2.1. Examples

Some well known examples of fields which are algebraically bounded structures as pure fields are: algebraically closed fields, p-adics and more generally Henselian fields (see [Reference Junker and Koenigsmann24, Theorem 5.5]), real closed fields, pseudo-finite fields; curve-excluding fields in the sense of [Reference Johnson and Ye22] are also algebraically bounded. Other examples of algebraically bounded structures which are not necessarily pure fields are:

• • Algebraically closed valued fields.

• • Henselian fields (of characteristic 0) with arbitrary relations on the value group and the residue field (see [Reference van den Dries10]).

• • All models of “open theories of topological fields”, as defined in [Reference Cubides Kovacsics and Point8].

• • The expansion of an algebraically bounded structure by a generic predicate (in the sense of [Reference Chatzidakis and Pillay7]) is still algebraically bounded (see [Reference Chatzidakis and Pillay7, Corollary 2.6]).

• • The theory of fields with several independent orderings and valuations has a model companion, whose models are algebraically bounded (see [Reference van den Dries and van Dalen9], [Reference Johnson21, Corollary 3.12]).

Johnson and Ye in a recent paper [Reference Johnson and Ye22] produced examples of an infinite algebraically bounded field with a decidable first-order theory which is not large (in the Pop sense), and of a pure field that is algebraically bounded but not very slim.

2.2. Assumptions

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 is its L-theory.

• • .

• • ${\mathbb K}$ is algebraically bounded over F.

• • $\dim $ is the dimension function on ${\mathbb K}$ (or on any model of T), $\operatorname {\mathrm {acl}}$ is the T-algebraic closure, and $\operatorname {\mathrm {rk}}$ the rank of the corresponding matroid.

3.1. Model completion

A. Robinson introduced the notion of model completion in relation with solvability of systems of equations. For convenience we recall the definition:

We give the following general criteria for model completion. In our context we use (3).

3.2. The axioms

We introduce the following notation:

Let $\delta : {\mathbb K} \to {\mathbb K}$ be some function, $n \in \mathbb {N}$ , $a \in {\mathbb K}$ and $\bar a$ tuple of ${\mathbb K}^m.$ We denote by:

Two possible axiomatizations for the model completion $T^{\delta }_g$ are given by $T^{\delta }$ and either of the following axiom schemas:

1. (Deep) For every $Z \subseteq {\mathbb K}^{n+1}$ $L({\mathbb K})$ -definable, if $\Pi _{n}(Z)$ is large, then there exists $c \in {\mathbb K}$ such that $\operatorname {\mathrm {Jet}}^n_{\delta }(c) \in Z$ .

2. (Wide) For every $W \subseteq {\mathbb K}^{n} \times {\mathbb K}^{n}$ $L({\mathbb K})$ -definable, if $\Pi _{n}(W)$ is large, then there exists $\bar c \in {\mathbb K}^{n}$ such that $\langle \bar c, \delta \bar c \rangle \in W$ .

Definition 3.4. We denote by

$$\begin{align*}{T^{\delta}_{\mathrm{deep}}} := T^{\delta} \cup (\texttt{Deep}), \qquad {T^{\delta}_{\mathrm{wide}}} := T^{\delta} \cup (\texttt{Wide}). \end{align*}$$

We will show that both ${T^{\delta }_{\mathrm {deep}}}$ and ${T^{\delta }_{\mathrm {wide}}}$ are axiomatizations for the model completion of $T^{\delta }$ . Notice that the axiom scheme $(\texttt {Wide})$ deals with many variables at the same time, but has only one iteration of the map $\delta $ , while $(\texttt {Deep})$ deals with only one variable at the same time, but many iteration of $\delta $ .

3.3. Proof preliminaries

In order to prove the main result we first introduce the following notation: given a polynomial $p(\bar x,y)$ we write

$$\begin{align*}p(\bar a, b) =^{y} 0 \iff\ p(\bar a, b) = 0 \wedge \frac{\partial p}{\partial y} (\bar a, b) \neq 0. \end{align*}$$

Let S be a field of characteristic 0 and $\varepsilon $ be a derivation on it. Let I be a set of indexes (possibly, infinite). Denote , and .

Definition 3.6. There exists a unique derivation $S[\bar x] \to S[\bar x,\bar y]$ , $p \mapsto p^{[\varepsilon ]}$ such that:

$$\begin{align*}\forall a \in S\quad a^{[\varepsilon]} = \varepsilon a,\quad \forall i \in I\quad x_{i}^{[\varepsilon]} = y_{i}; \end{align*}$$

such derivation extends uniquely to a derivation $S(\bar x) \to S(\bar x)[\bar y]$ , $q \mapsto q^{[\varepsilon ]}$ .

Moreover, the map $S(\bar x) \to S(\bar x)$ defined by

is the unique derivation on $S(\bar x)$ such that:

$$\begin{align*}\forall a \in S \quad a^{\varepsilon} = \varepsilon a, \qquad \forall i \in I \quad x_{i}^{\varepsilon} = 0; \end{align*}$$

when $p \in S[\bar x]$ , $p^{\varepsilon }$ is the polynomial obtained by p by applying $\varepsilon $ to each coefficient.

