Proof The equation in Statement 1 is easily seen to hold in the finite case. Each $A^\star _i \cdot |Z_i|$ is the cardinality of the set $\{(\bar {a},\bar {b}) \in M^{|\bar {x}| + |\bar {y}|} : \bar {b} \in Z_i$ and $M \models \varphi (\bar {a},\bar {b})$ }.

To prove Statement 2, note that under our special assumption, Statement 1, with the roles of $\bar {x}$ and $\bar {y}$ switched, tells us that $|\varphi (M^{|\bar {x}| + |\bar {y}|},\bar {c})| = B^\star \cdot |\{\bar {a} \in M^{|\bar {x}|} : |\varphi (\bar {a},M^{|\bar {y}|}, \bar {c}) = B^\star \}| = B^\star \cdot |\{\bar {a} \in M^{|\bar {x}|} : M \models \exists \bar {y} \varphi (\bar {a},\bar {y},\bar {c})\}|$ . Therefore $\sum _{i=1}^n A^\star _i \cdot |Z_i| = B^\star \cdot |\{\bar {a} \in M^{|\bar {x}|} : M \models \exists \bar {y} \varphi (\bar {a},\bar {y},\bar {c})\}|$ . Dividing both sides by $B^\star $ gives the desired result.

Proof

1. 1. Express $A^\star $ as an ultralimit $(a_\lambda )_{\lambda \to \mathcal {U}}$ , with each $a_\lambda \in \mathbb {R}$ . Then $F(A^\star ) = (F(a_\lambda ))_{\lambda \to \mathcal {U}}$ and $G(A^\star ) = (G(a_\lambda ))_{\lambda \to \mathcal {U}}$ . Since these two ultralimits are equal, we have $F(a_\lambda ) = G(a_\lambda )$ for almost all $\lambda \in \Lambda $ . Since $A^\star $ is infinite, for every standard $n \in \omega $ we have $a_\lambda> n$ for almost all $\lambda $ . It follows that for every n there is a real $a_\lambda> n$ such that $F(a_\lambda ) = G(a_\lambda )$ . Therefore the polynomial $F(x) - G(x)$ has arbitrarily large zeros in $\mathbb {R}$ . Hence it must be constant zero, and so $F(x) = G(x)$ for all x.

2. 2. In $\mathbb {R}$ , there is an n such that $F(x)$ is strictly increasing or decreasing on $(n,\infty )$ (in $\mathbb {R})$ ). In particular, the function $x \mapsto F(x)$ is injective on this open ray. This is first-order expressible, hence true in $\mathbb {R}^\star $ , and since $A^\star> n$ and $B^\star> n$ we get $A^\star = B^\star $ .

3. 3. If the leading coefficient of $F(x)$ were negative, then $F(a)$ is negative for sufficiently large real numbers a—that is, for $a> M$ for some real M. Therefore if $A^\star = (a_\lambda )_{\lambda \in \Lambda }$ is positive and infinite, the set $\{\lambda \in \Lambda : a_\lambda> M\}$ is $\mathcal {U}$ -large. Then the set $\{\lambda \in \Lambda : F(a_\lambda ) < 0\} = \{\lambda \in \Lambda : (F(A^\star ))_\lambda \}$ is $\mathcal {U}$ -large, whence $F(A^\star )$ is negative. Fact 3 follows contrapositively.

Fact 4.1 [Reference Buechler2, Lemma 3.4.10].

Let M be an uncountable model of an $\aleph _1$ -categorical theory T. Then M is saturated.

Fact 4.3 [Reference Pillay6, Chapter 1, Propositions 5.14 and 5.18].

Let T be an uncountably categorical theory. Then T has definable and finite Morley rank.

