GENERIC DERIVATIONS ON ALGEBRAICALLY BOUNDED STRUCTURES II: MODEL THEORETICAL PROPERTIES

ANTONGIULIO FORNASIERO; GIUSEPPINA TERZO

doi:10.1017/jsl.2026.10229

GENERIC DERIVATIONS ON ALGEBRAICALLY BOUNDED STRUCTURES II: MODEL THEORETICAL PROPERTIES

Part of: Model theory Connections with logic Differential and difference algebra

Published online by Cambridge University Press: 24 June 2026

ANTONGIULIO FORNASIERO

and

GIUSEPPINA TERZO

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ANTONGIULIO FORNASIERO: Affiliation:
DIPARTIMENTO DI MATEMATICA E INFORMATICA “U. DINI” UNIVERSITÀ DEGLI STUDI DI FIRENZE ITALY E-mail: antongiulio.fornasiero@gmail.com
GIUSEPPINA TERZO*: Affiliation:
DIPARTIMENTO DI MATEMATICA E APPLICAZIONI “R. CACCIOPPOLI” UNIVERSITÀ DEGLI STUDI DI NAPOLI “FEDERICO II” ITALY
*: E-mail: giuseppina.terzo@unina.it

Article contents

Abstract
Introduction
Preliminaries and conventions
Elimination of imaginaries
Independence relations
The extension theorem
Algebraic closure and independence relations
Simplicity
Distality
ω$\omega $-stability
The field of constants
Open core
Differential dimension
Genericity
Conjectures and open problems
Funding
Footnotes
References

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Abstract

Let T be an algebraically bounded theory. We consider the $L(\bar \delta )$-expansions of T by a tuple $\bar {\delta }$ of derivations (which may be commuting or not). We investigate the model completion of either of the above theories, whose existence has been established in [21], with particular attention to its model-theoretic properties, including $\omega $-stability, simplicity, open core, and elimination of imaginaries.

Keywords

derivation algebraically bounded model completion imaginaries simplicity open core

MSC classification

Primary: 03C60: Model-theoretic algebra 12H05: Differential algebra 12L12: Model theory

Secondary: 03C10: Quantifier elimination, model completeness and related topics

Information

Type: Article
Information: The Journal of Symbolic Logic , First View , pp. 1 - 32

DOI: https://doi.org/10.1017/jsl.2026.10229 [Opens in a new window]
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

1 Introduction

We study the model-theoretic properties of the model completion of the theory of algebraically bounded fields expanded with derivations, continuing the foundational work in [Reference Fornasiero and Terzo21]. Algebraically bounded fields, encompassing structures like algebraically closed, real closed, and p-adically closed fields, provide a rich framework for exploring the interplay between field-theoretic and model-theoretic algebraic notions. Throughout this article, we denote by $\mathbb K$ a structure that expands a field of characteristic 0. Recall that $\mathbb K$ is algebraically bounded if the model-theoretic algebraic closure and the field-theoretic algebraic closure coincide in every structure elementarily equivalent to $\mathbb K$ ; for more details, examples, and main properties (see [Reference van den Dries13]).

Let $\mathbb K$ be algebraically bounded, L be the language of $\mathbb K$ , and T be its theory. In order to study derivations on $\mathbb K$ , we denote by ${\bar \delta } = \{\delta _1, \ldots , \delta _k\}$ new unary functions symbols.

Assume that T is model complete. In [Reference Fornasiero and Terzo21] we proved that, for every $k \in \mathbb {N}$ , the ${L^{\bar {\delta }}}$ -expansions of T, ${T^{\bar {\delta },nc}}$ (saying that ${\bar \delta }$ is a k-tuple of derivations) and ${T^{\bar {\delta }}}$ (saying that ${\bar \delta }$ is a k-tuple of commuting derivations) have a model completion. We denote by $T^{\bar {\delta },?}_g$ the model completion of either of the above theories. In the preceding work [Reference Fornasiero and Terzo21], we began investigating the model-theoretic properties of $T^{\bar {\delta },?}_g$ : we showed that if T is stable (resp., dependent), then $T^{\bar {\delta },?}_g$ is also stable (resp., dependent).

The above results include and extend a long series of results in the literature (see [Reference Fornasiero and Terzo21] for a short survey: here we mention [Reference Tressl49] that shows that large model complete fields admit generic derivations). There are also examples of theories which do not admit generic derivations (see [Reference Fornasiero and Terzo22]).

In this article, we extend the investigation of $T^{\bar {\delta },?}_g$ , with a specific focus on model-theoretic properties such as stability, $\omega $ -stability, simplicity, and elimination of imaginaries (EI); we also address other questions raised in [Reference Fornasiero and Terzo21]. Multiple authors have examined these properties in several particular cases; here we consider general setting of algebraically bounded fields with several generic derivations. For instance, in [Reference Moosa and Scanlon38], the authors demonstrated that $ {\mathrm {DCF}}_{0, m, nc}$ , the model completion of the theory of fields with non-commuting derivations, is stable but not $\omega $ -stable, eliminates imaginaries, and is uniformly finite. Building upon these results, [Reference León Sánchez and Mohamed33, Reference Mohamed37] further generalized some of the properties established in [Reference Moosa and Scanlon38].

In this article, we pay particular attention to the following questions:

1) When is $T^{\bar {\delta },?}_g$ simple or totally transcendental?
2) Which are the imaginaries of $T^{\bar {\delta },?}_g$ ?
3) If T is equipped with a definable topology, is T the open core of $T^{\bar {\delta },?}_g$ ?
4) Is $T^{\bar {\delta },?}_g$ uniformly finite?
5) What is a “generic” object?
6) Does $T^{\bar {\delta },?}_g$ have a dimension function in the sense of [Reference van den Dries13]?
7) What is the theory of the restriction of $T^{\bar {\delta },?}_g$ where we forget the derivation but we keep the field of constants?

We will give a full answer to 1) (Sections 7 and 9 and [Reference Fornasiero, Kaplan and Matthews20]). In Section 7 we prove that if T is simple, then $T^{\bar {\delta },?}_g$ is simple. [Reference Moosa and Scanlon38] showed that the model completion of the theory of fields with non-commuting operators is simple, eliminates imaginaries, is uniformly finite, and under some additional assumptions it is stable.

In [Reference Fornasiero, Kaplan and Matthews20] the first author together with E. Kaplan and A. Matthews further extend the results exposed here, proving the following.

Theorem A.

• T is rosy iff it is super-rosy of thorn rank 1; in this case, T has Geometric EI;
• T is simple iff it is supersimple or SU-rank 1;
• T is stable iff it is equal to the theory of pure algebraically closed fields, expanded by some constants (we will simply write “ $T = {\mathrm {ACF}_0}$ ”).

Combining the above with the results in this article, we get that $T^{\bar {\delta },?}_g$ is stable iff T is stable iff $T={\mathrm {ACF}_0}$ . In the case the derivations commute, one can improve stability with $\omega $ -stability. Furthermore, in the commuting case when T is stable the theory $T^{\bar {\delta },?}_g$ coincides with the theory of differentially closed fields with several commuting derivations $ {\mathrm {DCF}}_{0,m}$ ; while in the noncommuting case it yields Moosa–Scanlon’s theory $ {\mathrm {DCF}}_{0,m,nc}$ .

We have some partial answers to 2). We show that if T has (Geometric) EI and it is either stable or supersimple, then $T^{\bar {\delta },?}_g$ also has (Geometric) EI: Theorems 7.5 and 7.9 and Corollary 7.10: together with Theorem A, we get that if T is simple then $T^{\bar {\delta },?}_g$ has GEI. We also show that if T has a definable topology satisfying some general conditions, then $T^{\bar {\delta },?}_g$ eliminates imaginaries relative to T: see Corollary 11.11; we generalize results in [Reference Kovacsics and Point32]. We conjecture that a stronger result holds than what we manage to prove (see Conjectures 3.7 and 6.7).

We give a complete answer to 3) (Section 11 in particular Corollary 11.11).

For 4) we will give a positive answer in a separate paper.

For 5) we give some possible answers (Section 13), where we study the Polish space of all derivations on a countable model $M \models T$ (under some “bigness” condition on M) and show that generic derivations form a dense $\mathcal G_{\delta }$ -set among all possible derivations.

For 6) we give a complete answer in Section 12: $T^{\bar {\delta },?}_g$ has a dimension function iff the derivations commute. We also generalize the result in [Reference Eleftheriou, Sánchez and Regnault14] on “coincidence of dimensions” (see also [Reference Brihaye, Michaux and Rivière6, Reference Guzy and Point24] for previous results in particular cases).

For 7), in Section 10 we study the field of constant $\mathcal C_{\bar {\delta }}$ of a model $\langle \mathbb K, \bar {\delta } \rangle \models T^{\bar {\delta },?}_g$ . We show that $\langle \mathbb K, \mathcal C_{\bar {\delta }} \rangle $ is a lovely pair of geometric structures (in the sense of [Reference Berenstein and Vassiliev4]), and we study the definable subsets of $\mathcal C_{\bar {\delta }}^{n}$ .

In Sections 5 and 6 we study independence relations on model of T and of $T^{\bar {\delta },?}_g$ . We introduce the independence relation and prove the fundamental Extension Theorem 5.3 and Independence Theorem 6.1: they will make much easier to prove the results in Sections 7 and 11. We also characterize the algebraic closure inside models of $T^{\bar {\delta },?}_g$ .

Let us mention here the recent work [Reference Pillay, Point and Rideau-Kikuchi42] where the authors study groups definable in models of $T^{\bar {\delta }}_g$ : they show that any such group can be embedded in a T-definable group.

We conclude the article with several open questions, conjectures, and announcements of further work.

2 Preliminaries and conventions

L is a language and T is a L-theory (later we will impose additional conditions on them).

When we say that $\mathfrak C$ is a “monster model”, we mean that $\mathfrak C$ is a $\lambda $ -saturated and $\lambda $ -homogeneous model of $T $ for a “sufficently big” cardinal $\lambda> \aleph _{0} + \lvert L\rvert $ (other authors mean that $\mathfrak C$ is a proper class, and it is $\lambda $ -saturated and $\lambda $ -homogeneous for every cardinal $\lambda $ ).

By “small” subset/tuple of $\mathfrak C$ we mean of cardinality less than $\lambda $ .

$\mathbb K$ will be an L-structure, and K will be either its domain, or, when $\mathbb K$ expands a field, its underlying field.

We say that $\mathbb K$ is algebraically bounded if it expands a field of characteristic 0 and, for every subset $A \subseteq K$ (not necessarily a substructure), the model theoretic closure of A (computed in a sufficiently saturated elementary extension of $\mathbb K$ ) coincides with the field theoretic algebraic closure of the field generated by A and $\mathrm {dcl}(\emptyset )$ .

2.1 Generic derivations

This work explores the model theory of differential fields, specifically focusing on expansions of a base theory T by derivations.

Our assumptions for the whole article are the following:

• $\mathbb K$ is a structure expanding a field of characteristic $0$ .
• L is the language of $\mathbb K$ and $T = Th(\mathbb K)$ is its model complete L-theory.
• $F := {\mathrm {dcl}}(\emptyset ) \subseteq \mathbb K$ .
• $\mathbb K$ is algebraically bounded over F.

By algebraic closure we always mean the T-algebraic closure; in particular, $ {\mathrm {acl}}$ denotes the T-algebraic closure, and algebraic independence is understood with respect to T (equivalently, over F in the field-theoretic sense).

Under our assumptions, $\mathbb K$ is geometric: in the monster model $\mathfrak C \succ \mathbb K$ , the operator $ {\mathrm {acl}}$ satisfies the exchange property, and thus induces a matroid structure.

Moreover, $\mathbb K$ is equipped with a dimension function $\dim $ , assigning to each definable set (with parameters) a natural number and satisfying the axioms of [Reference van den Dries13].

We also consider the rank function $ {\mathrm {rk}}$ associated with the matroid $ {\mathrm {acl}}$ : for sets V and B, $ {\mathrm {rk}}(V/B)$ is the cardinality of a basis of V over B. In particular, if $X \subseteq {\mathfrak{C}}^n$ is definable with parameters $\bar b$ , then

$$\begin{align*}\dim(X) = \max \{ {\mathrm{rk}}(\bar a / \bar b) : \bar a \in X \}. \end{align*}$$

Let $\bar {\delta } := \langle \delta _{1}, \dotsc , \delta _{k} \rangle $ . Let $\eta _{1}, \dotsc , \eta _{k}$ be derivations on F. We obtain two theories:

$T^{\bar \delta}:$ the expansion of T saying that the $\delta _{i}$ are derivations which commute with each other, i.e., for every i, $j \leq k$ , $\delta _{i} \circ \delta _{j} = \delta _{j} \circ \delta _{i}$ , and that $\delta _{i}$ extends $\eta _{i}$ for $i \leq k$ ;
$T^{\bar \delta, nc}:$ the expansion of T saying that the $\delta _{i}$ are derivations without any further conditions and that $\delta _{i}$ extends $\eta _{i}$ for $i \leq k$ .

Both theories have a model completion (remember that T is model complete) (see [Reference Fornasiero and Terzo21]). For convenience, we use $T^{\bar {\delta },?}_g$ to denote either of the model completions, both for commuting and non-commuting tuples of derivations.

$\langle \mathfrak C, {\bar \delta } \rangle $ is a monster model of $T^{\bar {\delta },?}_g$ .

$\langle \mathbb K, {\bar \delta } \rangle $ is some model of $T^{\bar {\delta },?}_g.$

2.2 Notations and know results

We introduce some notation that we use in the sequel. We denote by $\Gamma $ the free commutative (or non-commutative) monoid generated by $\bar {\delta },$ we can define the canonical partial order $\preceq $ given by $\beta \preceq \alpha \beta $ , for all $\alpha , \beta \in \Gamma $ .

Remark 2.1. If $\Gamma $ is the free non-commutative monoid generated by $\bar {\delta }$ then $\preceq $ is a well-founded partial order on $\Gamma $ , but it is not a well-partial-order (i.e., there exist infinite anti-chains):

1) $\emptyset $ (i.e., the empty word, corresponding to the identity function on $\mathbb K$ ) is the minimum of $\Gamma $ ;
2) if $\alpha \preceq \beta $ , then $\gamma \alpha \leq \gamma \beta $ and $\alpha \gamma \leq \beta \gamma $ .

Remark 2.2. If $\Gamma $ is the free commutative monoid generated by $\bar {\delta }$ , with the canonical partial order $\preceq $ , it is isomorphic to $\mathbb {N}^k$ .

