Let $T$ be a theory with a good notion of independence.  Generally, we have in mind forking independence, though something weaker would be acceptable.  In fact, for our proof, the only properties we use are that independence is invariant under replacing a set by its algebraic closure

• For $A \subseteq B$ and $A \subseteq C$, we have that $B$ is free from $C$ over $A$ if and only if $\operatorname{acl}(B)$ is free from $\operatorname{C}$ over $\operatorname{A}$.  Here, $\operatorname{acl}$ is the algebraic closure in the home sort.  We are not assuming elimination of imaginaries.

the fact that

• For any pair of sets $A \subseteq B$ we have that $B$ is free from $A$ over $A$.

and two monotonicity properties for independence

• If $A$ is independent over $B$ and $C \subseteq A$, then $C$ is independent over $B$.
• If $A \subseteq B \subseteq C$ and $C$ is independent over $A$, then $C \smallsetminus B$ is independent over $B$.

(To be honest we also use, very weakly, the hypothesis that $\operatorname{acl}(\varnothing) \neq \varnothing$.)

Let $n$ be a von Neumann natural number, that is, $n = \{0, 1, \ldots, n-1\}$.  We write $\mathcal{P}(n)^-$ for $\mathcal{P}(n) \smallsetminus n$.   Let $W \subseteq \mathcal{P}(n)$ be closed downwards.  That is, if $v \subseteq w \in W$, then $v \in W$.

A $W$-variable assignment is an association $w \mapsto x_w$ from $W$ to variables ranging over some (possibly infinite) product $S_w$ of sorts so that for $w \leq v \in W$, we require that $S_v$ is a subproduct of $S_w$ and if $\pi_{w,v}:S_w \to S_v$ is the natural projection, then $x_v = x_w \circ \pi_{w,v}$.     If $\{i\} \in W$, then we write $x_{\{i\}} =: x_i$.   Note our conventions:  $x_i$ may be naturally a tuple (even infinite) of variables.   In this way, if $i \in w \in W$, then $x_i = x_w \circ \pi_{w,\{i\}}$.

By an $W$-independent system of types we mean that we are given a $W$-variable assignment and then for each $w \in W$ a complete type $p_w$ relative to $T$ in the variable $x_w$ so that

1. If $v \subseteq w$, then $p_v = {\pi_{w,v}}_* (p_w)$.
2. $p_w \vdash \{ x_i : i \in w\}$ is independent.
3. $p_w \vdash x_w \subseteq \operatorname{acl}(\{ x_i : i \in w \})$.     Recall that $x_w$ ranges over some product of sorts $S_w$.  With this condition we are asking that $p_w$ include formulae explicitly requiring each component of $x_w$ to lie be algebraic over the components of the $x_i$ for $i \in w$.

We say that $T$ has $n$-amalgamation over algebraically closed sets if for every model $M \models T$, every $E \subseteq M$ with $\operatorname{acl}(E) = E$, every downward closed $W \subseteq \mathcal{P}(n)$, and every $W$-independent system $(p_w)_{w \in W}$ of types relative to $\operatorname{Th}(M_E)$, there is a $\mathcal{P}(n)$-independent system $(q_u)_{u \in \mathcal{P}(n)}$ with $q_w = p_w$ for $w \in W$.

Lemma: A theory $T$ has independent $n$-amalgamation over algebraically closed sets if and only if it has independent $m$-amalgamation for all $m < n$ and every $\mathcal{P}(n)^{-}$-independent system over an algebraically closed subset of a model of $T$ may be extended to a $\mathcal{P}(n)$-independent system.

Proof:  The left to right direction is immediate. For the other direction, we suppose that $T$  has independent $m$-amalgamation for all $m < n$ and every $\mathcal{P}(n)^{-}$-independent system over an algebraically closed subset of a model of $T$ may be extended to a $\mathcal{P}(n)$-independent system, that  $W \subseteq \mathcal{P}(n)$ is  downwards closed and that  $(p_w)_{w \in W}$ is a $W$-independent system over some algebraically closed subset $E$ of some model of $T$.

If $W = \mathcal{P}(n)$.

Working by induction on $|W|$, it suffices to show that we may extend the system to any given downwards closed $W’ \supseteq W$ with $|W’ \smallsetminus W| = 1$.    Let $w \in W’ \smallsetminus W$.

• If $w = n$, then $W = \mathcal{P}(n)^-$ so that by our hypotheses on $T$ we may extend the system.
• If $w = \{ i \}$ for some $i \in n$, then pick any $e \in E$ (this is where we are using the hypothesis that $\operatorname{acl}(\varnothing) \neq \varnothing$).  Set $p_i := \ulcorner x_i = e \urcorner$.  Then because $p_i \vdash x_i \in E$, this extension is $W’$-independent. Here we are using both of our hypotheses about the behavior of the notion of independence with respect to algebraic closure.
• If $1 < |w| < n$, then by our hypothesis that all independent systems over $\mathcal{P}(|w|)^-$ may be extended, we can find $p_w$ so that $(p_u)_{u \subseteq w}$ is a $\mathcal{P}(w)$-independent system.   The resulting system $\{ p_w \} \cup (p_u)_{u \in W}$ is a $W’$-independent system. $\Box$

Lemma: A theory $T$ has $n$-amalgamation over algebraically closed sets if and only if every independent system in which (3) is strengthened to (3)’: $x_w = \operatorname{acl}(\{x_i:i \in w\})$ may be extended to a $\mathcal{P}(n)$-independent system.

Proof:  Let us start with some downwards closed $W \subseteq \mathcal{P}(n)$ and a $W$-independent system satisfying the usual definition and show how to expand it to a $W$-independent system $(q_w)_{w \in W}$ in the variables $(y_w)_{w \in W}$ satisfying (3)’ so that for each $w$ the variable $x_w$ ranges over a subtuple of $y_w$ and $p_w = q_w \upharpoonright \mathcal{L}_{x_w}$.   We work by induction on the size of $B :=  \{ w \in W :  p_w \not \vdash x_w = \operatorname{acl}(\{ x_i : i \in w \}) \}$.  If $B = \varnothing$, we are done.  Note that if $w \in W \smallsetminus B$, then for every $u \subseteq w$ we have $u \notin B$ as well.  Thus, if $B \neq \varnothing$, there is some $w \in B$ so that $\{ u \in W : w \subseteq u \} = \{w\}$.  Pick such a $w$.  Let $a_w$ realize $p_w$ in some model $M \models T$.  Let $b_w$ be (an enumeration of) $\operatorname{acl}(a_w)$.   Let $y_w$ be the variable context corresponding to this enumeration of $b_w$.   For $u \subseteq w$, let $b_u := b_w \cap \operatorname{acl}(a_u)$ where $a_u$ is the restriction to $x_u$ of $b_u$ and where we choose the variable context $y_u$ for $b_u$ compatibly with $y_w$ and the pre-existing $x_u$.  Set $q_u := \operatorname{tp}(a_u/E)$.  For $u \in W$ with $u \not \subseteq w$, let $y_u$ be the concatenation of $x_u$ and $y_{u \cap w}$, identifying $x_{u \cap w}$ with the corresponding subtuple of $y_{u \cap w}$.  Let $q_u := p_u \cup q_{u \cap w}$.  Then by our first condition on the behavior of independence with respect to algebraic closure, $(q_u)_{u \in W}$ is still a $W$-independent system.  $\Box$.

