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Hensel minimality I

Published online by Cambridge University Press:  16 May 2022

Raf Cluckers
Affiliation:
Univ. Lille, CNRS, UMR 8524, Laboratoire Paul Painlevé, F-59000Lille, France; E-mail: Raf.Cluckers@univ-lille.fr KU Leuven, Department of Mathematics, B-3001Leuven, Belgium
Immanuel Halupczok*
Affiliation:
Lehrstuhl für Algebra und Zahlentheorie, Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225Düsseldorf, Germany
Silvain Rideau-Kikuchi
Affiliation:
Université de Paris and Sorbonne Université, CNRS, IMJ-PRG, F-75013Paris, France; E-mail: silvain.rideau@imj-prg.fr
*

Abstract

We present a framework for tame geometry on Henselian valued fields, which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show the existence of t-stratifications in Hensel minimal structures and Taylor approximation results that are key to non-Archimedean versions of Pila–Wilkie point counting, Yomdin’s parameterization results and motivic integration. In this first paper, we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.

Type
Algebra
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

1 Introduction

1.1 Background

Our goal is to understand what really lies behind the tameness of definable sets in valued fields and to describe this axiomatically, thereby providing a base for research in several directions of non-Archimedean geometry. In the present paper, we introduce the framework called ‘Hensel minimality’, which is very natural and readily verifiable from Ax–Kochen/Ershov type results. Moreover, it has strong consequences that were previously not expected to follow from such basic axioms.

Let us begin by describing some history of definable sets in valued fields and their uses for p-adic and motivic integration, geometry and diophantine applications. From the 1960s on, various results were obtained, providing rather precise control over semi-algebraic definable sets in the fields ${\mathbb Q}_{p}$ of p-adic numbers and $k\mathopen {(\!(} t\mathopen {)\!)}$ of formal Laurent series with coefficients in a field k of characteristic $0$ by Ax, Kochen, Ershov, Cohen, Macintyre and others [Reference Ax and Kochen1, Reference Ax and Kochen2, Reference Ax and Kochen3, Reference Eršov37, Reference Cohen25, Reference Eršov37, Reference Macintyre53]. Based on those, Denef [Reference Denef26] solved a rationality question of Serre [Reference Serre66] which established a strong connection between number theory and the geometry of p-adic definable sets. This in turn became a motivation for the further development of the model theory of valued fields [Reference Pas60, Reference Basarab4] and motivic integration [Reference Denef and Loeser28, Reference Denef and Loeser29] (an integration theory on arc spaces of varieties) and for the study of singularities and Berkovich spaces by Hrushovski and Loeser, among others [Reference Batyrev and Moreau5, Reference Denef and Loeser30, Reference Hrushovski and Loeser49]. The model theoretic approach also brought Fubini and Fourier to motivic integrals [Reference Cluckers and Loeser21, Reference Cluckers and Loeser22, Reference Hrushovski and Kazhdan47], which were key to applications to the Langlands program [Reference Cluckers, Hales and Loeser16, Reference Casselman, Cely and Hales8]. Recently, semi-algebraic and subanalytic p-adic geometry has led to point counting results analogous to Pila–Wilkie point counting and Yomdin’s parameterization results [Reference Cluckers, Comte and Loeser13, Reference Cluckers, Forey and Loeser14, Reference Pila and Wilkie62, Reference Yomdin72]. Applications to Lipschitz stratifications were also obtained for semi-algebraic and subanalytic sets from non-Archimedean geometry [Reference Halupczok43]. See [Reference Bélair6] for a more extensive panorama of results. These historical results are all based on concrete settings like semi-algebraic and subanalytic geometry; we open them up by providing a natural axiomatic and more general framework.

In the field of real numbers, a successful framework exists for tame geometry since work by van den Dries, Pillay and others [Reference van den Dries31, Reference van den Dries33, Reference Knight, Pillay and Steinhorn52, Reference Pillay and Steinhorn63]. Indeed, the tameness and geometry of real definable sets are very well captured by the notion of o-minimality, which is an axiomatic condition about definable sets. It became a central tool in real algebraic geometry and its generalizations on the one hand because of its beautiful naturality and on the other hand because of its vast consequences for the geometry of definable sets and its strong diophantine applications, for example to the André–Oort conjecture [Reference Pila61].

Soon after the first successes of o-minimality, the open quest for analogous frameworks in valued fields emerged as a central question in model theory. Several notions have been presented so far, each of which has some shortcomings or lack of generality. In analogy to o-minimality, there are P-minimality [Reference Haskell and Macpherson46] and C-minimality [Reference Macpherson and Steinhorn54, Reference Haskell and Macpherson45]; more motivated by applications are V-minimality [Reference Hrushovski and Kazhdan47] and b-minimality [Reference Cluckers and Loeser20]; motivated by classification theory is dp-minimality [Reference Onshuus and Usvyatsov59, Reference Jahnke, Simon and Walsberg51]. In the present paper, our new notion of Hensel minimality overcomes these shortcomings and provides a strong and natural framework for non-Archimedean tame geometry. Hensel minimality is easier to axiomatize and more natural than b-minimality and V-minimality; it applies more broadly than P-minimality, C-minimality, V-minimality or even dp-minimality; and it has stronger consequences than b-minimality, P-minimality and C-minimality.

In fact, we introduce two key notions of Hensel minimality: $1$ -h-minimality and $\omega $ -h-minimality. Most of our results are developed under the weaker assumption of $1$ -h-minimality; only some of the resplendency results from Section 4 need $\omega $ -h-minimality. As auxiliary notions, we define some intermediary variants of Hensel minimality, namely $\ell $ -h-minimality for integers $\ell>1$ . We use the term ‘Hensel minimality’ to talk about any of those variants in an implicit way. Most classically studied examples turn out to be $\omega $ -h-minimal, but for some interesting (somewhat less classical) examples, we were only able to prove $1$ -h-minimality (see Section 6).

In this paper, we work in equi-characteristic zero. The mixed characteristic case is derived from this in the sequel [Reference Cluckers, Halupczok, Rideau and Vermeulen17]. Indeed, the mixed characteristic version of Hensel minimality comes almost for free (including many of its consequences) from the equi-characteristic zero situation by using coarsened valuations. In Section 6, we briefly define the mixed characteristic variant of Hensel minimality for the sake of our examples.

We now sketch ideas, results and applications. In a way, o-minimality assumes that every unary definable set is controlled by a finite set, namely its set of boundary points; in this sense, Hensel minimality is very similar to o-minimality. To make this more precise, the definition of an o-minimal field R can be phrased as the following condition on definable subsets X of R: There exists a finite subset C of R such that, for any $x\in R$ , whether x lies in X depends only on the tuple $(\operatorname *{\mathrm {{sgn}}}(x-c))_{c\in C}$ , where $\operatorname *{\mathrm {{sgn}}}$ stands for the sign, which can be $-$ , $0$ or $+$ . The definition of Hensel minimality is similar, where the sign function is replaced by a suitable function adapted to the valuation, namely the leading term map $\operatorname {rv}$ . However, whereas in the o-minimal world, C is automatically definable over the same parameters as X, in the valued field setting, we need to impose precise conditions on the parameters over which C can be defined. This is also where the differences between $1$ -h-minimality and $\omega $ -h-minimality arise.

A large part of this paper consists of proving our main geometric results in Hensel minimal structures, which are similar to those in o-minimal structures, in particular cell decomposition, dimension theory, the ‘Jacobian Property’ (which plays a key role in constructing motivic integration and can be considered an analogue of the Monotonicity Theorem from the o-minimal context, where $\operatorname *{\mathrm {{sgn}}}$ is replaced by $\operatorname {rv}$ ), as well as higher-order and higher-dimension versions of the Jacobian Property, which state that definable functions have good approximations by their Taylor polynomials. Based on those properties, various recent results in the model theory of Henselian valued fields readily generalize to arbitrary Hensel minimal valued fields, like those on Lipschitz continuity [Reference Cluckers, Comte and Loeser11] and t-stratifications (which were introduced in [Reference Halupczok43] and studied further in [Reference García Ramírez42, Reference García Ramírez41, Reference Halupczok and Yin44]). As an extra upshot, Hensel minimality intrinsically has resplendency properties in the spirit of resplendent quantifier elimination: that is, it is preserved by different kinds of expansions of the structure. In particular, it should be considered a notion of tameness ‘relative to the leading term structure $\mathrm {RV}$ ’.

As first applications, we show the existence of t-stratifications in arbitrary Hensel minimal structures, and we use our results in valued fields to deduce a uniform Taylor approximation result in power-bounded o-minimal real closed fields, which strengthens results from [Reference Halupczok43, Reference Halupczok and Yin44]; here, the connection between valued fields and real closed fields is made using the notion of T-convexity from [Reference van den Dries and Lewenberg35, Reference van den Dries32]. Furthermore, our geometric results lay the ground for further generalizations of motivic integration from [Reference Cluckers and Loeser21, Reference Cluckers and Loeser22, Reference Hrushovski and Kazhdan47] and its use in, for example, [Reference Bilu7, Reference Chambert-Loir and Loeser9, Reference Cluckers, Comte and Loeser12, Reference Cluckers, Gordon and Halupczok15, Reference Cluckers and Loeser22, Reference Forey39, Reference Hrushovski and Kazhdan48, Reference Hrushovski, Martin and Rideau50, Reference Nguyen56, Reference Nicaise58, Reference Yin69, Reference Yin70]. Concretely, we show how Hensel minimality relates to the axiomatic frameworks under which motivic integration is developed in [Reference Cluckers and Loeser23] and [Reference Hrushovski and Kazhdan47]. Our results also lay the ground for $C^{r}$ parameterizations and bounds on the number of rational points as in [Reference Cluckers, Comte and Loeser13, Reference Cluckers, Forey and Loeser14], analogous to results by Yomdin [Reference Yomdin71] and Pila–Wilkie [Reference Pila and Wilkie62]. A first diophantine application to the counting of rational points of bounded height is given in [Reference Cluckers, Halupczok, Rideau and Vermeulen17].