Fact 4.4. Suppose T is $\aleph _1$ -categorical and $\varphi (\bar {x},\bar {y})$ is a formula. Then there is a natural number n such that in all models M of T and all $\bar {b} \in M^{|\bar {y}|}$ , either $\varphi (M^{|\bar {x}|},\bar {b})$ is infinite or $|\varphi (M^{|\bar {x}|}, \bar {b})| \leq n$ .

Proof

1. 1. We prove this by induction on S. At $S = 0$ , the statement is trivial. Specifically, let $\bar {b} \in B$ be any element. Since $f^{-1}(\bar {b}) \subseteq A$ we have ${MR(A) \geq MR(f^{-1}(\bar {b})) \geq R = R + S}$ .

   Suppose the statement is proven for $s < S$ and suppose $MR(B) \geq S$ . Let $B_1, B_2, \ldots $ be disjoint definable subsets of B of rank $S-1$ . For $i \in \omega $ let ${A_i = f^{-1}(B_i)}$ . Then $A_1, A_2, \ldots ,$ are disjoint definable subsets of A. Let $f_i$ be the restriction of f to $A_i$ . Then $f_i$ is a surjection of $A_i$ onto $B_i$ . For $\bar {b} \in B_i$ we have $f_i^{-1}(\bar {b}) = f^{-1}(\bar {b})$ which has rank $\geq R$ by assumption. Therefore, by the inductive hypothesis, $MR(A_i) \geq R + S - 1$ for each i. Hence ${MR(A) \geq R + S}$ .

2. 2. We prove Statement 2 by induction on $R + S$ , with base case $S = 0$ (and R equal to anything).

   If $S = 0$ then $MR(B) \leq S$ means B is finite. Let $B = \{b_1,\ldots ,b_p\}$ . Then $A = f^{-1}(b_1) \cup \dots \cup f^{-1}(b_p)$ . If each $f^{-1}(b_i)$ has Morley rank $\leq R$ , then as a finite union of such sets, A also has a Morley rank of $\leq R = R + S$ .

   Assume the statement is proven for all pairs $r,s$ such that ${r + s < R + S}$ . Suppose first that the Morley degree of B is 1, and suppose towards a contradiction that $MR(A)> R + S$ . Then we can find $A_1, A_2, A_3, \ldots $ which are disjoint subsets of A of Morley rank $R + S$ . For each i, let $B_i = f(A_i)$ . Then for each i, the map $f_i$ which is the restriction of f to $A_i$ is a surjection from $A_i$ onto $B_i$ . The fibers of $f_i$ are still of rank $\leq R$ , as they are subsets of the fibers of f. If $MR(B_i) < S$ then by induction we obtain $MR(A_i) \leq R + MR(B_i) < R + S$ , a contradiction. Therefore $MR(B_i) = S$ for each i, and since B has degree 1, we obtain $MR(B \setminus B_i) < S$ for each i, as well as the fact that the Morley degree of each $B_i$ is 1.

   In each $B_i$ , let $B_i^+ = \{\bar {b} \in B_i : MR(f_i^{-1}(\bar {b})) = R\}$ and let $B_i^- = \{\bar {b} \in B_i : MR(f_i^{-1}(\bar {b})) < R\} = B_i \setminus B_i^+$ . Each $B_i^+$ and $B_i^-$ is definable, by definability of Morley rank in $Th(M)$ . Let $A_i^+ = f_i^{-1}(B_i^+)$ and $A_i^- = f_i^{-1}(B_i^-) = A_i \setminus A_i^+$ . Then $f_i$ is a surjection of $A_i^+$ onto $B_i^+$ and of $A_i^-$ onto $B_i^-$ . If $R = 0$ then $B_i^-$ and $A_i^-$ are empty; if $R> 0$ then the fibers of $f_i$ restricted to $A_i^-$ have Morley rank $\leq R - 1$ , so by induction, $MR(A_i^-) \leq R - 1 + S$ . In either case, ${MR(A_i^+) = MR(A_i) = R + S}$ and $MR(A \setminus A_i^+) < R + S$ . If $MR(B_i^+) < S$ then as in the previous paragraph, since $f_i$ surjects $A_i^+$ onto $B_i^+$ with fibers of rank R, we would have $MR(A_i^+) < R + S$ , which is false. Therefore $MR(B_i^+) = S$ for each i, and since B has degree 1, we have $MR(B \setminus B_i^+) < S$ for each i. Therefore for every n, the set $B - (B^+_1 \cap \dots \cap B^+_n) = (B \setminus B_1^+) \cup \dots \cup (B \setminus B_n^+)$ has Morley rank $< S$ , and so $B^+_1 \cap \dots \cap B^+_n$ has Morley rank S.