Remark 3.7. For every $q \in S(\bar x)$ and $\bar a \in S^{n}$ ,

(1) $$ \begin{align} q^{[\varepsilon]}(\bar x, \bar y) &= q^{\varepsilon}(\bar x) + \sum_{i \in I} \frac{\partial q}{\partial x_{i}}(\bar x) y_{i}; \end{align} $$

(2) $$ \begin{align} \varepsilon(q(\bar a)) &= q^{[\varepsilon]}(\bar a, \varepsilon \bar a).\qquad\,\qquad \end{align} $$

If moreover $S'$ is a field containing S and $\varepsilon ': S(\bar x) \to S'$ is a derivation extending $\varepsilon $ , then

(3) $$ \begin{align} \varepsilon'(q) = q^{[\varepsilon]}(\bar x, \varepsilon'(\bar x)) = q^{\varepsilon}(\bar x) + \sum_{i \in I} \frac{\partial q}{\partial x_{i}}(\bar x) \varepsilon'(x_{i}). \end{align} $$

We will often also use the following fundamental fact, without further mentions.

Proof. W.l.o.g., $\bar a$ is a transcendence basis of $S'$ over S. By [Reference Zariski and Samuel57, Chapter II, Section 17, Theorem 39], there exists a unique derivation $\varepsilon ": S(\bar a) \to S'$ extending $\varepsilon $ and such that $\varepsilon "(\bar a) = \bar b$ ; we can also prove it directly, by defining, for every $q \in S(\bar x)$ ,

Given $c \in S'$ , let $p(y) \in S(\bar a)[y]$ be the (monic) minimal polynomial of c over S. Let . Let

(4)

Any derivation on $\varepsilon '$ on $S'$ extending $\varepsilon "$ must satisfy $\varepsilon '(c) = d$ , and by [Reference Zariski and Samuel57, Chapter II, Section 17, Theorem 39] again, there exists a unique derivation $\varepsilon "': S" \to S'$ extending $\varepsilon "$ and such that $\varepsilon "'(c) = d$ .

By iterating the above construction, we find a unique derivation $\varepsilon '$ on $S'$ extending $\varepsilon "$ .

We need the following preliminary lemmas.

Proof. Let $\alpha (x, \bar y)$ be an L-formula. Since ${\mathbb B}$ is algebraically bounded over F and of characteristic 0, there exist polynomials $p_1(x, \bar y), \ldots , p_k(x, \bar y) \in F[x,\bar y]$ associated with the formula $\alpha (x, \bar y)$ and formulas $\beta _i(x, \bar y) ="p_i(x, \bar y) =^{x} 0"$ such that $T \vdash (\alpha (x, \bar y) ) \wedge |\alpha (x, \cdot )| < \infty ) \rightarrow \bigvee _{i=1}^{k} \beta _i(x, \bar y)$ . Now we can associate to each polynomial $p_i$ the partial function

$$\begin{align*}f_i(x, \bar y, \delta \bar y) := \frac{\frac{\partial p_i}{\partial \bar y}\cdot \delta \bar y + p^{\eta}}{\frac{\partial p_i}{\partial x}}, \end{align*}$$

where $p^{\eta }$ is the polynomial defined in 3.6 obtained by p by applying $\eta $ to each coefficients.

So now we have a total T-definable function $f(x, \bar y, \delta \bar y)$ whose graph is defined in the following way:

$$ \begin{align*} z = &f(x, \bar y, \delta y ) \ \Leftrightarrow\ \big( \beta_1(x, \bar y) \wedge z=f_1(x, \bar y, \delta y) \big) \vee \big( \neg \beta_1(x, \bar y) \wedge \beta_2(x, \bar y) \wedge z=\\ &\qquad\qquad\qquad\qquad\qquad\quad =f_2(x, \bar y, \delta y) \big)\ \vee\\ &\ \vee\ \dots \vee \big( \neg \beta_1(x, \bar y) \wedge \dots \wedge \neg \beta_{k-1}(x, \bar y) \wedge \beta_{k}(x, \bar y) \wedge z=f_k(x, \bar y, \delta y) \big) \vee \\ &\qquad\qquad\qquad\quad \vee\ \big( \neg \beta_1(x, \bar y) \wedge \dots \wedge \neg \beta_{k-1}(x, \bar y) \wedge \neg \beta_{k}(x, \bar y) \wedge z= 0 \big). \end{align*} $$

3.4. Proof of Theorem 3.5

We can finally prove that both ${T^{\delta }_{\mathrm {deep}}}$ and ${T^{\delta }_{\mathrm {wide}}}$ axiomatize $T^{\delta }_g$ . The proof is in three steps: firstly we show that ${T^{\delta }_{\mathrm {wide}}} \vdash {T^{\delta }_{\mathrm {deep}}}$ , and later we prove that the conditions (3) of Proposition 3.2 hold for $U = T^{\delta }$ and, more precisely, (a) holds for $U^{*}$ equal to ${T^{\delta }_{\mathrm {wide}}}$ (i.e., that every model of $T^{\delta }$ can be embedded in a model of ${T^{\delta }_{\mathrm {wide}}}$ ), and (b) for $U^{*}$ equal ${T^{\delta }_{\mathrm {deep}}}$ (i.e., if $B \models T^{\delta }$ and $C \models {T^{\delta }_{\mathrm {deep}}}$ have a common substructure A, then every quantifier-free $L^\delta (A)$ -formula with one free variable having a solution in B also has a solution in C).