Fix $n \in \mathbb {N}$ , $\bar x = \langle x_{1}, \dotsc , x_{n} \rangle $ , and denote by $\Gamma _{n} := \{\gamma x_{i}: \gamma \in \Gamma , i \leq n\}$ : we consider $\Gamma _{n}$ as a set of some ${L^{\bar {\delta }}}$ -terms, as a set of functions from $K^{n}$ to K, and as a set of indices. For example, $\delta _1\delta _2x_3$ is in $\Gamma _{3}$ but $\delta _1\delta _2x_3 + \delta _1x_1$ is not. Given $\bar a \in \mathbb K^n$ , we denote by $ {\mathrm {Jet}}(\bar a) := \langle \gamma \bar a : \gamma \in \Gamma _n \rangle $ .

Moreover, we denote by $\bar x_{\Gamma } $ a set of formal variables indexed by $\Gamma _{n}$ : that is, for every $f \in \Gamma _{n}$ we have a variable $\bar x_{f}$ . Given an $L(A)$ -formula $\alpha (\bar x_{\Gamma })$ we denote by $\alpha (\Gamma )$ the ${L^{\bar {\delta }}}(A)$ -formula where we replaces each occurrence of a variable $\bar x_{f}$ with the corresponding term $f(\bar x) \in \Gamma _{n}$ .

Let $J \subseteq \Gamma _{n}$ and $A \subset K$ be a “small” subset of K with $\bar {\delta } A \subseteq A$ . Let $p(\bar x)$ be a complete ${L^{\bar {\delta }}}$ -type over A (in n variables). Define

$$\begin{align*}p_{J} := \{ \alpha: \alpha(\bar x_\Gamma)\ L(A)\text{-formula s.t. } \alpha(\Gamma) \in p\} \in S_{L}^{J}(A). \end{align*}$$

We say that some type $q \in S^{J}_{L}(A)$ is $\bar {\delta }$ -compatible if there exists $s \in S^{n}_{{L^{\bar {\delta }}}}(A)$ such that $q = s_{J}$ .

Many of the model-theoretic properties of T are inherited by $T^{\bar {\delta },?}_g$ . The following results from [Reference Fornasiero and Terzo21], which plays a crucial role in our subsequent analysis, is particularly useful.

Theorem 2.3. For every $\bar a$ tuple in $\mathfrak C$ and B subset of $\mathfrak C$ , the ${L^{\bar {\delta }}}$ -type of $\bar a$ over B is uniquely determined by the L-tuple of $ {\mathrm {Jet}}(\bar a)$ over $ {\mathrm {Jet}}(B)$ .

Theorem 2.4.

1) If T eliminates quantifiers, then $T^{\bar {\delta },?}_g$ eliminates quantifiers.
2) For every $L^{\bar {\delta }}$ -formula $\alpha (\bar x)$ there exists an L-formula $\beta (\bar x_\Gamma )$ such that
$$\begin{align*}T^{\bar{\delta},?}_g \models \forall \bar x\ \big( \alpha(\bar x) \leftrightarrow \beta(\Gamma) \big). \end{align*}$$

3 Elimination of imaginaries

EI is a powerful concept in model theory that simplifies arguments by allowing us to work directly with elements of the model, rather than with equivalence classes of definable sets.

We remind what EI and some of its variants are as follows.

Definition 3.1. T eliminates imaginaries (T has EI) if every imaginary is interdefinable with a real tuple: i.e., for every imaginary e there is a real tuple a such that ${\mathrm {dcl}^{\mathrm {eq}}}(e) = {\mathrm {dcl}^{\mathrm {eq}}}(a).$

Definition 3.2. T weakly eliminates imaginaries (T has WEI), if for every imaginary e there is a real tuple a such that e is definable over a and a is algebraic over e: that is, $e \in {\mathrm {dcl}^{\mathrm {eq}}}(a)$ and $a \in {\mathrm {acl}^{\mathrm {eq}}}(e).$

Remark 3.3. Equivalently, a theory T admits weak EI (WEI) in the sense of B. Poizat (see [Reference Casanovas and Farré10, Reference Yoneda53]) if, for any $\phi (\overline x, \overline a) \in L(\overline a)$ , we have the smallest algebraically closed set B such that $\phi (\overline x, \overline a) $ is definable over B.

Definition 3.4. T has geometric eliminates imaginaries (T has GEI), if for every imaginary e there is a real tuple a such that e is algebraic over a and a is algebraic over e: that is, $e \in {\mathrm {acl}^{\mathrm {eq}}}(a)$ and $a \in {\mathrm {acl}^{\mathrm {eq}}}(e)$ .

Remark 3.5. By definition, it is clear that EI implies WEI, and that WEI implies GEI. The converse implication, namely, that WEI implies EI, holds when working over a field as is the case here (see [Reference Marker, Messmer and Pillay35]).

Remark 3.6. In any structure containing at least two constants, as in our case, the definition of EI is equivalent to the Uniform EI (see [Reference Casanovas and Farré10] for definition and proof).

Regarding generic derivations, we formulated the following.

Conjecture 3.7 [Reference Fornasiero and Terzo21].

Let T be algebraically bounded. Then, $T^{\bar {\delta },?}_g$ has EI modulo $T^{eq}$ .

In particular: if T has GEI, then $T^{\bar {\delta },?}_g$ also has GEI, and if T has EI, then $T^{\bar {\delta },?}_g$ also has EI.

A few particular cases were already known, when T is one of the following:

• ${\mathrm {ACF}_0}$ : see [Reference McGrail36, Reference Moosa and Scanlon38];
• $ {\mathrm {RCF}}$ : see [Reference Fornasiero and Kaplan19, Reference Kovacsics and Point32] for a proof based on M. Tressl’s idea, see also [Reference Brouette, Kovacsics and Point7, Reference Point43] for different proofs.

We will prove two cases of the above conjecture:

• When T is simple: by Theorem A T has GEI; we then employ a result in [Reference Yoneda52] to show that $T^{\bar {\delta },?}_g$ also has GEI (see Section 7.1); if T has EI, we adapt a technique in [Reference Hrushovski and Chatzidakis28] to show that $T^{\bar {\delta },?}_g$ also has EI (see Section 7.2).
• When T has a suitable definable topology: we employ a technique by M. Tressl to show that $T^{\bar {\delta },?}_g$ has EI modulo $T^{eq}$ (see Section 11).

4 Independence relations

To establish the primary results of this article, we remind the concept of independence relations and revisit key theorems related to this notion. We fix a monster model $\mathfrak C \models T$ .

Let be a ternary relation on small subsets of $\mathfrak C$ . We mostly use the definitions and nomenclature from [Reference Adler1]: in particular, we are interested in the cases when is either a strict independence relation, or the relation defined in [Reference Adler1, Definition 1.26], or Shelah-forking .

We remind for completeness the definition introduced in [Reference Adler1].

Definition 4.1. The relation (M-dividing independence) is defined as:

iff for any $C'$ s.t $C \subseteq C' \subseteq {\mathrm {acl}}(BC)$ then $ {\mathrm {acl}}(AC') \cap {\mathrm {acl}}(BC') = {\mathrm {acl}} C'$ .

For the remainder of this section, is an independence relation on $\mathfrak C$ .

Remark 4.2. If , then .

Proof. By Existence, there exists $D \equiv _{BC}{ {\mathrm {acl}}(BC)}$ s.t. : notice that $D = {\mathrm {acl}}(BC)$ , and therefore . By transitivity, .

Notice that we don’t assume that T has some form of EI, nor that extends to $\mathfrak C^{eq}$ . However, the setting in [Reference Adler1] often assumes implicitly to work in $T^{eq}$ : to avoid confusion, we will introduce some additional nomenclature, following in part [Reference Yoneda52].

Definition 4.3. We say that an independence relation is:

real-strict: if, for every $a \in \mathfrak C$ and $B \subset \mathfrak C$ , iff $a \in {\mathrm {acl}}(B)$ ;
strict: if extends to an independence relation on $\mathfrak C^{eq}$ (which we also denote by ), and for every $a \in \mathfrak C^{eq}$ and $B \subset \mathfrak C^{eq}$ , iff $a \in {\mathrm {acl}^{\mathrm {eq}}}(B)$ ;
real-canonical: if it is real-strict and, for every $A, B, C, D \subset \mathfrak C$ such that $B \supseteq C, D$
canonical: if it is strict and, for every $A, B, C, D \subset \mathfrak C^{eq}$ , such that $B \supseteq C, D$

Notice that if is strict, then it is real-strict. If either $\mathfrak C = \mathfrak C^{eq}$ , or $\mathfrak C$ has geometric EI (GEI), then is real-strict iff it is strict; similarly, under the same assumption, is real-canonical iff it is canonical.

Fact 4.4.

1) There exist rosy theories with EI (even o-minimal) such that (thorn-forking independence) is not (real)-canonical: [Reference Adler1, Appendix A.3] explains an example from [Reference Loveys and Peterzil34]; see also [Reference Yoneda52, Remark 3.5].
2) There exists a theory with WEI (the “integral Urysohn space”) with more than one strict independence relation (see [Reference Conant11]).

Fact 4.5 [Reference Yoneda52, Lemma 3.1].

If is strict and real-canonical, then $\mathfrak C$ has GEI.

Corollary 4.6. Let be real-strict and real-canonical. T.f.a.e.:

1) $\mathfrak C$ has GEI;
2) is strict;
3) .

Lemma 4.7. Let be a real-strict independence relation on $\mathfrak C$ . If , then .

In particular, if T is simple and , then .

Proof. Let $A, B, C, C'$ be as in Definition 4.1. Assume that . Let $d \in {\mathrm {acl}}(AC') \cap {\mathrm {acl}}(BC')$ . We need to show that $d \in {\mathrm {acl}}(C')$ :

A independent of B over C implies A independent of acl(BC) over C implies A independent of BC prime over C implies A independent of B over C prime implies acl(AC prime) independent of acl(BC prime) over C prime implies d independent of d over C prime.

Since is real-strict, the latter implies that $d \in {\mathrm {acl}}(C')$ .

Fact 4.8 [Reference Adler1, Section 3.1],[Reference Buechler, Pillay and Wagner8].

1) If $\mathfrak C$ is either stable or super-simple, then it has elimination of hyperimaginaries (EHI);
2) if $\mathfrak C$ is simple with EHI, then is canonical;
3) if is canonical, then .

For more on EHI in (simple) theories, see, e.g., [Reference Buechler, Pillay and Wagner8, Reference Casanovas9, Reference Palacín and Wagner39]; there are no known examples of simple theories that do not eliminate hyperimaginaries; for us the relevant implication is the following fact.

Fact 4.9. If $\mathfrak C$ is either stable or super-simple, then is canonical and .

Definition 4.10. satisfies “Independence over Models” if:

For every $\bar a, \bar b \in \mathfrak C^{n}$ , $\bar a', \bar b' \in \mathfrak C^{m}$ , for every $M \prec \mathfrak C$ with M small, if

then there exists $\bar c \in \mathfrak C^{m}$ such that

satisfies “Independence over Algebraically Closed (real) Substructures” (IACS) if the above is true when we let M vary among algebraically closed substructure of $\mathfrak C$ . We say that T satisfies IACS if satisfies it.

Fact 4.11 (Kim–Pillay: see [Reference Kim and Pillay30], [Reference Tent and Ziegler48, Theorems 29.11 and 29.13]).

Let be an independence relation on $\mathfrak C$ that satisfies Independence over Models. Then, $\mathfrak C$ is simple and is Shelah-forking (on $\mathfrak C$ ).

Conversely, if $\mathfrak C$ is simple, then is a strict independence relation satisfying Independence over Models.

Fact 4.12 [Reference Buechler, Pillay and Wagner8].

If $\mathfrak C$ is simple with EHI and EI, then satisfies IACS.

can be extended to L-types: given $A \subseteq B$ small subsets of $\mathfrak C$ and an $L(B)$ -type (in a small number of variables) $q(\bar y)$ , we write

if for some $\bar c \in \mathfrak C^{\lvert \bar y\rvert }$ realizing $q(\bar y)$ we have that

Notice that the above is equivalent to:

“For every $\bar c \in \mathfrak C^{\lvert \bar y\rvert }$ realizing $q(\bar y)$ we have that ”.

4.1 Independence relations on algebraically bounded structures

We analyze now the case when L extends the language of rings and T is algebraically bounded.

Lemma 4.13. Let $A, B, C \subseteq \mathfrak C$ . T.f.a.e.:

1) ;
2) for every $A' \subseteq A$ , if $A'$ is algebraically independent over $CF$ (where $F = dcl(\emptyset )$ ), then it is still algebraically independent over $BC$ ;
3) for every $A' \subseteq A$ , if $A'$ generates A (in the sense of the matroid $ {\mathrm {acl}}$ ) over $BC$ , then it generates A over B;
4) for every $A' \subseteq A$ and $B' \subseteq B$ , $ {\mathrm {tr.deg.}}(A'B'/C) = {\mathrm {tr.deg.}}(A'/C) +$ $ {\mathrm {tr.deg.}}(B'/C)$ .

Moreover, is a real-strict independence relation on $\mathfrak C$ satisfying the strong finite character condition.

Proof. For the “moreover” part, notice that 4 implies that is symmetric, and the conclusion follows by [Reference Adler1, Theorem 2.39 and Exercise 2.6].

Fact 4.14. If $\mathfrak C$ is algebraically bounded and has GEI, then and $\mathfrak C$ is rosy, i.e., is an independence relation on $\mathfrak C^{eq}$ ; moreover, it is super-rosy of þ-rank 1.Footnote ¹

Proof. Since $\mathfrak C$ has GEI, is a strict independence relation on $\mathfrak C^{eq}$ , and therefore $\mathfrak C$ is rosy. For more details see [Reference Adler1]: in particular, [Reference Adler1, Definition 1.28].

$\mathfrak C$ is then super-rosy of rank 1 because , where is the þ-rank, $ {\mathrm {tr.deg.}}$ is the transcendence degree, and F is the model-theoretic algebraic closure of the empty set.

Remark 4.15. Assume that $\mathfrak C$ is algebraically bounded, with GEI, simple, and is canonical. Then, .

Proof. Since is canonical, , and by the Fact 4.14 we have the conclusion.

5 The extension theorem

The results presented in this section are of independent interest and will be utilized in the sequel. For now, K is some field of characteristic $0$ .

Definition 5.1. We denote by the ternary relation on subsets of K induced by $ {\mathrm {acl}}$ over F:

iff, for every $\bar a$ finite subset of A, if $\bar a$ is algebraically independent over $CF$ , then $\bar a$ remains algebraically independent over $CFB$ .Footnote ²

The following lemma is well-known; we provide a proof for completeness.