1.2 The notion of Hensel minimality

We start by giving the definition of Hensel minimality; see Section 2.3 for more details.

Let K be a non-trivially valued field of equi-characteristic $0$ , considered an ${\mathcal L}$ -structure for some language ${\mathcal L}$ containing the language ${\mathcal L}_{\mathrm {val}} = \{+,\cdot ,{\mathcal O}_{K}\}$ of valued fields (where ${\mathcal O}_{K}$ is a predicate for the valuation ring). We use multiplicative notation for valuations. We denote the value group by $\Gamma ^{\times }_{K}$ and the valuation map by

$$ \begin{align*} |\cdot|\colon K\to \Gamma_{K} := \Gamma_{K}^{\times} \cup\{0\}; \end{align*} $$

see Section 2.1 for more detailed definitions.

The analogue to the sign map from the o-minimal context will be the ‘leading term map’ $\operatorname {rv}\colon K \to \mathrm {RV}$ , defined as follows:

Definition 1.2.1 (Leading term structure $\mathrm {RV}_{\lambda }$ )

Let $\lambda \le 1$ be an element of $\Gamma _{K}^{\times }$ , and set $I := \{x \in K \mid |x| < \lambda \}$ . We define $\mathrm {RV}_{\lambda }^{\times }$ to be the quotient of multiplicative groups $K^{\times }/(1 + I)$ , and we let

$$ \begin{align*} \operatorname{rv}_{\lambda}\colon K \to \mathrm{RV}_{\lambda} := \mathrm{RV}_{\lambda}^{\times} \cup \{0\} \end{align*} $$

be the map extending the projection map $K^{\times }\to \mathrm {RV}_{\lambda }^{\times }$ by sending $0$ to $0$ . We abbreviate $\mathrm {RV}_{1}$ and $\operatorname {rv}_{1}$ by $\mathrm {RV}$ and $\operatorname {rv}$ , respectively. If several valued fields are around, we may also write $\mathrm {RV}_{K}$ and $\mathrm {RV}_{K,\lambda }$ .

We can now make precise in which sense a set $X\subset K$ can be controlled by a finite set $C \subset K$ .

Definition 1.2.2 (Prepared sets; see Definition 2.3.1 )

Let $\lambda \le 1$ be in $\Gamma _{K}^{\times }$ , let C be a finite non-empty subset of K, and let $X \subset K$ be an arbitrary subset. We say that $C \ \lambda $ -prepares X if whether some $x\in K$ lies in X depends only on the tuple $(\operatorname {rv}_{\lambda }(x-c))_{c\in C}$ . In other words, if $x, x^{\prime } \in K$ satisfy

(1.2.1) $$ \begin{align} \operatorname{rv}_{\lambda}(x-c) = \operatorname{rv}_{\lambda}(x^{\prime}-c) \mbox{ for each } c\in C, \end{align} $$

then either x and $x^{\prime }$ both are elements of X, or none of them are elements of X.

A key ingredient of Hensel minimality is that any A-definable set $X \subset K$ (for $A \subset K$ ) can be $1$ -prepared by a finite A-definable set C. This, however, is not yet strong enough a condition: We need some precise control of parameters from $\mathrm {RV}$ and $\mathrm {RV}_{\lambda }$ . This is where we obtain different variants of Hensel minimality, the difference consisting only in some number $\ell $ of allowed parameters.

Definition 1.2.3 ( $\ell $ -h-minimality; see Definition 2.3.3 )

Let $\ell \geq 0$ be either an integer or $\omega $ , and let ${\mathcal T}$ be a theory of valued fields of equi-characteristic $0$ , in a language ${\mathcal L}$ containing the language ${\mathcal L}_{\mathrm {val}}$ of valued fields. We say that ${\mathcal T}$ is $\ell $ -h-minimal if every model $K \models {\mathcal T}$ has the following property:

  • (1.2.2) For every $\lambda \le 1$ in $\Gamma _{K}^{\times }$ , for every set $A\subset K$ and for every set $A^{\prime } \subset \mathrm {RV}_{\lambda }$ of cardinality $\#A^{\prime } \le \ell $ , every $(A \cup \mathrm {RV} \cup A^{\prime })$ -definable set $X \subset K$ can be $\lambda $ -prepared by a finite A-definable set $C \subset K$ .

In this definition, the parameters from $\mathrm {RV}$ and $\mathrm {RV}_{\lambda }$ are imaginary elements (see Section 2.2).

The notion of $\ell $ -h-minimality seems most natural in the case $\ell = \omega $ , and indeed, this case is especially interesting in view of the resplendency results of Section 4. However, the case $\ell = 1$ plays an important role, too: most of the results in this paper follow from $1$ -h-minimality, and we obtain a rather different equivalent characterization of $1$ -h-minimality in terms of definable functions (see Theorem 2.9.1). This shows that $1$ -h-minimality is of a deeply geometric nature, even though its definition might appear somewhat contrived. Similar characterizations for integers $\ell \ge 2$ are developed in [Reference Vermeulen68], involving properties of definable functions on $\ell $ -dimensional sets.

Many structures on valued fields whose model theory is known to behave well are Hensel minimal. In Section 6, we provide examples of Hensel minimal structures on valued fields of mixed and equi-characteristic $0$ . Some are new and less expected; others incorporate more classical structures. We show that the following classical structures are Hensel minimal:

  1. 1. The theory of Henselian valued fields of equi-characteristic $0$ in the pure valued field language is $\omega $ -h-minimal.

  2. 2. The theory stays $\omega $ -h-minimal if we expand the valued field by analytic functions (forming an analytic structure as in [Reference Cluckers and Lipshitz18] or [Reference Cluckers and Lipshitz19]): For example, we can consider the theory of a field $K = k((t))$ with $\operatorname {char} k = 0$ , in a language containing all analytic functions ${\mathcal O}_{K}^{n} \to K$ for all n, where ‘analytic’ here means that the function is given by evaluating a t-adically converging power series in $x\in {\mathcal O}_{K}^{n}$ (see Section 3.4 of [Reference Cluckers and Lipshitz19]).

  3. 3. If ${\mathcal T}_{\mathrm {omin}}$ is the theory of a power-bounded o-minimal expansion of a real closed field, then the theory of ${\mathcal T}_{\mathrm {omin}}$ -convex valued fields in the sense of [Reference van den Dries and Lewenberg35] is $1$ -h-minimal. Recall that a ${\mathcal T}_{\mathrm {omin}}$ -convex valued field is obtained by taking a (sufficiently big) model K of ${\mathcal T}_{\mathrm {omin}}$ and turning it into a valued field using the convex closure of an elementary substructure $K_{0} \prec K$ as the valuation ring; see Section 6.3 for details.

  4. 4. Any expansion of the structure by predicates on Cartesian powers of $\mathrm {RV}$ preserves $0$ -, $1$ - and $\omega $ -h-minimality (Theorem 4.1.19).

  5. 5. The notion of $\omega $ -h-minimality is preserved under coarsening of the valuation (see Corollary 4.2.4). In [Reference Cluckers, Halupczok, Rideau and Vermeulen17, Theorem 2.2.7], it is proved that $1$ -h-min is also preserved under coarsening of the valuation.

By combining (5) and (2), one recovers the examples of coarsened analytic structures from [Reference Rideau64] and resplendently so by (4).

On the other hand, there are also some well-behaved structures on valued fields that are certainly not Hensel minimal: Since the very definition of Hensel minimality implies that every definable subset of K is either finite or contains a ball, one cannot add a section of the residue field or the value group to the language, and neither can one add an automorphism of K.

As a converse to (1), we show that a valued field that is Hensel minimal is automatically Henselian; see Theorem 2.7.2. In Proposition 6.4.2, we compare Hensel minimality to V-minimality from [Reference Hrushovski and Kazhdan47].

1.3 Description of the main results

We now list some of the consequences of $1$ -h-minimality (often in simpler forms than the versions in the main part of the paper). The first result is the ‘Jacobian Property’, which plays a crucial role for example in motivic integration, both in the Cluckers–Loeser version [Reference Cluckers and Loeser21, Reference Cluckers and Loeser22] and in the Hrushovski–Kazhdan version [Reference Hrushovski and Kazhdan47].

Theorem 1.3.1 (Jacobian Property; see Corollary 3.2.7 )

Let $f\colon K \to K$ be definable (with parameters). Then there exists a finite set $C \subset K$ such that the following holds: for every fiber B of the map sending $x \in K$ to the tuple $(\operatorname {rv}(x - c))_{c \in C}$ , there exists a $\xi _{B} \in \mathrm {RV}$ such that

(1.3.1) $$ \begin{align} \operatorname{rv}\left(\frac{f(x_{1}) - f(x_{2})}{x_{1} - x_{2}}\right) = \xi_{B} \end{align} $$

for all $x_{1}, x_{2} \in B$ , $x_{1} \ne x_{2}$ .

Remark 1.3.2. This formulation of the Jacobian Property becomes exactly the o-minimal Monotonicity Theorem if one replaces all occurrences of $\operatorname {rv}$ by $\operatorname *{\mathrm {{sgn}}}$ and if one adds the condition that f should be continuous on the fibers B (which, in the valued field setting, follows automatically).

Corollary 3.2.7 also includes the following strengthenings of Theorem 1.3.1:

  • The theorem still holds with $\operatorname {rv}$ replaced by $\operatorname {rv}_{\lambda }$ for any $\lambda \le 1$ in $\Gamma _{K}^{\times }$ (still only assuming $1$ -h-minimality), and one can moreover choose a single finite set C that works for all such $\lambda $ .

  • If f is definable with parameters from $A \cup \mathrm {RV}$ with $A \subset K$ , then C can taken to be A-definable. (Corollary 3.2.7 only speaks about $\emptyset $ -definable sets f; Remark 2.6.11 explains how to deduce this more general version.)