Definition:  For $K \models \text{ACFA}$ and $A \subseteq B$ and $A \subseteq C$, we say that $B$ is free from $C$ over $A$ if $\operatorname{acl}(B)$ is free from $\operatorname{acl}(C)$ over $\operatorname{acl}(A)$ in the sense of ACF.  Equivalently, the field $\operatorname{acl}(B)$ is linearly disjoint from the field $\operatorname{acl}(C)$ over the field $\operatorname{acl}(A)$.

Theorem:  For every natural number $n$, ACFA has independent $n$-amalgamation over algebraically closed sets.

Proof:  Note that the case of $n = 1$ is trivial – take $p_{0}$ to be any type over $E$.      Working by induction $n$, we may assume that we know ACFA has independent $m$-amalgamation of algebraically closed sets for $m < n$. By our lemmas, it suffices to consider $E \subseteq \mathbb{U} \models \text{ACFA}$ where $E = \operatorname{acl}(E)$, $K$ is $|E|^+$-saturated, and $(p_w)_{w \in \mathcal{P}(n)^-}$ is an independent $\mathcal{P}(n)^-$-system over $E$ in which $p_w \vdash x_w = \operatorname{acl}(E \cup \{ x_i : i \in w\})$.  Let $W’ := \{ w \in \mathcal{P}(n)^- : n-1 \in w \}$.   Let $a_n \models p_n$.  For $u \in \mathcal{P}(n-1)^-$, let $y_u = x_{u \cup \{n-1\}}$ and set $q_u := p_{u \cup \{ n \}} \cup x_n = a_n \in S_{y_u}(a_n)$.  The system $(q_u)_{u \in \mathcal{P}(n-1)^-}$ is a $\mathcal{P}(n-1)^-$-independent system over the algebraically closed $a_n$.  Hence, by induction, it may be extended to a $\mathcal{P}(n-1)$-independent system $(q_u)_{u \in \mathcal{P}(n-1)^-}$.  Let $b$ realize $q_{\{0, \ldots, n-2\}}$ and let $L := \operatorname{acl}(b)$.  (If we were to apply our lemma about requiring (3)’ to hold, then we would already have $b = L$.). Let $K$ be the compositum of $b_u := x_u(b)$ for $u \in \mathcal{P}(n-1)$.  Let $M$ be (generated by) a realization of $p_{\{0, 1, \ldots, n-2\}}$.  We have a copy of $K$ embedded in $M$ so that $M = K^\text{alg}$ (though it need not be the case that $M = K^\text{alg}$ as a sub-difference field of $\mathbb{U}$.

Because $K$ is a compositum of algebraically closed fields, and both $M$ and $L$ are perfect, check that $M \otimes_K L$ is a domain, it suffices to show that $M \cap L = K$.   Note that if we succeed in showing that $M \otimes_K L$ is a domain, then embedding this domain (with the endomorphisms coming from $\sigma$ on each tensor factor) into $\mathbb{U}$ over $E$, we could complete the system by taking $a_{\{0, \ldots, n-1\}}$ to be the image of  $ a_{\{0, \ldots, n-2\}} \otimes 1 \frown 1 \otimes b$ and $p_{\{0, \ldots, n-1\}} := \operatorname{tp}(a_{\{0, \ldots, n-1\}}/E)$, we complete the system.

Let us check now that $M \cap L = K$.  Let $c \in M \cap L$.  Any element $c$ of $L$ may be expressed as $c = \sum_{i=1}^m \prod_{u \in \mathcal{P}(n-1)^-} c_{i,u}$ where $c_{i,u} \in \operatorname{acl}(a_{n-1},b_w)$.   Let $Q_{i,w}(x,y) \in K[x,y]$ be a polynomial so that $Q_{i,w}[x,a_{n-1}]$ is the minimal monic polynomial over $K (a_{n-1})$ of $b_{i,w}$.  Let $\phi(x,y) := (\exists z_{i,u})_{i=1,u \in \mathcal{P}(n-1)^-}^m \left( x = \sum_i \prod_w z_{i,w} \land \bigwedge_{i,w} Q_{i,w}(z_{i,w},y) \land Q_{i,w}(X,y) \not \equiv 0 \right)$.

Now, $c$ satisfies $\phi(x,a_{n-1})$ and because $c \in M$, which is the algebraic closure of $\{ a_i : 0 \leq i < n-1\}$, $c$ is free from $a_{n-1}$ over $E$ in the sense of ACF.  Hence,  $\phi(c,y)$ defines a cofinite set and there is some $a \in E$ so that $\phi(c,a)$ holds.  But then $\phi(c,a)$ says that $c$ lies in the compositum of the $b_w$ for $w \in \mathcal{P}(n-1)^-$, that is, in $K$. $\Box$

Our proof of model completeness for ACFA yields  strong quantifier simplification results.

Let us start with a relative completeness result.   If a theory $T$ is model complete, then whenever $E \models T$ and $E \subseteq M_i \models T$ for $i = 1$ and $2$, then $M_1 \equiv_E M_2$.  For ACFA we can replace the condition $E \models T$ with just that $E$ is a substructure which is algebraically closed as a field.

Proposition:  Let $(E,\sigma)$ be a difference field which is algebraically closed as a field.  Suppose that $(E,\sigma) \subseteq (M_i,\sigma_i) \models \text{ACFA}$ are two extensions to existentially closed difference fields for $i = 1$ and $2$, then $M_1 \equiv_E M_2$.