Lemma 5.2 (Amalgamation).

Let $B_0$ , $B_1$ , and $B_2$ be subrings of K containing F. Let $\bar {\delta }_i$ be k-tuples of derivations on $B_i$ , $i=0, 1, 2$ . Assume:

1) $\langle B_0, \bar {\delta }_0 \rangle $ is a common substructure of $\langle B_1, \bar {\delta }_1 \rangle $ and $\langle B_2, \bar {\delta }_2 \rangle $ ;
2)

Then, there exists a k-tuple of derivations $\bar {\delta }$ on K which extends all the $\bar {\delta }_i$ . Moreover, if each $\bar {\delta }_i$ commutes, then $\bar {\delta }$ commutes.

Proof. Let $\bar b_i$ be a transcendence basis of $B_i$ over $B_0$ , $i = 1,2$ . By 2), $\bar b_1$ and $\bar b_2$ are disjoint, and $\bar b_1 \cup \bar b_2$ is algebraically independent over $B_0$ . Thus, there exists a k-tuple of derivations $\bar {\delta }$ on K extending $\bar {\delta }_{0}$ and such that

$$\begin{align*}\bar{\delta}(\bar b_i) = \bar{\delta}_i(\bar b_i), \qquad i= 1, 2. \end{align*}$$

But then $\bar {\delta }$ extends also $\bar {\delta }_1$ and $\bar {\delta }_2$ . If moreover each $\bar {\delta }_i$ commutes, let $\bar c$ be a transcendence basis of K over $B_0 \cup B_1 \cup B_2$ . Let $\bar {\delta }$ be the unique k-tuple of derivations on K such that:

1) $\bar {\delta }$ extends $\bar {\delta }_{0}$ ;
2) $\bar {\delta }(\bar b_i) = \bar {\delta }_i(\bar b_i)$ , $i= 1, 2$ ;
3) $\bar {\delta } (\bar c )$ = 0.

Then, $\bar {\delta }$ commutes.

Now $\langle \mathbb K, \bar {\delta } \rangle $ is a monster model of $T^{\bar {\delta },?}_g$ .

The following technical result has several applications: we will see some later; in [Reference Peterzil, Pillay and Point40, Lemma 5.8] the authors prove a version of it in order to study definable groups.

Theorem 5.3 (Extension).

Let $A \subseteq B$ be “small” subsets of K with $\bar {\delta } A \subseteq A$ and $\bar {\delta } B \subseteq B$ . Let $p(\bar x)$ be a complete ${L^{\bar {\delta }}}$ -type over A (in n variables). Let $q \in S_{L}^{J}(B)$ be some extension of $p_{J}$ . If

then q is $\bar {\delta }$ -compatible (see Section 2.2).

Proof. W.l.o.g., A and B are also subrings of K. Let $\bar a \in K^{n}$ be a realization of p and $\bar b \in K^{J}$ be a realization of q. Since $ {\mathrm {Jet}}(\bar a)$ and $\bar b$ have the same L-type q over A, there exists $\phi $ an L-automorphism of $\mathbb K$ fixing A point-wise and mapping $\bar b$ to $ {\mathrm {Jet}}(\bar a)$ .

Define $\bar \varepsilon := \phi ^{-1} \circ \bar {\delta } \circ \phi $ . Let

• ${\bar \delta }_{0}$ be the restriction of ${\bar \delta }$ to A;
• ${\bar \delta }_1$ be the restriction of ${\bar \delta }$ to B;
• ${\bar \delta }_{2}$ be the restriction of $\bar \varepsilon $ to $A[\bar b]$ .

Notice that $\bar \varepsilon $ is a tuple of derivations on K which extends $\bar {\delta }_{0}$ and such that $\bar \varepsilon $ commutes if $\bar {\delta }$ commutes. Thus, by the amalgamation lemma, there exists a k-tuple $\bar \lambda $ of derivations on K which extends ${\bar \delta }_0$ , ${\bar \delta }_{1}$ , and ${\bar \delta }_{2}$ , and such that $\bar \lambda $ commutes if ${\bar \delta }$ commutes. Thus, $\langle \mathbb K, \bar \lambda \rangle \models T^{\bar {\delta },?}$ ( $\bar \lambda $ is not generic, in general). Let $ \bar b_0 \in K^{n}$ , interpreted in $\langle \mathbb K, \bar \lambda \rangle $ , we have that $ {\mathrm {Jet}}( \bar b_0) = \bar b$ , and therefore $ {\mathrm {Jet}}( \bar b_0)$ satisfies q. Thus, the partial ${L^{\bar {\delta }}}(B)$ -type

$$\begin{align*}u(\bar x) := T^{\bar{\delta},?}_g \cup {\mathrm{Diag}}_{{L^{\bar{\delta}}}}(B) \cup q( {\mathrm{Jet}}(\bar x)) \end{align*}$$

is consistent. Since $\langle \mathbb K, \bar {\delta } \rangle $ is existentially closed and $\lvert B\rvert ^{+}$ -saturated, there exists $\bar c \in K^{n}$ satisfying u.

5.1 Relation with the framework of [Reference León Sánchez and Mohamed33]

It is worth noting that, in view of Lemma 5.2, the theory $T^{\bar \delta }$ with respect to fits into the framework of derivation-like expansions introduced in [Reference León Sánchez and Mohamed33].

Indeed, Lemma 5.2 yields the amalgamation property required in part (1) of the definition of derivation-like expansions. Moreover, since $\mathbb K$ is algebraically bounded, algebraic closure defines an independence relation, and, in characteristic $0$ , derivations extend uniquely to algebraic extensions. This immediately implies part (2) of the definition.

Furthermore, when $T = \mathrm {Th}(\mathbb K)$ is simple, forking independence implies algebraic independence, and hence $T^{\bar \delta }$ is derivation-like also with respect to .

As a consequence, several of our results can alternatively be obtained from the general theory developed in [Reference León Sánchez and Mohamed33], and our Theorems 5.3, 6.1, 7.1, and 7.4 (as well as Corollary 6.2) can be recovered from their corresponding general results in that setting.

Nevertheless, we have chosen to include self-contained proofs of these results. This is partly for the convenience of the reader, and partly to maintain a uniform approach and notation throughout the article, avoiding the need to repeatedly translate between our setting and the abstract framework of [Reference León Sánchez and Mohamed33].

6 Algebraic closure and independence relations

In this section we examine the relations between algebraic closure, independence relations, and (geometric) EI. The results in this section will be used in Sections 7 and 12.

We fix $\langle \mathfrak C; \bar {\delta } \rangle $ monster model of $T^{\bar {\delta },?}_g$ .

Let be some ternary relation on subsets of $\mathfrak C$ . We define the following ternary relation on subsets of $\mathfrak C$ :

Notice that we are not assuming that T eliminates imaginaries, or that extends to $\mathfrak C^{eq}$ ).

Theorem 6.1. Assume that is a real-strict independence relation on $\mathfrak C$ (see Def. 4.3). Then, is a real-strict independence relation on $\langle \mathfrak C, \bar {\delta } \rangle $ .

If moreover is real-canonical, then is real-canonical.

Proof. In [Reference Adler1], a list of axioms for independence relations is presented. The author subsequently demonstrates that the Extension Axiom is equivalent to the conjunction of the Existence Axiom and the Symmetry Axiom. Notably, among the axioms for independence ( ) outlined in [Reference Adler1], only the Existence Axiom requires non-trivial verification. Thus, let $\bar a, \bar b, \bar c$ be small tuples in $\mathfrak C$ . Since is an independence relation, there exists a small tuple $\bar d \subset \mathfrak C$ s.t. . Let $p := {\mathrm {tp}}^{{L^{\bar {\delta }}}}( {\mathrm {Jet}} (\bar a)/ {\mathrm {Jet}} (\bar c))$ and $q := {\mathrm {tp}}^{L}(\bar d / {\mathrm {Jet}}(\bar c) {\mathrm {Jet}} (\bar b))$ . By assumption, q is a complete L-type extending p, and

Since is real-strict, Lemma 4.7 implies that

and therefore, by the extension theorem, there exists $\bar a' \subset \mathfrak C$ s.t. $ {\mathrm {Jet}} (\bar a')$ realizes q: hence,

proving that satisfies Existence.

The fact that is real-strict is clear.

Finally, assume that is real-canonical. Assume that

We have to prove that

W.l.o.g., we may assume that $E = {\mathrm {acl}^{{\bar {\delta }}}} E$ . Thus, we have

Since is real-canonical, we have

$ {\mathrm {acl}}(E {\mathrm {Jet}}(\bar b)) = {\mathrm {acl}^{{\bar {\delta }}}}(E \bar b)$ , and similarly for $\bar a$ , we are done.

In particular, we can choose , and obtain that the induced relation is a real-strict independence relation on $\langle \mathfrak C; \bar {\delta } \rangle $ .

Corollary 6.2 (cf. [Reference León Sánchez and Mohamed33, Corollary 3.16], [Reference Pillay, Point and Rideau-Kikuchi42, Corollary 1.4]).

1) The algebraic closure on $\langle \mathfrak C; \bar {\delta } \rangle $ is given by
$$\begin{align*} {\mathrm{acl}}( {\mathrm{Jet}}(A)), \end{align*}$$
for every $A \subseteq \mathfrak C$ .Footnote ³
2) If $\langle \mathfrak C; \bar {\delta } \rangle $ has GEI, then it is rosy, and coincides with þ-forking (for $\langle \mathfrak C; \bar {\delta } \rangle $ ).

Proof. Let ${\mathrm {acl}^{{\bar {\delta }}}}$ be the algebraic closure according to $T^{\bar {\delta },?}_g$ .

1) If $a \in {\mathrm {acl}^{{\bar {\delta }}}} B$ , then, since is an independence relation,

and in particular,

that is, $a \in {\mathrm {acl}}( {\mathrm {Jet}}(B))$ .

Conversely, it is clear that $ {\mathrm {acl}}( {\mathrm {Jet}}(B))$ must be contained in ${\mathrm {acl}^{{\bar {\delta }}}}(B)$ .

2) Since is a real-strict independence relation on $\langle \mathfrak C, {\bar \delta } \rangle $ , we have that the latter is real-rosy; if moreover $\langle \mathfrak C, {\bar \delta } \rangle $ has GEI, it is rosy (see [Reference Adler1] for definitions and proofs: in particular, [Reference Adler1, Definition 1.26]).

6.1 GEI in algebraically bounded structures

Proposition 6.3. Let K be any field of characteristic $0$ . Define if, for every $X \subseteq A$ which is algebraically independent over C (in the field-theoretic sense), X remains algebraically independent over $BC$ . Then, is real-canonical.

Proof. Let $\tilde K$ be the field-theoretic algebraic closure of K. Given $C \subseteq K$ , let $\tilde C$ be the field-theoretic algebraic closure of C inside $\tilde K$ , and $ {\mathrm {acl}}(C)$ be the field-theoretic algebraic closure of C inside K.

Let $D := C_{1} \cap C_{2}$ .

Claim 1.

$$\begin{align*}\tilde{C_{1}} \cap \tilde{C_{2}} = \widetilde { {\mathrm{acl}}(C_{1}) \cap {\mathrm{acl}}(C_{2})}. \end{align*}$$

First, w.l.o.g. we may assume that $C_{i} = {\mathrm {acl}}(C_{i})$ , for $i = 1, 2$ . It is clear that

$$\begin{align*}\tilde D \subseteq \tilde{C_{1}} \cap \tilde{C_{2}}; \end{align*}$$

we have to prove the opposite inclusion. Let C be either $C_{1}$ or $C_{2}$ . Let $a \in \tilde C_{1} \cap \tilde C_{2}$ and $a_{1}, \dotsc , a_{\ell }$ be the conjugates of a over K. Then, $a_{j} \in \tilde C$ , $j= 1, \dotsc , \ell $ . Let $\sigma _{t}(x_{1}, \dotsc , x_{\ell })$ be one of the symmetric polynomials: we have that $\sigma _{t}(a_{1}, \dotsc , a_{\ell }) \in \tilde C \cap K = C$ . Thus, if $p(x) \in K[x]$ is the minimal polynomial of a over K, we have $p \in C[x]$ , and therefore $p \in D[x]$ .

Thus, $a \in \tilde D$ , proving the claim.

Let be the analogue of inside $\tilde K$ .

Claim 2. is real-canonical.

In fact, coincides with on $\tilde K$ seen as a pure field, and, since $\tilde K$ (using the fact that it is an algebraically closed field) is stable with EI, is real-canonical (see [Reference Adler1, Section 3.1]).

Assume now that $C_{i} \subseteq B$ for $i = 1, 2$ , and and $C_{i} \subseteq B$ , for $i = 1, 2$ .

We have for $i = 1, 2$ . Since is real-canonical, we have that

and therefore, by Claim 1,

which is equivalent to .

Corollary 6.4. is real-strict and real-canonical. is also real-strict and real-canonical.

Proof. This is an immediate consequence of Theorem 6.1 and Proposition 6.3.

Thus, by Corollary 4.6, we have the following theorem.

Theorem 6.5. T.f.a.e.:

1) $\mathfrak C$ has GEI;
2) is strict;
3) ;
4) $\mathfrak C$ is super-rosy of þ-rank 1.

Theorem 6.6. T.f.a.e.:

1) $\langle \mathfrak C; \bar {\delta } \rangle $ has GEI;
2) is strict;
3) is equal to þ-forking on $\langle \mathfrak C; \bar {\delta } \rangle $ .

Conjecture 6.7. Assume that $\mathfrak C$ has GEI. Then, $\langle \mathfrak C; \bar {\delta } \rangle $ also has GEI.

Proposition 6.8. Let $\mathfrak C$ be a monster model of some theory. Assume that is an independence relation on $\mathfrak C$ satisfying the following conditions:

I) is real-canonical;
II) is “controlled by a rank”: there exist a linearly ordered set $\langle I; \leq \rangle $ and a function $\rho $ associating an element of I to each type (over some small subset of the monster model $\mathfrak C$ in finitely many real variables), such that:
1. a) $\rho $ is invariant under automorphisms;
2. b) $\rho $ for every $\bar a$ , B, C, $\rho (\bar a/B) \leq \rho (a/BC)$ , with equality iff ;
3. c) for every non-empty set X which is type-definable over A, $\rho (\bar x/A)$ attains a maximum on X (e.g., $\rho $ is upper semi-continuous).

