \{C(x)\mid x\in K\}!$ and define the map f as
If x and $x'$ are in the same ball $\lambda $ next to C, then $C(x)=C(x')$ and so $a(x) = a(x')$ . If x and $x'$ are in the same fibre of f and we are in the first case, then the fact that $\operatorname {rv}_{m\lambda }(xa(x)) = \operatorname {rv}_{m\lambda }(x'a(x'))$ implies that $\operatorname {rv}_{\lambda }(xc) = \operatorname {rv}_{\lambda }(x'c)$ for any $c\in C$ . Thus, any fibre of f is contained in a ball $\lambda $ next to C.
If $\lambda = n'$ for some integer $n'$ , then one can take $p = n'\cdot m$ . Then f will be constant on any ball $p$ next to C.
The following proposition states that we can also prepare families, similarly to [Reference Cluckers, Halupczok and Rideau10, Proposition 2.6.2]. That proposition will also be needed for $0$ h $^{\mathrm {mix}}$ minimality, so we formulate it more generally.
Proposition 2.3.2 (Preparing families).
Assume $\ell $ h $^{\mathrm {mix}}$ minimality for ${\mathcal T}$ and some $\ell \ge 0$ . Let K be a model of ${\mathcal T}$ . Let $A\subset K$ and let
be $(A \cup \mathrm {RV}_{n})$ definable for some integers k and $n \ge 1$ . Then there exists a finite Adefinable set $C \subset K$ and a positive integer m such that for any $\lambda \leq 1$ and any ball B which is $m\lambda $ next to C, the set $W_{x, \lambda }:= \{(\xi , \xi ') \in \mathrm {RV}^k_{n} \times \mathrm {RV}_{\lambda }^{\ell } \mid (x, \xi , \xi ') \in W\}$ is independent of x as x runs over B.
Proof. For each $\lambda $ and each $(\xi , \xi ') \in \mathrm {RV}^k_{n} \times \mathrm {RV}_{\lambda }^{\ell }$ , let $C_{\lambda ,\xi ,\xi '}$ be a finite Adefinable set $m_{\lambda ,\xi ,\xi '}\lambda $ preparing the fiber $W_{\xi ,\xi '} \subset K$ . Using compactness, we may assume that for varying $\lambda ,\xi ,\xi '$ , there are only finitely many different sets $C_{\lambda ,\xi ,\xi '}$ and integers $m_{\lambda ,\xi ,\xi '}$ . Let C be the union of the $C_{\lambda ,\xi ,\xi '}$ and m be the least common multiple of the $m_{\lambda ,\xi ,\xi '}$ .
Remark 2.3.3. In that proposition, instead of $\mathrm {RV}^k_{n}$ , we can have any product Z of sorts from $\mathrm {RV}_{n}^{\mathrm {eq}}$ (including, in particular, $\Gamma _K$ ). Indeed, given such a $W \subset K \times Z \times \bigcup _{\lambda \le 1}\mathrm {RV}_{\lambda }^{\ell }$ , we can apply Proposition 2.3.2 to the preimage of W in $K \times \mathrm {RV}^k_{n} \times \bigcup _{\lambda \le 1}\mathrm {RV}_{\lambda }^{\ell }$ under some quotient map $\mathrm {RV}^k_{n} \to Z$ . The same also applies to Corollary 2.3.4 below.
We will mostly apply the following special case of Proposition 2.3.2:
Corollary 2.3.4. Let K be a model of a $0$ h $^{\mathrm {mix}}$ minimal theory ${\mathcal T}$ , let $A\subset K$ and let
be $(A \cup \mathrm {RV}_{n})$ definable, for some integers k and $n \ge 1$ . Then there exists a finite Adefinable set $C \subset K$ and a positive integer m such that for any ball B which is $m$ next to C, the set $W_{x}:= \{\xi \in \mathrm {RV}^k_{n} \mid (x, \xi ) \in W\}$ is independent of x as x runs over B.
Note that in the corollary, instead of saying that $W_x$ is constant on each B, one could equivalently also say that for each $\xi \in \mathrm {RV}^k_{n}$ , the set $W_{\xi }:= \{x \in K \mid (x, \xi ) \in W\}$ is $m$ prepared by C. In applications, we will sometimes use this point of view without further notice.
Yet another point of view of the corollary is obtained if W is the graph of a function $f\colon K \to \mathrm {RV}^k_{n}$ . In that case, the conclusion is that f is constant on each ball $m$ next to C.
We also obtain the following corollary about preparing families in all models K of the (possibly noncomplete) theory ${\mathcal T}$ . The point here is that the integer m can be taken uniformly over all models.
Corollary 2.3.5 (of Proposition 2.3.2).
Assume that ${\mathcal T}$ is $\ell $ h $^{\mathrm {mix}}$ minimal for some $\ell \ge 0$ , and suppose that $\phi $ is an ${\mathcal L}$ formula such that for every model $K \models {\mathcal T}$ , $W_K := \phi (K)$ is a subset of $K\times \mathrm {RV}^k_{n}\times \mathrm {RV}_{\lambda _K}^{\ell }$ for some k, n and some $\lambda _K \le 1$ in $\Gamma _K^{\times }$ . Then there exists an ${\mathcal L}$ formula $\psi $ and an integer $m \ge 1$ such that for every model $K \models {\mathcal T}$ , $C_K := \psi (K)$ is a finite subset of K which $\lambda _K\cdot m$ prepares $W_K$ in the following sense: for every ball $B \subset K$ which is $\lambda _K\cdot m$ next to $C_K$ , the fiber
does not depend on x when x runs over B.
Proof. Let $\phi $ be given as in the statement. Whether a pair $(m, \psi )$ works as desired in a model K can be expressed by an ${\mathcal L}$ sentence, by Lemma 2.3.1. By compactness and $\ell $ h $^{\mathrm {mix}}$ minimality, we deduce that there exist finitely many pairs $(m_i, \psi _i)$ which cover all models. We may furthermore assume that the sets $\psi _i(K)$ are finite for each model K. We are done by letting m be the least common multiple of the $m_i$ (so that $m \le m_i$ for each i) and $\psi $ be the disjunction of the $\psi _i$ .
Proposition 2.3.2 also implies the following corollaries. Denote by $\mathrm {RV}_{\bullet }$ the disjoint union of all $\mathrm {RV}_{\lambda }$ for $\lambda \leq 1$ .
Corollary 2.3.6. Assume 1h $^{\mathrm {mix}}$ minimality for ${\mathcal T}$ . The following hold for any model K of ${\mathcal T}$ .