Proof:  Since $E$ is algebraically closed as a field, the ring $R := M_1 \otimes_E M_2$ is an integral domain (see Corollary 2 on page 202 of Nathan Jacobson, Lectures in Abstract Algebra III. Theory of Fields and Galois Theory, GTM 32.)  and has a distinguished endomorphism $\sigma$ defined on tensors via $a \times b \mapsto \sigma_1(a) \otimes \sigma_2(b)$.    Let $K := \mathcal{Q}(R)$ be the field of fractions of $R$ with $\sigma:K \to K$ the extension of $\sigma$ on $R$ defined by $\sigma(\frac{a}{b}) = \frac{\sigma(a)}{\sigma(b)}$.  Let $(L,\sigma) \models \text{ACFA}$ be an existentially closed extension of $(K,\sigma)$.  Then because ACFA is model complete, we have two elementary exrtenions $M_1 \preceq L$ and $M_2 \preceq L$.  As $E$ is a common substructure of both $M_1$ and $M_2$, a fortiori, we have $M_1 \equiv_E L \equiv_E M_2$.   $\Box$

Corollary:  The completions of ACFA are in correspondence with pairs $(p,[\sigma])$ where $p$ is a prime number or $0$ and $[\sigma]$ is a conjugacy class in $\operatorname{Gal}(k_0^\text{alg}/k_0)$ with $k_0 = \mathbb{F}_p$ if $p$ is prime and  $k_0 = \mathbb{Q}$ if $p = 0$.

Proof:   For $(K,\sigma) \models \text{ACFA}$, let $\chi((K,\sigma)) := (p,[\sigma])$ where $p$ is the characteristic of $K$ and $[\sigma]$ is the conjugacy class of $\sigma \upharpoonright_{k_0^\text{alg}}$ with $k_0$ the prime field in $K$.  As $K$ is algebraically closed, $k_0^\text{alg} \subseteq K$.

If $(K_i,\sigma_i) \models \text{ACFA}$ are models of ACFA for $i = 1$ and $2$ and $\chi((K_1,\sigma_1)) = \chi((K_2,\sigma_2))$, then $K_1$ and $K_2$ have a common prime field $k_0$ (as they have the same characteristic) and there is some $\tau \in \operatorname{Gal}(k_0^\text{alg}/k_0)$ with $\sigma_2 \upharpoonright_{k_0^\text{alg}}  = \tau (\sigma_2 \upharpoonright_{k_0^\text{alg}} ) \tau^{-1}$.   That is, $\tau:(k_0^\text{alg},\sigma_1 \upharpoonright_{k_0^\text{alg}} ) \to (k_0^\text{alg},\sigma_2 \upharpoonright_{k_0^\text{alg}} )$ is an isomorphism of difference fields.  Set $E := (k_0^\text{alg},\sigma_1 \upharpoonright_{k_0^\text{alg}} )$.  Then via the usual inclusion, $E$ is an algebraically closed substructure of $K_1$ and via $\tau$, it is an algebraically closed substructure of $K_2$.  By the above proposition, $K_1 \equiv_E K_2$, which implies that upon forgetting $E$, we have $K_1 \equiv K_2$.

On the other hand, if $K_1$ and $K_2$ are models of ACFA with $K_1 \equiv K_2$, then clearly they have the same characteristic, and thus the same prime field $k_0$.  As they are both algebraically closed, they both contain (copies of) $k_0^\text{alg}$.  Thus to see that $\chi(K_1) = \chi(K_2)$ it suffices to show that for each finite Galois extension $L/k_0$ of $k_0$ there is an isomorphism $\tau:(L,\sigma_1 \upharpoonright_L) \to (L,\sigma_2 \upharpoonright_L)$, where here “$\sigma_i \upharpoonright_L$” means the restriction $\sigma_i$ to $L$ under some embedding of $L$ into $K_i$ for $i = 1$ and $2$.   By the primitive element theorem, we may express $L = k_0[a]$ for some element $a \in L$.  Let $P(x) \in k_0[x]$ be the minimal monic polynomial of $a$ over $k_0$.  Then  is a polynomial $Q \in k_0[x]$ so that $\sigma_1(a) = Q(a)$.  As $K_1 \equiv K_2$, $K_2 \models (\exists b) (P(b) = 0 \land \sigma(b) = Q(b))$.  Define $\tau:(L,\sigma_1 \upharpoonright_L) \to (K_2,\sigma)$ by $a \mapsto b$.   A general element of $L$ may be expressed as $R(a)$ for some $R \in k_0[x]$.  We compute that $\tau \sigma_1 (R(a)) = \tau (R(\sigma(a)) = \tau R(Q(a)) = R(Q(b)) = R(\sigma_2(b)) = \sigma_2(R(b))$.  Thus, $\tau$ is an isomorphism of difference fields and in the direct limit we have an isomorphism $\tau:(k_0^\text{alg},\sigma_1 \upharpoonright_{k_0^\text{alg}}) \to (k_0^\text{alg},\sigma_2 \upharpoonright_{k_0^\text{alg}})$.  That is, $\sigma_2 \upharpoonright_{k_0^\text{alg}} = \tau \circ \sigma_1 \upharpoonright_{k_0^\text{alg}} \circ \tau^{-1}$.  $\Box$

Remark:  It follows from the fact that Frobenius automorphism topologically generates $\operatorname{Gal}(\mathbb{F}_p^\text{alg}/\mathbb{F}_p)$ in characteristic $p$ and the Chebotarev density theorem in characteristic zero that for $k_0$ the prime field every difference field of the form $(k_0^\text{alg},\sigma)$ is isomorphic to the the algebraic closure of the prime field in $\prod_\mathcal{U} (\mathbb{F}_q^\text{alg},x \mapsto x^q)$ where $\mathcal{U}$ is a nonprincipal ultrafilter on the set of prime powers.  In fact, in characteristic zero, we may assume that $\mathcal{U}$ is concentrated on the primes.

Proposition:  If $(K,\sigma) \models \text{ACFA}$, $E \subseteq K$ is an algebraically closed substructure, and $a$ and $b$ are two tuples from $K$ of the same length, then $\operatorname{tp}(a/E) = \operatorname{tp}(b/E)$ if and only if there is an isomorphism of difference fields over $E$ between $E \langle a \rangle_{\sigma,\sigma^{-1}}^\text{alg}$ and $E \langle b \rangle_{\sigma,\sigma^{-1}}^\text{alg}$ taking $a$ to $b$ where the notation $E \langle a \rangle_{\sigma,\sigma^{-1}}$ means the inversive difference field generated by $a$ over $E$, that is, the field $E( \{ \sigma^j (a) : j \in \mathbb{Z} \})$ with the evident difference field structure.