   Consider the partial type $\pi (\bar {y}) = \{"\bar {y} \in B^+_i" : i \in \omega \}$ , which is definable with parameters from the set of parameters which define the sets $A_1, A_2, \ldots $ . Since $B^+_1 \cap \dots \cap B^+_n$ has Morley rank S and is in particular non-empty for every n, the type $\pi $ is consistent. Because M is $\omega _1$ -saturated, $\pi $ is realized by some $\bar {b} \in B$ . Then $MR(f_i^{-1}(\bar {b})) = R$ for all $i \in \omega $ . But then $f_1^{-1}(\bar {b}), f_2^{-1}(\bar {b}), \ldots $ are disjoint subsets of $f^{-1}(\bar {b})$ of Morley rank R, hence $MR(f^{-1}(\bar {b}))> R$ , a contradiction.

   This proves that $MR(A) \leq R + S$ when the Morley degree of B is 1. If the degree of B is $D> 1$ , then B is the disjoint union of definable sets ${B_1, \ldots , B_D \subseteq B}$ , each of Morley rank S and Morley degree 1. Letting $A_ i = f^{-1}(A_i)$ we get by the $D=1$ case that $MR(A_i) \leq R + S$ , and since $A = A_1 \cup \dots \cup A_i$ this completes the proof.

Fact 4.9 [Reference Zil’ber10, Lemma 1].

Let T be an uncountably categorical theory and let $M \models T$ be uncountable. Let $A, B \subset M$ be infinite definable sets, with B strongly minimal. Then there exists a proper stratification of A with respect to B.

Proof Consider the partial type $\Pi (\bar {y},\bar {w})$ consisting of the formulas

$$\begin{align*}\forall \bar{z} [Mor_{\theta(x,\bar{w}) \wedge \pi(x,\bar{z}), 0}(\bar{z},\bar{w}) \vee Mor_{\theta(x,\bar{w}) \wedge \neg \pi(x,\bar{z}), 0}(\bar{z},\bar{w})]\end{align*}$$

and

$$\begin{align*}\forall{\bar{z}} [\neg PrStrat_{\psi,\varphi, \theta}(\bar{y},\bar{z},\bar{w})]\end{align*}$$

for all L-formulas $\pi (x,\bar {z})$ and $\psi (x,v,\bar {z})$ (recall $Mor_{\phi ,n})$ from Definition 4.2 and $PrStrat_{\varphi , \psi , \theta }$ from Definition 4.8.) If $M \models \Pi (\bar {b},\bar {d})$ , the first formula schema tells us that $\theta (M,\bar {d})$ is strongly minimal, and the second formula schema tells us that there is no proper stratification of $\varphi (M,\bar {b})$ with respect to $\theta (M,\bar {d})$ .