Another point of view of the Jacobian Property is that on each fiber B (using notation from the Theorem), f has a good approximation by its first-order Taylor series. One of the deepest results of this paper is a similar result for higher-order Taylor approximations:

Theorem 1.3.3 (Taylor approximations; see Theorem 3.2.2 )

Let $f\colon K \to K$ be a definable function, and let $r \in {\mathbb N}$ be given. Then there exists a finite set C such that for every fiber B of the map sending $x \in K$ to the tuple $(\operatorname {rv}(x - c))_{c \in C}$ , f is $(r+1)$ -fold differentiable on B and we have

(1.3.2) $$ \begin{align} \left|f(x) - \sum_{i = 0}^{r} \frac{f^{(i)}(x_{0})}{i!}(x - x_{0})^{i} \right| \le |f^{(r+1)}(x_{0})\cdot (x-x_{0})^{r+1}| \end{align} $$

for every $x_{0}, x \in B$ .

As in Theorem 1.3.1, if f is $(A \cup \mathrm {RV})$ -definable for $A \subset K$ , then C can be taken A-definable (again using Remark 2.6.11).

Using that ${\mathcal T}_{\mathrm {omin}}$ -convex valued fields are $1$ -h-minimal (see (3) above, on p. 4), this Taylor approximation result implies a uniform Taylor approximation result in power-bounded o-minimal real closed fields; see Corollary 6.3.7.

As in the o-minimal context, the preparation of unary sets and the Jacobian Property lend themselves well (by logical compactness) to obtain results about higher-dimensional objects (namely, definable subsets of $K^{n}$ and definable functions on $K^{n}$ ). Indeed, this can be pursued, mutatis mutandis, in the style of how cell decomposition and dimension theory are built up from b-minimality in [Reference Cluckers and Loeser20]. In particular, we obtain results about

  • almost everywhere differentiability (Subsection 5.1),

  • cell decomposition in two variants (Subsections 5.2 and 5.7),

  • dimension theory (Subsection 5.3), and

  • higher-dimensional versions of the Jacobian Property and Taylor approximations (Subsections 5.4 and 5.6).

For cell decomposition, Subsection 5.2 provides a new approach specific to Hensel minimality: Usually, the notion of cells in valued fields is a lot more technical than in o-minimal structures, partly due to the lack of (certain) Skolem functions. Item (4) above (on p. 4) allows us to add the missing Skolem functions to the language without destroying Hensel minimality, and there are also some tools enabling us to get back to the original language afterward (see Subsection 4.3). This allows us, in Subsection 5.2, to work with a notion of a cell, which is much less technical than most previous ones in valued fields.

One of our original goals was to deduce the Jacobian Property starting from abstract conditions on unary sets only. Previous proofs of the Jacobian Property either use piecewise analyticity arguments, even in the semi-algebraic case (like in [Reference Cluckers and Lipshitz18]), or are restricted to rather specific situations (like [Reference Hrushovski and Kazhdan47] for V-minimal valued fields and [Reference Yin70] for power-bounded T-convex valued fields). Our new proof is more general and works for arbitrary Henselian valued fields, which are $1$ -h-minimal, and goes without analyticity arguments.

The remainder of this paper is organized as follows. In Section 2, after fixing notation and terminology, we develop many basic tools that are useful for proofs in Hensel minimal theories, and we obtain first elementary results like a key ingredient to dimension theory (Lemma 2.8.1) and a weak version of the Jacobian Property (Lemma 2.8.5). We also show that those two results are essentially equivalent to $1$ -h-minimality (Theorem 2.9.1).

The deepest results of this paper are contained in Section 3 about definable functions from K to K, namely almost everywhere differentiability and Taylor approximation.

Section 4 is devoted to understanding in which sense Hensel minimality is a notion ‘relative to $\mathrm {RV}$ ’ and to proving that various modifications of the language preserve Hensel minimality.

The geometric results in $K^{n}$ (like cell decomposition, dimension theory) are collected in Section 5. That section also contains our main application of t-stratifications, our higher-dimensional Taylor approximation results, and Cluckers–Loeser style motivic integration.

Finally, in Section 6, we show that many previously studied structures are Hensel minimal, and we give some new examples as well, namely coarsened valued fields as variants of classical analytic structures. Since some of those examples are of mixed characteristics, at the beginning of Section 6 we briefly define the mixed characteristic variant of Hensel minimality, which is treated in full in the sequel [Reference Cluckers, Halupczok, Rideau and Vermeulen17] to this paper.

As an application, we show in Section 6.3 how Hensel minimality results yield corresponding results in power-bounded real closed fields. At the end, we compare our notion to V-minimality from [Reference Hrushovski and Kazhdan47].

2 Hensel minimality

2.1 Notation and terminology for valued fields and balls

In the entire paper, K will denote a valued field of characteristic zero, with valuation ring ${\mathcal O}_{K} \subset K$ and maximal ideal ${\mathcal M}_{K} \subset {\mathcal O}_{K}$ . In this paper, we only consider non-trivially valued fields: that is, ${\mathcal O}_{K} \ne K$ . Moreover, apart from Section 6, K will always be of equi-characteristic $0$ , meaning that both K and the residue field ${\mathcal O}_{K}/{\mathcal M}_{K}$ have characteristic $0$ .

Note that we allow Krull-valuations (and thus valuations of arbitrary rank): that is, we allow ${\mathcal O}_{K}$ to be an arbitrary (proper, by the non-triviality) subring of K such that for every element $x\in K^{\times }$ , at least one of x or $x^{-1}$ belongs to ${\mathcal O}_{K}$ . The value group is then defined to be the quotient $\Gamma _{K}^{\times } := K^{\times } / {\mathcal O}_{K}^{\times }$ of multiplicative groups.

We denote the valuation map by $|\cdot |\colon K^{\times } \to \Gamma _{K}^{\times }$ and use multiplicative notation for the value group. We write $\Gamma _{K}$ for the disjoint union $\Gamma _{K}^{\times } \cup \{0\}$ , we extend the valuation map to $|\cdot |\colon K \to \Gamma _{K}$ by setting $|0| := 0$ , and we define the order on $\Gamma _{K}$ in such a way that ${\mathcal O}_{K} = \{x \in K \mid |x| \le 1\}$ and $|x|<|y|$ whenever $x/y\in {\mathcal M}_{K}$ for x and nonzero y in K.

For $x = (x_{1}, \dots , x_{n}) \in K^{n}$ , we set $|x| := \max _{i} |x_{i}|$ .

We use the (generalized) leading term structures $\mathrm {RV}_{\lambda }$ (for $\lambda \le 1$ in $\Gamma _{K}^{\times }$ ) that have already been introduced in Definition 1.2.1, and we also denote the natural map $\mathrm {RV}_{\lambda } \to \Gamma _{K}$ by $|\cdot |$ . Note that $\mathrm {RV}_{\lambda }$ is a semi-group for multiplication.

Remark 2.1.1. Write $\mathrm {RV}^{\times }$ for $\mathrm {RV}\setminus \{0\}$ . Recall that one has a natural short exact sequence of multiplicative groups $({\mathcal O}_{K}/{\mathcal M}_{K})^{\times } \to \mathrm {RV}^{\times } \to \Gamma ^{\times }_{K}$ . (So $\mathrm {RV}$ combines information from the residue field and value group.)

Example 2.1.2. In the case $K = k((t))$ , the above short exact sequence naturally splits, giving an isomorphism $\mathrm {RV}^{\times } \to ({\mathcal O}_{K}/{\mathcal M}_{K})^{\times } \times \Gamma ^{\times }_{K}$ , which, for $a = \sum _{i=N}^{\infty } a_{i} t^{i} \in K^{\times }$ with $a_{N} \ne 0$ , sends $\operatorname {rv}(a)$ to $(a_{N}, N)$ .

Remark 2.1.3. For $a, a^{\prime } \in K$ , one has $\operatorname {rv}_{\lambda }(a) = \operatorname {rv}_{\lambda }(a^{\prime })$ if and only if either $a = a^{\prime } = 0$ or $|a - a^{\prime }| < \lambda \cdot |a|$ .

We consider several kinds of balls:

Definition 2.1.4 (Balls)

  1. 1. We call a subset $B \subset K^{n}$ a ball if B is infinite, $B \ne K^{n}$ , and for all $x,x^{\prime }\in B$ and all $y\in K^{n}$ with $|x-y|\leq |x-x^{\prime }|$ , one has $y\in B$ .

  2. 2. By an open ball, we mean a set of the form

    $$ \begin{align*}B = B_{<\gamma}(a) := \{x\in K^{n}\mid | x - a | < \gamma \} \end{align*} $$
    for some $a\in K^{n}$ and some $\gamma \in \Gamma _{K}^{\times }$ .
  3. 3. By a closed ball, we mean a set of the form

    $$ \begin{align*}B = B_{\le\gamma}(a) := \{x\in K^{n}\mid | x - a | \leq \gamma \} \end{align*} $$
    for some $a\in K^{n}$ and some $\gamma \in \Gamma _{K}^{\times }$ .
  4. 4. For B as in (2) or (3), we call $\gamma $ the radius of the open (respectively, closed) ball and denote it by $\operatorname {rad}_{\mathrm {op}}(B)$ (respectively, $\operatorname {rad}_{\mathrm {cl}}(B)$ ).

We call the valuation on K discrete if there is a uniformizing element $\varpi $ in ${\mathcal O}_{K}$ , namely satisfying $\varpi {\mathcal O}_{K}={\mathcal M}_{K}$ .

Remark 2.1.5. When $\Gamma _{K}^{\times }$ is discrete, a ball B can be open and closed at the same time, but with $\operatorname {rad}_{\mathrm {cl}}(B) = |\varpi |\cdot \operatorname {rad}_{\mathrm {op}}(B) < \operatorname {rad}_{\mathrm {op}}(B)$ .