Proof:  For the forward direction, let $L \succeq K$ be an $|E|^+$-strongly homogeneous elementary extension of $K$.  By strong homogeneity, if  $\operatorname{tp}(a/E) = \operatorname{tp}(b/E)$, then there is an automorphism $\tau:L \to L$ fixing $E$ and taking $a$ to $b$.  As $\tau$ is a map of difference fields, it takes $\sigma^j(a)$ to $\sigma^j(b)$ for all $j \in \mathbb{Z}$.  Thus, $\tau$ maps $E \langle a \rangle_{\sigma,\sigma^{-1}}$ to $E \langle a \rangle_{\sigma,\sigma^{-1}}$, and thus anything algebraic over $E \langle a \rangle_{\sigma,\sigma^{-1}}$ to something algebraic over $E \langle b \rangle_{\sigma,\sigma^{-1}}$. That is, the restriction of $\tau$ to $E \langle a \rangle_{\sigma,\sigma^{-1}}^\text{alg}$ is an isomorphism of difference fields between $E \langle a \rangle_{\sigma,\sigma^{-1}}^\text{alg}$ and $E \langle b \rangle_{\sigma,\sigma^{-1}}^\text{alg}$ taking $a$ to $b$.

In the other direction, fix some isomorphism of difference fields $\tau:E \langle a \rangle_{\sigma,\sigma^{-1}}^\text{alg} \to E \langle b \rangle_{\sigma,\sigma^{-1}}^\text{alg}$ fixing $E$ and taking $a$ to $b$.  Let $B := E \langle a \rangle_{\sigma,\sigma^{-1}}^\text{alg}$.  Let $L_1 := L_2 := K$.  Embed $B$ into $K_1$ via the usual inclusion and into $L_2$ via $\tau$.  Then by our main proposition, $L_1 \equiv_B L_2$.   Thus, for any formula $\phi(x) \in \mathcal{L}_E(x)$ where $x$ is a free variable of length equal to that of $a$, we have $K \models \phi(a) \Longleftrightarrow L_1 \models \phi(a) \Longleftrightarrow L_2 \models \phi(a) \Longleftrightarrow K \models \phi(\tau(a)) \Longleftrightarrow K \models \phi(b)$.  That is, $\operatorname{tp}(a/E) = \operatorname{tp}(b/E)$. $\Box$

Corollary:  If $K \models \text{ACFA}$, $k_0$ is the prime field in $K$, and $A \subseteq K$ is any subset, then $\operatorname{acl}(A) = k_0 \langle A \rangle_{\sigma,\sigma^{-1}}^\text{alg}$.

Proof:  Set $E := k_0 \langle A \rangle_{\sigma,\sigma^{-1}}^\text{alg}$.    Clearly, every element of $E$ is algebraic over $A$.    Suppose that $b \in K \smallsetminus E$.  Let $L := E \langle b \rangle_{\sigma,\sigma^{-1}}^\text{alg}$.  Because $E$ is algebraically closed, for every $n$, the algebra $L^{\otimes n} = L \otimes_E L  \otimes_E  \cdots \otimes_E L$ is an integral domain with an endomorphism $\sigma$ defined on tensors by $a_1 \otimes \cdots \otimes a_n \mapsto \sigma(a_1) \otimes \cdots \otimes \sigma(a_n)$.  Hence, it may be embedded into a difference field, and then into a model $(M,\sigma)$ of ACFA.  The images of $b \otimes 1 \otimes \cdots \otimes 1$, $1 \otimes b \otimes 1 \cdots \otimes 1$, $\ldots$, $1 \otimes 1 \otimes \cdots \otimes b$ are distinct but the isomorphism between $L$ and its image where $x$ is mapped to the tensor $1 \otimes \cdots \otimes 1 \otimes x \otimes 1 \otimes \cdots \otimes 1$ where $x$ appears as the $m^\text{th}$ component shows by our last proposition that all of those elements have the same type over $E$ as $b$ itself.  Thus, $b \notin \operatorname{acl}(E)$; showing that $E$ is model theoretically algebraically closed.  $\Box$

The theory of difference fields admits a model companion, ACFA, often called “algebraically closed fields with a generic automorphism”.    To prove this, we need to show that the class of existentially closed difference fields is first-order axiomatizable.

We have already seen that every existentially closed differrnce field $(K,\sigma)$ is algebraically closed and that the distinguished endomorphism $\sigma$ is actually an automorphism.  Thus, ACFA is given by at least ACF together with the sentence $\forall y \exists x \sigma(x) = y$.   To complete the axiomatization we add the following schema of axioms:

For any absolutely irreducible embedded affine algebraic variety  $X \subseteq \mathbb{A}_K^n$ and irreducible subvariety $Y \subseteq X \times X^\sigma$ for which both projections $Y \to X$ and $Y \to X^\sigma$ are dominant there is a point $a \in X(K)$ with $(a,\sigma(a)) \in Y(K)$.    As we discussed when showing that $\operatorname{Fix}(\sigma)$ is PAC, these conditions can be expressed by a set of first-order sentences.   The key points are the following.

1. For $g_1, \ldots, g_m \in \mathbb{Z}[x_1, \ldots, x_n; y_1, \ldots, y_\ell]$ polynomials with integer coefficients there is a quantifier-free formula $\theta_g$ so that for any field $L$ and tuple $b \in L^y$ we have that the ideal $(g_1(x,b), \ldots, g_m(x,b)) \subseteq L^\text{alg}[x_1, \ldots, x_n]$ is prime if and only if $L \models \theta_g(b)$.   Hence, we can quantify over the absolutely irreducible $X \subseteq \mathbb{A}^n_K$ of bounded complexity and then the irreducible subvarieties $Y \subseteq X \times X^\sigma$.
2. Morley rank is definable in families in strongly minimal theories.  From this, we can definably in parameters determine where $Y \to X$ and $Y \to X^\sigma$ are dominant.

We call this axiom scheme “the geometric axioms”.  Let us note that they are equivalent to the ostensibly weaker system of axioms in which we require that $\dim X = \dim Y$.  Indeed, given an instance $(X,Y)$ of the usual geometric axioms, by Bertini’s Theorem we may find an affine space $V \subseteq \mathbb{A}_K^n \times \mathbb{A}_K^n$ with $\dim V = 2n + \dim X – \dim Y$ so that $Z := V \cap Y$ is (absolutely) irreducible and both projections $Z \to X$ and $Z \to X^\sigma$ are dominant.  Applying the restricted geometric axiom to $(X,Z)$ gives $a \in X(K)$ with $(a,\sigma(a)) \in Z(K) \subseteq Y(K)$. Hence, the geometric axiom for $(X,Y)$ also holds.