Proof. Let $Z \subseteq {\mathbb K}^{n+1}$ be $L({\mathbb K})$ -definable such that $\Pi _{n}(Z)$ is large. Define

$$\begin{align*}W := \{\langle \bar x, \bar y \rangle \in {\mathbb K}^{n} \times {\mathbb K}^{n}: \langle \bar x, y_{n} \rangle \in Z \wedge \bigwedge_{i=1}^{n-1} y_{i} = x_{i+1}\}. \end{align*}$$

Clearly, $\Pi _{n}(W) = \Pi _{n}(Z)$ , and therefore $\Pi _{n}(W)$ is large. By $(\texttt {Wide})$ , there exists $\bar c \in {\mathbb K}^{n}$ such that $\langle \bar c, \delta \bar c \rangle \in W$ . Then, $\operatorname {\mathrm {Jet}}^n_{\delta }(c_{1}) \in Z$ .

Proof. By Lemma 3.12 there exist $n \in \mathbb {N}$ and an $L(A)$ -formula $\psi $ such that $\theta (x) = \psi (\operatorname {\mathrm {Jet}}^n_{\delta }(x))$ .

Let $Y^{B} := \psi (B) = \{\bar d \in B^{n+1}: B \models \psi (\bar d)\}$ , and $Y^{C} := \psi (C)$ .

Let d be the smallest integer such that $\delta ^d (b)$ is algebraically dependent over $\operatorname {\mathrm {Jet}}^{d-1}(b) \cup A$ (or $d = + \infty $ if $\operatorname {\mathrm {Jet}}^{\infty }_{\delta }(b)$ is algebraically independent over A). We distinguish two cases:

(1) $d \geq n$ : in this case, $\Pi _{n}(Y^C)$ is large because $\operatorname {\mathrm {Jet}}^{n-1}(b) \in \Pi _{n}(Y^{B})$ , therefore, by $(\texttt {Deep})$ , there exists $c \in C$ such that $C \models \theta (\operatorname {\mathrm {Jet}}^n_{\delta }(c)).$

(2) $d < n$ : this means that $\delta ^{d}b \in \operatorname {\mathrm {acl}}(\operatorname {\mathrm {Jet}}^{d-1}(b)),$ so there exists a polynomial $p(\bar y, x) \in A[\bar y,x]$ such that $p(\operatorname {\mathrm {Jet}}^{d-1}(b), \delta ^{d} b) =^{x} 0$ . By Lemma 3.9 there exist $L(A)$ -definable functions $f_{d+1}, f_{d+2}, \ldots , f_n$ such that $\delta ^{i} = f_i(\operatorname {\mathrm {Jet}}^{d}(b))$ where $i = d + 1, d + 2, \ldots , n.$ Let

$$\begin{align*}Z^{B} := \{\bar y \in B^{d+1} : p(\bar y) = ^{y_{d+1}} 0 \wedge \theta(\bar y, f_{d+1}(\bar y), \ldots, f_{n}(\bar y))\}. \end{align*}$$

Notice that $\Pi _{d}(Z^{C})$ is large, because $\operatorname {\mathrm {Jet}}^{d-1}(b) \in \Pi _{d}(Z^{B})$ , and therefore by axiom $(\texttt {Deep})$ there exists $c \in C$ such that $\operatorname {\mathrm {Jet}}^{d}(c) \in Z^{C}$ and so $\operatorname {\mathrm {Jet}}^{n}(c) \in Y^{C}$ .

3.5. Corollaries

Corollary 3.16. Assume that T eliminates quantifiers. Then, ${T^{\delta }_{\mathrm {deep}}}$ and ${T^{\delta }_{\mathrm {wide}}}$ are axiomatizations for the model completion $T^{\delta }_g$ of $T^{\delta }$ .

Moreover, $T^{\delta }_g$ admits elimination of quantifiers, and for every $L^\delta $ -formula $\alpha (\bar x)$ there exists a quantifier-free L-formula $\beta (\bar y)$ such that

$$\begin{align*}T^{\delta}_g \models \forall \bar x \, \bigl( \alpha(\bar x) \leftrightarrow \beta(\operatorname{\mathrm{Jet}} (\bar x) ) \bigr). \end{align*}$$

Finally, $T^{\delta }_g$ is complete.

The next corollary is without any further assumptions on T.

Corollary 3.18. ${T^{\delta }_{\mathrm {deep}}}$ and ${T^{\delta }_{\mathrm {wide}}}$ are equivalent consistent theories (which we denote by $T^{\delta }_g$ ).

Moreover, for every $L^\delta $ -formula $\alpha (\bar x)$ there exists an L-formula $\beta (\bar y)$ such that

$$\begin{align*}T^{\delta}_g \models \forall \bar x \, \bigl( \alpha(\bar x) \leftrightarrow \beta(\operatorname{\mathrm{Jet}} \bar x) \bigr). \end{align*}$$

Finally, $T^{\delta }_g$ is complete.

5.1. Configurations

In order to give axioms for $T^{\bar \delta }_g$ , we first need some more definitions and notations. We fix $\langle {\mathbb K}, \bar \delta \rangle \models T^{\bar \delta }$ . We denote by K the field underlying  ${\mathbb K}$ .