Note also that for any $\lambda \le 1$ in $\Gamma _{K}^{\times }$ and any $\xi \in \mathrm {RV}_{\lambda } \setminus \{0\}$ , the preimage $\operatorname {rv}_{\lambda }^{-1}(\xi )$ is an open ball satisfying $\operatorname {rad}_{\mathrm {op}}(\operatorname {rv}_{\lambda }^{-1}(\xi )) = |\xi | \cdot \lambda $ .

Definition 2.1.6 ( $\lambda $ -next balls)

Fix $\lambda \le 1$ in $\Gamma ^{\times }_{K}$ .

  1. 1. We say that a ball B is $\lambda $ -next to an element $c \in K$ if

    $$ \begin{align*}B= \{x\in K\mid \operatorname{rv}_{\lambda}(x-c) = \xi \} \end{align*} $$
    for some (nonzero) element $\xi $ of $\mathrm {RV}_{\lambda }$ .
  2. 2. We say that a ball B is $\lambda $ -next to a finite non-empty set $C\subset K$ if B equals $\bigcap _{c\in C} B_{c}$ with $B_{c}$ a ball $\lambda $ -next to c for each $c\in C$ .

Remark 2.1.7. Using that the intersection of the finitely many balls $B_{c}$ is either empty or equal to one of the $B_{c}$ , one deduces that every ball B that is $\lambda $ -next to C is in particular $\lambda $ -next to one element $c \in C$ .

Remark 2.1.8. Given a finite non-empty set $C \subset K$ , the fibers of the map $x \mapsto (\operatorname {rv}_{\lambda }(x - c))_{c \in C}$ are exactly the singletons consisting of one element of C and the balls $\lambda $ -next to C. In particular, the balls $\lambda $ -next to C form a partition of $K \setminus C$ , and a subset X of K is $\lambda $ -prepared by C (as in Definition 2.3.1) if and only if every ball B that is $\lambda $ -next to C is either contained in X or disjoint from X.

Example 2.1.9. The balls $1$ -next to an element $c \in K$ are exactly the maximal balls in K not containing c. From this, one deduces that a ball $1$ -next to a finite set C is exactly a maximal ball disjoint from C. This means that a set $X \subset K$ is $1$ -prepared by C if and only if every ball disjoint from C is either contained in X or disjoint from X. Note again how closely ‘every definable set can be $1$ -prepared’ resembles o-minimality (where ‘balls disjoint from C’ becomes ‘intervals disjoint from C’).

Given a subset A of a Cartesian product $B\times C$ and an element $b\in B$ , we write $A_{b}$ for the fiber $\{c\in C\mid (b,c)\in A\}$ . Also, for a function g on A, we write $g(b,\cdot )$ for the function on $A_{b}$ sending c to $g(b,c)$ .

2.2 Model theoretic notations and conventions

As already stated, in the entire paper, K is a non-trivially valued field of characteristic zero, and outside of Section 6, K is moreover of equi-characteristic $0$ . In the entire paper, we fix a language ${\mathcal L}$ containing the language ${\mathcal L}_{\mathrm {val}}$ of valued fields, and we consider the valued field K as an ${\mathcal L}$ -structure. More precisely, as ‘language of valued fields’, it suffices for us to take ${\mathcal L}_{\mathrm {val}} = \{+,\cdot , {\mathcal O}_{K}\}$ , where $\mathcal {O}_{K}$ is a predicate for the valuation ring; in any case, we only care about which sets are definable (and not how they are definable). For that reason, we will often specify languages only up to interdefinability.

If not specified otherwise, ‘definable’ always refers to the fixed language ${\mathcal L}$ . As usual, ‘ ${\mathcal L}$ -definable’ means definable (in ${\mathcal L}$ ) without additional parameters, ‘A-definable’ means ${\mathcal L}(A)$ -definable, and ‘definable’ means ${\mathcal L}(K)$ -definable (i.e., with arbitrary parameters).

Sometimes, we will also consider K a structure in other languages (e.g., ${\mathcal L}^{\prime }$ ); in that case, we may specify the language as an index, writing for example $\operatorname {Th}_{{\mathcal L}^{\prime }}(K)$ for the theory of K considered an ${\mathcal L}^{\prime }$ -structure.

In almost the entire paper (more precisely, everywhere except in parts of Section 4.1), ${\mathcal L}$ will be a one-sorted language. Nevertheless, we often work with imaginary sorts of K: that is, quotients $K^{n}/\mathord {\sim }$ , where $\sim $ is a $\emptyset $ -definable equivalence relation. In particular, we consider imaginary definable sets and imaginary elements. As usual, this can be made formal either by working in the ${\mathcal L}^{\mathrm {eq}}$ -structure $K^{\mathrm {eq}}$ (see for example [Reference Tent and Ziegler67, Proposition 8.4.5]) or, equivalently, by ‘simulating’ imaginary objects in ${\mathcal L}$ , namely as follows:

  • By a ‘definable subset X of $K^{n}/\mathord {\sim }$ ’, we really mean its preimage in $K^{n}$ : that is, a definable set $\tilde X \subset K^{n}$ that is a union of $\sim $ -equivalence classes.

  • If, in a formula $\phi (x, \dots )$ , the variable x runs over an imaginary sort $K^{n}/\mathord {\sim }$ , this means that we really have an n-tuple $\tilde {x}$ of variables (running over $K^{n}$ ) and that the truth value of $\phi (\tilde {x}, \dots )$ only depends on the equivalence class of $\tilde x$ modulo $\sim $ .

  • If A is a set of potentially imaginary elements, then by ‘A-definable’, we mean definable in the expansion of K by predicates for the equivalence classes (in $K^{n}$ for some n) corresponding to the imaginary elements $a \in A$ ; the corresponding extension of the language ${\mathcal L}$ by predicate symbols is denoted by ${\mathcal L}(A)$ .

  • A (potentially imaginary) element b is said to be in the definable closure of a set A (of potentially imaginary elements) if the equivalence class in $K^{n}$ corresponding to b is ${\mathcal L}(A)$ -definable. If X is any imaginary sort (or even more generally an arbitrary set of imaginary elements), we write $\mathrm {dcl}_{X}(A)$ for the set of elements from X that are in the definable closure of A. Being in the algebraic closure, with notation $\mathrm {acl}_{X}(A)$ , is defined accordingly.

The value group $\Gamma _{K}$ is, of course, an imaginary sort. In general, $\mathrm {RV}_{\lambda }$ (for $\lambda \le 1$ in $\Gamma ^{\times }_{K}$ ) is, by itself, not an imaginary sort, since the equivalence relation used to define it may not be $\emptyset $ -definable. However, the disjoint union

$$ \begin{align*} \mathrm{RV}_{\bullet} := \bigcup_{\lambda \le 1} \mathrm{RV}_{\lambda} \end{align*} $$

is an imaginary sort, and $\mathrm {RV}_{\lambda }$ is a definable subset of $\mathrm {RV}_{\bullet }$ ; in particular, it makes sense to use elements from $\mathrm {RV}_{\lambda }$ as parameters.

2.3 Hensel minimality

We first restate the definitions of prepared sets and Hensel minimality from the introduction. We use $\mathrm {RV}_{\lambda }$ and $\operatorname {rv}_{\lambda }$ from Definition 1.2.1.

Definition 2.3.1 (Prepared sets)

Let K be a non-trivially valued field of equi-characteristic $0$ . Let $\lambda \le 1$ be in $\Gamma _{K}^{\times }$ , let C be a finite non-empty subset of K, and let $X \subset K$ be an arbitrary subset. We say that $C \ \lambda $ -prepares X if there exists a set $\Xi \subset \mathrm {RV}_{\lambda }^{\#C}$ such that X is the preimage of $\Xi $ under the map $K\to \mathrm {RV}_{\lambda }^{\#C}, x \mapsto (\operatorname {rv}_{\lambda }(x-c))_{c\in C}$ .

The condition that C is non-empty is essentially irrelevant, since one could always add $0$ to C, but it will sometimes avoid pathologies.

Example 2.3.2. A subset of K is $\lambda $ -prepared by the set $C = \{0\}$ if and only if it is the preimage under $\operatorname {rv}_{\lambda }$ of a subset of $\mathrm {RV}_{\lambda }$ .

Definition 2.3.3 ( $\ell $ -h-minimality)

Let $\ell \geq 0$ be either an integer or $\omega $ , and let ${\mathcal T}$ be a (possibly non-complete) theory of valued fields of equi-characteristic $0$ , in a language ${\mathcal L}$ containing the language ${\mathcal L}_{\mathrm {val}}$ of valued fields. We say that ${\mathcal T}$ is $\ell $ -h-minimal if every model $K \models {\mathcal T} $ has the following property:

  • (2.3.1) For every $\lambda \le 1$ in $\Gamma _{K}^{\times }$ , for every set $A\subset K$ and for every set $A^{\prime } \subset \mathrm {RV}_{\lambda }$ of cardinality $\#A^{\prime } \le \ell $ , every $(A \cup \mathrm {RV} \cup A^{\prime })$ -definable set $X \subset K$ can be $\lambda $ -prepared by a finite A-definable set $C \subset K$ .

Note that a theory ${\mathcal T}$ is $\ell $ -h-mininimal if and only if each of its completions is. Conversely, compactness arguments show that most of our results hold uniformly in a theory ${\mathcal T}$ if and only they hold in every completion (see Proposition 2.6.2, for example). For this reason, we will often work with complete theories in this paper.

Remark 2.3.4. In the case $\ell = 0$ , $\lambda $ plays no role, and the definition simplifies to: Any $(A \cup \,\mathrm {RV})$ -definable set (for $A \subset K$ ) can be $1$ -prepared by a finite A-definable set.