(1) ( $\mathrm {RV}$ unions stay finite [Reference Cluckers, Halupczok and Rideau10, Corollary 2.6.7]). Let $W\subset K \times \mathrm {RV}_{\bullet }^{k}$ be $\emptyset $ definable such that $W_{\xi }$ is finite for any $\xi \in \mathrm {RV}_{\bullet }^{k}$ . Then the union $\bigcup _{\xi } W_{\xi }$ is also finite.

(2) (Finite image in K [Reference Cluckers, Halupczok and Rideau10, Corollary 2.6.8]). The image of any $\emptyset $ definable $f: \mathrm {RV}_{\bullet }^{k}\to K$ is finite.

(3) (Removing $\mathrm {RV}$ parameters [Reference Cluckers, Halupczok and Rideau10, Corollary 2.6.10]). Let C be a finite $A\cup \mathrm {RV}_{n}^{\mathrm {eq}}$ definable set, for some $A\subset K$ and some integer $n>0$ . Then there exists a finite Adefinable set $C'$ containing C.
Proof. (1) The case $k = 1$ follows by applying Proposition 2.3.2. (Each $W_{\xi }$ is contained in the set C one obtains.) For $k \ge 2$ , use induction (adding parameters to the language using Lemma 2.3.1 (1)).
(2) Apply (1) to the graph of f.
(3) C is a fiber of an Adefinable subset $W \subset K \times \mathrm {RV}_{\bullet }^k$ , for some k. Apply (1) to W.
To obtain (T1,T2) from 1h $^{\mathrm {mix}}$ minimality for ${\mathcal T}$ , we follow [Reference Cluckers, Halupczok and Rideau10, Section 2.8]. Here we have to be slightly more careful in our formulations and proofs. First, we set some notation.
Definition 2.3.7. If B is an open ball in K, and $\lambda $ is in $\Gamma _K^{\times }$ , $\lambda \leq 1$ , then a $\lambda $ shrinking of B is an open ball $B'\subset B$ with
Lemma 2.3.8. Assume 1h $^{\mathrm {mix}}$ minimality for ${\mathcal T}$ . Let K be a model of ${\mathcal T}$ and let $f: K\to K$ be an $\emptyset $ definable function. Then the following hold.

(1) (Basic preservation of dimension [Reference Cluckers, Halupczok and Rideau10, Lemma 2.8.1]). The set of $y\in K$ for which $f^{1}(y)$ is infinite, is finite.

(2) (Piecewise constant or injective [Reference Cluckers, Halupczok and Rideau10, Lemma 2.8.2]). There exists a finite $\emptyset $ definable set C and a positive integer m such that on any ball $B \ m$ next to C, f is either constant or injective.