Let $\Theta $ be the free commutative monoid generated by $\bar \delta $ , with the canonical partial order $\preceq $ (notice that $\Theta $ is isomorphic to $\mathbb {N}^{k}$ ).

Notice that $\Theta $ is, canonically, a quotient of the free monoid $\Gamma $ . For every $v \in \Gamma $ we denote by $[v]\in \Theta $ the equivalence class of v.

We fix the total order on $\Theta $ , given by

$$\begin{align*}\theta \leq \theta' \mbox{ iff } \lvert\theta\rvert < \lvert\theta'\rvert \ \vee\ \bigl( \lvert\theta\rvert = \lvert\theta'\rvert \ \wedge\ \theta <_{lex} \theta' \bigr), \end{align*}$$

where $<_{lex}$ is the lexicographic order, and .

Given $a \in {\mathbb K}$ and $\theta \in \Theta $ , we denote by , and similarly , and . Moreover, for each $\theta \in \Theta $ we have a variable $x_{\theta }$ , and we denote . Moreover, given a set $A \subseteq \Theta $ , we denote , and . Given a rational function $q \in K(x_{\Theta })$ and $\bar a \in K^{\Theta }$ , we denote by

Let $K_{0}$ be a differential subfield of K (i.e., such that $\bar \delta (K_{0}) \subseteq K_{0}$ ).

A configuration $\mathfrak S$ with parameters in $K_{0}$ is given by the following data.

(1) A $\preceq $ -anti-chain $\mathcal P \subset \Theta $ . Notice that, by Dickson’s Lemma, $\mathcal P$ must be finite.

We distinguish two sets:

• • , the set of leaders.Footnote ²

• • , the set of free elements. Moreover, we define:

• • .

Notice that $\mathcal P$ is the set of $\preceq $ -minimal elements of $\mathcal B$ . We assume that $\mathcal F$ is non-empty (equivalently, that $0 \in \mathcal F$ ).

(2) For every $\pi \in \mathcal P$ we are given a nonzero polynomial $p_{\pi } \in K_{0}[x_{\pi }, x_{\mathcal F < \pi }]$ which depends on $x_{\pi }$ (i.e., its degree in $x_{\pi }$ is nonzero).

Consider the quasi-affine variety (defined over $K_{0}$ )

(by quasi-affine variety we simply mean a subset of $K^{n}$ which is locally closed in the Zariski topology. We don’t consider its spectrum).

(3) Finally, we are given $W \subseteq W_{0}$ , which is Zariski closed in $W_{0}$ , such that W is defined (as a quasi-affine variety) over $K_{0}$ and such that $\Pi _{\mathcal F}(W)$ is large (where $\Pi _{\mathcal F}: K^{\mathcal V} \to K^{\mathcal F}$ is the canonical projection).

For the remainder of this section, we are given a configuration:

$$\begin{align*}\mathfrak S := (W; p_{\pi}: \pi \in \mathcal P). \end{align*}$$

We want to impose some commutativity on $\mathfrak S$ .

We define now some induced data.

For every $\alpha \in \Theta $ we define a rational function $f_{\alpha } \in K_{0}(x_{\mathcal V})$ and a tuple of rational functions $F_{\alpha }$ .

• • If $\alpha \in \mathcal V$ , then and $F_{\alpha } = \{f_{\alpha }\}$ .

• • For every $\pi \in \mathcal P$ and $v \in \Gamma $ , we define $f_{v, \pi }$ by induction on v, and then

  $$\begin{align*}F_{\alpha} := \{f_{v, \pi}: v \in \Gamma, \pi \in \mathcal P, [v]\pi = \alpha\}, \end{align*}$$
  and $f_{\alpha }$ is an arbitrary function in $F_{\alpha }$ .

  If $v = 0$ , then .

  If $v = \delta $ for some $\delta \in \bar \delta $ , define

  (5)

  
  If $v = \delta w$ with $0 \neq w \in \Gamma $ , define
  
  where when $\mu \in \mathcal F$ .

We also define $g_{\alpha }$ and $g_{w, \pi }$ as the restriction to W of $f_{\alpha }$ and $f_{w, \pi }$ , respectively, and $G_{\alpha }$ the family $\{ g_{w, \pi }: w \in \Gamma , \pi \in \mathcal P, [w]\pi = \alpha \}$ .

Let $\theta \in \Theta $ be the $\preceq $ -l.u.b. of $\mathcal P$ and d be the dimension of W. We say that $\mathfrak S$ commutes at $\alpha \in \Theta $ if, for every $g,g' \in G_{\alpha }$ , g and $g'$ coincide outside a subset of W of dimension less than d (i.e., they coincide “almost everywhere” on W). We say that $\mathfrak S$ commutes locally if it commutes at every $\alpha \leq \theta $ , and it commutes globally if it commutes at every $\alpha \in \Theta $ .

The main result that makes the machinery work is the following.

We will say then that $\mathfrak S$ commutes if it commutes locally (equivalently, globally).

In order to prove the above theorem, we need some preliminary definitions and results. It suffices to prove the theorem for every $K_{0}$ -irreducible component of W of dimension d. Thus, w.l.o.g. we may assume for the remainder of this subsection that W is $K_{0}$ -irreducible. Then, the condition that W commutes at a certain $\alpha $ becomes that $G_{\alpha }$ is a singleton.