Remark 2.3.5. In the case $\ell = \omega $ , we can more generally allow X to use parameters $\xi _{i} \in \mathrm {RV}_{\lambda _{i}}$ for different $\lambda _{i}$ (using that we have surjections $\mathrm {RV}_{\lambda } \to \mathrm {RV}_{\lambda ^{\prime }}$ for $\lambda ^{\prime }> \lambda $ ); in that case, C is required to $\lambda $ -prepare X for $\lambda := \min _{i} \lambda _{i}$ .

Remark 2.3.6. The assumption that C is definable using only the parameters from K will enable us to simultaneously prepare families of sets parametrized by $\mathrm {RV}$ (and $\mathrm {RV}_{\lambda }$ ). This plays a central role in making Hensel minimality independent of the structure on $\mathrm {RV}$ .

2.4 Basic model theoretic properties of Hensel minimality

Let us now verify that the notion of Hensel minimality (in all its variants) has some basic properties one would expect from any good model theoretic notion.

First of all, note that it is preserved under expansions of the structure by constants; more precisely:

Lemma 2.4.1 (Adding constants)

Let $\ell \geq 0$ be an integer or $\omega $ , suppose that $\operatorname {Th}(K)$ is $\ell $ -h-minimal, and let A be any subset of $K \cup \mathrm {RV}^{\mathrm {eq}}$ . Then $\operatorname {Th}_{{\mathcal L}(A)}(K)$ is also $\ell $ -h-minimal.

Here, by $\mathrm {RV}^{\mathrm {eq}}$ , we mean imaginary sorts of the form $\mathrm {RV}^{n}/\mathord {\sim }$ for some n and some $\emptyset $ -definable equivalence relations $\sim $ . In particular, the lemma allows us to add constants from $\Gamma _{K}$ . Note, however, that adding parameters from other sorts than K and $\mathrm {RV}^{\mathrm {eq}}$ may destroy Hensel minimality.

The lemma should be clear in the case $A \subset K \cup \mathrm {RV}$ ; we mainly give the following proof to show that parameters from $\mathrm {RV}^{\mathrm {eq}}$ are not a problem either.

Proof of Lemma 2.4.1

We verify Definition 2.3.3: Let $K^{\prime }\models \operatorname {Th}_{{\mathcal L}(A)}(K)$ , and let $X\subset K^{\prime }$ be ${\mathcal L}(A\cup A^{\prime }\cup \mathrm {RV}_{K^{\prime }}\cup A^{\prime \prime })$ -definable for some $A^{\prime } \subset K^{\prime }$ and some $A^{\prime \prime } \subset \mathrm {RV}_{K^{\prime },\lambda }$ satisfying $\#A^{\prime \prime } \le \ell $ , with $\lambda \le 1$ in $\Gamma _{K^{\prime }}$ . Choose $\tilde A \subset \mathrm {RV}_{K^{\prime }}$ such that every element of $A \cap \mathrm {RV}_{K^{\prime }}^{\mathrm {eq}}$ is ${\mathcal L}(\tilde A)$ -definable. Then X is ${\mathcal L}((A \cap K) \cup \tilde {A}\cup A^{\prime }\cup \mathrm {RV}_{K^{\prime }}\cup A^{\prime \prime })$ -definable, so by $\ell $ -h-minimality of $\operatorname {Th}_{{\mathcal L}}(K^{\prime })$ , X can be $\lambda $ -prepared by a finite ${\mathcal L}((A \cap K) \cup A^{\prime })$ -definable set C. In particular, C is ${\mathcal L}(A\cup A^{\prime })$ -definable, as desired.

With this lemma in mind, many results in this paper are formulated for $\emptyset $ -definable sets; those results then automatically also hold for A-definable sets, when $A \subset K \cup \mathrm {RV}^{\mathrm {eq}}$ , and using a compactness argument (given in Remark 2.6.3), one then often obtains family versions of the results.

As so often in model theory, it is sufficient to verify Hensel minimality in sufficiently saturated models. To see this, we first prove that preparation is a first-order property in the following sense:

Lemma 2.4.2 (Preparation is first order)

Let $X_{q}$ and $C_{q}$ be $\emptyset $ -definable families of subsets of K, where q runs over a $\emptyset $ -definable subset Q of an arbitrary possibly imaginary sort. Suppose that $C_{q}$ is finite for every $q \in Q$ . Then the set of pairs $(q,\lambda ) \in Q \times \Gamma ^{\times }_{K}$ with $\lambda \le 1$ such that $C_{q} \ \lambda $ -prepares $X_{q}$ is $\emptyset $ -definable.

Proof. If $\phi (x, q)$ defines $X_{q}$ and $\psi (z, q)$ defines $C_{q}$ , then the above set of pairs $(q,\lambda )$ is defined by the following ${\mathcal L}$ -formula:

$$ \begin{align*} \forall x,x^{\prime}\colon \big(\underbrace{\forall z\colon (\psi(z, q) \to \operatorname{rv}_{\lambda}(x - z) = \operatorname{rv}_{\lambda}(x^{\prime} - z))}_{\text{i.e., }(\operatorname{rv}_{\lambda}(x-c))_{c\in C_{q}} = (\operatorname{rv}_{\lambda}(x^{\prime}-c))_{c\in C_{q}}} \to (\phi(x, q) \leftrightarrow \phi(x^{\prime}, q))\big). \end{align*} $$

Lemma 2.4.3 (Saturated models suffice)

Let $\ell \geq 0$ be either an integer or $\omega $ , and suppose that K is $\aleph _{0}$ -saturated. Then the theory $\operatorname {Th}(K)$ of K is $\ell $ -h-minimal if and only if K satisfies Condition (2.3.1) from Definition 2.3.3.

Proof. We need to show that if K satisfies (2.3.1), then so does any other model $K^{\prime }$ of $\operatorname {Th}(K)$ .

Suppose for contradiction that $K^{\prime }$ is a model not satisfying (2.3.1): that is, there exist a $\lambda \le 1$ in $\Gamma ^{\times }_{K^{\prime }}$ , tuples $a \in (K^{\prime })^{n}$ , $\zeta \in \mathrm {RV}_{K^{\prime }}^{n^{\prime }}$ , $\xi \in \mathrm {RV}_{K^{\prime },\lambda }^{n^{\prime \prime }}$ with $n, n^{\prime }$ arbitrary and $n^{\prime \prime } \le \ell $ , and an $(a, \zeta , \xi )$ -definable set $X = \phi (K^{\prime }, a, \zeta , \xi ) \subset K^{\prime }$ such that no finite non-empty a-definable set $C \subset K \ \lambda $ -prepares X.

For fixed $\phi $ , the non-existence of C can be expressed by an infinite conjunction of ${\mathcal L}$ -formulas in $(\lambda , a, \zeta , \xi )$ . Indeed, for every formula $\psi (z, a)$ that could potentially define C and for every integer $k \ge 1$ , there is (by Lemma 2.4.2) an ${\mathcal L}$ -formula $\chi _{\psi }(\lambda , a, \zeta , \xi )$ expressing ‘ $\psi (K^{\prime }, a)$ has cardinality k and $\psi (K^{\prime }, a)$ does not $\lambda $ -prepare $\phi (K^{\prime }, a, \zeta , \xi )$ .’

The fact that this partial type $\{\chi _{\psi } \mid \psi \text { as above}\}$ is realized in $K^{\prime }$ implies that it is also realized in K, so that Condition (2.3.1) fails in K.

Whether a structure is o-minimal can also be characterized via its $1$ -types. For $0$ -h-minimality, we have a similar characterization:

Lemma 2.4.4 ( $0$ -h-minimality in terms of types)

Suppose that K is $\aleph _{0}$ -saturated. The theory $\operatorname {Th}(K)$ is $0$ -h-minimal if and only if, for every parameter set $A \subset K$ and every ball $B \subset K \setminus \mathrm {acl}_{K}(A)$ , any two elements of B have the same type over $A \cup \mathrm {RV}$ .

Remark 2.4.5. One could also formulate similar conditions for $\ell $ -h-minimality, but it would be more technical.

Proof of Lemma 2.4.4

$\Rightarrow $ ’: Suppose for contradiction that B contains two elements $x, x^{\prime }$ with $\mathrm {tp}(x/A \cup \mathrm {RV}) \ne \mathrm {tp}(x^{\prime }/A \cup \mathrm {RV})$ . This means that there exists an $(A \cup \mathrm {RV})$ -definable set X containing x but not $x^{\prime }$ . By $0$ -h-minimality, there exists a finite A-definable set $C \ 1$ -preparing X. In particular, $C \subset \mathrm {acl}_{K}(A)$ and hence $C \cap B = \emptyset $ . However, by Example 2.1.9, this implies that B is either contained in X or disjoint from X, contradicting the properties of x and $x^{\prime }$ .

$\Leftarrow $ ’: Let X be $(A \cup \mathrm {RV})$ -definable for some parameter set $A \subset K$ that we may assume to be finite, and suppose that no finite A-definable $C \subset K \ 1$ -prepares X. This means (using Example 2.1.9 again) that for every finite A-definable $C \subset K$ , there exists an open ball $B \subset K$ that is disjoint from C and such that neither $B \subset X$ nor $B \subset K \setminus X$ . Taking all those conditions on B together (for all finite A-definable C), we obtain a (finitely satisfiable) type in an imaginary variable running over the open balls in K. A realization of this type is a ball B that is disjoint from $\mathrm {acl}_{K}(A)$ on the one hand but, on the other hand, contains elements x and $x^{\prime }$ that satisfy $x \in X$ and $x^{\prime } \notin X$ and hence have different types over $A \cup \mathrm {RV}$ .

Yet another way to characterize o-minimality is: Every unary definable set is already quantifier-free definable in the language $\{<\}$ . Using Lemma 4.1.10, one obtains a similar kind of characterization of $0$ -h-minimality and $\omega $ -h-minimality.