Let use now demonstrate that ACFA is the model companion of the theory of difference fields.

Theorem: ACFA, the theory described above, is the theory of the class of existentially closed difference fields.  That is, ACFA is the model companion of the theory of difference fields.

Proof:   First, we check that if $(K,\sigma)$ is an existentially closed difference field, then $(K,\sigma) \models \text{ACFA}$.  We have already seen that it $K$ is algebraically closed and that $\sigma$ is an automorphism.  For the geometric axioms, as we have seen it suffices to consider the case where $Y \subseteq X \times X^\sigma \subseteq \mathbb{A}_K^n \times \mathbb{A}_K^n$ with $\dim Y = \dim X$ and $X$ and $Y$ are absolutely irreducible.   Let us write the coordinate ring of $\mathbb{A}_K^n \times \mathbb{A}^n_K$ as $K[x_1, \ldots, x_n, y_1, \ldots, y_n]$.  Let $L := \mathcal{Q}(K[x_1,\ldots,x_n]/I(X))^\text{alg} = K(X)^\text{alg}$ be the algebraic closure of the function field of $X$.  Since $Y \to X$ is dominant and generically finite, the extension of functions field $K(Y)/K(X)$ is finite.  That is, $K(Y)$ embeds into $L$ over $K(X)$.  Let $a_i$ be the image of $x_i$ in $L$ and let $b_i$ be the image of $y_i$ in $L$ under the embedding of $K(Y)$ for $1 \leq i \leq n$.  We define an extension $\tau:K[x_1, \ldots, x_n] \to L$ of $\sigma$ by $x_i \mapsto b_i$.  Since $Y \to X^\sigma$ is dominant and $\sigma$ is an automorphism, the kernel of $\tau$ is $I(X^\sigma)^{\sigma^{-1}} = I(X)$.  Hence, $\tau$ extends to map, which we will still call $\tau$, $\tau:K(X) \to L$.  On general grounds, every map from a field to its algebraic closure extends to an endomorphism of the algebraic closure itself.  Hence, $\tau$ extends to a field endomorphism $\rho:L \to L$.   This gives an extension of difference fields $(K,\sigma) \subseteq (L,\rho)$ and $(L,\rho)$ satisfies that there is a point $a \in X(L)$ with $(a,\sigma(a)) \in Y(L)$.  Hence, by existential closedness of $(K,\sigma)$, the same is true of $(K,\sigma)$.

Now we need to show that every model of ACFA is existentially closed as a difference field. Let $(K,\sigma) \models \text{ACFA}$ and let $(L,\sigma)$ be an extension as a difference field.  Consider a quantifier-free formula $\theta(x,y)$ in the free variables $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_m)$.  Suppose that $b \in K^y$ and $a \in L^x$ with $L \models \theta(a,b)$.

Expressing $\theta$ in disjunctive normal form, we may write it as $$\theta = \bigvee_i (\bigwedge_{j=1}^{s_i} f_{i,j}(x,y) = 0 \land \bigwedge_{\ell=1}^{t_i} g_{i,\ell}(x,y) \neq 0)$$ where each $f_{i,j}$ and $g_{i,\ell}$ is a difference polynomial.  Since $L \models \theta(a,b)$, for some $i$, it satisfies one disjunct.  To prove that $K \models \exists x \theta(x,b)$, it suffices to show that there is a witness to that disjunct in $K$ as well.  So, we may assume that $$\theta = \bigwedge_{j=1}^s f_{j}(x,y) = 0 \land \bigwedge_{\ell=1}^t g_{\ell}(x,y) \neq 0)$$.

For each $\ell$, we may add a new variable $z_\ell$, and setting $z = (z_1, \ldots, z_s)$, if we let $c_\ell = \frac{1}{g_\ell(a,b)}$,  and set $$\vartheta(x,y,z) := \bigwedge_{i=1}^s f_j(x,y) = 0 \land \bigwedge_{\ell=1}^t  (z_\ell g_\ell(x,y) – 1) = 0$$ then $L \models \vartheta(a,b,c)$ and $\text{ACFA} \vdash \vartheta(x,y,z) \to \theta(x,y)$.  Thus, it suffices to show that $\vartheta(x,b,z)$ may be realized in $K$.  That is,  replacing $\theta$ by $\vartheta$, $x$ by $(x,z)$, and $a$ by $(a,c)$, we may assume that no inequalities appear in the formula.

A difference polynomial $f(x)$ may be expressed in the form $f(x) = g(x,\sigma(x), \ldots, \sigma^m(x))$ for some ordinary polynomial $g$ in the variables $x_0, \ldots, x_m$.   Then $a$ satisfies $f(x) =0$ if and only if $a’ := (a_0, a_1, \ldots, a_{m-1}) := (a,\sigma(a), \ldots, \sigma^{m-1}(a))$ satisfies the system of equations $g(x_0, \ldots, x_{m-1}, \sigma(x_{m-1})) = 0$ and $\sigma(x_i) = x_{i+1}$ for $0 \leq i < m-1$.  We repeat this process with each equation in the formula $\theta$.  That is, we find $m$ big enough so that each $f_j$  may be expresses as $f_j(x,y) = g_j(x,\sigma(x),\ldots,\sigma^m(x),y,\sigma(y), \ldots, \sigma^m(y))$, then if we set  $$\eta := \bigwedge_j g_j(x_0, \ldots, x_{m-1},\sigma(x_{m-1}),y_0, \ldots, y_{m-1},\sigma(y_{m-1})) = 0 \land \bigwedge_{i=0}^{m-1} \sigma(x_i) = x_{i+1} \land \bigwedge_{i=0}^{m-1} \sigma(y_i) = y_{i+1}$$ and we set $\widetilde{b}_i := \sigma^i(b)$ for $0 \leq i < m$, we have $$\text{ACFA} \vdash (\exists x \theta(x,b)) \leftrightarrow (\exists x_0  \cdots \exists x_{m-1} \eta(x_0, \ldots, x_{m-1}, \widetilde{b})$$. Thus, we replacing $\theta$ by $\eta$ and $b$ by $\widetilde{b}$ , we may assume that $\sigma$ is unnested in $\theta$.