Suppose the first paragraph of the corollary were false. Then for any finite collection of L-formulas $\psi _1(x,v,\bar {z}), \ldots , \psi _n(x,v,\bar {z})$ , there is an $M \models T$ and $\bar {b},\bar {d} \in M$ such that $\theta (M,\bar {d})$ is strongly minimal and such that for each i, there is no $\bar {c} \in M$ such that $\psi _i(x,v,\bar {c})$ is a proper stratification of $\varphi (M,\bar {b})$ with respect to $\theta (M,\bar {d})$ . Then $\Pi (\bar {y},\bar {w})$ is consistent: we can realize any finite subset of $\Pi (\bar {y},\bar {w})$ with such a $\bar {b},\bar {d}$ . Choose an uncountable M which is a model of T. Then M is a saturated model, by categoricity. Therefore $\Pi $ is realized by some $\bar {b},\bar {d} \in M$ . Let $A = \varphi (M,\bar {b})$ and $B = \theta (M,\bar {d})$ . Then B is strongly minimal, and there is no proper stratification of A with respect to B, contradicting Fact 4.9.

Therefore the first paragraph of the corollary holds. We can force the index i to be unique by modifying our formulas $\psi _i$ so that, for instance, $\psi _2(x,v,\bar {c})$ implies that $\psi _1(x,v,\bar {c})$ does not properly stratify $\varphi (M,\bar {b})$ with respect to $\theta (M,\bar {b'})$ .

Fact 4.11 [Reference Zil’ber10, Theorem 3].

Let $\Delta (x),\Phi (x)$ be L-formulas over $A \subset M$ with $MR(\Delta (M)) = m$ and let $\psi (v_0,v_1)$ be a stratification of $\Phi (M)$ with respect to $\Delta (M)$ of rank at most n. Then $MR(\Phi (M)) \leq m + n$ .

Proof

1. 1. As D is strongly minimal, there is an $N \in \omega $ such that for all $a \in M$ we have that either $|\psi (a,D)| \leq N$ or $|\neg \psi (a,D)| \leq N$ . Let $d_1, \ldots , d_N, d_{N+1} \in D$ be distinct elements. For each $i = 1, \ldots , N+1$ let $X_i = \psi (X,d_i)$ . As $\psi $ is a proper stratification, we have $MR(X_i) < MR(X)$ for each i. Therefore $MR(X_1 \cup \dots \cup X_{N+1}) < MR(X)$ . If $a \in X$ and $|\psi (a,D)|> N$ then $\psi (a,D)$ is cofinite in D with $|D - \psi (a,D)| \leq N$ . Therefore at least one of $d_1,\ldots ,d_{N+1}$ must be an element of $\psi (a,D)$ , that is, $M \models \psi (a,d_i)$ for some i. Then $a \in \varphi (X,d_i) = X_i$ . Hence $\{a \in X : |\psi (a,D)|> N\} \subseteq X_1 \cup \ldots \cup X_{N+1}$ , and therefore $MR(\{a \in X : |\psi (a,D)|> N\}) \leq MR(X_1 \cup \dots X_{N+1}) < MR(X)$ .

2. 2. Suppose otherwise. Then $\{d \in D : MR(\psi (M,d)) = MR(X) - 1\}$ is finite and $\{d \in D : MR(\psi (M,d)) \leq MR(X) - 2\}$ is cofinite. Let $\{d \in D : MR(\psi (M,d)) \kern1.3pt{=}\kern1.3pt MR(X) - 1\} \kern1.3pt{=} \{d_1,\ldots ,d_n\}$ . For $i \kern1.3pt{=}\kern1.3pt 1, \ldots , n$ let ${X_i \kern1.3pt{=}\kern1.3pt \psi (M,d_i)}$ and let $X' = \bigcup _{d \in D - \{d_1,\ldots ,d_n\}} \psi (M,d)$ . Then $X = X_1 \cup \dots \cup X_n \cup X'$ . Each $X_i$ has Morley rank $MR(X) - 1$ . $X'$ is stratified by $\psi (x,y)$ over $D - \{d_1,\ldots ,d_n\}$ , and this stratification has rank at most $MR(X) - 2$ . Therefore $MR(X') \leq MR(X) - 2 + MR(D - \{d_1,\ldots ,d_n\}) = MR(X) - 1$ . So X is the union of finitely many sets of Morley rank at most $MR(X) - 1$ , an impossibility. Therefore the set $\{d \in D : rk(\psi (X,d)) = rk(X) - 1\}$ must be cofinite in D.