2.5 Basic properties under weaker assumptions

Recall (from the beginning of Section 2.1) that K is a non-trivially valued field of equi-characteristic zero, considered a structure in a language ${\mathcal L} \supset {\mathcal L}_{\mathrm {val}} = \{+, \cdot ,{\mathcal O}_{K}\}$ . In this subsection, we assume $\operatorname {Th}(K)$ to be ‘Hensel minimal without control of parameters’, namely:

Assumption 2.5.1. For every $K^{\prime } \equiv K$ , every definable (with any parameters) subset of $K^{\prime }$ can be $1$ -prepared by a finite set C. (We do not impose definability conditions on C.)

Note that this assumption is preserved under adding constants to ${\mathcal L}$ (even from arbitrary imaginary sorts), so below, every occurrence of ‘ $\emptyset $ -definable’ can also be replaced by ‘A-definable’.

Lemma 2.5.2 ( $\exists ^{\infty }$ -elimination)

Under Assumption 2.5.1, every infinite definable set $X \subset K$ contains an (open) ball. In particular, if $\{X_{q}\mid q\in Q\}$ is a $\emptyset $ -definable family of subsets $X_{q}$ of K for some $\emptyset $ -definable set Q in an arbitrary possibly imaginary sort, then the set $Q^{\prime } := \{q\in Q \mid X_{q}\text { is finite}\}$ is a $\emptyset $ -definable set, and there exists a uniform bound $N \in {\mathbb N}$ on the cardinality of $X_{q}$ for all $q \in Q^{\prime }$ .

Proof. If X is infinite, it is not contained in the finite set C preparing it, which implies that it contains a ball. The definability of $Q^{\prime }$ then follows since the condition that a set contains an open ball can be expressed by a formula. The existence of the bound N then follows by compactness: If no bound would exist, then in a sufficiently saturated model, we would find a $q \in Q^{\prime }$ with $\#X_{q}> N$ for every N.

Lemma 2.5.3 (Finite sets are $\mathrm {RV}$ -parametrized)

Under Assumption 2.5.1, let $C_{q}\subset K$ be a $\emptyset $ -definable family of finite sets, where q runs over some $\emptyset $ -definable set Q in an arbitrary possibly imaginary sort. Then there exists a $\emptyset $ -definable family of injective maps $f_{q}\colon C_{q} \to \mathrm {RV}^{k}$ (for some k).

Proof of Lemma 2.5.3

Using Lemma 2.5.2, we can bound the cardinality $\#C_{q}$ and then assume that it is constant. We do an induction over $\#C_{q}$ .

If $C_{q}$ is always a singleton or empty, we can define $f_{q}$ to always be constant. Otherwise, the lemma is obtained by repeatedly taking averages of the elements of $C_{q}$ and subtracting. More precisely, setting $a_{q} := \frac 1{\#C_{q}}\sum _{x \in C_{q}} x$ , we get that the map $\hat f_{q}\colon C_{q}\to \mathrm {RV}, x \mapsto \operatorname {rv}(x - a_{q})$ is not constant on $C_{q}$ . Therefore, each fiber $\hat f^{-1}_{q}(\xi )$ of $\hat f$ (for $\xi \in \hat f_{q}(C_{q})$ ) has cardinality less than $C_{q}$ , so by induction, we obtain a definable family of injective maps $g_{q,\xi }\colon \hat f^{-1}_{q}(\xi ) \to \mathrm {RV}^{k}$ . Now set $f_{q}(x) := (\hat f_{q}(x), g_{q,\hat f_{q}(x)}(x))$ .

The family of balls $1$ -next to some finite set $C \subset K$ can be parameterized by $\mathrm {RV}$ -variables; more precisely (and more generally), we have the following:

Lemma 2.5.4 ( $\lambda $ -next balls as fibers)

Under Assumption 2.5.1, let $\lambda \le 1$ be an element of $\Gamma ^{\times }_{K}$ , let A be any set of possibly imaginary parameters containing $\lambda $ , and let $C\subset K$ be a finite non-empty A-definable set. Then there exists an A-definable map $f\colon K \to \mathrm {RV}^{k} \times \mathrm {RV}_{\lambda }$ (for some k) such that each nonempty fiber of f is either a singleton contained in C or contained in a single ball $\lambda $ -next to C. In the case $\lambda = 1$ , we may even obtain that each fiber of f that is not a singleton is equal to a ball $1$ -next to C.

Proof. Given $x \in K$ , let $\mu (x) := \min \{|x-c| \mid c \in C\}$ be the minimal distance to elements of C, let $C(x) := \{c \in C \mid |x-c| = \mu (x)\}$ be the set $c \in C$ realizing that distance, and let $a(x) := \frac 1{\#C(x)}\sum _{c \in C(x)} c$ be the average of those elements. Note that the map $a\colon K \to K$ has a finite image. Using Lemma 2.5.3, we find an injective map $\alpha $ from the image of a to $\mathrm {RV}^{k}$ . If $\lambda < 1$ , we define

$$ \begin{align*} f(x) := (\alpha(a(x)), \operatorname{rv}_{\lambda}(x - a(x))). \end{align*} $$

In the case $\lambda = 1$ , to obtain the more precise statement, we define

$$ \begin{align*} f(x) := \begin{cases} (\alpha(a(x)), \operatorname{rv}(x - a(x))) & \text{if } |x - a(x)| \ge \mu(x)\\ (\alpha(a(x)), \operatorname{rv}(0)) & \text{if } |x - a(x)| < \mu(x). \end{cases} \end{align*} $$

It is now just a computation to check that the lemma holds. For the first part, suppose that $f(x_{1}) = f(x_{2}) = (\zeta , \xi ) \in \mathrm {RV}^{k} \times \mathrm {RV}_{\lambda }$ for some $x_{1}, x_{2} \in K$ ; our aim is to show that either $x_{1} = x_{2} \in C$ or they both lie in the same ball $\lambda $ -next to C. Set $a_{0} := a(x_{1}) = a(x_{2})$ .

If $\xi = 0$ , then either $x_{i} = a_{0}$ or $|x_{i} - a_{0}| < \mu (x_{i})$ . The latter is only possible if $a_{0} \notin C$ , and it implies that $x_{i}$ and $a_{0}$ lie in the same ball $1$ -next to C. This shows: Either $x_{1} = x_{2} \in C$ or $\lambda = 1$ , and both lie in the same ball $1$ -next to C.

If $\xi \ne 0$ , then the fact that $|x_{i} - a(x)| \le \mu (x_{i})$ shows that $\operatorname {rv}_{\lambda }(x_{i} - a(x))$ determines $\operatorname {rv}_{\lambda }(x_{i} - c)$ for each $c \in C$ , so we are done with the first part of the lemma.

For the second part, pick $x_{1}, x_{2}$ in the same ball $\lambda $ -next to C. Then one obtains $\mu (x_{1}) = \mu (x_{2}) =: \mu _{0}$ , $C(x_{1}) = C(x_{2}) =: C_{0}$ and $a(x_{1}) = a(x_{2}) =: a_{0}$ . If $|x_{i} - a_{0}| = \mu $ , then $\operatorname {rv}(x_{i} - a_{0})$ is determined by $\operatorname {rv}(x_{i} - c)$ for any $c \in C_{0}$ , so $f(x_{1}) = f(x_{2})$ . Otherwise, if $|x_{i} - a_{0}| < \mu $ , then $f(x_{i}) = (\alpha (a_{0}), 0)$ for $i = 1,2$ by definition.

Remark 2.5.5. In Lemma 2.5.4, we can also find a map f with codomain $\mathrm {RV}_{\lambda }^{k+1}$ instead of $\mathrm {RV}^{k} \times \mathrm {RV}_{\lambda }$ ; indeed, in Lemma 2.5.3 and its proof, $\mathrm {RV}$ can be replaced by $\mathrm {RV}_{\lambda }$ everywhere.

2.6 Preparing families

In this subsection, we show that $\ell $ -h-minimality implies that we can prepare not only definable subsets of K (by finite sets C) but also various other kinds of definable objects that ‘live in $K \times \mathrm {RV}^{k} \times \mathrm {RV}_{\lambda }^{\ell }$ ’.

Definition 2.6.1 (Preparing families)

Let $C \subset K$ be a non-empty set, and let $\lambda \le 1$ be an element of $\Gamma _{K}^{\times }$ .

  1. 1. Suppose that W is, up to permutation of coordinates, a subset of $K \times Q$ , where Q is a Cartesian product of some (possibly imaginary) sorts. We say that $C \ \lambda $ -prepares W if for every ball $B \subset K \ \lambda $ -next to C, the fiber $W_{x} \subset Q$ does not depend on x when x runs over B. (The terminology will only be applied when exactly one of the coordinates of W runs over K; the fibers $W_{x}$ always are over that coordinate.)

    We synonymously also say that $C \ \lambda $ -prepares the family $(W_{\xi })_{\xi \in Q}$ of subsets of K. Indeed, the above condition is equivalent to $C \ \lambda $ -preparing $W_{\xi } \subset K$ for each $\xi \in Q$ .

  2. 2. If f is a definable function whose graph W lives in a Cartesian product as in (1), we say that $C \ \lambda $ -prepares f if it $\lambda $ -prepares W.

  3. 3. We say that C prepares a set

    $$ \begin{align*}W \subset K \times Q \times \bigcup_{\lambda \in \Gamma_{K}^{\times}, \lambda \le 1}\mathrm{RV}_{\lambda}^{k},\end{align*} $$

    uniformly in $\lambda $ , where Q is a product of sorts and $k \ge 0$ is an integer, if it $\lambda $ -prepares $W \cap (K \times Q \times \mathrm {RV}_{\lambda }^{k})$ for each $\lambda \le 1$ in $\Gamma ^{\times }_{K}$ . As in (1), we allow the coordinates of W to be in a different order, but making sure that this is unambiguious.