We denote by $K_{0}[W]$ the ring of regular functions on W (that is, the restriction to W of polynomial maps $K_{0}^{\mathcal V} \to K_{0}$ ). Since we are assuming that W is $K_{0}$ -irreducible, $K_{0}[W]$ is an integral domain. Therefore, we can consider its fraction field $K_{0}(W)$ . An element of $K_{0}(W)$ is the restriction to W of a rational function in $K_{0}(x_{\mathcal V})$ ; in particular, the functions $g_{\alpha }$ and $g_{w, \pi }$ are in $K_{0}(W)$ .

Given $\delta \in \bar \delta $ , we define the following function:

$$ \begin{align*} {R^{\delta}_{0}} : {K_0(x_{\mathcal V})} & \longrightarrow {K_0(x_{\mathcal V})} \\ h &\longmapsto\! {h^{\delta} + \sum_{\mu \in \mathcal V} \frac{\partial h}{\partial \mu} \cdot f_{\delta, \mu}.} \end{align*} $$

Notice that $R^{\delta }_{0}$ is the unique derivation extending $\delta $ such that $R^{\delta }_{0}(x_{\mu }) = f_{\delta , \mu }$ , for every $\mu \in \mathcal V$ . Given $w = w_{1} \dots w_{\ell } \in \Gamma $ , we can define $R^{w}_{0}: K_{0}(x_{\mathcal V}) \to K_{0}(x_{\mathcal V})$ as the composition .

We have, for every $v,w \in \Gamma $ and every $\pi \in \mathcal P$ ,

(6) $$ \begin{align} R^{w}_{0}(f_{v, \pi}) = f_{wv, \pi}. \end{align} $$

If we restrict $R^{\delta }_{0}$ to $K_{0}(x_{\mathcal F}) $ and compose with the restriction to W, we obtain a derivation $R^{\delta }_{1}: K_{0}(x_{\mathcal F}) \to K_{0}(W)$ . An equivalent definition is that $R^{\delta }_{1}$ is the unique derivation extending $\delta $ such that $R^{\delta }_{1}(x_{\mu }) = g_{\delta \mu }$ for every $\mu \in \mathcal F$ . Finally, observe that $K_{0}(W)$ is an algebraic extension of $K_{0}(x_{\mathcal F})$ . So, $R^{\delta }_{1}$ extends uniquely to a derivation $R^{\delta }$ from $K_{0}(W)$ to the algebraic closure of $K_{0}(W)$ .

We will consider the objects $x_{\mu }$ both as variables and as functions. Observe that, for every $\mu \in \mathcal V$ , $g_{\mu }$ is the restriction of $x_{\mu }$ to W.

Remark 5.3. Let $f \in K_{0}(x_{\mathcal V})$ and $h \in K_{0}(W)$ be the restriction of f to W. Then, we have $h = f(g_{\mathcal V})$ (we are seeing f as a rational function). Therefore,

$$\begin{align*}R^{\delta}(h) = f^{\delta}(g_{\mathcal V}) + \sum_{\mu \in \mathcal V} \frac{\partial f}{\partial \mu}\upharpoonright_{W} \cdot R^{\delta}(g_{\mu}) = f^{\delta} \upharpoonright_W + \sum_{\mu \in \mathcal V} \frac{\partial f}{\partial \mu}\upharpoonright_{W} \cdot g_{\delta, \mu}. \end{align*}$$

Lemma 5.4. Let $\mu \in \mathcal V$ . Then,

(7) $$ \begin{align} R^{\delta}(g_{\mu}) = g_{\delta, \mu}. \end{align} $$

Proof. If $\mu \in \mathcal F$ , the conclusion follows by definition of $R^{\delta }_{1}$ .

If $\mu = \pi \in \mathcal P$ , observe that $g_{\pi }$ satisfies the algebraic condition

$$\begin{align*}p_{\pi}(g_{\pi}, g_{\mathcal F<\pi}) =^{\pi} 0. \end{align*}$$

Therefore,

$$\begin{align*}R^{\delta}(g_{\pi}) = {{-}\frac{p_{\pi}^{\delta} + \sum_{\beta \in \mathcal F < \pi} \frac{\partial p_{\pi}}{\partial \beta} \cdot R^{\delta}(g_{\beta})} {\frac{\partial p_{\pi}}{\partial \pi}}}(g_{\pi}, g_{\mathcal F<\pi}) = f_{\delta, \pi} \upharpoonright_{W} = g_{\delta, \pi}.\\[-42pt] \end{align*}$$

Lemma 5.7. Let $\delta , \varepsilon \in \bar \delta $ , $f \in K_{0}(x_{\mathcal V})$ , and $h \in K_{0}(W)$ be the restriction of f to W. Then,

(8) $$ \begin{align} [R^{\varepsilon}, R^{\delta}]h = \sum_{\mu \in \mathcal V} \frac{\partial f }{\partial \mu}\upharpoonright _{W} \cdot (R^{\varepsilon} g_{\delta, \mu} - R^{\delta} g_{\varepsilon, \mu}). \end{align} $$