Proof Let $n\ =\ Num_T(\varphi (x,\bar {y}))$ . Then for $M \ \models\ T$ and $\bar {b} \ \in\ M$ , if ${RK(\varphi (M,\bar {b})) = 0}$ —which means $\varphi (M,\bar {b})$ is finite—then $|\varphi (M,\bar {b})| < n$ . We take our $F_1(x),\ldots ,F_{n}(x)$ to be the constant degree-0 polynomials $0,\ldots ,n-1$ and take each $\pi _i(\bar {y})$ to be an L-formula expressing “ $|\varphi (M,\bar {b})| = i-1$ ”—note that this is a formula over $\emptyset $ .

Proof By the inductive hypothesis, there are polynomials $G_1,\ldots ,G_m \in \mathbb {Q}[x]$ of degree $< R$ and formulas $\rho _1(v,\bar {z},\bar {w}),\ldots ,\rho _m(v,\bar {z},\bar {w})$ such that for all models $M \models T$ , all $\bar {d} \in M^{|\bar {w}|}$ such that $D_{\bar {d}} = \theta (M,\bar {d})$ is strongly minimal, and all ${(e,\bar {c}) \in M^{1+|\bar {z}|}}$ ,

• • if $MR(\psi (M,e,\bar {c})) < R$ then $|\psi (M,e,\bar {c})| = G_i(|D_{\bar {d}}|)$ for some i, and

• • each $\rho _i$ defines the set $\{(e,\bar {c},\bar {d}) : MR(\psi (M,e,\bar {c})) < R$ and $|\psi (M,e,\bar {c})| = G_i(|D_{\bar {d}}|)\}$ , and

• • if $M \models \rho _i(e,\bar {c},\bar {d})$ then $\deg G_i = MR(\psi (M,e,\bar {c}))$ .

Let N be the maximum of the $2m$ numbers given by $Num_T(\xi _i(v;\bar {z}\bar {w}))$ and $Num_T(\xi ^{\prime }_i(v;\bar {z}))$ for $i = 1, \ldots , m$ , where $\xi _i(v;\bar {z} \bar {w})$ is $\theta (v,\bar {w}) \wedge \rho _i(v,\bar {z},\bar {w})$ and $\xi ^{\prime }_i(v;\bar {z} \bar {w})$ is $\theta (v,\bar {w}) \wedge \neg \rho _i(v,\bar {z},\bar {w})$ . Then if $\theta (M,\bar {d})$ is strongly minimal, we have that for each $i = 1, \ldots , m$ and for all $\bar {c}' \in M^{|\bar {z}|}$ , either $|\rho _i(D,\bar {c}')| < N$ or ${|D - \rho _i(D,\bar {c}')| < N}$ .

Let $\bar {b} \in Mor_{\varphi , R}(M^{|\bar {y}|})$ and $\bar {d} \in M^{|\bar {w}|}$ such that $D_{\bar {d}} = \theta (M,\bar {d})$ is strongly minimal. By the assumptions of our lemma, there is $\bar {c} \in M^{|\bar {z}|}$ such that $\psi (x,v,\bar {c})$ is a proper stratification of $\varphi (M,\bar {b})$ with respect to $D_{\bar {d}}$ —i.e., $M \models PrStrat_{\psi ,\varphi ,\theta }(\bar {b},\bar {c},\bar {d})$ (this formula is defined in the remarks before Lemma 5.4).