(3) (Images of most balls are almost balls [Reference Cluckers, Halupczok and Rideau10, Lemma 2.8.3]). There exists a finite $\emptyset $ definable set C and positive integers $m,n$ such that for any open ball B contained in a ball $m$ next to C, either $f(B)$ is a singleton, or for any $y\in f(B)$ , there are open balls $B', B"$ for which $y\in B'\subset f(B)\subset B"$ and
$$\begin{align*}\operatorname{rad}_{\mathrm{op}} B' \geq n \operatorname{rad}_{\mathrm{op}} B". \end{align*}$$ 
(4) (Preservation of scaling factor [Reference Cluckers, Halupczok and Rideau10, Lemma 2.8.4]). Suppose that there are $\alpha , \beta $ in $\Gamma _K^{\times }$ , $\alpha < 1$ , such that for every open ball $B\subset {\mathcal M}_K$ of radius $\alpha $ , the image $f(B)$ is contained in an open ball of radius $\beta $ . Assume, moreover, that there is an integer p and open balls $B', B"$ such that $B'\subset f({\mathcal M}_K)\subset B"$ and
$$\begin{align*}\operatorname{rad}_{\mathrm{op}} B' \geq p \operatorname{rad}_{\mathrm{op}} B". \end{align*}$$Then $f({\mathcal M}_K)$ is contained in an open ball of radius at most $\frac {\beta }{n\alpha }$ for some positive integer n.
Proof. (1) Suppose for contradiction that f has infinitely many infinite fibers. Let $X \subset K$ be the subset of the domain of f where f is locally constant (i.e, of points $x \in K$ such that f is constant on $B_{<\lambda }(x)$ for some $\lambda \in \Gamma _K^{\times }$ ). Since each infinite fiber of f contains a ball (by preparation of the fiber), the restriction of f to X still has infinite image.
Let $C \subset K$ be a finite set $m$ preparing X, for some integer $m \ge 1$ . Then enlarge C and m in such a way that for each $\lambda \in \Gamma _K^{\times }$ , whether f is constant on $B_{<\lambda }(x)$ only depends on the ball $m$ next to C containing x. This is possible by applying Corollary 2.3.4 (and Remark 2.3.3) to the set $W \subset K \times \Gamma _K^{\times }$ of those $(x,\lambda )$ for which f is constant on $B_{<\lambda }(x)$ .
By Corollary 2.3.6 (1), there exists a ball $B_0 \subset X \ m$ next to C such that $f(B_0)$ is still infinite. Fix such a $B_0$ , and also fix a $\lambda _0 \in \Gamma _K^{\times }$ such that f is constant on every open ball of radius $\lambda _0$ contained in $B_0$ .
The family of all those balls can definably be parametrized by a subset $Z \subset \mathrm {RV}_{\lambda _1}$ , for $\lambda _1 = \lambda _0/\operatorname {rad}_{\mathrm {op}} B_0$ . Now f induces a map from Z to K with infinite image, contradicting Corollary 2.3.6 (1).
(2) First, find C and m such that $C \ m$ prepares every infinite fiber of f (using (1) to see that there are only finitely many infinite fibers). Then apply Lemma 2.3.1 (4) to the family of finite fibers of f to obtain an injective map from each finite fiber of f to $\mathrm {RV}^k_{n}$ for some n and k. We put all those maps together to one single map $g\colon Y \to \mathrm {RV}^k_{n}$ , where Y is the union of all finite fibers of f. Enlarge C and m in such a way that g is constant on each ball $B \subset Y \ m$ next to C (by applying Corollary 2.3.4 to the graph of g). Then, for each ball $B \ m$ next to C, either B is entirely contained in an infinite fiber of f (and hence f is constant on B, as desired) or $B \subset Y$ and g is constant on B. In that case, f is injective on B since $f(x_1) = f(x_2)$ for $x_1, x_2 \in B$ would imply that $x_1$ and $x_2$ lie in the same fiber of f, contradicting that g is injective on each fiber of f.
(3) Use (2) to obtain a finite $\emptyset $ definable set C and an integer m such that on any ball $m$ next to C, f is either constant or injective. Let $W_0\subset K\times (\Gamma _K^{\times })^2$ consist of those $(x, \lambda , \mu )$ with $\mu \leq 1$ such that for every $y\in f(B_{<\lambda }(x))$ , there are open balls $B', B"$ with $y\in B'\subset f(B_{<\lambda }(x))\subset B"$ and
We enlarge C and m, such that C also $m$ prepares this set $W_0$ .
The set C is already as desired, but m will later be enlarged to some $m\cdot m"$ . Note that we already simplified the statement we will need to prove. First, it suffices to consider balls B which are contained in a ball $B_1 \ m$ next to C on which f is injective, and secondly, for each $\lambda \le 1$ , it suffices to find a single $\lambda \cdot m"$ shrinking B of $B_1$ for which the lemma holds (using some n which we still need to specify). Indeed, the fact that $C \ m$ prepares $W_0$ implies that then the lemma also holds for all translates of B within $B_1$ (using the same n).
Before we can continue, we need to do some preparation on the range side of f. We want to find a finite $\emptyset $ definable D and a positive integer p such that for every $\lambda \le 1$ and every ball $B \ \lambda m$ next to C, the set $f(B)$ is $\lambda p$ prepared by D. To see that such a D exists, first note that by Lemma 2.3.1 (5), there exists a $\lambda $ definable map $g_{\lambda }\colon K \to \mathrm {RV}_{n'}^k\times \mathrm {RV}_{m'\cdot \lambda }$ such that every ball $\lambda m$ next to C is a union of fibers of $g_{\lambda }$ . We may assume (by compactness) that the maps $g_{\lambda }$ form a $\emptyset $ definable family. Now apply Proposition 2.3.2 to the set
and let D and p be the result. To see that this works, let $\lambda \le 1$ and $B \ \lambda m$ next to C be given. We have $B = g_{\lambda }^{1}(\Xi )$ for $\Xi := g_{\lambda }(B)$ , and the image $f(B)$ consists of those $y \in K$ for which the fiber $W_{(y, \lambda )}$ is not disjoint from $\Xi $ . Since this fiber is constant when y runs over a ball $B' \ \lambda p$ next to D, we either have $B' \subset f(B)$ or $B' \cap f(B) = \emptyset $ , as desired.
Now that we constructed D and p (on the range side of f), we construct another set $C'$ and integer $m'$ on the domain side of f. For $\lambda \in \Gamma _K^{\times }$ , $\lambda \leq 1$ , use Lemma 2.3.1 to get a map $h_{\lambda }: K\to \mathrm {RV}_{p}^k\times \mathrm {RV}_{\lambda n"}$ such that each fibre is contained in a ball $\lambda p$ next to D. We may assume that $h_{\lambda }$ is a $\emptyset $ definable family of maps, with a parameter $\lambda $ . Let $C'$ be a finite $\emptyset $ definable set which $\lambda m'$ prepares every map $h_{\lambda }\circ f$ . This can again be done using Proposition 2.3.2.
We need one last ingredient before we can verify that the lemma holds. Since $C'$ is finite, there exists an integer $p' \ge 1$ such that every ball $B_1 \ m m'$ next to C has a $p'$ shrinking $B_1^{\prime } \subset B_1$ disjoint from $C'$ .
We now claim that the lemma holds using C, $mm'p'$ and $n = m'p'$ . According to the beginning of the proof, it suffices to check that given a ball $B_1 \ m$ next to C on which f is injective, and given a $\lambda \le 1$ , there exists a $\lambda \cdot p' m'$ shrinking B of $B_1$ for which the claim holds. Choose a $p'$ shrinking $B_1^{\prime }$ of $B_1$ disjoint from $C'$ and choose B to be any $\lambda m'$ shrinking of $B_1^{\prime }$ . To finish the proof, we show that the lemma holds for this B.
On the one hand, since B is a $\lambda \cdot m'$ shrinking of $B_1^{\prime }$ (and $B_1^{\prime }$ is disjoint from $C'$ ), B is contained in a ball $\lambda \cdot m'$ next to $C'$ . By definition of $C'$ , this means that $h_{\lambda } \circ f$ is constant on B, and hence (by definition of $h_{\lambda }$ ), $f(B)$ is contained in a ball $B" \ \lambda p$ next to D.
On the other hand, B is $\lambda mm'p'$ next to C, so that $f(B)$ is $\lambda pm'p'$ prepared by D (by definition of D). Thus, for any $y\in f(B)$ , we obtain that the entire ball $B' \ \lambda pm'p'$ next to D containing y is contained in $f(B)$ . This ball $B'$ is just the (unique) $m'p'$ shrinking of $B"$ containing y; in particular, $\operatorname {rad}_{\mathrm {op}} B' = m'p'\operatorname {rad}_{\mathrm {op}} B"$ , as desired.
(4) If the residue field of K has characteristic zero, then this is exactly [Reference Cluckers, Halupczok and Rideau10, Lemma 2.8.4]. So we may assume that we are in mixed characteristic (though the following proof easily also adapts to the equicharacteristic zero case.) The family of radius $\alpha $ open balls in ${\mathcal M}_K$ can be definable parametrized by the set $\Lambda = \operatorname {rv}_{\alpha }({\mathcal M}_K)$ . Namely, for $\xi \in \Lambda $ , let $B_{\xi }\subset {\mathcal M}_K$ be the unique open ball of radius $\alpha $ containing $\operatorname {rv}_{\alpha }^{1}(\xi )$ . Using 1h $^{\mathrm {mix}}$ minimality, we can find a finite set $C\subset K$ and a positive integer m such that $C \ m\alpha $ prepares every set $f(B_{\xi })$ , for every $\xi \in \Lambda $ . In other words, if $\xi \in \Lambda $ , and if $B'$ is an open ball $m\alpha $ next to C then either $B'\subset f(B_{\xi })$ or $B'\cap f(B_{\xi }) = \emptyset $ . Now, if $B'$ has radius strictly larger than $\beta $ , then it follows that $B'\cap f(B_{\xi }) = \emptyset $ . Therefore, $f({\mathcal M}_K)$ is contained in the union of C and all balls $m\alpha $ next to C of radius at most $\beta $ . This union is equal to the finite union of all closed balls of the form $B_{\leq \beta /(m\alpha )}(c), c\in C$ . In particular the open ball $B'$ is contained in this finite union of closed balls of radius $\beta /(m\alpha )$ . Since we are in mixed characteristic, there exists some positive integer q such that $\operatorname {rad}_{\mathrm {op}} B'\leq \beta /(q\alpha )$ . But then $B"$ is an open ball containing $f({\mathcal M}_K)$ of radius at most $\beta / (pq\alpha )$ , finishing the proof.
Finally, we can prove an approximate valuative Jacobian property in mixed characteristic. The lemma and its proof are similar to [Reference Cluckers, Halupczok and Rideau10, Lemma 2.8.5] and are sharpened to an actual Jacobian property below in Corollary 3.1.3. Note that the sharpened version is obtained only using a huge detour, involving approximations by second degree Taylor polynomials. We do not see a more direct proof of this sharpened version.
Lemma 2.3.9 (Approximate valuative Jacobian property).
Assume that ${\mathcal T}$ is 1h $^{\mathrm {mix}}$ minimal. Let K be a model of ${\mathcal T}$ and let $f: K\to K$ be an Adefinable function, for some $A\subset K\cup \mathrm {RV}_{n}$ . Then there exists a finite Adefinable set C and a positive integer m such that for every ball $B \ m$ next to C, there exists a $\mu _B\in \Gamma _K$ such that for all $x, y\in B$ we have
Proof. We may assume that $A=\emptyset $ by Lemma 2.3.1. Using Lemma 2.3.8 (3), we can find a finite $\emptyset $ definable set $C_0$ and positive integers $m,n$ such that