Let $e \in \mathfrak C^{eq}$ , where $e = f(\bar a)$ for some (real finite tuple) $\bar a \in \mathfrak C^{\ell }$ and some function f which is definable without parameters. Let $E := {\mathrm {acl}^{\mathrm {eq}}}(e) \cap \mathfrak C$ and P be the set of realization of $ {\mathrm {tp}}(a/E)$ (inside $\mathfrak C$ ). Then there exists $\bar c \in P$ such that $f (\bar c) = e$ and .

Proof. Proposition 6.8 follows the proof in [Reference Hrushovski and Chatzidakis28, (1.10)] and [Reference Moosa and Scanlon38, Theorem 5.12]. As noticed in [Reference Hrushovski and Chatzidakis28], Neumann’s Lemma implies that there exists $\bar b$ conjugate of $\bar a$ over $E \cup \{e\}$ such that

(1) $$ \begin{align} {\mathrm{acl}}(E\bar a) \cap {\mathrm{acl}}(E\bar b) = {\mathrm{acl}^{\mathrm{eq}}}(E e) \cap \mathfrak C = E \end{align} $$

(see [Reference Evans and Hrushovski15]). Let Q be the set of $\bar b \in P$ satisfying (1).

Claim 3. Q is type-definable over $E \bar a$ .Footnote ⁴

Since P is type-definable, it suffices to show that the set

$$\begin{align*}X := \{\bar x \in \mathfrak C^{\ell}: {\mathrm{acl}}(E\bar a) \cap {\mathrm{acl}}(E\bar x) \supset E\} \end{align*}$$

is $\mathord {\vee }$ -definable. For every $d \in {\mathrm {acl}}(E\bar a) \setminus E$ , let $X_{d} := \{\bar x \in \mathfrak C^{\ell }: d \in {\mathrm {acl}}(E \bar x)\}$ . $X_{d}$ is $\mathord {\vee }$ -definable (over $E d$ ) because it is a union of sets defined by formulae of the form $\alpha (\bar c,d, \bar x) \ \wedge \ \lvert \alpha (\bar c, \mathfrak C, x)\rvert \leq n $ , as $\alpha $ varies among all L-formulae and n varies in $\mathbb {N}$ . Therefore, $X = \bigcup _{d \in {\mathrm {acl}}(E\bar a) \setminus E} X_d$ is also $\mathord {\vee }$ -definable (over $ {\mathrm {acl}}(E \bar a)$ ).

By assumption, there exists $\bar b \in Q$ such that $\rho (\bar b/E \bar a)$ is maximum (inside Q).

Let $\bar c \models {\mathrm {tp}}(\bar b/E\bar a)$ such that . As in [Reference Hrushovski and Chatzidakis28, (1.10)] (cf. [Reference Moosa and Scanlon38, Theorem 5.12]), we see that

Since is real-canonical, and $ {\mathrm {acl}}(E \bar a) \cap {\mathrm {acl}}(E \bar b) = E$ , we have that

and therefore , proving Proposition 6.8.

7 Simplicity

We will now prove that if T is simple, then $T^{\bar {\delta },?}_g$ is also simple. Our proof relies on the crucial notion of “Independence over Algebraically Closed Substructures” (IACS), which we define and explore in this section.

Theorem 7.1 (Simplicity of $T^{\bar {\delta },?}_g$ ).

If T is simple, then $T^{\bar {\delta },?}_g$ is also simple.

The above is non-trivial, but thanks to the preliminary work we can give an easy proof.

We fix $\langle \mathfrak C, \bar {\delta } \rangle $ monster model of $T^{\bar {\delta },?}_g$ . Assume that $\mathfrak C$ is simple. Let be Shelah-forking relation on $\mathfrak C$ , and be the induced independence relation on $\langle \mathfrak C, \bar {\delta } \rangle $ as in Section 6.

Since we want to prove a stronger version of Theorem 7.1, we need the following definition.

Definition 7.2. An independence relation satisfies “Independence over Algebraically Closed Substructures” (IACS) if:

(IACS) For every $\bar a, \bar b \in \mathfrak C^{n}$ , $\bar a', \bar b' \in \mathfrak C^{m}$ , for every $M \subset \mathfrak C$ with M small and algebraically closed, if

then there exists $\bar c \in \mathfrak C^{m}$ such that

We say that T satisfies IACS if satisfies it.

As we will see later, IACS is closely related to (weak) EI (cf. [Reference Hrushovski and Chatzidakis28]).

Remark 7.3. Obviously, IACS implies Independence over Models; the converse is not true: for instance, let $L = \{E\}$ , where E is a binary relation, and T be the theory saying that E has two equivalence classes, both infinite; then, T is $\omega $ -stable and hence simple, therefore satisfies Independence over Models, but it is easy to see that it does not satisfy IACS (take $M = \emptyset $ ). Indeed it is enough to consider four distinct elements $a, a', b, b'$ with $a', b'$ in two different equivalence classes.

Theorem 7.4. Assume that T is simple. Then, is a real-strict independence relation on $\langle \mathfrak C, {\bar \delta } \rangle $ and it satisfies independence over models.

Thus, $T^{\bar {\delta },?}_g$ is simple, and is Shelah-forking on $\langle \mathfrak C, \bar {\delta } \rangle $ .

If moreover T satisfies IACS, then also $T^{\bar {\delta },?}_g$ satisfies IACS.

Proof. From Theorem 6.1 we have that is a strict independence relation.

We have to show that satisfies independence over models (resp., over algebraically closed substructures) if satisfies it. Let $\bar a$ , $\bar b$ , $\bar a'$ , $\bar b'$ be as above and M be a small elementary ${L^{\bar {\delta }}}$ -substructure (resp., small ${L^{\bar {\delta }}}$ -algebraically closed substructure) of $\langle \mathfrak C, {\bar \delta } \rangle $ . The assumptions become

Jet(a-bar) independent of Jet(b-bar) over M with f. Jet(a-bar prime) independent of Jet(a-bar) over M. Jet(b-bar prime) independent of Jet(b-bar) over M with f. Jet(a-bar prime) is equivalent to Jet(b-bar prime) over M with L delta-bar.

By Fact 4.11 (applied to $\mathfrak C$ ), satisfies independence over models (resp., by assumption, satisfies IACS), and therefore there exists $\bar d \in M^{\Gamma _{n}}$ such that

Let $q := {\mathrm {tp}}^{L}(\bar d/ M {\mathrm {Jet}}(\bar a) {\mathrm {Jet}}(\bar b))$ , and p be restriction of q to $\mathfrak C$ . Notice that p is realized by $\delta \bar a$ , and it is therefore ${\bar \delta }$ -compatible. Moreover, and therefore , and therefore

Thus, by the extension theorem, q is ${\bar \delta }$ -compatible, and it is therefore realized by $ {\mathrm {Jet}}(\bar c)$ for some $\bar c \in \mathfrak C^{m}$ . Thus, we have

which implies the conclusion.

7.1 GEI and simplicity

Theorem 7.5. Assume that T is algebraically bounded, simple, with GEI and is real-canonical (e.g., T is either stable or supersimple). Then:

1) ;
2) T is super-simple of $SU$ -rank 1;
3) is strict and real-canonical;
4) Shelah forking on $T^{\bar {\delta },?}_g$ is equal to and is also strict and real-canonical;
5) $T^{\bar {\delta },?}_g$ has GEI.

Proof. by Fact 4.14 and it is therefore strict. by Remark 4.15 is real-canonical by Corollary 6.4.

is Shelah’s forking by Theorem 7.4; it is therefore strict; it is real-canonical by Theorem 6.1.

$T^{\bar {\delta },?}_g$ has GEI by 3) and Fact 4.5.

By Theorem A, we can deduce that the assumptions of Theorem 7.5 can be weakened to “T is algebraically bounded and simple.”

7.2 Elimination of imaginaries and simplicity

In this section we give some sufficient conditions for $T^{\bar {\delta },?}_g$ to have EI when T is simple (see Theorem 7.9 and Corollary 7.10); for previous results see [Reference Mohamed37, Reference Moosa and Scanlon38].

Given a monster model $\mathfrak C$ , we use the following convention: $\bar a, \bar b, \bar c, \bar d$ are finite tuples of real elements, $A, B,C, D, E$ are small sets of real elements, while e will be an imaginary element; $ {\mathrm {acl}}$ will always denote the algebraic closure inside $\mathfrak C$ (and not $\mathfrak C^{eq}$ ); when we consider two theories, T and $T^{\bar {\delta },?}_g$ , we will use $ {\mathrm {acl}}$ for the algebraic closure according to T, and ${\mathrm {acl}^{{\bar {\delta }}}}$ for the algebraic closure according to $T^{\bar {\delta },?}_g$ .

Definition 7.6. We say that T is “good” if:

1) T is algebraically bounded;
2) T has EI;
3) T is supersimple.

Remark 7.7. If T is good then T has EHI and (Fact 4.8 and Theorem 6.5).

Moreover conditions 1) and 2) imply that (Theorem 6.5).

Conditions 2) and 3) imply that satisfies IACS (Fact 4.12).

Finally, if T is good, then T is supersimple of SU rank 1.

Remark 7.8. If we consider the conditions 1) and 2) of Definition 7.6 together with the assumption that the theory T is stable, then it is easy to see that which implies that T is superstable of SU rank 1 (a particular case of supersimplicity), and therefore it is good. By Theorem A, we can further weaken the assumption to conditions 1) and 2) plus T is simple.

Theorem 7.9. If T is good, then $T^{\bar {\delta },?}_g$ is simple, is real-canonical and satisfies IACS, and $T^{\bar {\delta },?}_g$ has EI.

We will prove the above theorem at the end of this section, after some consequences and preliminary results.

Corollary 7.10. If T is algebraically bounded and stable with EI, then $T^{\bar {\delta },?}_g$ has EI.

Proof. If T is stable with EI, then it is superstable (of U-rank 1). (Theorem A gives that $T = {\mathrm {ACF}_0}$ ).

We prove a more general result that we will use for the proof of Theorem 7.9.

Proposition 7.11. Let $\mathfrak C$ be a monster model satisfying the following conditions:

1) $\mathfrak C$ is simple;
2) satisfies IACS;
3) is real-canonical;
4) is controlled by a rank $\rho $ , as defined in II of Proposition 6.8.

Then, $\mathfrak C$ has WEI. If moreover $\mathfrak C$ expands a field, then $\mathfrak C$ has EI.

Proof. The proof keeps following the one in [Reference Hrushovski and Chatzidakis28, (1.10)] and [Reference Moosa and Scanlon38, Theorem 5.12].Footnote ⁵ Here are some more details.

Let e be an imaginary element. Write $e = f(\bar a)$ for some (real finite tuple) $\bar a \in \mathfrak C^{\ell }$ and some function f which is definable without parameters. Let $E := {\mathrm {acl}^{\mathrm {eq}}}(e) \cap \mathfrak C$ . Our thesis that U has WEI is equivalent to $e \in {\mathrm {dcl}^{\mathrm {eq}}}(E)$ (when $\mathfrak C$ expands a field, WEI implies EI see Remark 3.6).

Let P be the set of realization of $ {\mathrm {tp}}(\bar a/E)$ (inside $\mathfrak C$ ). By Proposition 6.8, there exists $\bar c \in P$ such that $f (\bar c) = e$ and $\bar a$ and $\bar c$ are (forking)-independent over E.

Claim 4. f is constant on P.

Otherwise, there exists $\bar d' \in P$ such that $f(\bar a) \neq f(\bar d')$ . Let $\bar d", \bar a"$ be such that and $\bar d" \bar a" \equiv _{E} \bar d' \bar a$ . Thus, $f(\bar d") \neq f(\bar a")$ , $\bar d" \equiv _{E} \bar a$ , and $\bar a" \equiv _{E} \bar a$ . Thus, choosing either $\bar d := \bar d"$ or $\bar d := \bar a"$ we get $f(\bar d) \neq f(\bar a) $ and and .

By IACS (since E is algebraically closed), (by taking $\bar a := \bar a$ , $\bar b := \bar d$ , $\bar a' := \bar c$ , and $\bar b' := \bar a$ in Definition. 7.2) there exists $\bar c' \in \mathfrak C^{\ell }$ such that $\bar c' \equiv _{E \bar d} \bar a$ and $\bar c' \equiv _{E \bar a} \bar c$ . Since $e = f(\bar a) \neq f(\bar d)$ , we get $f(\bar c') \neq e$ ; but since $e = f(\bar a) = f(\bar c)$ , we get $f(\bar c') = e$ , absurd.

Claim 5. $e \in {\mathrm {dcl}^{\mathrm {eq}}}(E)$ .

By Claim 4 and compactness, there exists an $L(E)$ -definable set X such that $P \subseteq X$ and f is constant on X. Thus, e is defined by the $L(E)$ -formula $\exists \bar x \in X f(\bar x) = y$ .

Proof of Theorem 7.9.

We have to show that T and $T^{\bar {\delta },?}_g$ satisfy the assumptions of Proposition 7.11.

For T, we define $\rho (\bar a/B)$ to be the rank given by the algebraic closure (notice that under our assumptions T is super-simple of rank $1$ , and $\rho $ is equal to the $SU$ -rank). Since $\rho (a_{1}, \dotsc , a_{m}) / B \leq m$ , condition IIc of Proposition 6.8 is automatically true, and therefore T satisfies the assumptions of Proposition 7.11.

We consider now $T^{\bar {\delta },?}_g$ : we have to show that the various properties of T that we need are inherited by $T^{\bar {\delta },?}_g$ .

We have seen before that $T^{\bar {\delta },?}_g$ is simple with IACS and real-canonical.

We need to introduce a suitable function $\rho $ and show that it satisfies the conditions in Proposition 7.11. We can use the analogue of the dimension function $\dim _{\mathcal D}$ in [Reference Moosa and Scanlon38]. For every $r \in \mathbb {N}$ , define $\rho (\bar a/B)(n)$ as the transcendence degree of $ {\mathrm {Jet}}_n(\bar a)$ over $ {\mathrm {acl}}( {\mathrm {Jet}}(B))$ , where

$$\begin{align*} {\mathrm{Jet}}_n(\bar a) = \{\mu \bar a: \mu \in \Gamma \text{ and } \lvert\mu\rvert \leq n\}. \end{align*}$$

Thus, $\rho (\bar a/B) \in \omega ^{\omega }$ . If we endow $\omega ^{\omega }$ with the lexicographic ordering, then $\rho $ is upper semi-continuous, and satisfies the conditions in Proposition 7.11.

8 Distality

This section is based on a suggestion by Elliot Kaplan. Distality is a stronger condition than NIP, it characterizes NIP theories that are as far from stability as possible. For a formal definition and key properties of distal theories, see [Reference Aschenbrenner, Chernikov, Gehret and Ziegler2, Reference Simon47]. In [Reference Fornasiero and Terzo21] we showed that if T is NIP, then $T^{\bar {\delta },?}_g$ is NIP. A similar results hold when T is distal.