Let $X := \text{loc}(a/K)$ and $Y := \text{loc}((a,\sigma(a))/K)$.  Here “loc” refers to the Zariski locus, the intersection of all $K$-algebraic varieties containing $a$, respectively $(a,\sigma(a))$.  Since $K$ is algebraically closed, these loci are absolutely irreducible.   As $a$ is the generic point (relative to $K$) of $X$, $\sigma(a)$ is generic in $X^\sigma$.  Thus, the projections $Y \to X$ and $Y \to X^\sigma$ are dominant.  The geometric axioms apply and we find $c \in X(K)$ with $(c, \sigma(c)) \in Y(K)$.  The point $(c,\sigma(c))$ satisfies the same polynomial equations over $K$ that $(b,\sigma(b))$ does.  In particular, it satisfies all of the equations in $\theta$.  This yields that $K \models \exists x (x \in X \land (x,\sigma(x)) \in Y)$.   $\Box$

Before we get to the scientfic content, allow me to remind you that the location for the seminar has changed to 891 Evans and we meet Wednesdays, 3-4pm.

Definition:  A difference field $(K,\sigma)$ is a field given together with a distinguished field endomorphism $\sigma:K \to K$.

The class of difference fields is naturally axiomatized by a first-order theory in the language $\mathcal{L}(+,\cdot,-,0,1,\sigma)$ of rings augmented by a unary function symbol for the endormorphism.

Theorem: The theory of difference fields has a model companion called $\operatorname{ACFA}$ for “Algebraically closed fields with a (generic) automorphism”.

Before we discuss the proof and attributions of this theorem, let us note how this theorem suggests a more general problem on theories of expansions by endomorphisms.

Let $\mathcal{L}$ be some first-order language and let $T$ be a theory which is assumed to be the model companion of it universal consequences $T_\forall$.  The example to keep in mind is $T = \operatorname{ACF}$.    In the language $\mathcal{L}(\sigma)$ expanding $\mathcal{L}$ by a unary function symbol $\sigma$ let $T_\sigma$ be the theory $T$ together with axioms expressing that $\sigma$ is an $\mathcal{L}$-automorphism.     We say that “$T_A$ exists” if $T_\sigma$ has a model companion. On general grounds, if $T_A$ exists, then it is the theory of the class of existentially closed models of $T_\sigma$.     The theory on $\operatorname{ACFA}$ says that $\operatorname{ACF}_A$ exists.   However, there are many cases in which it is known that $T_A$ does not exist.

• Kikyo, Hirotaka, Model companions of theories with an automorphism, JSL 65 (2000), no. 3, 1215 – 1222 shows that if $T$ is unstable and NIP or stable with the finite cover property, or if $T$ is unstable and $T_\sigma$ has the amalgamation property, then $T_A$ does not exist.
• Hirotaka Kikyo and Saharon Shelah, The strict order property and generic automorphisms, JSL 67 (2002), no. 1, 214 – 216 show that $T_A$ does not exist when $T$ has the strict order property.
• John Baldwin and Saharon Shelah, Model companions of $T_{\operatorname{Aut}}$ for stable $T$, Notre Dame J. Formal Logic 42 (2001), no. 3, 129 – 142 determine for which stable theories $T_A$ exists.

In the positive direction, Zoé Chatzidakis and Anand Pillay, Generic structures and simple theories, APAL 95 (1998), pp. 71 – 92 show that $T_A$ exists, for instance, when $T$ is strongly minimal and has the definable multiplicity property.

The exact dividing line for when $T_A$ exists remains unknown and there are many cases in which something like $T_A$ may be shown to exist when $T_\sigma$ is strengthened to require further universal conditions on $\sigma$. For example, if $T$ is the theory of nontrivially valued algebraically closed fields of residue characteristic zero and we define $T’_\sigma$ to be $T_\sigma$ together with the axiom that $(\forall x) v(\sigma(x)) = v(x)$, then $T’_\sigma$ has a model companion.

Harper Wells has offered to present details about $T_A$ later this term.

The name “algebraically closed fields with a generic automorphism” suggests that in every model of ACFA, the endomorphism $\sigma$ is an automorphism and that the underlying field is algebraically closed.  Both of these asssertions are true, though these two conditions do not suffice to axiomatize ACFA.

Proposition:  If $(K,\sigma)$ is an existentially closed difference field, then $K$ is algebraically closed.

Proof:  Let $(K,\sigma)$ be an existentially closed difference field.  Let $L := K^\text{alg}$ be an algebraic closure of $K$.  Then there is an extension $\tau:L \to L$ of $\sigma$ to $L$.  To show this it suffices to see (by topological compactness of $\text{Hom}(L,L)$ or the compactness theorem of first-order logic) that for each finite subextension $K \leq M \leq L$ that there is an extension $\tau:M \to L$ of $\sigma$.   Working recursively, it suffices to consider the case that $M = K[a]$ is a simple extension.  Let $P(x) \in K[x]$ be the minimal monic polynomial of $a$ over $K$.  Then $P^\sigma(x) \in K[x]$ is also a monic polynomial and thus has a zero $b$ in $L$. Using the universal mapping property of the polynomial ring, there is a unique homomorphism $\rho:K[x] \to L$ which restricts to $\sigma$ on $K$ and takes $x$ to $b$.   The image of $P(x)$ under $\rho$ is $P^\sigma(b) = 0$.  Hence, $\rho$ induces a map $\tau:L \cong K[x]/(P) \to L$.    Thus, $(K,\sigma)$ extends to an algebraically closed difference field $(K,\tau)$.  For any nonconstant polynomial $Q \in K[x]$ there is some $a \in L$ with $Q(a) = 0$.  That is, $(L,\tau) \models (\exists x) Q(x) = 0$.  As $(K,\sigma)$ is existentially closed, $(K,\sigma) \models  (\exists x) Q(x) = 0$.  That is, $K$ is algebraically closed. $\Box$

Proposition:  If $(K,\sigma)$ is an existentially closed difference field, then $\sigma$ is an automorphism.

Proof:    The map $\iota:(K,\sigma) \to (L,\tau) := (K,\sigma)$ given $x \mapsto \sigma(x)$ is an embedding of difference fields.   For each $a \in K$, we have that $(L,\tau) \models (\exists x) \sigma(x) = a$ as in $L$, $a$ is interpreted as $\sigma(a)$ and $\tau(a) = \sigma(a) = \iota(a)$.  Thus, as $(K,\sigma)$ is existentially closed, $(K,\sigma) \models (\exists x) \sigma(x) =a$.  That is, $\sigma(K) = K$ so that on $K$, $\sigma$ is an automorphism.  $\Box$

Early in the development of the model theory of difference fields, it was observed (by Lou van den Dries, I think, though let me confirm!) the fixed field $F$ of $\sigma$ in an existentially closed difference field $(K,\sigma)$ is pseudofinite.