○ f is constant or injective on balls $m$ next to $C_0$ , and

○ if B is an open ball contained in a ball $m$ next to $C_0$ , then either $f(B)$ is a singleton or there are open balls $B'\subset f(B)\subset B"$ such that the radii of $B'$ and $B"$ differ by at most $n$ .
For an open ball $B = B_{<\alpha }(x)$ contained in a ball $m$ next to C on which f is injective, define $\mu (x, \alpha )$ to be the (convex) set of $\mu \in \Gamma _K^{\times }$ for which $f(B)$ is contained in an open ball of radius $\mu $ and contains an open ball of radius $n\mu $ . Note that we have $n\mu (x,\alpha )\leq \mu (x,\alpha )$ , in the sense that for every $\nu \in n\mu (x,\alpha )$ and every $\nu ' \in \mu (x,\alpha )$ , we have $\nu \leq \nu '$ . (In the following, inequalities between convex subsets of $\Gamma _K$ are always meant in this sense.) Also define
(An element $\nu \in s(x,\alpha )$ is a kind of ‘scaling factor’: the ball $B_{<\alpha }(x)$ is sent into a ball of radius $\alpha \nu $ .)
We will now enlarge C and m to obtain the following claim. As a side remark, note that we will keep n fixed for the entire proof so that the definitions of $\mu (x,\alpha )$ and $s(x, \alpha )$ are unaltered.
Claim 1. By possibly enlarging C and m, we can achieve that

(1) $\mu (x,\alpha )$ and $s(x,\alpha )$ are independent of x as x runs over a ball $m$ next to C,