Theorem 8.1. If T is distal, then $T^{\bar {\delta },?}_g$ is distal.

Proof. Immediate from Theorem 2.4 and [Reference Aschenbrenner, Chernikov, Gehret and Ziegler2, Proposition 7.1].

9 $\omega $ -stability

In [Reference Fornasiero and Terzo21], we showed that if the theory T is stable, then $T^{\bar {\delta },?}_g$ retains stability. The analogous result for $\omega $ -stability is false: here we give necessary and sufficient conditions for $T^{\bar {\delta },?}_g$ to be $\omega $ -stable. Recall that countable theory is totally transcendental iff the theory is $\omega $ -stable (see [Reference Sacks46, p. 125]).

Theorem 9.1. $T^{\bar {\delta },?}_g$ is totally transcendental iff $T = {\mathrm {ACF}_0}$ (i.e., T is the theory of pure algebraically closed fields, possibly with some constants) and ${\bar \delta }$ commute.

We first need the following lemma (cf. Theorem A).

Lemma 9.2. T is algebraically bounded and strongly minimal iff $T = {\mathrm {ACF}_0}$ .

Proof. It is clear that, if $T = {\mathrm {ACF}_0}$ , then T is algebraically bounded and strongly minimal.

Conversely, assume that T is algebraically bounded and strongly minimal. Let $\mathbb K \models T$ . Since $\mathbb K$ is strongly minimal, as a field K is algebraically closed (see Macintyre’s Theorem [Reference Poizat44, Theorem 3.1]). By [Reference Hrushovski27, Theorem 1], every $\mathbb K$ -definable set is definable in the field language.

Proof of Theorem 9.1

If $T = {\mathrm {ACF}_0}$ , then $T^{\bar {\delta }}_g$ is $\omega $ -stable [Reference McGrail36].

Let us now consider the converse. Let $\langle \mathbb K, \bar {\delta } \rangle \models T^{\bar {\delta },?}_g$ .

1) Assume that ${\bar \delta }$ do not commute. Let $C_{1}$ be the fixed field of $\delta _{1}$ . Notice that $\delta _{2}(C_{1}) = K$ , since $\mathbb K$ is existentially closed. Thus, $ {\mathrm {MR}}(C_{1}) = {\mathrm {MR}}(K)$ . However, $C_{1}$ is a subgroup of K of infinite index, and therefore $ {\mathrm {MR}}(K) = \infty $ , proving that $\langle \mathbb K, \bar {\delta } \rangle $ is not $\omega $ -stable.
2) We have to prove that $T = {\mathrm {ACF}_0},$ so by Lemma 9.1 we assume that K is not strongly minimal. We want to show that $\langle \mathbb K, \bar {\delta } \rangle $ is not $\omega $ -stable. It suffices to show that, assuming that $L(K)$ is countable, there exist $2^{\omega } {L^{\bar {\delta }}}$ -1-types over K. Then, let $X \subset K$ which is $\mathbb K$ -definable and such that both X and $\mathbb K \setminus X$ are infinite. Fix $J \subseteq \mathbb {N}$ and consider the following partial ${L^{\bar {\delta }}}(K)$ -type:
$$\begin{align*}p_{J}(x) = \big( \delta_{1}^{i} x \in X: i \in J \big) \cup \big( \delta_{1}^{i} x \notin X: i \notin J \big). \end{align*}$$

Notice $p_{J}$ is indeed a partial type and, for every $J \neq J'$ , $p_{J}$ and $p_{J'}$ are incompatible. Thus, $\lvert S^{1}_{{L^{\bar {\delta }}}}(K)\rvert \geq 2^{\omega }$ .

10 The field of constants

We give some interesting results on the field of constants. Before we introduce some notations and definitions.

Definition 10.1. The field of constants is the set

$$\begin{align*}\mathcal C_{\bar{\delta}} := \{a \in K: \delta_{1}(a) = \dots = \delta_{k}(a) = 0\}. \end{align*}$$

We consider the reduct $\langle \mathbb K,\mathcal C_{\bar {\delta }} \rangle $ (i.e., the expansion of $\mathbb K$ with a unary predicate for $\mathcal C_{\bar {\delta }}$ ).

Observe that $\mathcal C_{\bar {\delta }}$ is dense in $\mathbb K$ w.r.t. the matroid $ {\mathrm {acl}}$ : that is, for every $Z \subseteq K$ which is L-definable (remember that L is the signature of $\mathbb K$ ) with parameters in $\mathbb K$ and large, Z intersects $\mathcal C_{\bar {\delta }}$ ; moreover, $\mathcal C_{\bar {\delta }}$ is also L-algebraically closed in $\mathbb K$ . Thus, $\langle \mathbb K,\mathcal C_{\bar {\delta }} \rangle $ is a lovely pair of geometric structures (in the sense of [Reference Berenstein and Vassiliev4, Reference Boxall5]: see also [Reference Fornasiero16]). Thus, we can apply the known results (see [Reference Berenstein and Vassiliev4, Reference Boxall5, Reference Fornasiero16]).

Lemma 10.2. $\bar b \in K^{\ell }$ and $B := {\mathrm {dcl}}_L( {\mathrm {Jet}}(\bar b)= {\mathrm {dcl}}_{{L^{\bar {\delta }}}}( {\mathrm {Jet}}(\bar b))$ . Then, B is P-independent: that is, , where $A := B \cap P$ .

Proof. Assume not: then, there exists $\bar d \in {\mathrm {Jet}}(\bar b)^n$ s.t. $\bar d$ is L-algebraically independent over A but the following system has a solution $\bar c \in (K^n)^m$ :

(1) $$ \begin{align} \left\{\begin{aligned} \delta \bar x &= 0\\ \bar x &\neq 0\\ \sum_{i,j} x_{ij}d_j^i &= 0. \end{aligned}\right. \end{align} $$ StartLayout Enlarged left brace 1st Row 1st Column delta x overbar 2nd Column equals 0 2nd Row 1st Column x overbar 2nd Column not equals 0 3rd Row 1st Column sigma summation Underscript i comma j Endscripts x Subscript i j Baseline d Subscript j Superscript i 2nd Column equals 0 period EndLayout

StartLayout Enlarged left brace 1st Row 1st Column delta x overbar 2nd Column equals 0 2nd Row 1st Column x overbar 2nd Column not equals 0 3rd Row 1st Column sigma summation Underscript i comma j Endscripts x Subscript i j Baseline d Subscript j Superscript i 2nd Column equals 0 period EndLayout

W.l.o.g., we may assume that $\langle \mathbb {K}, {\bar \delta } \rangle $ is a monster model. Let $\bar b' \equiv _A^{L^{\bar {\delta }}} \bar b$ s.t. . Let $\sigma $ be an ${L^{\bar {\delta }}}$ -automorphism of $\mathbb K$ over A mapping $\bar b$ to $\bar b'$ and $\bar d' := \sigma (\bar d)$ . Then, $\bar d'$ is L-algebraically independent over A and therefore (1) (with $\bar d'$ in place of $\bar d$ ) cannot have a solution in K. But $\bar b$ and $\bar b'$ have the same ${L^{\bar {\delta }}}$ -type, absurd.

Theorem 10.3. $\mathcal C_{\bar {\delta }}$ is an elementary L-substructure of $\mathbb K$ .

Let $\bar b \in K^{\ell }$ and $X \subseteq \mathcal C_{\bar {\delta }}^{n}$ be ${L^{\bar {\delta }}}$ -definable with parameters $\bar b$ . Then, there exists $Y \subseteq K^{n}$ which is L-definable (in $\mathbb K$ ) with parameters $ {\mathrm {Jet}}(\bar b)$ such that $X = Y \cap \mathcal C_{\bar {\delta }}^{n}$ . If moreover $\bar b \in \mathcal C_{\bar {\delta }}$ , then X is L-definable in $\mathcal C_{\bar {\delta }}$ with parameters $\bar b$ : equivalently, there exists $Y \subseteq K^{n}$ which is L-definable in $\mathbb K$ with parameters $\bar b$ such that $X = Y \cap K^{n}$ .

Proof. The theorem is a reformulation of [Reference Fornasiero16, Lemma 8.22] (see also [Reference Berenstein and Vassiliev4, Proposition 3.4]), using Lemma 10.2.

Notice that, by Corollary 6.2, $\mathcal C_{\bar {\delta }}$ is algebraically closed in $\mathbb K$ w.r.t. the ${L^{\bar {\delta }}}$ -structure.

Definition 10.4. A basic formula is a formula of the form

$$\begin{align*}\exists \bar y\, \big( \bar y \in K^{\ell} \wedge \psi(\bar x, \bar y) \big), \end{align*}$$

where $\psi $ is an L-formula. A basic set is a set definable by a basic formula (with parameters from $\mathbb K$ ).

Theorem 10.5. Let $Z \subseteq K^{n}$ be definable in $\langle \mathbb K,\mathcal C_{\bar {\delta }} \rangle $ with parameters from $\mathbb K$ . Then, Z is a finite Boolean combination of basic sets, with the same parameters as Z.

The Theorem is a reformulation of [Reference Berenstein and Vassiliev4, Corollary 3.2] and [Reference Fornasiero16, Theorem 8.5].

Remark 10.6. Let $\langle A,B \rangle $ be a lovely pair of geometric structures, with $A \models T$ . Then, there exists $B^{*} \succeq B$ and a derivation $\delta ^*$ on $B^{*}$ such that $\langle B^{*}, \delta ^{*} \rangle \models T^{\delta }_g$ and $\langle B^{*}, A^{*} \rangle \succeq \langle B,A \rangle $ , where $A^{*} := \mathfrak C_{\delta ^{*}} $

Proof. The theory $T^{lovely}$ of lovely pairs of models of T is complete (see [Reference Berenstein and Vassiliev4]). Let $\langle B^{*}, A^{*} \rangle \succeq \langle B,A \rangle $ be a 0-big (a.k.a. “splendid”: see [Reference Hodges26]). By bigness, there exists a derivation $\delta ^{*}$ on $B^{*}$ satisfying the conclusion.

[Reference Kovacsics and Point32, Section 5] use a particular case of the above remark to (re-)prove some results about lovely pairs.

For more results (in particular on imaginaries in $\langle \mathbb K,\mathcal C_{\bar {\delta }} \rangle $ ) see [Reference Berenstein and Vassiliev4, Reference Boxall5, Reference Fornasiero16].

11 Open core

The open core of a topological structure is the structure generated by the sets which are both open and definable. We investigate expansions of algebraically bounded topological structures with generic derivations: more precisely we prove that, under very weak assumptions, T is the open core of $T^{\bar {\delta },?}_g$ (Corollary 11.11).

By a topology $\tau $ on T we mean an assignment, to every $\mathbb K \models T$ and every $n \in \mathbb {N}$ , of a topology $\tau _{n, \mathbb K}$ on $K^{n}$ .

Definition 11.1 [Reference Pillay41].

$\tau $ is definable if, for every $n\in \mathbb {N}$ , there exists an L-formula (without parameters) $\alpha _{n}(\bar x,\bar y)$ , where $\bar x$ is an n-tuple and $\bar y$ is a finite tuple, such that, for every $\mathbb K \models T$ , we have that

$$\begin{align*}\{\alpha(\mathbb K,\bar b): \bar b \in K^{\lvert\bar y\rvert}\} \end{align*}$$

is a basis of open sets for $\tau _{n, \mathbb K}$ . The family $\{\alpha (\mathbb K,\bar b): \bar b \in K^{\lvert \bar y\rvert }\}$ is a definable basis for the topology.

If we give only a topology on K, we assume that the topology on $K^{n}$ is the product topology.

Definition 11.2. T is an open core of $T^{\bar {\delta },?}_g$ if, for every $\langle \mathbb K, \bar {\delta } \rangle \models T^{\bar {\delta },?}_g$ and every $n \in \mathbb {N}$ , every $\tau $ -open ${L^{\bar {\delta }}}(K)$ -definable subset of $K^{n}$ is already $L(K)$ -definable.

We give: sufficient conditions for T being an open core of $T^{\bar {\delta },?}_g$ (see Assumption 11.7 and Theorem 11.9) and for $T^{\bar {\delta },?}_g$ to have EI (see Theorem 11.10).

In an article in preparation we will present another approach, with slightly different assumptions and a very different proof.

Let $\mathbb K \models T$ and $n \in \mathbb {N}$ . Let $X \subseteq K^{n}$ be an $L(K)$ -definable set.

Definition 11.3. Given $\bar a \in K^{n}$ , the Local Dimension of X at $\bar a$ is

$$\begin{align*} {\mathrm{loc-dim}}_{\bar a}(X) := \min\{ \dim(X \cap U): U \text{ open definable set containing } {\bar a}\}. \end{align*}$$

The Local Dimension of X is

$$\begin{align*} {\mathrm{loc-dim}}(X) := \max\{ {\mathrm{loc-dim}}_{\bar a}(X): \bar a \in X \}. \end{align*}$$

We say that X has constant local dimension d if, for every $\bar a \in X$ , $ {\mathrm {loc-dim}}_{\bar a}(X) = d$ .

Remark 11.4. $ {\mathrm {loc-dim}}(X) \leq \dim (X)$ .

Definition 11.5. $\dim $ is local (w.r.t. to $\tau $ ) if, for every $\mathbb K \models T$ , for every $n \in \mathbb {N}$ , and every $X \subseteq K^{n} L(K)$ -definable, $\dim (X) = {\mathrm {loc-dim}}(X)$ .

[Reference Fornasiero and Halupczok18] gives sufficient conditions for dim to be local. It gives also an example where it is not local (the Sorgenfrey plane).

Fact 11.6 [Reference Fornasiero and Halupczok18].

Assume that $\tau $ is definable, the topology on $K^{n}$ is the product topology (for every $n \in \mathbb {N}$ ), $\langle K,+ \rangle $ is a Hausdorff topological group (for every $\mathbb K \models T$ ), and every open definable set is large, i.e., it has the same dimension of ambient space.

Then, $\dim $ is local.

Assumptions 11.7. We assume that, for every $n,m \in \mathbb {N}$ and $\mathbb K \models T$ ,

1) $\tau $ is definable;
2) the projection map $K^{n} \times K^{m} \to K^{n}$ is continuous;
3) $\tau _{n, \mathbb K}$ is invariant under permutation of coordinates;
4) $\dim $ is local.

Examples 11.8. The following topologies satisfy the assumptions:

1) $T = {\mathrm {RCF}}$ with the usual Euclidean topology.
2) T is a theory of Henselian valued fields with the valuation topology.