Proposition 2.6.2 (Preparing families)

Assume that $\operatorname {Th}(K)$ is $\ell $ -h-minimal and that A is a subset of K. For any integer $k\ge 0$ and any $(A \cup \mathrm {RV})$ -definable set

$$ \begin{align*}W\subset K\times \mathrm{RV}^{k}\times \bigcup_{\lambda \le 1}\mathrm{RV}_{\lambda}^{\ell}, \end{align*} $$

there exists a finite non-empty A-definable set C that prepares W uniformly in $\lambda $ .

Proof. For each $\lambda $ and each $\xi \in \mathrm {RV}^{k}\times \mathrm {RV}_{\lambda }^{\ell }$ , let $C_{\xi }$ be a finite A-definable set $\lambda $ -preparing $W_{\xi }$ . By a usual compactness argument (see Remark 2.6.3 below), we may suppose that $C := \bigcup _{\xi } C_{\xi }$ is finite and A-definable. It prepares each $W_{\xi }$ for each $\xi $ and hence also W.

Remark 2.6.3. In the above proof, we used a compactness argument that we will be using (in variants) many times in this paper. We give some details once. First, recall that by Lemma 2.4.2, ‘preparing’ is a definable condition. In particular, the set

$$ \begin{align*} \Xi_{\xi} :=\{\xi^{\prime} \in \mathrm{RV}^{k}\times \bigcup_{\lambda \le 1}\mathrm{RV}_{\lambda}^{\ell} \mid C_{\xi}\text{ }\lambda\text{-prepares }W_{\xi^{\prime}} \} \end{align*} $$

is A-definable. Since $\xi \in \Xi _{\xi }$ , the union of all $\Xi _{\xi }$ is equal to $\mathrm {RV}^{k}\times \bigcup _{\lambda \le 1}\mathrm {RV}_{\lambda }^{\ell }$ , and then compactness implies that finitely many sets $\Xi _{\xi _{i}}$ suffice to cover everything. Now $C := \bigcup _i C_{\xi _i}$ is a finite A-definable set that prepares every $W_\xi $ .

Remark 2.6.4. A variant of the compactness argument shows that Propsosition 2.6.2 holds even more uniformly, namely: Given an $\ell $ -h-minimal theory ${\mathcal T}$ (possibly non-complete) and a formula $\phi $ defining a set

$$ \begin{align*} W_{K,a} := \phi(K,a) \subset K\times \mathrm{RV}_{K}^{k}\times \bigcup_{\lambda \le 1}\mathrm{RV}_{\lambda}^{\ell} \end{align*} $$

for $K \models {\mathcal T}$ and $a \in K^{\nu } \times \mathrm {RV}_{K}^{\nu ^{\prime }}$ , there exists a formula $\psi $ defining a set $C_{K,a} := \psi (K,a) \subset K$ such that for each model $K \models {\mathcal T}$ and each $a \in K^{\nu } \times \mathrm {RV}_{K}^{\nu ^{\prime }}$ , this set $C_{K,a}$ uniformly prepares $W_{K,a}$ .

Remark 2.6.5. In Proposition 2.6.2, we can also replace $\mathrm {RV}^{k}$ by any Cartesian product Q of sorts from $\mathrm {RV}^{\mathrm {eq}}$ . Indeed, in that case, just apply the original version of the proposition to the preimage of W under some quotient map $\mathrm {RV}^{k} \to Q$ . In particular, W can additionally use (arbitrarily many) $\Gamma _{K}$ -coordinates, since $\Gamma _{K}$ is a quotient of $\mathrm {RV}$ . (One can of course also allow $\mathrm {RF}$ -coordinates, given that $\mathrm {RF}$ can be considered a $\emptyset $ -definable subset of $\mathrm {RV}$ .)

In almost all applications of Proposition 2.6.2, we will only need the following special case:

Corollary 2.6.6 (Preparing families)

Assume that $\operatorname {Th}(K)$ is $0$ -h-minimal and that A is a subset of K. For any $k>0$ and any $(A \cup \mathrm {RV})$ -definable set

$$ \begin{align*}W\subset K\times \mathrm{RV}^{k}, \end{align*} $$

there exists a finite non-empty A-definable set $C \subset K$ $1$ -preparing W (in the sense of Definition 2.6.1).

Proof of Corollary 2.6.6

Clear.

Recall that we set $\mathrm {RV}_{\bullet } := \bigcup _{\lambda \le 1} \mathrm {RV}_{\lambda }$ .

Corollary 2.6.7 ( $\mathrm {RV}$ -unions stay finite)

  1. 1. Assume that $\operatorname {Th}(K)$ is $1$ -h-minimal. For any $k \ge 0$ and any definable (with parameters) set $W\subset \mathrm {RV}_{\bullet }^{k} \times K$ such that the fiber $W_{\xi }\subset K$ is finite for each $\xi \in \mathrm {RV}_{\bullet }^{k}$ , the union $\bigcup _{\xi } W_{\xi }$ is also finite.

  2. 2. Under the (weaker) assumption that $\operatorname {Th}(K)$ is $0$ -h-minimal, the previous statement still holds if we assume $W\subset \mathrm {RV}^{k} \times K$ .

Proof. We suppose that $W\subset \mathrm {RV}^{k} \times \mathrm {RV}_{\bullet }^{k^{\prime }} \times K$ and proceed by induction on $k^{\prime }$ . If $k^{\prime } \le 1$ , we obtain both claims of the corollary from Proposition 2.6.2 applied with $\ell = k^{\prime }$ , namely: We find a finite set C such that, for all $\lambda $ and $\xi \in \mathrm {RV}^k\times \mathrm {RV}_\lambda ^{k'}$ , $W_{\xi }$ is $\lambda $ -prepared by C. Since $W_{\xi }$ is finite, we have $W_\xi \subset C$ , and hence $\bigcup _\xi W_\xi \subset C$ is finite.

The case of $k^{\prime }>1$ now follows by induction on $k^{\prime }$ and the case $k^{\prime }=1$ .

Corollary 2.6.8 (Finite image in K )

  1. 1. Assume that $\operatorname {Th}(K)$ is $1$ -h-minimal. Then the image of any definable (with parameters) function $f\colon \mathrm {RV}_{\bullet }^{k} \to K$ for any $k \ge 0$ is finite.

  2. 2. Under the (weaker) assumption that $\operatorname {Th}(K)$ is $0$ -h-minimal, the previous statement still holds for functions $f\colon \mathrm {RV}^{k} \to K$ .

Proof. Apply Corollary 2.6.7 to the graph of f.

Remark 2.6.9. Remark 2.6.5 also applies to Corollaries 2.6.6, 2.6.7 and 2.6.8. In particular, we can additionally allow (arbitrarily many) $\Gamma _{K}$ -coordinates in W (in 2.6.6, 2.6.7) and the domain of f (in 2.6.8).

Corollary 2.6.10 (Removing $\mathrm {RV}$ -parameters)

Assume that $\operatorname {Th}(K)$ is $0$ -h-minimal. For any $A \subset K$ and any finite $(A \cup \mathrm {RV}^{\mathrm {eq}})$ -definable set $C \subset K$ , there exists a finite A-definable set $C^{\prime } \subset K$ containing C. In other words, $\mathrm {acl}_{K}(A \cup \mathrm {RV}^{\mathrm {eq}}) = \mathrm {acl}_{K}(A)$ .

Proof. Add constants for A to the language. We have $C = W_ {\xi _{0}}$ for some $\emptyset $ -definable $W \subset K \times \mathrm {RV}^{k}$ and some $\xi _{0} \in \mathrm {RV}^{k}$ . We may assume that all fibers $W_{\xi }$ have cardinality at most the cardinality of C. Let $C^{\prime }$ be their union, which is finite by Corollary 2.6.7.

Remark 2.6.11. Many results in this paper are stated in the form:

  • (⋆) For every $\emptyset $ -definable object X of a certain kind, there exists a finite $\emptyset $ -definable set $C \subset K$ that ‘prepares’ X in some sense (depending on the context).

By Lemma 2.4.1, we get for free that $(\star )$ holds more generally, namely if X is $(A \cup \mathrm {RV})$ -definable, for $A \subset K$ , we get an $(A \cup \mathrm {RV})$ -definable C ‘preparing’ X. By applying Corollary 2.6.10, we then may even assume that C is A-definable. (It will always be the case that if C prepares X, then so does any set containing C.) Finally, using that the notions of preparation under consideration will be definable, we can apply compactness to get for free that this works uniformly in all models of a non-complete theory, in the same style as in Remark 2.6.4.

We end this subsection by noting that $\mathrm {RV}$ is stably embedded in a strong sense (namely, with the $\mathrm {RV}$ -parameters being in the definable closure of the original parameters):

Proposition 2.6.12 (Stable embeddedness of $\mathrm {RV}$ )

Assume that $\operatorname {Th}(K)$ is $0$ -h-minimal. Then $\mathrm {RV}$ is stably embedded in the following strong sense: Given any $A \subset K$ , every A-definable set $X \subset \mathrm {RV}^{n}$ is already $\mathrm {dcl}_{\mathrm {RV}}(A)$ -definable.

Proof. We may assume that A is finite; we do an induction on the cardinality of A.

Let $A = \hat A \cup \{a\}$ . Then we have an $\hat A$ -definable set $Y \subset K \times \mathrm {RV}^{n}$ such that X is equal to the fiber $Y_{a} \subset \mathrm {RV}^{n}$ . By applying Corollary 2.6.6 to Y, we find a finite $\hat A$ -definable set $C \subset K$ such that either $a \in C$ or, for every $a^{\prime } \in K$ in the same ball $1$ -next to C as a, we have $Y_{a^{\prime }} = Y_{a}$ . Using Lemma 2.5.4, we find an $\hat A$ -definable map $f\colon K \to \mathrm {RV}^{k}$ (for some k) whose fibers are exactly the elements of C and the balls $1$ -next to C. In particular, the set $X = Y_{a}$ is definable using $\hat A$ and $f(a)$ as parameters. Thus we have $X = Z_{f(a)}$ for some $\hat A$ -definable set $Z \subset \mathrm {RV}^{k} \times \mathrm {RV}^{n}$ . By induction, Z is $\mathrm {dcl}_{\mathrm {RV}}(\hat A)$ -definable, so X is $\mathrm {dcl}_{\mathrm {RV}}(\hat A)\cup \{f(a)\}$ -definable and hence $\mathrm {dcl}_{\mathrm {RV}}(A)$ -definable.