(2) if $B_{<\alpha }(x)$ and $B_{<\beta }(y)$ are open balls contained in the same ball $m$ next to C and $\alpha \leq \beta $ , then
$$\begin{align*}s(y,\beta) \leq s(x,\alpha)/m. \end{align*}$$
To prove Claim 1, let $W\subset K\times (\Gamma _K^{\times })^2$ consist of those $(x, \lambda , \mu )$ such that $f(B_{<\lambda }(x))$ is contained in a ball $B'$ of radius $\mu $ , and contains a ball of radius $n\mu $ . Enlarge C and m such that $C \ m$ prepares this set W. Note that item (1) of the claim then holds by preparation. Let B be $m$ next to C. If f is constant on B, then there is nothing to check, so we can assume that f is injective on B. By item (1), $\mu (x,\alpha )$ and $s(x,\alpha )$ are constant when x runs over B and $\alpha \leq \operatorname {rad}_{\mathrm {op}} B$ is fixed, so we write simply $\mu (\alpha )$ and $s(\alpha )$ . It remains to prove item (2) of Claim 1 (after possibly enlarging m once more). Fix $\mu \in \mu (\alpha )$ and take $\alpha \leq \beta \leq \operatorname {rad}_{\mathrm {op}} B$ . Then any ball of radius $\alpha $ inside B has image under f contained in a ball of radius at most $\mu $ . Hence, by a rescaled version of Lemma 2.3.8(4), there exists an integer $p \ge 1$ such that for $x \in B$ , the image $f(B_{<\beta }(x))$ is contained in an open ball of radius $\frac {\mu \cdot \beta }{p \cdot \alpha }$ . In particular, we have
But this means precisely that
which proves Claim 1 (after replacing m by, for example, $mnp$ ).
To prove the lemma, we also need an inequality opposite to the one of Claim 1 (2).
Claim 2. After possibly further enlarging C and m, in Claim 1 (2), we can additionally obtain
The idea of the proof of Claim 2 is to apply Claim 1 to $f^{1}$ (which, in reality, only exists piecewise). This is made precise as follows:
Denote by Y the set of $y\in K$ for which $f^{1}(y)$ is finite. This is a cofinite $\emptyset $ definable set in K, by Lemmas 2.3.1 and 2.3.8 (1). Use Lemma 2.3.1 (4) to obtain a $\emptyset $ definable family of injections
for $y\in Y$ . For $\eta \in \mathrm {RV}_{n_0}^k$ , define
Then we have that
For each $\eta $ , apply Lemma 2.3.8 (3) to $g_{\eta }$ (extended by $0$ outside of $Y_{\eta }$ ) to obtain a finite $\eta $ definable set $D_{\eta }$ and integers $m_{\eta }, n_{\eta }$ . By compactness, we may take $n' := n_{\eta }$ independent of $\eta $ . Now enlarge $D_{\eta }$ and $m_{\eta }$ using Claim 1, so that (1) and (2) hold for $g_{\eta }$ (where the corresponding $\mu $ and s are defined using $n'$ ). We may, moreover, assume that $D_{\eta } \ m_{\eta }$ prepares $Y_{\eta }$ (enlarging $D_{\eta }$ and $m_{\eta }$ once more, if necessary). After that, apply compactness once more to make $m':=m_{\eta }$ independent of $\eta $ and to turn $(D_{\eta })_{\eta }$ into a $\emptyset $ definable family. By Corollary 2.3.6, the union $D=\bigcup _{\eta } D_{\eta }$ is a finite $\emptyset $ definable set.
Using Lemma 2.3.1 (5), take a $\emptyset $ definable function $\chi : K\to \mathrm {RV}_{n_1}^{k'}$ such that every ball $m'$ next to D is a union of fibres of $\chi $ . Then choose $C_2$ and $C_3$ such that, after possibly enlarging m, $C_2 \ m$ prepares the family of sets $f^{1}(\chi ^{1}(\eta ))$ for $\eta \in \mathrm {RV}_{n_1}^{k'}$ , and $C_3 \ m$ prepares the family of images $g_{\eta }(Y_{\eta })$ . We will now prove that Claim 2 holds after replacing C by $C \cup C_2 \cup C_3$ (and some further enlargement of m).
So suppose that we have open balls $B_1, B$ with $\operatorname {rad}_{\mathrm {op}} B_1 = \alpha \leq \operatorname {rad}_{\mathrm {op}} B = \beta $ which are contained in the same ball $m$ next to C. If f is constant on B, then we are done, so assume that f is injective on B. We may assume that $B_1\subset B$ since $\mu (x, \alpha )$ is independent of x as x runs over B. Let $B'\subset f(B)\subset B"$ be open balls whose radii differ by at most $n$ . By definition of $C_2$ , $\chi \circ f$ is constant on B, so (by definition of $\chi $ ), $f(B)$ is contained in a ball $m'$ next to D. Perhaps after shrinking $B"$ , we can assume that also $B"$ is contained in a ball $m'$ next to D. By definition of $C_3$ , there is a (unique) $\eta \in \mathrm {RV}_{n_0}^k$ such that $B \subset g_{\eta }(Y_{\eta })$ ; this implies that $f_B$ and $g_{\eta }_{f(B)}$ are mutually inverse bijections between B and $f(B)$ . Using that we applied Lemma 2.3.8 to $g_{\eta }$ , take open balls $\tilde {B}'\subset g_{\eta }(B')\subset \overline {B}'$ whose radii differ by at most $n'$ and do the same with open balls $\tilde {B}"\subset g_{\eta }(B")\subset \overline {B}"$ . Note that we have a chain of inclusions
Choose similar balls corresponding to $B_1$ : $B_1^{\prime }, B_1^{\prime \prime }, \tilde {B}_1^{\prime }, \dots $ . We may certainly assume that $\operatorname {rad}_{\mathrm {op}} B_1^{\prime } \le \operatorname {rad}_{\mathrm {op}} B"$ . Therefore, our application of Claim 1 to $g_{\eta }$ yields that
Combining this with $\operatorname {rad}_{\mathrm {op}} B \le \operatorname {rad}_{\mathrm {op}}\overline {B}"$ , $n'\operatorname {rad}_{\mathrm {op}} \overline {B}^{\prime }_1 \le \operatorname {rad}_{\mathrm {op}}\tilde {B}^{\prime }_1 \le \operatorname {rad}_{\mathrm {op}} B_1$ and $n\operatorname {rad}_{\mathrm {op}} B^{\prime \prime }_1 \le \operatorname {rad}_{\mathrm {op}} B^{\prime }_1$ , we obtain
Since $\frac {\operatorname {rad}_{\mathrm {op}} B^{\prime \prime }_1}{\operatorname {rad}_{\mathrm {op}} B_1} \in s(\alpha )$ and $\frac {\operatorname {rad}_{\mathrm {op}} B"}{\operatorname {rad}_{\mathrm {op}} B} \in s(\beta )$ , we deduce
(where the additional factor $n^2$ takes into account the length of the convex sets $s(\alpha )$ and $s(\beta )$ ). This finishes the proof of Claim 2 (where we take m to be a multiple of $n^3n'm'$ ).
We are now ready to prove the lemma itself. Take $x,y$ in the same ball $B \ m$ next to C. Denote by $\beta $ the open radius of B. Let $\mu \in \mu (x, \beta )$ , so that $f(B)$ is contained in a ball of radius $\mu $ and contains a ball of radius $n\mu $ . We will show that we can take $\mu _B = \mu /\beta $ .
If we choose $\alpha> xy$ , then x and y are contained in an open ball of radius $\alpha $ . If we, moreover, choose $\mu '\in \mu (x,\alpha )$ , then $f(x), f(y)$ are contained in an open ball of radius $\mu '$ . Thus,
Since this holds for any $\alpha> xy$ , this gives that
For the other inequality, set $\alpha = xy$ and denote by $B'$ the open ball of radius $\alpha $ around x. By the injectivity of f on B, $f(y)$ is not in $f(B')$ . But $f(B')$ contains an open ball of radius $n\mu '$ (for $\mu ' \in \mu (x, \alpha )$ , as before) and so
Thus,
This proves the Lemma.
We now already obtain a part of Theorem 2.2.8.
Corollary 2.3.10. Any 1h $^{\mathrm {mix}}$ minimal theory ${\mathcal T}$ satisfies (T1,T2).
2.4 Basic results under (T1,T2)
We now prove some consequences of ${\mathcal T}$ satisfying (T1,T2) in the sense of Definition 2.2.7. First note that by applying (T1 $^{\mathrm {mix}}$ ) to the characteristic function of an Adefinable set $X \subset K$ , for some $A \subset K \cup \mathrm {RV}_{n}$ , we obtain a finite Adefinable set $C \subset K \ m$ preparing X for some $m\ge 1$ . In Lemma 2.4.1, we will see that in (T1 $^{\mathrm {mix}}$ ), we can even take C to be $(A \cap K)$ definable, so that ${\mathcal T}$ is $0^{\mathrm {mix}}$ hminimal. For the moment, note that the above weaker statement already suffices to obtain $\exists ^{\infty }$ elimination (namely, (3) of Lemma 2.3.1).
Lemma 2.4.1. Assume that ${\mathcal T}$ satisfies (T1,T2). Let K be a model of ${\mathcal T}$ . Then