Theorem 11.9 (Open Core).

T is the open core of $T^{\bar {\delta },?}_g$ .

More precisely: let $\langle \mathbb K, \bar {\delta } \rangle \models T^{\bar {\delta },?}_g$ and $A \subseteq K$ s.t. ${\bar \delta }(A) \subseteq A$ . Let $X \subseteq K^{n}$ be ${L^{\bar {\delta }}}(A)$ -definable and closed. Then, X is $L(A)$ -definable.

We give a proof later in Section 11.1. Particular cases of Theorem 11.9 were already known (see [Reference Kovacsics and Point32, Section 6]).

Theorem 11.10. Assume moreover that $\tau $ satisfies the following condition:

5) If X is $L(\mathbb K)$ -definable and nonempty, then $\dim (\overline X \setminus X) < \dim (X)$ , where $\overline X$ is the topological closure of X.

Then, $T^{\bar {\delta },?}_g$ has EI modulo $T^{eq}$ .

Proof. With trivial modifications, the proof suggested by M. Tressl in [Reference Fornasiero and Kaplan19] works.

For the following corollary, we spell all conditions explicitly.

Corollary 11.11. Let T be algebraically bounded, with a definable topology $\tau $ . Assume that $\tau $ is a non-trivial and non-discrete ring topology. Then, T is the open core of $T^{\bar {\delta },?}_g$ .

If moreover $\tau $ and satisfies Assumption 5) in Theorem 11.10, then $T^{\bar {\delta },?}_g$ has EI modulo $T^{eq}$ .

Proof. Since $\tau $ is a non-trivial ring topology, it is Hausdorff (see, e.g., [Reference Warner51, Theorem 11.10]).

We only need to show that $\dim $ is $\tau $ -local.

By Fact 11.6, it suffices to show that every nonempty open subset of $K^{n}$ is large.

For $n=1$ , since $\tau $ is Hausdorff and non-discrete, every nonempty open subset of K is infinite and hence large.

For $n> 1$ , let $U \subseteq K^{n}$ be open and nonempty. Let $B_{1}, \dotsc , B_{n} \subseteq K$ be open and nonempty s.t. $B := B_{1} \times \dots \times B_{n} \subseteq K^{n}$ . Since, by the case $n=1$ , each $B_{i}$ is large, then B is large.

11.1 Proof of Theorem 11.9

We present some preliminary results before giving the proof.

Fix $n \in \mathbb {N}$ and $\bar x = \langle x_{1}, \dotsc , x_{n} \rangle $ . We use the same notation as in Section 2.

We need to endow $K^{\Gamma _{n}}$ with a topology. For every J finite subset of $\Gamma _{n}$ of cardinality $m \in \mathbb {N}$ , $\tau _{m, \mathbb K}$ is a topology on $K^{m}$ and hence on $K^{J}$ (it does not depend on how we enumerate the elements of J, since we assumed that $\tau _{m, \mathbb K}$ is invariant under permutation of coordinates). We have a natural map $\Pi _{J}: K^{\Gamma _{n}} \to K^{J}$ . We define the topology $\tau _{\mathbb K,\Gamma _{n}}$ as the coarsest topology making all the maps $\Pi _{J}$ continuous: equivalently, a basis of $\tau _{\mathbb K, \Gamma _{n}}$ is given by

$$\begin{align*}\{\Pi_{J}^{-1}(U): U \subseteq K^{J} \text{ open}, J \subset_{fin} \Gamma_{n}\}. \end{align*}$$

Let $\mathbb K \models T$ be a monster model (i.e., $\lambda $ -saturated and $\lambda $ -homogeneous for some sufficiently large cardinal $\lambda> \aleph _{0} + \lvert L\rvert $ ) and $A \subset K$ be a “small” subset (i.e., $\lvert A\rvert < \lambda $ ).

Let $J \subseteq \Gamma _{n}$ (J could be finite or infinite); again, we consider J as a set of indices and $x_{J}$ as a tuple of variables (see Section 2.2)

A partial type $p(x_{J})$ over A determines a (type-definable) subset $Z \subseteq K^{J}$ . If J is finite, we can define the dimension of Z as

$$\begin{align*}\dim(Z) := \min \{\dim(\alpha(\mathbb K): \alpha \in p\} \in \mathbb{N}. \end{align*}$$

If however J is infinite, the dimension of Z (defined in the “obvious” way) may be infinite (e.g., $\dim (K^{J})$ is infinite). We need to define when $Z'$ is a large subset of Z even when $\dim (Z)$ is infinite.

Definition 11.12. Let $Y \subseteq Z$ be subsets of $K^{J}$ which are type-definable (over some small $A \subset K$ ). We say that Y is large in Z if, for every $J' \subseteq J$ with $J'$ finite, $\dim (\Pi _{J'}(Y)) = \dim (\Pi _{J'}(Z))$ .

Given a partial type $p(x_J)$ , we denote by $p(\mathbb K)$ the set of realizations of p in $\mathbb K^{J}$ , and $\dim (p(x_J)) := \dim (p(\mathbb K)$ ). Given $J' \subseteq J$ , we denote by $p(J')$ the partial type corresponding to the set $\Pi _{J'}(p(\mathbb K))$ .

Let $q \in S^{J}(A)$ be a complete type, of dimension d.

Lemma 11.13. Assume that J is finite.

Then, there is a family $\{X_{i}: i \in I\}$ of $L(A)$ -definable sets such that:

1) $q(\mathbb K) = \bigcap _{i \in I} X_{i}$ ;
2) each $X_{i}$ has constant local dimension d.

Proof. Let $\{Y_{i}: i \in I\}$ be a family of $L(A)$ -definable sets such that: $q(\mathbb K) = \cap \{Y_{i}: i \in I\}$ . Let Z be some $L(A)$ -definable set of dimension d containing $q(\mathbb K)$ : by replacing $Y_{i}$ with $Y_{i} \cap Z$ , w.l.o.g. we may assume that all $Y_{i}$ have dimension d. Given an $L(A)$ -definable set $Y \subseteq K^{J}$ of dimension d, let

$$\begin{align*} {\mathrm{Sing}}(Y) := \{\bar a \in Y: {\mathrm{loc-dim}}_{\bar a}(Y) < d\}\\ \text{ and } {\mathrm{Reg}}(Y) := Y \setminus {\mathrm{Sing}}(Y). \end{align*}$$

Since $\dim $ is definable, $ {\mathrm {Sing}}(Y)$ and $ {\mathrm {Reg}}(Y)$ are $L(A)$ -definable. Since $\dim $ is local, $\dim ( {\mathrm {Sing}}(Y)) < d$ , and therefore $\dim ( {\mathrm {Reg}}(Y)) = d$ and $ {\mathrm {Reg}}(Y)$ has constant local dimension d.

Claim 6. For every $i \in I$ , $q(\mathbb K) \subseteq {\mathrm {Reg}}(Y_{i})$ .

Assume not: then, since q is a complete type over A, $q(\mathbb K) \subseteq {\mathrm {Sing}}(Y_{i})$ , but then $\dim (q) \leq \dim ( {\mathrm {Sing}}(Y_{i})) < d$ , absurd.

Thus, we can define $X_{i} := {\mathrm {Reg}}(Y_{i})$ .

Notice that in the above lemma we used that $\dim $ is definable.

Lemma 11.14. Let $U \subseteq K^{J}$ be open and definable (with parameters in some small set B with $A \subseteq B \subset K$ ). If $U \cap q(\mathbb K^{J})$ is nonempty, then it is large inside $q(\mathbb K^{J})$ .

Proof. Since U is definable, there is some $J_{0} \subseteq J$ finite and $V \subseteq K^{J_{0}} L(A)$ -definable and open, such that $U = \pi _{J_{0}}^{-1}(V)$ . Let $J_{0} \subseteq J' \subseteq J$ , with $J'$ finite. We need to prove that $\Pi _{J'}(U \cap q(K^{J'}))$ is large inside $\Pi _{J'}(q(\mathbb K))$ .

The former is equal to $U' \cap q(\mathbb K(J'))$ , where $U' := \Pi _{J'}(U')$ : thus, by replacing U with $U'$ , J with $J'$ , and $q(x_{J})$ with $q(x_{J'})$ , we may assume that $J'$ is finite. Thus, by Lemma 11.13, we may assume that $q(\mathbb K) = \bigcap _{i}X_{i}$ , where each $X_{i}$ is an $L(A)$ -definable set of constant local dimension d. Thus, since $X_{i} \cap U$ is nonempty, $X_{i} \cap U$ has (local) dimension d, and therefore $\dim (\bigcap _{i} X_{i} \cap U) = d$ .

Proof of Theorem 11.9.

W.l.o.g., we may assume that $\langle \mathbb K, \bar {\delta } \rangle $ is a monster model and A is “small”. By Beth definability, it suffices to show that X is $L(A)$ -invariant (i.e., that X is set-wise invariant under automorphisms of $\mathbb K$ as L-structure fixing A point-wise).

Let $Z := {\mathrm {Jet}}(X) \subseteq K^{\Gamma _{n}}$ and Y be the closure of Z (according to the topology $\Pi _{\mathbb K, \Gamma _{n}}$ ). Since X is closed, we have that $X = \Pi _{n}(Y)$ (where $\Pi _{n}$ is the projection onto the first n coordinates). Thus, it suffices to show that Y is $L(A)$ -invariant.

Let $\bar a, \bar a' \in K^{\Gamma _{n}}$ , with the same $L(A)$ -type, and such that $\bar a \in Y$ . We need to show that $\bar a' \in Y$ . Since $\tau _{\mathbb K, n}$ is definable and Y is closed, it suffices to prove the following.

Claim 7. Let $U' \subseteq K^{\Gamma _{n}}$ be an $L(\mathbb K)$ -definable open set containing $\bar a'$ . Then, $U'$ intersects Y.

Let $\phi $ be an $L(A)$ -automorphism of $\mathbb K$ such that $\phi (\bar a') = \bar a$ . Let $U := \phi (U')$ : notice that U is an open $L(A)$ -definable set containing $\bar a$ ; therefore, there exists $\bar b \in X$ such that $ {\mathrm {Jet}}(\bar b) \in U$ . Let $\bar c':= \phi ^{-1}( {\mathrm {Jet}}(\bar b))$ .

Let $q(\bar x_{\Gamma })$ be the $L(A)$ -type of $ {\mathrm {Jet}}(\bar b)$ : by definition, it is also the $L(A)$ -type of $\bar c'$ . By Lemma 11.14, $q(\mathbb K) \cap U'$ is large inside $q(\mathbb K)$ . Thus, by the Extension Theorem, there exists $\bar b' \in K^{n}$ such that

$$\begin{align*} {\mathrm{Jet}}(\bar b') \in q(\mathbb K) \cap U'. \end{align*}$$

Therefore, $\bar b' \in X$ and therefore $ {\mathrm {Jet}}(\bar b') \in Z \cap U' \subseteq Y \cap U'$ .

12 Differential dimension

12.1 The commutative case

Let $\langle A, \bar {\delta } \rangle $ be a field (of characteristic $0$ ) with k commuting derivations. The derivations induce a matroid on A. Given $a \in A$ , $Y \subseteq A$ , and $X\subseteq A$ , we define $a \in {{\bar {\delta }}\text{-}\mathrm {acl}}_{Y}(X)$ if $ {\mathrm {Jet}}(a)$ is not algebraically independent over $ {\mathrm {Jet}}(X \cup Y)$ .

As shown in [Reference Fornasiero and Kaplan19], ${{\bar {\delta }}\text{-}\mathrm {acl}}_{Y}$ is a matroid on A.Footnote ⁶

Fix $\langle \mathfrak C, \bar {\delta } \rangle $ monster model of $T^{\bar {\delta }}_g$ . We have the corresponding matroid ${{\bar {\delta }}\text{-}\mathrm {acl}}(X) := {{\bar {\delta }}\text{-}\mathrm {acl}}_{F}(X)$ .

Theorem 12.1. ${{\bar {\delta }}\text{-}\mathrm {acl}}$ is an existential matroid (in the sense of [Reference Fornasiero16]).

Proof. We have to prove that ${{\bar {\delta }}\text{-}\mathrm {acl}}$ is definable and it satisfies existence (see [Reference Fornasiero16, Section 3]).

The fact that ${{\bar {\delta }}\text{-}\mathrm {acl}}$ is definable means that, for every $A \subseteq \mathfrak C$ and $b \in {{\bar {\delta }}\text{-}\mathrm {acl}}(A)$ there exists an $L^{\bar {\delta }}$ -formula $\phi (\bar x, y)$ and $\bar a \in A^{n}$ such that $\langle \mathfrak C, \bar {\delta } \rangle \models \phi (\bar a,b)$ and, for every $\bar a',b'$ in $\mathfrak C$ , if $\langle \mathfrak C, \bar {\delta } \rangle \models \phi (\bar a', b')$ , then $b' \in {{\bar {\delta }}\text{-}\mathrm {acl}}(\bar a')$ . We can take as $\phi $ any formula witnessing that $b^{ {\mathrm {Jet}}}$ is not algebraically independent over A.

For existence, let $A \subseteq B \subset \mathfrak C$ be subsets of small cardinality. Let $c \in \mathfrak C$ such that $c \notin {{\bar {\delta }}\text{-}\mathrm {acl}}(A)$ . We have to show that there exists $d \in \mathfrak C$ such that c and d have the same $L^\delta $ -type over A and $d \notin {{\bar {\delta }}\text{-}\mathrm {acl}}(B)$ .

Since satisfies existence, there exists $d \in \mathfrak C$ such that c and d have the same $L^\delta $ -type over A and . Then, $ {\mathrm {Jet}}(d)$ is algebraically independent over $ {\mathrm {Jet}}(B)$ : therefore, $d \notin {{\bar {\delta }}\text{-}\mathrm {acl}}(B)$ , proving that ${{\bar {\delta }}\text{-}\mathrm {acl}}$ is an existential matroid.

Thus, ${{\bar {\delta }}\text{-}\mathrm {acl}}$ induces a dimension function ${{\bar {\delta }}\text {-}\mathrm {dim}}$ on models of $T^{\bar {\delta }}_g$ (see [Reference Fornasiero16]; see also [Reference Guzy and Point24]).

Remark 12.2.

1) ${{\bar {\delta }}\text{-}\mathrm {acl}}$ is not the $T^{\bar {\delta }}_g$ -algebraic closure: the former only contains the latter. For instance, the whole field of constants $\mathcal C_{\bar {\delta }}$ is in ${{\bar {\delta }}\text{-}\mathrm {acl}}(\emptyset )$ .
2) is not the independence relation induced by ${{\bar {\delta }}\text{-}\mathrm {acl}}$ , because is strict. For instance, if $a \in {\mathcal {C}}_{\bar {\delta }} \setminus {\mathrm {acl}}(F)$ , then .