Definition:  A field $F$ is pseudofinite if it is an infinite model of $\text{Th}(\{ K~:~K \text{ a finite field } \})$, the theory of all finite fields.  Equivalently, $F$ is pseudofinite if there is a nonprincipal ultrafilter $U$ on the set of prime powers for which $F \equiv \prod_U \mathbb{F}_q$.

James Ax produces a first-order axiomatization of the theory of pseudofinite fields in The Elementary Theory of Finite Fields, Annals of Mathematics, 88, no. 2, September 1968, pp. 239 – 271.

Theorem (Ax):  A field $F$ is pseudofinite if and only if

• $F$ is perfect,
• $\text{Gal}(F^\text{sep}/F) \cong \widehat{\mathbb{Z}}$, and
• $F$ is PAC: pseudo-algebraically closed

The first of these conditions is in the usual sense of field theory:  either $F$ has characteristic zero or it has characteristic $p > 0$ and $F^p = F$.   In the presence of perfection, the second condition may be re-expressed by saying that for each positive integer $n \in \mathbb{Z}_+$ there is a unique extension of fields $F_n/F$ with $[F_n:F] = n$.  Each of these conditions is naturally expressed by a first-order theory.  For example, perfection is given by the countably many sentences $$\overbrace{1 + 1 + \cdots + 1}^{p \text{ times }} = 0 \to (\forall x)(\exists y) y^p = x$$ We may describe field extensions of a fixed degree $n$ by quantifying over the coefficients of monic polynomials of degree $n$.  For example, to say that there is an extension of degree $n$ we assert that there is an irreducible monic polynomial of degree $n$ which could be expressed as $$(\exists a_0) \cdots (\exists a_{n-1}) \left(  \bigwedge_{1 \leq m < n} (\forall b_0) \cdots (\forall b_m) (\forall c_0) \cdots  (\forall c_{n-m-1}) \bigvee_{\ell=0}^{n-1}  \sum_{i+j=\ell} b_i c_j \neq a_\ell      \right)$$  Alternatively, one could more directly work with the family of $F$-algebras of dimension $n$ by describing these with $n \times n \times n$ arrays from $F$ giving the multiplication and then relativize the field axioms to these interpreted structures to isolate the fields.  Maps between these field extensions are described by linear maps, that is, by matrices.  So, one could express that all degree $n$ extensions are isomorphic.

It is less clear how to express the PAC condition in first-order logic.  In the definition of PAC we were a little sloppy about the meaning of “variety”.  Nothing would be lost by taking this term to mean “embedded affine variety”.   Such varieties are described by systems of polynomial equations, which may be taken to be finite systems of equations by Noetherianity of the polynomial rings.  For the natural statement of the PAC condition, one should show that given variables $\mathbf{x} = (x_1, \ldots, x_n)$ and $\mathbf{y} = (y_1, \ldots, y_m)$ and polynomials $F_1(\mathbf{x},\mathbf{y}), \ldots, F_\ell(\mathbf{x},\mathbf{y}) \in \mathbb{Z}[\mathbf{x},\mathbf{y}]$ there is a formula $\theta(\mathbf{y}) = \theta_{\mathbf{F}}$ so that for any field $F$ and tuple $\mathbf{b} \in F^{\mathbf{y}}$ we have $F \models \theta(\mathbf{b})$ if and only if the ideal generated by $F_1(\mathbf{x},\mathbf{b}), \ldots, F_\ell(\mathbf{x},\mathbf{b})$ in $F^\text{alg}[\mathbf{x}]$ is prime.   If one proves this assertion in the case that $F = F^\text{alg}$, then using quantifier elimination for algebraically closed fields, one sees that $\theta$ may be taken to be quantifier-free and then it will continue to have the requisite properties for arbitrary fields.   That such formulas exist was shown by Lou van den Dries and Karsten Schmidt in Bounds in the theory of polynomial rings over fields: A nonstandard approach, Inventiones Math. 76 (1984), no. 1, 77 – 91.   The theorem is proven as a corollary of general result that if $K$ is any field and $U$ is an ultrafilter on some set $I$, then the algebra $(K[x_1,\ldots,x_n])^U$ is a faithfully flat extension of $K^U[x_1, \ldots, x_n]$.  An alternative more elementary proof was given by Will Johnson in the appendix to Differential Chow varieties exist, J. LMS (2), 95 (2017), no. 1, 128 – 156.

Theorem:  If $(K,\sigma)$ is an existentially closed difference field and $F := \operatorname{Fix}(\sigma,K) := \{ x \in K : \sigma(x) = x \}$, then $F$ is pseudofinite.

Proof: First, we check that $F$ is perfect. If the characteristic of $K$ is zero, there is nothing to check.  Suppose that the characteristic of $K$ is $p > 0$.  Let $a \in F$.  Since $K$ is algebraically closed, there is some $b \in K$ with $b^p = a$.  We compute that $\sigma(b)^p = \sigma(b^p) = \sigma(a) = a = b^p$.  Since the map $x \mapsto x^p$ is injective on fields of characteristic $p$, we conclude that $\sigma(b) = b$.  That is, $b \in F$.  As $a \in F$ was arbitrary, we see that $F = F^p$.  That is, $F$ is perfect.

For the second condition, let $n \in \mathbb{Z}_+$ and set $R := K[x_0,\ldots,x_{n-1}]$.  Let $L := \mathcal{Q}(R) = K(x_0,\ldots,x_{n-1})$ be the field of fractions of $R$.  From the universal mapping property of the polynomial ring, there is a unique map $\tau:R \to L$ for which $\tau \upharpoonright K = \sigma$ and $\tau(x_i) = x_{i+1 \mod n }$ for $0 \leq i < n$.   More concretely, if $P(x_0, \ldots, x_{n-1}) \in R$, then $\tau(P) = P^\sigma (x_1, x_2, \ldots, x_{n-1}, x_0)$.   In particular, if $P \neq 0$, then $\tau(P) \in L^\times$.  Thus, from the universal property of the localization, $\tau$ extends uniquely to a map $\tau:L \to L$.  That is, $(L,\tau)$ is a difference field extension of $(K,\sigma)$.