Remark 2.6.13. If $\operatorname {Th}(K)$ is $\omega $ -h-minimal, there are also various variants of Proposition 2.6.12 involving $\mathrm {RV}_{\lambda }$ , with similar proofs. For example, building on Remark 2.5.5 instead of Lemma 2.5.4, one obtains that any A-definable subset of $\prod _{i} \mathrm {RV}_{\lambda _{i}}$ (for $A \subset K$ ) is $\mathrm {dcl}_{\mathrm {RV}_{\lambda }}(A)$ -definable with $\lambda = \min _{i} \lambda _{i}$ .

2.7 Henselianity of the valued field K

As an analogue of o-minimal fields being real closed, in this subsection, we prove that any equi-characteristic zero-valued field that is Hensel minimal (in any language containing ${\mathcal L}_{\mathrm {val}}$ ) is Henselian. This is one reason we call our notion ‘Hensel minimality’.

A collection of balls is called nested if, for any two balls in the collection, one is contained in the other.

Lemma 2.7.1 (Definable spherical completeness)

Assuming $0$ -h-minimality, let $\{B_{q} \mid q\in Q\}$ be a definable family of nested balls in K, for some non-empty definable set Q in an arbitrary, possibly imaginary sort. Then the intersection $\bigcap _{q \in Q} B_{q}$ is non-empty.

Proof. First we suppose that $Q \subset \Gamma ^{\times }_{K}$ and that each $B_{q}$ is an open ball of radius q. By Corollary 2.6.6 (and Remark 2.6.9), there exists a finite set $C \ 1$ -preparing the family of balls $B_{q}$ . Now one checks that at least one of the following two situations occurs. First: For each $q\in Q$ , the intersection of C with $B_{q}$ is non-empty. Second: The set Q has a minimum $q_{0}\in Q$ . (Indeed, suppose that the intersection of C with $B_{q_{0}}$ is empty for some $q_{0}\in Q$ ; then Q contains no $q < q_{0}$ , since $B_{q}$ would not be $1$ -prepared by C.) In both situations, the lemma follows.

Finally, we reduce the general case to the case that $Q\subset \Gamma ^{\times }_{K}$ and that each $B_{q}$ is an open ball of radius q. To this end, for $\gamma \in \Gamma ^{\times }_{K}$ , let $B^{\prime }_{\gamma }$ be the (necessarily unique) open ball of radius $\gamma $ containing some $B_{q}$ ( $q \in Q$ ) if such a ball exists, and let it be the empty set otherwise. Then it is clear that the non-empty $B^{\prime }_{\gamma }$ form a nested definable family of open balls. Moreover, the intersection of the non-empty $B^{\prime }_{\gamma }$ equals the intersection of the $B_{q}$ (since each $B_{q}$ is equal to the intersection of all open balls containing $B_{q}$ ).

Theorem 2.7.2 (Hensel minimality implies Henselian)

Suppose that K is a valued field of equi-characteristic $0$ with $0$ -h-minimal theory (in a language ${\mathcal L} \supset {\mathcal L}_{\mathrm {val}}$ ). Then K is Henselian.

If ${\mathcal L}$ is the pure valued field language, Corollary 6.2.6 implies the converse. Combining, we have, for K of equi-characteristic $0$ : K is Henselian if and only if $\operatorname {Th}_{{\mathcal L}_{\mathrm {val}}}(K)$ is $0$ -h-minimal, if and only if $\operatorname {Th}_{{\mathcal L}_{\mathrm {val}}}(K)$ is $\omega $ -h-minimal.

Proof of Theorem 2.7.2

Let $P\in {\mathcal O}_{K}[X]$ be a polynomial such that $P(0)\in {\mathcal M}_{K}$ and $P^{\prime }(0)\in {\mathcal O}^{\times }_{K}$ ; we need to prove that P has a root in ${\mathcal M}_{K}$ . (Note that the uniqueness of such a root then follows automatically.) The idea is to use ‘Newton approximation’ as in the usual proof of Hensel’s lemma for complete discretely valued fields, but where complete and discretely valued is replaced by definably spherically complete.

To make this formal, we suppose that P has no root in ${\mathcal M}_{K}$ and we set $B_{x} := B_{\le |P(x)|}(x)$ for $x \in {\mathcal M}_{K}$ . Note that the $B_{x}$ form a definable family of balls, parameterized by $x \in {\mathcal M}_{K}$ . We will prove that (a) all these balls form a chain under inclusion and (b) that an element in the intersection of all those balls (which is non-empty by Lemma 2.7.1) is, after all, a root of P.

(a) Let $x_{1}, x_{2} \in {\mathcal M}_{K}$ be given, and set $\varepsilon := x_{2} - x_{1}$ . To see that the balls $B_{x_{1}}$ and $B_{x_{2}}$ are not disjoint, we verify that $|\epsilon | \le \max \{|P(x_{1})|, |P(x_{2})|\}$ . Taylor expanding P around $x_{1}$ yields

(2.7.1) $$ \begin{align} |P(x_{1}+\varepsilon) - P(x_{1}) - \varepsilon P^{\prime}(x_{1})| \le |\varepsilon^{2}|, \end{align} $$

which, together with $|P^{\prime }(x_{1})| = 1$ , implies $|\epsilon | \le \max \{|P(x_{1})|, |P(x_{1} + \epsilon )|\}$ .

(b) Let $x_{1}$ be in the intersection $\bigcap _{x \in {\mathcal M}_{K}} B_{x}$ , and suppose that $P(x_{1}) \ne 0$ . Then equation (2.7.1) with $\varepsilon := - \frac {P(x_{1})}{P^{\prime }(x_{1})}$ implies $|P(x_{1} + \varepsilon )| \le |\varepsilon ^{2}| < |\varepsilon |$ and hence $x_{1} \notin B_{x_{1} + \varepsilon }$ , contradicting our choice of $x_{1}$ .

Remark 2.7.3. Lemma 2.7.1 implies a ‘definable Banach Fixed Point Theorem’ (exactly in the form of [Reference Halupczok43, Lemma 2.32], and with the same proof). The above proof of Theorem 2.7.2 can be considered applying that Fixed Point Theorem to the map ${\mathcal M}_{K} \to {\mathcal M}_{K}, x \mapsto x - P(x)/P^{\prime }(x)$ .

2.8 Definable functions

We continue assuming that K is an equi-characteristic $0$ valued field, and we now assume that $\operatorname {Th}(K)$ is $1$ -h-minimal (unless specified otherwise). Under those assumptions, we now prove the first basic properties of definable functions in one variable; in particular, we already obtain a weak version of the Jacobian Property (Lemma 2.8.5) and simultaneous domain and image preparation (Proposition 2.8.6).

The first result is key to dimension theory (although in our proofs of dimension theory, this will only be used indirectly, namely in the proof of Proposition 5.3.3).

Lemma 2.8.1 (Basic preservation of dimension)

Assume (as convened for the whole Section 2.8) that $\operatorname {Th}(K)$ is $1$ -h-minimal. Let $f\colon K \to K$ be a definable function. Then there are only finitely many function values that are taken infinitely many times.

Proof. We may assume f to be $\emptyset $ -definable (say, after adding enough parameters from K to the language). Suppose f takes infinitely many values y infinitely many times. Then for each such y, $f^{-1}(y)$ contains a ball (by Lemma 2.5.2). Thus, letting $X \subset K$ be the set of points where f is locally constant, $f(X)$ is still infinite.

Let $W\subset K\times \Gamma ^{\times }_{K}$ consist of those $(x,\lambda )$ such that f is constant on $B_{<\lambda }(x)$ . This set W is $\emptyset $ -definable, so we find a finite set $C \ 1$ -preparing W (by Corollary 2.6.6). By enlarging C, we may moreover assume that C also $1$ -prepares X.

From the fact that $f(X)$ is infinite, we can deduce that there exists a ball $B_{0} \subset X \ 1$ -next to C such that $f(B_{0})$ is still infinite. Indeed, letting $g\colon K \to \mathrm {RV}^{k}$ be a map whose fibers are the singletons in C and the balls $1$ -next to C (using Lemma 2.5.4), if $f(g^{-1}(\xi ))$ would be finite for every $\xi \in g(X)$ , then so would be $f(X)$ (by Corollary 2.6.7).

Choose $x \in B_{0}$ and $\lambda _{0} \in \Gamma ^{\times }_{K}$ such that f is constant on $B_{<\lambda _{0}}(x)$ . Since $C \ 1$ -prepares W, f is constant on $B_{<\lambda _{0}}(x^{\prime })$ for every $x^{\prime } \in B_{0}$ .

Set $\lambda _{1} := \lambda _{0}/\operatorname {rad}_{\mathrm {op}}(B_{0})$ . Then the family $F_{1}$ of open balls of radius $\lambda _{0}$ contained in $B_{0}$ can be definably parametrized by a subset of $\mathrm {RV}_{\lambda _{1}}$ (using some parameters). Indeed, if we fix $c \in K$ such that $B_{0}$ is $1$ -next to c, then each member of $F_{1}$ is of the form $c + \operatorname {rv}_{\lambda _{1}}^{-1}(\xi )$ for some $\xi \in \mathrm {RV}_{\lambda _{1}}$ . We define $F_{2}$ to be the family of $f(B)$ , for B in $F_{1}$ . Then each family member of $F_{2}$ is a singleton, yet their union is infinite, contradicting Corollary 2.6.7.

Using this, we obtain that definable functions are (in a strong sense) locally constant or injective:

Lemma 2.8.2 (Piecewise constant or injective)

For every