Lemma 12.3 (See [Reference Eleftheriou, Sánchez and Regnault14]).

Let $\langle \mathbb K, \bar {\delta } \rangle \models T^{\bar {\delta }}_g$ . Let $Y \subseteq K^{n}$ be L-definable (with parameters). Then, $\dim (Y) = {{\bar {\delta }}\text {-}\mathrm {dim}}(Y)$ .

Proof. By the properties of dimension functions (see [Reference van den Dries13]) it suffices to treat the case when $n = 1$ (the general case follows by induction on n). If $\dim (X) = 0$ , then X is finite, and therefore ${{\bar {\delta }}\text {-}\mathrm {dim}}(X) = 0$ . If $\dim (X) = 1$ , then $(X - X)/(X - X) = \mathbb K$ , and therefore ${{\bar {\delta }}\text {-}\mathrm {dim}}(X) = 1$ .

The same proof gives a more general result.

Proposition 12.4 (Invariance of dimension for fields).

Let L be a language expanding the language of rings, and $L^{*}$ be an expansion of L. Let $A^{*}$ be an $L^{*}$ -structure expanding a field, and A be its restriction to the language L. Assume that $\dim ^{*}$ and $\dim $ be dimension functions on $A^{*}$ and A, respectively. Then, for every $X \subseteq A^{n}$ which is L-definable (with parameters), $\dim ^{*}(X) = \dim (X)$ .

Unlike in the case of lovely pairs, we cannot approximate $L^\delta $ -definable sets with L-definable sets.

Remark 12.5. Let $\langle \mathbb K, \delta \rangle \models T^{\delta }_g$ . Let $X \subseteq K$ be $L^\delta $ -definable (with parameters). If X is definable in the lovely pair $\langle \mathbb K, \mathcal C_{\bar {\delta }} \rangle $ (see Section 10), then there exists $Y \subseteq K$ which is L-definable and such that ${{\bar {\delta }}\text {-}\mathrm {dim}}(X \Delta Y) < 1$ ([Reference Fornasiero16, Proposition 8.36]). If not, such Y might not exist: for instance, let $\mathbb K$ be a real closed field, and $X := \{x \in K: \delta x> 0\}$ .

12.2 The non-commutative case

The assumption that the derivations commute cannot be dropped.

Lemma 12.6. If the derivations do not commute, then ${{\bar {\delta }}\text{-}\mathrm {acl}}_{Y}$ is not a matroid, because it is not transitive.Footnote ⁷ In fact, let $k = 2$ and $\langle \mathbb K, \bar {\delta } \rangle \models T^{\bar {\delta },nc}_g$ . Then, there exist $a, b, c \in K$ such that:

1) $a^{\Gamma }$ is algebraically independent over F;
2) $\delta _{2} b = 0$ and $\delta _{1} b = \delta _{1} a$ ;
3) $c = a - b$ .

Notice that $\delta _{1}c = 0$ . Then, $a \notin {{\bar {\delta }}\text{-}\mathrm {acl}}(F)$ , $b,c \in {{\bar {\delta }}\text{-}\mathrm {acl}}(F)$ , but $a \in {{\bar {\delta }}\text{-}\mathrm {acl}}_{F}(b,c)$ : thus, transitivity fails.

Lemma 12.7. For $ k\geq 2$ , models of $T^{\bar {\delta },nc}_g$ do not have a dimension function.

Proof. For simplicity, we do the case when $k = 2$ . Let $\langle \mathbb K, \bar {\delta } \rangle \models T^{\bar {\delta },nc}_g$ . Assume, by contradiction, that $\dim '$ is a dimension function on $\langle \mathbb K, \bar {\delta } \rangle $ . Let $X := \{b \in K: \delta _{1} b = 0\}$ . Let $Y := \{c \in K: \delta _{2} c = 0\}$ . Notice that X and Y are $L^\delta $ -definable subfields of K of infinite index inside $\mathbb K$ : thus, $\dim '(X) = \dim '(Y) = 0$ .

Claim 8. $X + Y = K$ .

Let $a \in K$ . Let $b \in K$ such that $\delta _{1} b = 0$ and $\delta _{2} b = \delta _{2} a$ , and let $c = a - b$ . Notice that $b \in X$ and $c \in Y$ . Thus $X + Y = K$ .

But $\dim '(X) = \dim '(Y) = 0$ , and therefore $\dim '(K) = 0$ , while the axioms of dimension require that $\dim '(K) = 1$ .

13 Genericity

We denote by $K^{K}$ the set of all functions from K to K, and by ${\mathrm {Der}_{\mathrm {K}}} \subset K^{K}$ the set of derivations on K extending $\eta $ (remember that $\eta $ is a fixed derivation on the field $F = dcl(\emptyset )$ : see Section 2.1). The main references for this section is [Reference Hjorth25], from which our presentation is heavily inspired; for the background notions of descriptive set theory see [Reference Kechris29].

For every $\bar a, \bar b \in K^{n}$ , we define

$$\begin{align*}B_{\bar a, \bar b} := \{\delta \in K^{K}: \delta(\bar a) = \bar b\}. \end{align*}$$

For every $L^\delta $ -sentence $\phi $ with parameters in $\mathbb K$ , we define

$$\begin{align*}U_{\phi} := \{\delta \in K^{K}: \langle \mathbb K, \delta \rangle \models \phi\}. \end{align*}$$

The set $K^{K}$ has two “canonical” topologies:

• The pro-discrete topology, whose basis of open sets is given by
$$\begin{align*}\{B_{\bar a,\bar b}: \bar a, \bar b \in K^{n}, n \in \mathbb{N}\}, \end{align*}$$
and which we denote by $\tau _d$ , which is the topology induced by the product topology on $K^{K}.$
• The “first-order” topology, whose basis of open sets is given by
$$\begin{align*}\{U_{\phi}: \phi\ L^\delta\text{-}\text{sentence with parameters in } \mathbb{K}\}, \end{align*}$$
and which we denote by $\tau _{FO}$ .

Remark 13.1. Another basis for $\tau _d$ is

$$\begin{align*}\{U_{\phi}: \phi \text { quantifier-free } L^\delta\text{-}\text{sentence with parameters in } \mathbb {K}\}. \end{align*}$$

In fact,

$$\begin{align*}B_{\bar a,\bar b} = U_{(\delta a_{1} = b_{1} \wedge \dotsb \wedge \delta a_{n} = b_{n})}. \end{align*}$$

For the remainder of this section, when we don’t specify the topology, we mean $\tau _d$ . The following notions concerning specific formulas are useful for obtaining information about sets with respect to the topology.

Definition 13.2. We say that an $L^\delta $ -sentence $\phi $ with parmeters $\bar a$ is “relatively quantifier free” if $\phi = \alpha ( {\mathrm {Jet}}_{\delta }^{}(\bar a))$ for some L-formula without parameters $\alpha $ . We say $\phi $ is “relatively existential” if

$$\begin{align*}\phi = \exists \bar x\ \alpha( {\mathrm{Jet}}(\bar a), {\mathrm{Jet}}(\bar x)), \end{align*}$$

for some L-formula without parameters $\alpha $ . We can define “relatively universal” and “relatively $\forall \exists $ ” $L^\delta $ -sentences with parameters.

Lemma 13.3. Let $\phi $ be an $L^\delta $ sentence with parameters.

• If $\phi $ is relatively quantifier free, then $U_{\phi }$ is clopen.
• If $\phi $ is relatively existential, then $U_{\phi }$ is open.
• If $\phi $ is relatively universal, then $U_{\phi }$ is closed.
• If $\phi $ is relatively $\forall \exists $ , then $U_{\phi }$ is $\mathcal G_{\delta }$ .

Proof. We do only the case when $\phi $ is relatively existential: the others are similar. Write $\phi = \exists \bar y\ \alpha (\bar a, \delta \bar a, \dotsc , \delta ^{n}\bar a, \bar b, \delta \bar b, \dotsc , \delta ^{m} \bar b)$ , for some L-formula $\phi $ . Then,

$$ \begin{align*} &U_{\phi} = \bigcup \big( B_{\bar a,\bar a_{1}} \cap B_{\bar a_{1},\bar a_{2}} \cap \dotsb \cap B_{\bar a_{n-1},\bar a_{n}} \cap B_{\bar b,\bar b_{1}} \cap B_{\bar b_{1},\bar b_{2}} \cap \dotsb \cap B_{\bar b_{m-1},\bar b_{m}}:\\ &\bar a_{1}, \dotsc, \bar a_{n}, \bar b_{1}, \dotsc, \bar b_{m} \in K^{< \omega} \wedge \langle \mathbb K, \delta \rangle \models \alpha(\bar a, \bar a_{1}, \dotsc, \bar a_{n}, \bar b, \bar b_{1}, \dotsc, \bar b_{m}) \big). \end{align*} $$

StartLayout 1st Row 1st Column Blank 2nd Column upper U Subscript phi Baseline equals union left parenthesis upper B Subscript a overbar comma a overbar Sub Subscript 1 Subscript Baseline intersection upper B Subscript a overbar Sub Subscript 1 Subscript comma a overbar Sub Subscript 2 Subscript Baseline intersection midline horizontal ellipsis intersection upper B Subscript a overbar Sub Subscript n minus 1 Subscript comma a overbar Sub Subscript n Subscript Baseline intersection upper B Subscript b overbar comma b overbar Sub Subscript 1 Subscript Baseline intersection upper B Subscript b overbar Sub Subscript 1 Subscript comma b overbar Sub Subscript 2 Subscript Baseline intersection midline horizontal ellipsis intersection upper B Subscript b overbar Sub Subscript m minus 1 Subscript comma b overbar Sub Subscript m Subscript Baseline colon 2nd Row 1st Column Blank 2nd Column a overbar Subscript 1 Baseline comma ellipsis comma a overbar Subscript n Baseline comma b overbar Subscript 1 Baseline comma ellipsis comma b overbar Subscript m Baseline element of upper K Superscript less than omega Baseline and left angle bracket double struck upper K comma delta right angle bracket true alpha left parenthesis a overbar comma a overbar Subscript 1 Baseline comma ellipsis comma a overbar Subscript n Baseline comma b overbar comma b overbar Subscript 1 Baseline comma ellipsis comma b overbar Subscript m Baseline right parenthesis right parenthesis period EndLayout

For the remainder of this section, we assume that $\mathbb K$ and L are countable.

Thus, ${\mathrm {Der}_{\mathrm {K}}}$ is $\tau _{FO}$ -closed and it is a $\tau _d$ - $\mathcal G_{\delta }$ inside $K^{K}$ ; we use the same names for the induced topologies on ${\mathrm {Der}_{\mathrm {K}}}$ . Notice that $K^{K}$ is a Polish space: therefore, ${\mathrm {Der}_{\mathrm {K}}}$ is also a Polish space (see [Reference Kechris29]). Thus, any two dense $\mathcal G_{\delta }$ subsets of ${\mathrm {Der}_{\mathrm {K}}}$ always intersect.

Given $Z \subseteq K^{n} \times K^{n}$ , we define

$$\begin{align*}I_{Z} := \{ \delta \in {\mathrm{Der}_{\mathrm{K}}}: \exists \bar b \in K^{n}: \langle \bar b,\delta \bar b \rangle \in Z \}. \end{align*}$$

Lemma 13.4. For every $Z \subseteq K^{n} \times K^{n}$ , $I_{Z}$ is an open subset of ${\mathrm {Der}_{\mathrm {K}}}$ .

Proof.

$$\begin{align*}I_{Z} = \bigcup\big( B_{\bar a,\bar b} : \bar a \in K^{n}, \bar b \in K^{n}, \langle \bar a, \bar b \rangle \in Z \big). \end{align*}$$

Let $\mathbb G$ be the family of derivations $\delta \in {\mathrm {Der}_{\mathrm {K}}}$ such that $\langle \mathbb K, \delta \rangle \models T^{\delta }_g$ . Let $\mathcal L$ be the family of the sets $Z \subseteq K^{n + n}$ definable with parameters, such that $\Pi _{n}(Z)$ is large (for some $n \in \mathbb {N}$ ).

Lemma 13.5. $\mathbb G(M) = \bigcap _{Z \in \mathcal L} I_{Z}$ . Moreover, $\mathbb G$ is a $\mathcal G_{\delta }$ -subset of ${\mathrm {Der}_{\mathrm {K}}}$ .

Proof. By the axiomatization ${T^{\delta }_{\mathrm {wide}}}$ (where we introduced in [Reference Fornasiero and Terzo21, Definition 3.3]), $\mathbb G(M) = \bigcap _{Z \in \mathcal L} I_{Z}$ . Each $I_{Z}$ is open. By our assumptions, $\mathcal L$ is countable.

Lemma 13.6. On $\mathbb G$ , $\tau _{FO}$ and $\tau _d$ coincide.

Proof. By elimination of quantifiers, every $L^\delta $ -sentence is equivalent, modulo $T^{\delta }_g$ , to a relatively quantifier-free sentence. The conclusion follows from Lemma 13.3.

Lemma 13.7. Assume that $ {\mathrm {rk}}(K / F)$ is infinite. Then, for every $\bar a$ finite tuple in K and every W large subset of $K^{n}$ which is L-definable with parameters, there exists $\bar b \in W$ which is algebraically independent over $F\bar a$ .

Proof. By induction on n, it suffices to treat the case when $n = 1$ . Let $b \in K \setminus {\mathrm {acl}}(F\bar a)$ . Since $W \subseteq K$ is large, then there exists $b_1, b_2, b_3, b_4 \in W$ such that $(b_{1} - b_{2})/(b_{3} - b_{4}) = b$ and $b_3 \not = b_4$ . Therefore, at least one of the $b_{i}$ is not in $ {\mathrm {acl}}(F\bar a)$ .

Theorem 13.8. There exists $\mathbb K \models T$ which is countable and of infinite rank over F. For any such $\mathbb K$ , the set $\mathbb G$ is a dense subset of ${\mathrm {Der}_{\mathrm {K}}}$ .

Thus, in a precise topological sense, $\mathbb G$ is a generic set (notice that $\mathbb G$ is $\tau _{FO}$ -closed in ${\mathrm {Der}_{\mathrm {K}}}$ ).

Proof. We have seen that each $I_{Z}$ is open. It suffices to prove the following claim.

Claim 9. For every $Z \in \mathcal L$ , $I_{Z}$ is dense.