Proof. (First proof) Let $\lambda = [\varepsilon , \delta ]$ be the Lie bracket of $\varepsilon $ and $\delta $ , and $R^{\lambda } = [R^{\varepsilon }, R^{\delta }]$ be corresponding Lie bracket. We can write $h = f(g_{\mathcal V})$ , where we see f as a rational function: therefore, since $R^{\lambda }$ is a derivation, we have

$$\begin{align*}R^{\lambda} h = h^{\lambda} + \sum_{\mu \in \mathcal V} \frac{\partial f}{\partial \mu}(g_{\mathcal V}) \cdot R^{\lambda}(g_{\mu}) = \sum_{\mu \in \mathcal V} \frac{\partial f }{\partial \mu}\upharpoonright _{W} \cdot (R^{\varepsilon} g_{\delta, \mu} - R^{\delta} g_{\varepsilon, \mu}). \end{align*}$$

(Second proof) Since both the LHS and the RHS of (8) define a derivation on $K_{0}(W)$ , it suffices to show that they are equal when $f = x_{\mu }$ for some $\mu \in \mathcal F$ .

Then, RHS is equal to $R^{\varepsilon } g_{\delta , \mu } - R^{\delta } g_{\varepsilon , \mu }$ , which is equal to $R^{\varepsilon } R^{\delta } g_{\mu } - R^{\delta } R^{\varepsilon } g_{\mu }$ , i.e., the LHS.

We can finally prove the theorem.

Proof of Theorem 5.2

Assume that $\mathfrak S$ commutes locally (and that W is $K_{0}$ -irreducible). Let $\alpha \in \mathcal B$ : we show, by induction on $\alpha $ , that $\mathfrak S$ commutes at $\alpha $ . Let $\pi , \pi ' \in \mathcal P$ and $w, w' \in \Gamma $ be such that $[w] \pi = [w'] \pi ' = \alpha $ : we need to show that $g_{w, \pi } = g_{w', \pi '}$ .

If $\pi = \pi '$ , we can reduce to the case when $w= v \delta \varepsilon u$ and $w' = v \varepsilon \delta u$ for some $\delta , \varepsilon \in \bar \delta $ , $v, u \in \Gamma $ .

If $u \neq 0$ , we have, by inductive hypothesis, that $g_{v \delta \varepsilon , \pi } = g_{v \varepsilon \delta , \pi }$ and therefore

$$\begin{align*}g_{w, \pi} = R^{u} g_{v \delta \varepsilon, \pi} = R^{u} g_{v \varepsilon \delta, \pi} = g_{w', \pi}, \end{align*}$$

which is the thesis.

If instead $u = 0$ , we have

$$\begin{align*}g_{w, \pi} - g_{w', \pi} = [R^{\varepsilon}, R^{\delta}] g_{v, \pi} = \sum_{\mu \in \mathcal V} \frac{\partial f_{v, \pi} }{\partial \mu}\upharpoonright _{W} \cdot (R^{\varepsilon} g_{\delta, \mu} - R^{\delta} g_{\varepsilon, \mu}). \end{align*}$$

Fix some $\mu $ in the above sum. Notice that $\mu < [v]\delta $ , and therefore $\delta \varepsilon \mu = \varepsilon \delta \mu < \alpha $ . Therefore, by inductive hypothesis, $R^{\varepsilon } g_{\delta , \mu } = g_{\varepsilon \delta \mu } = R^{\delta } g_{\varepsilon , \mu }$ . Thus, all summands are $0$ and $g_{w, \pi } - g_{w', \pi } = 0$ , and we are done.

If $\pi \neq \pi '$ , let ; by definition, $\beta \leq \theta $ , and therefore, by assumption, $G_{\beta } = \{g_{\beta }\}$ . Let $u, u', w \in \Gamma $ be such that

$$\begin{align*}\beta = [u] \pi = [u']\pi', \quad \alpha = [w] \beta. \end{align*}$$

By the previous case, we have

$$\begin{align*}g_{v, \pi} = g_{w u, \pi} = R^{w} g_{u, \pi} = R^{w} g_{\beta} = R^{w} g_{u', \pi'} = g_{w u', \pi'} = g_{v' \pi'}.\\[-33pt] \end{align*}$$

The following remark motivates the definition of the functions $g_{\mu }$ .

Remark 5.9. Let $b \in K$ be such that $b^{\mathcal V} \in W$ . Then, for every $w \in \Gamma $ and $\pi \in \mathcal P,$

$$\begin{align*}b^{[w]\pi} = g_{w, \pi}(b). \end{align*}$$

Therefore, for every $\mu \in \Theta $ ,

$$\begin{align*}b^{\mu} = g_{\mu}(b). \end{align*}$$

5.2. The axioms

Notice that the above is the analogue of the axiom scheme $(\texttt {Deep})$ . We don’t have an analogue for the axiom scheme $(\texttt {Wide})$ . Notice also that the axiom scheme (k -Deep) is first-order expressible thanks to Theorem 5.2.

We assume that T eliminates quantifiers. We use the criterion in Proposition 3.2(3) to show that $T^{\bar \delta }_g$ is the model completion of $T^{\bar \delta }$ and it eliminates quantifiers. We will do it in two lemmas.

