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.
- 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 $Y \subseteq X \times X^\sigma$.
- 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 $K \models \exists x (x \in X \land (x,\sigma(x)) \in Y)$. $\Box$