Let $Z \subseteq K^{n} \times K^{n}$ . Let $B_{\bar a, \bar b}$ be a nonempty basic open set. We have to verify that $I_{Z} \cap B_{\bar a, \bar b}$ is nonempty. Let $\varepsilon \in B_{\bar a, \bar b}$ : that is, $\varepsilon \in {\mathrm {Der}_{\mathrm {K}}}$ and $\varepsilon \bar a = \bar b$ . Let $\varepsilon _{0}$ be the restriction of $\varepsilon $ to $ {\mathrm {acl}} (F\bar a)$ . Let $\bar c \in \Pi _{n}(Z)$ be algebraically independent over $F\bar a$ . We can extend $\varepsilon _{0}$ arbitrarily to $\bar c$ ; in particular, there exists $\delta \in {\mathrm {Der}_{\mathrm {K}}}$ such that $\delta $ extends $\varepsilon _{0}$ and $\delta \bar c \in Z$ . Thus, $\delta \in I_{Z} \cap B_{\bar a, \bar b}$ .

The following theorem gives a “topological” criterion for when a differential system has a solution in models of $T^{\delta }_g$ .

Theorem 13.9. Let $\langle \mathbb K, \varepsilon \rangle \models T^{\delta }$ . Assume that $\mathbb K$ countable and of infinite rank over F. Let $\bar a \in K^{\ell }$ . Let ${\mathrm {Der}_{\mathrm {K}}}(\bar a, \varepsilon )$ be the set of derivations $\delta $ on K extending $\eta $ and such that $\varepsilon $ and $\delta $ coincide on $ {\mathrm {Jet}}_{\epsilon }(\bar a)$ , the Jet relative to the derivation $\epsilon .$

Let $Z \subseteq K^{n} \times K^{n}$ be L-definable with parameters $\bar a$ . Let $\delta $ be a derivation in K such that $\langle \mathbb K, \delta \rangle \models T^{\delta }_g$ . T.f.a.e.:

1) $I_{Z}$ is dense in ${\mathrm {Der}_{\mathrm {K}}}(\bar a, \varepsilon )$ ;
2) $I_Z$ is nonempty;
3) $I_{Z} \cap \mathbb G$ is nonempty;
4) $\delta \in I_{Z}$ .

Proof. First of all it is easy to see that $(1) \implies (2),$ and $(4) \implies (3)$ are obvious. Moreover (2) is equivalent to (3), since $I_{Z}$ by Lemma 13.4 is open and $\mathbb G$ by Theorem 13.8 is dense in ${\mathrm {Der}_{\mathrm {K}}}$ .

We prove first the case when $\bar a$ is empty (i.e., Z is L-definable without parameters and ${\mathrm {Der}_{\mathrm {K}}}(\bar a, \varepsilon ) = {\mathrm {Der}_{\mathrm {K}}}$ ).

In this case, we can add another equivalent formulation to 4:

5) $\mathbb G \subseteq I_{Z}$ .

Since $T^{\delta }_g$ is complete, and “ $\delta \in I_{Z}$ ” can be expressed as a first-order sentence (without parameters), we have that $(4) \implies (5),$ the converse is trivial so (4) and (5) are equivalent. Therefore, (1) is equivalent to (2).

Let us consider now the case when $\bar a$ is non-empty. Let $F' := F[ {\mathrm {Jet}}_{\epsilon }(\bar a)]$ and let $\eta '$ be the restriction of $\varepsilon $ to $F'$ . We denote by $L' := L(F)$ , and $T' := T \cup {\mathrm {Diag}}(F')$ .

We can consider the theory ${T'}^{\delta }_{g}$ of generic derivations on K extending $\eta '$ : notice that $\langle \mathbb K,\delta \rangle \models {T'}^{\delta }_{g}$ . We can apply the previous proof to $T'$ , since Z is now $L'$ -definable in K without parameters (notice that we need to modify the definition of $\mathbb G$ , since we are restricting the space of derivations to those extending $\eta '$ : however, we already proved the equivalence between (2) and (3)). We conclude in this way the proof.

Barbina and Zambella [Reference Barbina and Zambella3] deal with a similar situation: however, we cannot use their results, since to apply them to our setting we would need that $\mathbb K$ is countable and saturated. Maybe there could be a common refinement if one could weaken their assumption to $\mathbb K$ resplendent (since every countable consistent theory has a countable resplendent model: see [Reference Hodges26]).

14 Conjectures and open problems

In the article [Reference Fornasiero and Terzo22], we posed several conjectures. We now conclude the article with a list of additional open problems, remarks, and some further ideas.

14.1 Definable types

Let $\langle \mathbb K, \bar {\delta } \rangle \models T^{\bar {\delta },?}_g$ . Given a type $p \in S_{{L^{\bar {\delta }}}}^{n}(\mathbb K)$ , let $\bar a$ be a realization of p; we define $\tilde p \in S_{L}^{\Gamma _n}(\mathbb K)$ as the L-type of $ {\mathrm {Jet}}(\bar a)$ over $\mathbb K$ .

Open problem 14.1. Is it true that p is definable iff $\tilde p$ is definable? We conjecture that it is true when $T^{\bar {\delta },?}_g = T^{\bar {\delta }}_g$ and T has NIP.

14.2 Zariski closure

Given $X \subseteq K^{n}$ , denote by $X^{Zar}$ be the Zariski closure of X.

Questions 14.2 (See [Reference Freitag, Sánchez and Li23]).

1) Let $\big ( X_{i}: i \in I \big )$ be an L-definable family of subsets of $K^{n}$ . Is $\big ( X_{i}^{Zar}: i \in I \big )$ also L-definable?
2) Assume that 1) holds for $\mathbb K$ . Let $\langle \mathbb K, \bar {\delta } \rangle \models T^{\bar {\delta },?}_g$ . Let $\big ( X_{i}: i \in I \big )$ be an $L^\delta $ -definable family of subsets of $K^{n}$ . Is $\big ( X_{i}^{Zar}: i \in I \big )$ also $L^\delta $ -definable?

We have affirmative answers to both questions, which will be presented in a paper in preparation.

14.3 Kolchin polynomial

Let $\langle \mathfrak C, \bar {\delta } \rangle $ be a monster model of $T^{\bar {\delta }}_g$ . Let $\bar a \in \mathfrak C^{n}$ , $B \subseteq \mathfrak C$ such that $\bar {\delta } B \subseteq B$ . There exists a polynomial $\omega _{\bar a \mid B}(t)$ such that, for n large enough, $ {\mathrm {rk}}( {\mathrm {Jet}}_n(\bar a) \mid B) = \omega _{\bar a \mid B}(n)$ , where $ {\mathrm {Jet}}_n(\bar a) = \{\mu \bar a: \mu \in \Gamma \text { and } \lvert \mu \rvert \leq n\}$ (see [Reference Kolchin31]). The degree of the polynomial is at most k; denote by $\mu (\bar a)$ the leading monomial of $\omega _{\bar a \mid K}$ (including its coefficient). Let $X \subseteq K^{n}$ be ${L^{\bar {\delta }}}$ -definable with parameters $\bar b$ : define

$$\begin{align*}\begin{aligned} \mu(X) &:= \sup \big( \mu(\bar a \mid { {\mathrm{Jet}}}(\bar b)): \bar a \in X \big)\\ \omega_{X} &:= \sup \big( \omega_{\bar a \mid { {\mathrm{Jet}}(\bar b)}}: \bar a \in X \big), \end{aligned} \end{align*}$$

where the supremum for $\omega $ is taken inside $\mathbb {R}[t]$ w.r.t. the order $p>q$ if $\lim _{t \to + \infty } p(t)> q$ (cf. [Reference Freitag, Sánchez and Li23]), while the supremum of $\mu $ is taken w.r.t. the order $r t^d \leq r' t^{d'}$ if $d < d'$ or $d = d'$ and $r \leq r'$ (cf. [Reference Fornasiero17]). Notice that, by Section 6, $\mu (X)$ and $\omega _{X}$ are well-defined (i.e., they not depend on the choice of the parameters $\bar b$ ).

Conjecture 14.3 (See [Reference Freitag, Sánchez and Li23, Reference Rivière45]).

The suprema in the definitions of $\omega _X$ and $\mu (X)$ are maxima. Moreover, $\omega $ and $\mu $ are definable in families: that is, for every ${L^{\bar {\delta }}}$ -definable family $\big ( X_{i}: i \in I \big )$ there exists a partition of I into finitely many definable set $I = I_{1} \sqcup \dotsb \sqcup I_{m}$ such that $\mu (X_{i})$ and $\omega _{X_{i}}$ are constant on each $I_{j}$ .

Open problem 14.4. What is the “geometric” meaning of $\mu (X)$ ? Notice that, up to a multiplicative constant, the $k^{th}$ coefficent of $\omega _{X}$ is equal to ${{\bar {\delta }}\text {-}\mathrm {dim}}(X)$ .

If Conjecture 14.3 is true, then the function $X \mapsto \mu (X)$ behaves like a dimension on $L^\delta $ -definable sets (with the difference that the values of $\mu $ are not natural numbers, but monomials: cf. [Reference van den Dries13, Section 2.5]).

Conjecture 14.5. Assume that $\mathfrak C$ is endowed with a topology $\tau $ satisfying some suitable conditions. Let $\tau _{\bar {\delta }}$ be the topology on $\mathfrak C$ induced by the embedding $\mathfrak C \to \mathfrak C^{\Gamma }$ , $x \mapsto {\mathrm {Jet}}(x)$ (where $\mathfrak C^{\Gamma }$ is endowed with the product topology induced by $\tau $ ). Denote by $\overline X^{\tau _{\bar {\delta }}}$ the $\tau _{\bar {\delta }}$ -closure of X. Then, for every $X \subseteq \mathfrak C^{n}$ which is ${L^{\bar {\delta }}}$ -definable and nonempty, $\overline X^{\tau _{\bar {\delta }}}$ is also ${L^{\bar {\delta }}}$ -definable, and $\mu (\overline X^{\tau _{\bar {\delta }}}\setminus X) < \mu (X)$ .

Acknowledgments

The authors thank Elliot Kaplan, Itay Kaplan, Noa Lavi, Silvain Rideau-Kikuchi, and Marcus Tressl. The authors thank the anonymous referee for the careful reading of the manuscript and for the constructive suggestions.

Funding

Both authors are members of the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA-INDAM). The authors acknowledge financial support from Istituto Nazionale di Alta Matematica (INdAM), particularly through the Intensive Period in Model Theory, “Model Theory and tame expansions of topological Fields,” Naples, May19– July 18, 2025.

Footnotes

1 Later, we will prove a kind of converse to this fact.

2 When $\mathbb K$ is algebraically bounded, the above definition coincides with the one in Section 4.

3 $T^{\bar {\delta },?}_g$ is differentially algebraically bounded (see [Reference Wang50, Definition 4.1, Remark 4.21]; cf. [Reference van den Dries13, Section 2.25]).

4 This step was not clear to us: thanks to Itay Kaplan for explaining it.

5 Notice that in their proof [Reference Moosa and Scanlon38] do not mention that $\mathfrak C$ should have IACS: but this holds in their setting. [Reference d’Elbée12, Fact 4.14] gives a similar result that might be easier to prove in this context.

6 A more general result is true. Let $\mathbb A := \langle A, {\mathrm {cl}} \rangle $ be a finitary matroid. Let $\bar {\delta }$ be a tuple of commuting quasi-endomorphisms of $\mathbb A$ , in the sense of [Reference Fornasiero and Kaplan19]. Given $X, Y \subseteq A$ , define $\bar {\delta }$ - $cl_{Y}(X)$ as the set of $a \in A$ such that $a^{ {\mathrm {Jet}}}$ is not $ {\mathrm {cl}}$ -independent over $ {\mathrm {Jet}} (X) {\mathrm {Jet}}(Y)$ . Then, $\bar {\delta }$ - $ {\mathrm {cl}}$ is a finitary matroid on A.

7 Naturally, we use the free monoid $\Gamma $ instead of the free commutative monoid $ {\mathrm {Jet}}$ to define ${{\bar {\delta }}\text{-}\mathrm {acl}}$ in this situation.

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Article contents

GENERIC DERIVATIONS ON ALGEBRAICALLY BOUNDED STRUCTURES II: MODEL THEORETICAL PROPERTIES

Abstract

Keywords

MSC classification

Information

1 Introduction

2 Preliminaries and conventions

2.1 Generic derivations

2.2 Notations and know results

3 Elimination of imaginaries

Conjecture 3.7 [Reference Fornasiero and Terzo21].

4 Independence relations

Fact 4.5 [Reference Yoneda52, Lemma 3.1].

Fact 4.8 [Reference Adler1, Section 3.1],[Reference Buechler, Pillay and Wagner8].

Fact 4.11 (Kim–Pillay: see [Reference Kim and Pillay30], [Reference Tent and Ziegler48, Theorems 29.11 and 29.13]).

Fact 4.12 [Reference Buechler, Pillay and Wagner8].

4.1 Independence relations on algebraically bounded structures

5 The extension theorem

Lemma 5.2 (Amalgamation).

Theorem 5.3 (Extension).

5.1 Relation with the framework of [Reference León Sánchez and Mohamed33]

6 Algebraic closure and independence relations

Corollary 6.2 (cf. [Reference León Sánchez and Mohamed33, Corollary 3.16], [Reference Pillay, Point and Rideau-Kikuchi42, Corollary 1.4]).

6.1 GEI in algebraically bounded structures

7 Simplicity

Theorem 7.1 (Simplicity of $T^{\bar {\delta },?}_g$ ).

7.1 GEI and simplicity

7.2 Elimination of imaginaries and simplicity

Proof of Theorem 7.9.

8 Distality

9 $\omega $ -stability

Proof of Theorem 9.1

10 The field of constants

11 Open core

Definition 11.1 [Reference Pillay41].

Fact 11.6 [Reference Fornasiero and Halupczok18].

Theorem 11.9 (Open Core).

11.1 Proof of Theorem 11.9

Proof of Theorem 11.9.

12 Differential dimension

12.1 The commutative case

Lemma 12.3 (See [Reference Eleftheriou, Sánchez and Regnault14]).

Proposition 12.4 (Invariance of dimension for fields).

12.2 The non-commutative case

13 Genericity

14 Conjectures and open problems

14.1 Definable types

14.2 Zariski closure

Questions 14.2 (See [Reference Freitag, Sánchez and Li23]).

14.3 Kolchin polynomial

Conjecture 14.3 (See [Reference Freitag, Sánchez and Li23, Reference Rivière45]).

Acknowledgments

Funding

Footnotes

References

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