Proof. Let $B \succeq A$ be $\lvert A\rvert ^{+}$ -saturated. Let $\bar b = (b_{\nu }: \nu \in \mathcal V) \in X_{B}$ be such that $b_{\mathcal F} := \Pi _{\mathcal F}(\bar b)$ is algebraically independent over A. Let $I \subset B$ be such that I is disjoint from $b_{\mathcal F}$ and is a transcendence basis of B over A. We extend each derivation $\delta \in \bar \delta $ to B in the following way. It suffices to specify the value of $\delta $ on each $c \in D$ .

If $c \in I$ , we define .

If $c = b_{\mu }$ for some $\mu \in \mathcal F$ , we define .

By definition, it is clear that $b_{0}^{\mathcal V} = \bar b \in X$ . Thus, it suffices to show that the extensions of $\bar \delta $ commute on all B. Again, it suffices to show that, for every $c \in D$ and every $\delta , \varepsilon \in \bar \delta $ ,

$$\begin{align*}\delta \varepsilon c= \varepsilon \delta c. \end{align*}$$

If $c \in I$ , then both sides are equal to $0$ , and we are done.

Suppose now that $c = b_{\mu }$ for some $\mu \in \mathcal F$ , we have to show that

(9) $$ \begin{align} \delta \varepsilon b_{\mu}= \varepsilon \delta b_{\mu}. \end{align} $$

Since the argument is delicate we prefer to give two different proofs with two different approaches.

(First proof) For this first proof, we replace W with its A-irreducible component containing $\bar b$ . Thus, we may assume that W is A-irreducible. Since $\bar b$ is generic in W, the map $A(W) \to A(\bar b)$ , $h \mapsto h(\bar b)$ is an isomorphism of A-algebras. Via the above isomorphism, the derivation $R^{\delta }$ on $A(W)$ corresponds to the derivation $\delta $ on $A(\bar b)$ . The assumptions plus Theorem 5.2 and Remark 5.8 imply that $R^{\delta }$ and $R^{\varepsilon }$ commute: therefore, also $\delta $ and $\varepsilon $ commute on $A(\bar b)$ .

(Second proof) Since $\bar b \in W$ , we have that $\delta b_{\pi } = f_{\delta , \pi }(\bar b)$ for every $\pi \in \mathcal P$ . Thus, by definition, for every $\nu \in \mathcal V$ ,

$$\begin{align*}\delta b_{\nu} = f_{\delta, \nu}(\bar b). \end{align*}$$

Denote . By definition, the LHS of (9) is equal to

(10) $$ \begin{align} \begin{aligned} \delta g_{\varepsilon \mu}(\bar b)= \delta f(\bar b)= f^{\delta}(\bar b) &+ \sum_{\nu \in \mathcal V} \frac{\partial f}{\partial \nu}(\bar b) \cdot \delta b_{\nu} = \\ &= f^{\delta}(\bar b) + \sum_{\nu \in \mathcal V} \frac{\partial f}{\partial \nu}(\bar b) \cdot f_{\delta, \nu}(\bar b) = f_{\delta, \varepsilon \mu}(\bar b) = g_{\delta, \varepsilon \mu}(\bar b). \end{aligned}\\[-15pt]\nonumber \end{align} $$

Similarly, the RHS of (9) is equal to $g_{\varepsilon , \delta \mu }(\bar b)$ . Finally, since $\mathfrak S$ commutes, $g_{\delta , \varepsilon \mu } = g_{\varepsilon , \delta \mu }$ , and we are done.

Proof. W.l.o.g., we may assume that the only constants in the language L are the elements of F, the only function symbols are $+$ and $\cdot $ , thus, an $L(\bar \delta )$ -substructure is a differential subring containing F. Let $A'$ (resp., $A"$ ) be the relative algebraic closure (as fields, or equivalently as L-structures) of A inside B (resp., C). Since A has the same L-type in B and C, there exists an isomorphism $\phi $ of L-structures between of $A'$ and $A"$ extending the identity on A. Moreover, any derivations on A extends uniquely to $A'$ and $A"$ : thus, $\phi $ is also an isomorphism of $L^{\bar \delta }$ -structures. Thus, w.l.o.g. we may assume that A is relatively algebraically closed in B and in C.

Let

and . If $\mathcal F$ is empty, then $b \in \operatorname {\mathrm {acl}}(A) = A$ , and we are done.

Otherwise, let $\mathcal P$ be the set of $\preceq $ -minimal elements of $\mathcal B$ . For every $\pi \in \mathcal P$ , let $p_{\pi } \in A[x_{\pi }, x_{\mathcal F< \pi }]$ be such that $p_{\pi }(\pi b, b^{\mathcal F < \pi }) =^{\pi } 0$ . Let W be the A-irreducible component of

containing $b^{\mathcal V}$ .

Claim 2. is a commutative configuration (over A).

Again, we prefer to give two different proofs.

(First proof) Since $b^{\mathcal V}$ is generic (over A) in the A-irreducible variety W, the map $\Phi : A(W) \mapsto A(b^{\mathcal V})$ , $h \mapsto h(b^{\mathcal V})$ is an isomorphism of A-algebrae. Via the above isomorphism, $R^{\delta }$ corresponds to the derivation $\delta $ on $A(b^{\mathcal V})= A(b^{\Theta })$ : thus, the maps $R^{\delta }$ commute with each other. Therefore, for every $\pi , \pi ' \in \mathcal P$ and $w,w' \in \Gamma $ with $[w] \pi = [w'] \pi '$ ,

$$\begin{align*}g_{w, \pi} = R^{w} g_{\pi} = \Phi((b^{\pi})^{[w]}) = \Phi((b^{\pi'})^{[w']}) = g_{w', \pi'}. \end{align*}$$

(Second proof) By Corollary 5.10.