The element $x_0 \in L$ satisfies $\tau^n(x_0) = x_0$.  Let $P(X) := \prod_{i=0}^{n-1} (X – x_i)$.  Then $P \in \operatorname{Fix}(\tau,L)[X]$ and its roots, $\{ x_i : 0 \leq i < n \}$,  form one orbit under the action of the subgroup of $\operatorname{Gal}(\operatorname{Fix}(\tau,L)^\text{alg}/\operatorname{Fix}(\tau,L))$.  Hence, $[\operatorname{Fix}(\tau,L)(x_0):\operatorname{Fix}(\tau,L)] = n$.   That is, $$(L,\tau) \models (\exists x) [\operatorname{Fix}(\sigma)(x):\operatorname{Fix}(\sigma)] = n$$ which implies that the same is true of $(K,\sigma)$ as it is existentially closed in $(L,\tau)$.   On general grounds, if $a \in K$ and $[F(a):F] = n$, then $\sigma^n(a) = a$.  Indeed, if $P(X) \in F[X]$ is the minimal monic polynomial of $a$, then $0 = \sigma(0) = \sigma(P(a))= P^\sigma(\sigma(a)) = P(\sigma(a))$ where the last equality holds as $P$ is defined over $F$.   Thus, the set $\{ \sigma^i(a) : i \in \mathbb{Z} \}$ has size exactly $n$, which implies in particular that $a = \sigma^n(a)$.  Therefore, $\operatorname{Fix}(\sigma^n)$ is the unique extension of $F$ of degree $n$.

Finally, let us check that $F$ is PAC.  Let $X \subseteq \mathbb{A}_F^n$ be an absolutely irreducible embedded affine variety over $F$.  Let $g_1, \ldots, g_m$ be generators of $I(X) \subseteq \mathbb{A}_F^n$.  Since $X$ is absolutely irreducible, $(g_1, \ldots, g_m) \subseteq K[x_1, \ldots, x_n]$ remains prime.  Let $L := \mathcal{Q}(K[x_1, \ldots, x_n]/(g_1, \ldots, g_m))$ be the field of the ring $K[x_1, \ldots, x_n]/(g_1, \ldots, g_m)$.  Define $\tau:K[x_1, \dots, x_n] \to L$ using the universal mapping property of polynomial rings via $\tau \upharpoonright K := \sigma$ and $\tau(x_i) = \frac{x_i + (g_1, \ldots, g_m)}{1}$ for $1 \leq i \leq n$.   The kernel of $\tau$ is $(g_1, \ldots, g_m)$ (NOTE: we are using the fact that $X$ is defined over $F$: for $h = \sum h_\alpha x^\alpha \in K[x_1, \ldots, x_n]$, the image of $h$ under $\tau$ is $h^\sigma + (g_1, \ldots, g_m) = \sum \sigma(h_\alpha) x^\alpha + (g_1, \ldots, g_m)$.  Thus, $\tau(h) = 0$ if and only if $h^\sigma \in (g_1, \ldots, g_m)$, which, because $\sigma$ is an automorphism of $K$, is equivalent to $h \in (g_1^{\sigma^{-1}}, \ldots, g_m^{\sigma^{-1}}) = (g_1, \ldots, g_m)$.).  Thus, $\tau$ extends to a map of fields $\tau:L \to L$.  That is, $(K,\sigma) \subseteq (L,\tau)$ is an extension of difference fields. $(L,\sigma) \models (\exists x)  x \in X(\operatorname{Fix}(\sigma))$.  Hence, as $(K,\sigma)$ is existentially closed, it also satisfies this conclusion.  $\Box$

This proposition suggested an ambitious conjecture which was subsequently proven to be true.

Theorem:  If $U$ is a nonprincipal ultrafilter on the set of prime power, then $\prod_U (\mathbb{F}_q^\text{alg},x \mapsto x^q)$ is an existentially closed difference field.  In fact, if $(K,\sigma)$ is an existentially closed difference field, then there is a nonprincipal ultrafilter $U$ on the set of prime powers for which $\prod_U (\mathbb{F}_q^\text{alg},x \mapsto x^q) \equiv (K,\sigma)$.

This theorem was announced by Macintyre in the late 1990s, though his proof has not been made available.  An arXiv posting of Hrushovski, The Elementary Theory of the Frobenius, is devoted to its proof through a twisted version of Deligne’s theorem on rational points on varieties in finite fields proven with a mix of difference geometry, the model theory of valued fields, and étale cohomology, amongst other methods.   Shuddhodan and Varshavsky, The Hrushovski-Lang-Weil estimates, Alg. Geom. 9 (2022), no. 6. 651 – 687, prove the estimates with arguments closer to Deligne’s method.

Our seminar this semester will focus on the model theory of difference fields.  We meet in 891 (note the change in location!) Evans, Wednesdays 3-4pm.  The Course Control Number is 15443.

The main sources are the following research papers.

• Zoé Chatzidakis and Ehud Hrushovski, Model Theory of Difference Fields, Trans. AMS, 351, no. 8, pp. 2997 – 3071, April 8, 1999
• Zoé Chatzidakis, Ehud Hrushovski, and Ya’acov Peterzil, Model theory of difference fields II: Periodic ideals and the trichotomy in all characteristics, Proc. LMS (3) 85 (2002) 257 – 311
• Zoé Chatzidakis and Ehud Hrushovski, Revisiting virtual difference ideals, Model Theory 3 no. 2 (2024) 285 – 304.

The model companion ACFA of the theory of a fields with a distinguished endomorphism was one the first non-trivial examples of a simple unstable theory to be studied in detail.  We will recount some of the development of simplicity through the lens of difference fields.

Looking to applications outside of model theory, ACFA has been used in the study of special point problems and in algebraic dynamics.  We will look at some of these applications.  References include the following papers.

• Zoé Chatzidakis and Ehud Hrushovski, Difference fields and descent in algebraic dynamics I, JIMJ 7 (2008) no. 4, 653 – 686.
• Zoé Chatzidakis and Ehud Hrushovski, Difference fields and descent in algebraic dynamics II, JIMJ 7 (2008) no. 4, 687 – 704.
• Ehud Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, APAL 112 (2001), no. 1, 43 – 115.
• Alice Medvedev and Thomas Scanlon, Invariant varieties for polynomial dynamical systems, Annals of Mathematics (2) 179 (2014), no. 1, 81 – 177
• Thomas Scanlon, A positive characteristic Manin-Mumford theorem, Compos. Math. 141:6 (2005) 1351 – 1364
• Thomas Scanlon, Local André-Oort conjecture for the universal abelian variety, Inv. Math. 163 (2006) no. 1, 191 – 211

Learning seminar on o-minimality (Autumn 2024)

Research seminar (Autumn 2024)

Geometric Stability Theory learning seminar