(1) ( $\mathrm {RV}_{\lambda }$ unions stay finite [Reference Cluckers, Halupczok and Rideau10, Lemma 2.9.4]). If $C_{\xi }\subset K$ is a definable family (with parameters) of finite sets, parametrized by $\xi \in \mathrm {RV}_{\lambda }^k$ , then $\bigcup _{\xi } C_{\xi }$ is still finite.

(2) (Eliminating $\mathrm {RV}$ parameters [Reference Cluckers, Halupczok and Rideau10, Lemma 2.9.5]). If $f: K\to K$ is Adefinable for some $A\subset K\cup \mathrm {RV}_{n}$ , then we can find a finite $(A\cap K)$ definable set C and an integer m such that f satisfies property (T1 $^{\mathrm {mix}}$ ) with respect to C and m. In particular, ${\mathcal T}$ is 0h $^{\mathrm {mix}}$ minimal.
The following proof is the same as in [Reference Cluckers, Halupczok and Rideau10].
Proof of Lemma 2.4.1.
(1) We induct on k, so first assume that $k=1$ . Using $\exists ^{\infty }$ elimination, we may assume that the cardinality of $C_{\xi }$ is constant, say, equal to m. Let $\sigma _1, \dots , \sigma _m$ be the elementary symmetric polynomials in m variables, considered as functions on melementsubsets of K. Then the map $K^k \to K, x \mapsto \sigma _i(C_{\operatorname {rv}_{\lambda }(x)})$ is locally constant everywhere except possibly at $0$ , so by (T2), it has finite image. Since $\sigma _1(C_{\xi }), \dots , \sigma _m(C_{\xi })$ together determine $C_{\xi }$ , there are only finitely many different $C_{\xi }$ .
For arbitrary k, consider $C_{\xi }$ as a definable family $C_{\xi _1, \xi _2}$ with $\xi _1\in \mathrm {RV}_{\lambda }$ and $\xi _2\in \mathrm {RV}_{\lambda }^{k1}$ . Then by induction on k, the union $\cup _{\xi } C_{\xi } = \cup _{\xi _1\in \mathrm {RV}_{\lambda }}\cup _{\xi _2\in \mathrm {RV}_{\lambda }^k} C_{\xi _1, \xi _2}$ is finite.
(2) Using (T1 $^{\mathrm {mix}}$ ), we find an Adefinable C. We consider C as an $(A\cap K)$ definable family $C_{\xi }$ , parametrized by $\xi \in \mathrm {RV}_{n}^k$ . Using (1), the union $C' = \cup _{\xi \in \mathrm {RV}_{n}^k} C_{\xi }$ is $(A\cap K)$ definable, finite, and f satisfies (T1 $^{\mathrm {mix}}$ ) with respect to $C'$ .
Now that we know that any theory ${\mathcal T}$ satisfying (T1,T2) is $0^{\mathrm {mix}}$ hminimal, we can use Corollary 2.3.4 and Lemma 2.3.1.
The following is a first adaptation of [Reference Cluckers, Halupczok and Rideau10, Lemma 2.9.6]. In Corollary 3.1.3, we will obtain an adaptation which is better in the sense that it has a more precise conclusion (and more similar to the equicharacteristic $0$ case).
Lemma 2.4.2 (Images of balls).
Assume that ${\mathcal T}$ satisfies (T1,T2). Let $f: K\to K$ be Adefinable, for some $A\subset K\cup \mathrm {RV}_{n}$ . Then there exists a finite Adefinable C and a positive integer m such that (T $1^{\mathrm {mix}}$ ) holds for f and C and m and such that the following holds for any ball $B \ m$ next to C. Let $B'\subset B$ be an open ball and let $\mu _B$ be as in (T $1^{\mathrm {mix}}$ ). Then for every $x\in B'$ , there are open balls $B_1, \tilde {B}_1$ such that $f(x)\in \tilde {B}_1\subset f(B')\subset B_1$ , and moreover,
Proof. Using Lemma 2.3.1 we may as well assume that $A=\emptyset $ . Take a finite $\emptyset $ definable set C and an integer m such that (T $1^{\mathrm {mix}}$ ) holds for f with respect to C and m. Let $\chi : K\to \mathrm {RV}_{p}^k$ be a $\emptyset $ definable map coming from Lemma 2.3.1(5) for $C,m$ . So every ball $m$ next to C is a union of fibres of $\chi $ and every ball $n'$ next to C is contained in a fibre of $\chi $ . Let D be a finite $\emptyset $ definable set $n$ preparing the family $(f(\chi ^{1}(\eta )))_{\eta \in \mathrm {RV}_{p}^k}$ ; here, we use Corollary 2.3.4. Let $\psi : K\to \mathrm {RV}_{p'}^{k'}$ be a $\emptyset $ definable map such that every ball $n$ next to D is a union of fibres of $\psi $ and every ball $n"$ next to D is contained in a fibre of $\psi $ . Finally, use Corollary 2.3.4 again to $q$ prepare the family $(f^{1}(\psi ^{1}(\eta )))_{\eta \in \mathrm {RV}_{p'}^{k'}}$ with a finite $\emptyset $ definable set $C_0$ .
We claim that $C'=C\cup C_0$ suffices. So let $B'$ be a ball $q$ next to $C'$ and let B be the ball $m$ next to C containing $B'$ . We can assume that $\mu _B\neq 0$ , for else f is constant on B. Fix any open ball $B"$ in $B'$ and let $x,y\in B"$ . Then
Therefore, if we denote by $B_1^{\prime \prime }$ the open ball of radius $\frac {\mu _B}{m}\operatorname {rad}_{\mathrm {op}} B"$ around $f(x)$ , then $f(B")$ is contained in $B_1^{\prime \prime }$ . By definition of $C_0$ , $f(B')$ is contained in a ball $B_1$ $n$ next to D. However, $f(B)$ is $n$ prepared by D. Since $B_1\cap f(B)\neq \emptyset $ , we have
Using property (T $1^{\mathrm {mix}}$ ), we see that
Now take $z\in K$ such that $f(x)z < m\mu _B\operatorname {rad}_{\mathrm {op}} B"$ . Then we have $z\in B_1\subset f(B)$ , and so there is some $x'\in B$ with $f(x')=z$ . Applying (T $1^{\mathrm {mix}}$ ) one more time yields that
We conclude that $f(B")$ contains the open ball of radius $m\mu _B\operatorname {rad}_{\mathrm {op}} B"$ around $f(x)$ .
2.5 Basic results under 1h $^{\mathrm {ecc}}$ minimality
We provide the tools necessary to transfer preparation results to mixed characteristic, starting from $1$ h $^{\mathrm {ecc}}$ minimality. We prove part of Theorem 2.2.8 which states that 1h $^{\mathrm {ecc}}$ minimality implies 1h $^{\mathrm {mix}}$ minimality.
Notation 2.5.1. The notion of balls $\lambda $ next to a finite set $C \subset K$ now has different meanings for $\cdot $ and for $\cdot _{\mathrm {ecc}}$ , with notation from Definition 2.1.3. To make clear which of the valuations we mean, we either write $1$ next or $1_{\mathrm {ecc}}$ next (instead of just $1$ next).
Remark 2.5.2. Suppose that $\cdot _{{\mathrm {ecc}}}$ is nontrivial on K. For any $x, x' \in K$ , we have
and given a finite set $C \subset K$ , the points x and $x'$ lie in the same ball $1_{\mathrm {ecc}}$ next to C if and only if for every integer $m \ge 1$ , they lie in the same ball $m$ next to C.
Assuming $1$ h $^{\mathrm {ecc}}$ minimality of K as an ${\mathcal L}_{\mathrm {ecc}}$ structure, we will be able to find finite ${\mathcal L}_{\mathrm {ecc}}$ definable sets. To get back to the smaller language ${\mathcal L}$ , we will use the following lemma:
Lemma 2.5.3 (From ${\mathcal L}_{\mathrm {ecc}}$ definable to ${\mathcal L}$ definable).
Let ${\mathcal T}$ be $1$ h $^{\mathrm {ecc}}$ minimal. Let K be a model of ${\mathcal T}$ which is $\aleph _0$ saturated and strongly $\aleph _0$ homogeneous as an ${\mathcal L}^{\mathrm {eq}}$ structure. (Note that this, in particular, implies that $\cdot _{\mathrm {ecc}}$ is nontrivial.) Then, any finite ${\mathcal L}_{\mathrm {ecc}}$ definable set, $C \subset K$ is already ${\mathcal L}$ definable.
Remark 2.5.4. It is a standard result that every model K has an elementary extension satisfying the properties of the lemma. Indeed, any model which is special in the sense of [Reference Tent and Ziegler22, Definition 6.1.1] is strongly $\aleph _0$ homogeneous by [Reference Tent and Ziegler22, Theorem 6.1.6], and it is easy to construct $\aleph _0$ saturated special models.
Proof of Lemma 2.5.3.
It suffices to prove that for any $a \in C$ , all realizations of $p:=\mathrm {tp}_{{\mathcal L}}(a/\emptyset )$ lie in C; indeed, by $\aleph _0$ saturation, this then implies that p is algebraic (using that C is finite) and hence isolated by some formula $\phi _p(x)$ . Therefore, C is defined by the disjunction of finitely many such $\phi _p(x)$ .
So now suppose for contradiction that there exist $a \in C$ and $a' \in K \setminus C$ which have the same ${\mathcal L}$ type over $\emptyset $ . Then by our homogeneity assumption, there exists an ${\mathcal L}$ automorphism of K sending a to $a'$ (and hence not fixing C setwise). Such an automorphism also preserves ${\mathcal O}_{{\mathrm {ecc}}}$ and hence is an ${\mathcal L}_{\mathrm {ecc}}$ automorphism, but this contradicts C being ${\mathcal L}_{\mathrm {ecc}}$ definable.
We obtain yet another part of Theorem 2.2.8.
Corollary 2.5.5. Assume that ${\mathcal T}$ is $1$ h $^{\mathrm {ecc}}$ minimal. Then ${\mathcal T}$ is 1h $^{\mathrm {mix}}$ minimal.
Proof. By Remark 2.5.4, we may consider a model K of ${\mathcal T}$ which is sufficiently saturated and sufficiently homogeneous (as in Lemma 2.5.3). Consider an integer n, $\lambda \in \Gamma _K^{\times }$ , $A\subset K$ , $\xi \in \mathrm {RV}_{\lambda }$ , $\eta \in \mathrm {RV}_{n}^k$ for some integers $k,n>0$ , and an ${\mathcal L}(A\cup \{\eta ,\xi \})$ definable set $X\subset K$ . We have to show that X can be $m\cdot \lambda $ prepared by some finite ${\mathcal L}(A)$ definable set C for some integer $m>0$ .
Let $\lambda _{\mathrm {ecc}}$ be the image of $\lambda $ in $\Gamma _{K,{\mathrm {ecc}}}$ . Since $B_{<\lambda _{\mathrm {ecc}}}(1) \subset B_{<\lambda }(1)$ , we have a canonical surjection $\mathrm {RV}_{\lambda _{\mathrm {ecc}}} \to \mathrm {RV}_{\lambda }$ . Similarly, there is a canonical surjection $\mathrm {RV}_{{\mathrm {ecc}}}\to \mathrm {RV}_{n}$ . We fix any preimage $(\xi _{\mathrm {ecc}}, \eta _{\mathrm {ecc}}) \in \mathrm {RV}_{\lambda _{\mathrm {ecc}}} \times \mathrm {RV}_{\mathrm {ecc}}^k$ of $(\xi ,\eta )$ , so that X is ${\mathcal L}(A \cup \{\xi _{\mathrm {ecc}}, \eta _{\mathrm {ecc}}\})$ definable. By $1$ hminimality for the ${\mathcal L}_{\mathrm {ecc}}$ structure on K, there exists a finite ${\mathcal L}_{\mathrm {ecc}}(A)$ definable set C such that for every pair $x, x'$ in the same ball $\lambda _{\mathrm {ecc}}$ next to C, we have $x \in X \iff x' \in X$ . By Lemma 2.5.3, C is already ${\mathcal L}(A)$ definable; we claim that it is as desired. Suppose for contradiction that there exists no m as in the corollary; that is, for every integer $m \ge 1$ , there exists a pair of points $(x, x') \in K^2$ which lie in the same ball $\lambda \cdot m$ next to C such that $x \in X$ but $x' \notin X$ . By $\aleph _0$ saturation (in the language ${\mathcal L}$ ), we find a single pair $(x, x') \in K^2$ of points with $x \in X$ but $x' \notin X$ and which lie in the same ball $\lambda \cdot m$ next to C for every $m \ge 1$ . The latter implies that x and $x'$ lie in the same ball $\lambda _{\mathrm {ecc}}$ next to C (by Remark 2.5.2), so we get a contradiction to our choice of C.
2.6 Resplendency
In order to prove that (3) implies (1) in Theorem 2.2.8, we will need a way to add the coarsened valuation ring to the language. This is made possible via a mixed characteristic resplendency result, as in [Reference Cluckers, Halupczok and Rideau10, Section 4]. All proofs in this subsection work exactly as in [Reference Cluckers, Halupczok and Rideau10]; one just needs to replace $\mathrm {RV}_{I}$ by $\bigcup _n\mathrm {RV}_{nI}$ everywhere (where I is an ideal of ${\mathcal O}_K$ ). For completeness, we nevertheless give most of the details.
Fix a ∅definable ideal I of ${\mathcal O}_K$ which is neither $\{0\}$ nor equal to ${\mathcal O}_K$ . Call a language ${\mathcal L}'$ an $\bigcup _n\mathrm {RV}_{nI}$ expansion of ${\mathcal L}$ if ${\mathcal L}'$ is obtained from ${\mathcal L}$ by adding (any) predicates which live on Cartesian products of the (imaginary) definable sets $\mathrm {RV}_{nI}$ for some $n>0$ . (Recall that $\mathrm {RV}_I$ has been defined in 2.1.)
In the following, we assume that K is $\kappa $ saturated for some $\kappa> {\mathcal L}$ ; we call a set ‘small’ if its cardinality is less than $\kappa $ , and ‘large’ otherwise.
Definition 2.6.1 (MixedIpreparation).
We say that K has mixed Ipreparation if for every integer n, every set $A\subset K$ and every $(A\cup \mathrm {RV}_{nI})$ definable subset $X\subset K$ , there exists a finite Adefinable set $C \subset K$ and an integer m such that X is $mI$ prepared by C.
We say that K has resplendent mixed Ipreparation if for every set $A\subset K$ , for every $\bigcup _n\mathrm {RV}_{nI}$ expansion ${\mathcal L}'$ of ${\mathcal L}$ and for every ${\mathcal L}'(A)$ definable subset $X\subset K$ , there exists an integer m and a finite ${\mathcal L}(A)$ definable set $C \subset K$ such that X is $mI$ prepared by C.
Clearly, resplendent mixedIpreparation implies mixedIpreparation.
We consider the language ${\mathcal L}$ as having the sorts K and RV_{ nI } for each $n> 0$ . Moreover, let L_{ I } be the language with the same sorts but consisting only of the additive group on K and the maps rv_{ nI }.
Lemma 2.6.2 (Preparation and partial isomorphisms).
Let $A \leq K$ be a small ${\mathbb Q}$ subvector space. The following are equivalent:

(i) Any ${\mathcal L}(A\cup \bigcup _n \mathrm {RV}_{nI})$ definable set $X\subset K$ can be $mI$ prepared, for some integer m, by some finite set $C\subset A$ .

(ii) For every small subset $A_2\subset K$ , $c_1, c_2 \in K$ and all (potentially large) sets $B_1, B_2\subset \bigcup _n\mathrm {RV}_{nI}$ with $\bigcup _n\operatorname {rv}_{nI}(\langle A,c_1\rangle ) \subset B_1$ , if $f\colon A B_1 c_1 \to A_2 B_2 c_2$ is a partial ${\mathcal L}_I$ isomorphism sending $c_1$ to $c_2$ whose restriction ${\left .{f}\right {}_{AB_1}}$ is a partial elementary ${\mathcal L}$ isomorphism, then the entire f is a partial elementary ${\mathcal L}$ isomorphism.

(iii) For all $c_1, c_2 \in K$ and all (potentially large) sets $B\subset \mathrm {RV}_{nI}$ which contain $\bigcup _n\operatorname {rv}_{nI}(\langle A,c_1\rangle )$ , any partial ${\mathcal L}_I(A\cup B)$ isomorphism $f\colon \{c_1\} \to \{c_2\}$ , is a partial elementary ${\mathcal L}(A\cup B)$ isomorphism.
Proof. (i) $\Rightarrow $ (iii): Let f be as in (iii). We have to check that for every ${\mathcal L}(A\cup B)$ definable set $X\subset K$ , $c_1 \in X$ if and only if $c_2\in X$ . By (i), there exists a finite $C \subset A$ and an integer m such that X is $mI$ prepared by C. Since f is an