Let $\beta (z_{\Theta })$ be a quantifier-free $L(A)$ -formula such that

$$\begin{align*}T^{\bar \delta} \cup \operatorname{\mathrm{Diag}}_{L}(A) \vdash \gamma(x) \leftrightarrow \beta(x^{\Theta}). \end{align*}$$

Let

Since $b^{\mathcal V} \in X$ , and X is $L(A)$ -definable, we have that $\Pi _{\mathcal F}(X)$ is large. Thus, there exists $c \in C$ such that $c^{\mathcal V} \in X$ . Since $c^{\mathcal V} \in W$ , by Remark 5.9 for every $\mu \in \Theta $ we have $c^{\mu } = g_{\mu }(c^{\mathcal V})$ . Therefore, $\langle C, \bar \delta \rangle \models \beta ( c^{\Theta })$ .

5.2.1. Addendum

Call a configuration $\mathfrak S =(W; p_{\pi }: \pi \in \mathcal P)$ “ $K_0$ -irreducible” if W is $K_{0}$ -irreducible. In the definition of a configuration we did not impose that it is irreducible. However, by the proof of the Amalgamation Lemma 5.14, it seems that it would suffice to impose in Axiom (k -Deep) that only irreducible configurations need to be satisfied. The reason why we did not restrict ourselves to irreducible configurations is that we don’t know if we can impose irreducibility in a first-order way.

Question 5.15. Let $(W_{i}: i \in I)$ be an L-definable family of varieties in $K^{n}$ . Is the set

$$\begin{align*}\{i \in I: W_{i} \text{ is } K\text{-irreducible}\} \end{align*}$$

definable?

8.1. Elimination of imaginaries

A few particular cases are known, when $T^{\bar \delta ,?}_g$ is one of the following:

• • $\textrm {DCF}_{0,\textrm {m}}$ : see [Reference McGrail32].

• • $\textrm {RCF}$ or certain theories of Henselian valued fields, endowed with m commuting generic derivations: see [Reference Cubides Kovacsics and Point8, Reference Fornasiero and Kaplan14] for a proof based on M. Tressl’s idea; see also [Reference Brouette, Kovacsics and Point5, Reference Montenegro and Rideau-Kikuchi34, Reference Point41] for different proofs.

• • $\textrm {DCF}_{0,\textrm {m,nc}}$ (see [Reference Moosa and Scanlon36]).

We have also established the validity of the above conjecture for certain topological structures. Drawing upon established techniques, it is probable that the conjecture can be proven for T simple (as proved in [Reference Mohamed33, Reference Moosa and Scanlon36]). However, for the general case, we believe that novel approaches are required (although some progress has been made in [Reference Brouette, Kovacsics and Point5]).

8.2. Definable types

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

8.3. Zariski closure

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

8.4. Monoid actions

Let $\Lambda $ be a monoid generated by a k-tuple $\bar \delta $ : we consider $\Lambda $ as a quotient of the free monoid $\Gamma $ . We can consider actions of $\Lambda $ on models of T such that each $\delta _{i}$ is a derivation: we have a corresponding theory $T^{\Lambda }$ whose language is $L^\delta $ and with axioms given by T, the conditions that each $\delta _{i}$ is a derivation, and, for every $\gamma , \gamma ' \in \Gamma $ which induce the same element of $\Lambda $ , the axiom $\forall x\, \gamma x = \gamma 'x$ .

As a consequence of [Reference León Sánchez and Moosa28], when $T = ACF_{0}$ and $\Lambda = \Gamma _{\ell }\times \Theta _{k}$ , $T^{\Lambda }$ has a model companion.

Maybe the following conditions on $\Lambda $ suffice for $T^{\Lambda }$ to have a model completion:

Let $\preceq $ be the canonical quasi ordering on $\Lambda $ given by $\alpha \preceq \beta \alpha $ for every $\alpha ,\beta \in \Lambda $ ; we assume that:

• • $\preceq $ is a well-founded partial ordering;

• • for every $\lambda \in \Lambda $ , the set $\{\alpha \in \Lambda : \alpha \preceq \lambda \}$ is finite;

• • for every $\alpha , \beta \in \Lambda $ , if they have an upper bound, then they have a least upper bound;

• • let $X \subset \Lambda $ be finite; assume that X is $\preceq $ -initial in $\Lambda $ ; then, $\Lambda \setminus X$ has finitely many $\preceq $ -minimal elements;

• • if $\alpha _{1} \delta _{1} = \alpha _{2} \delta _{2}$ for some $\alpha _{i} \in \Lambda $ and $\delta _{i} \in \bar \delta $ , then $\delta _{1}$ and $\delta _{2}$ commute with each other; moreover, there exists $\beta \in \Lambda $ such that $\alpha _{1} = \delta _{2} \beta $ and $\alpha _{2} = \delta _{1} \beta $ .