If $\bar {c}$ is such that $M \models PrStrat_{\psi ,\varphi ,\theta }(\bar {b},\bar {c},\bar {d})$ then since $\psi (x,w,\bar {c})$ is a proper stratification of $\varphi (M,\bar {b})$ , the x-fibers $\psi (M,e,\bar {c})$ have rank $< R$ for all $e \in D$ . Therefore for each $e \in D_{\bar {d}}$ there is a unique i such that $M \models \rho _i(e,\bar {c},\bar {d})$ . So the sets $\rho _1(D_{\bar {d}},\bar {c}), \ldots , \rho _m(D_{\bar {d}},\bar {c})$ partition $D_{\bar {d}}$ . Therefore all but one of these sets is finite of cardinality $\leq N$ , and one of these sets is cofinite of pseudofinite cardinality at least $|D_{\bar {d}}| - N$ (in fact, exactly equal to $|D_{\bar {d}}| - K$ where K is the sum of the remaining finite cardinalities, which must therefore be no greater than N).

Let $\Sigma _{N,m}$ be the finite set $\{(\sigma _1,\ldots ,\sigma _m) \in \{0,1,\ldots ,N,\infty \}^m : \sigma _i = \infty $ for exactly one i, and $\sum _{j \neq i} \sigma _j \leq N\}$ .

For $\bar {b} \in Mor_{\varphi , R}(M^{|\bar {y}|})$ and $\bar {c} \in PrStrat_{\psi ,\varphi ,\theta }(\bar {b},M^{|\bar {z}|},\bar {d})$ , let $\sigma (\bar {b},\bar {c},\bar {d})$ be the tuple $(\sigma _1,\ldots ,\sigma _m) \in \Sigma _{N,m}$ such that $\sigma _i = |\rho _i(D_{\bar {d}},\bar {c},\bar {d})|$ if this set is finite, and $\sigma _i = \infty $ if i is the unique index such that $\rho _i(D_{\bar {d}},\bar {c},\bar {d})$ is infinite. Note that for any $\sigma \in \Sigma _{N,m}$ , the set $\{(\bar {b},\bar {c},\bar {d}) : \bar {b} \in Mor_{\varphi , R}(M^{|\bar {y}|})$ and $\bar {c} \in PrStrat_{\psi ,\varphi ,\theta }(\bar {b},M^{|\bar {z}|},\bar {d})$ and $\sigma (\bar {b},\bar {c},\bar {d}) = \sigma \}$ is definable by a formula $\pi _\sigma (\bar {y},\bar {z},\bar {w})$ over $\emptyset $ .

Suppose $\sigma (\bar {b},\bar {c},\bar {d}) = \sigma = (\sigma _1,\ldots ,\sigma _m)$ . Let i be the unique index such that $\sigma _i = \infty $ , and let $n_\sigma := \sum _{j \neq i} \sigma _j$ . Noting that for $j \neq i$ the set $\rho _j(D_{\bar {d}},\bar {c},\bar {d})$ is the set $\{b \in D_{\bar {d}} : |\psi (M,b,\bar {c})| = G_j(|D_{\bar {d}}|)\}$ , we obtain by Lemma 3.4 the equation

$$\begin{align*}|\varphi(M,\bar{b})| = \frac{G_i(|D_{\bar{d}}|) \cdot (|D_{\bar{d}}| - n_\sigma) + \sum_{j \neq i} G_j(|D_{\bar{d}}|) \cdot \sigma_j}{n}.\end{align*}$$

Therefore $|\varphi (M,\bar {b})| = F_\sigma (|D_{\bar {d}}|)$ , where

$$\begin{align*}F_\sigma(X) = \frac{G_i(X) \cdot (X - n_\sigma) + \sum_{j \neq i} G_j(X) \cdot \sigma_j}{n} \in \mathbb{Q}[X].\end{align*}$$

By induction, each $G_j(X)$ has degree $< R$ . Moreover, $G_i(X)$ has degree exactly $R - 1$ , which follows from Lemma 4.12.2, since i is the unique i such that $\{b \in D_{\bar {d}} : |\psi (M,b,\bar {c})| = G_i(D_{\bar {d}})\}$ is cofinite. Therefore $F_\sigma (X)$ is a polynomial of degree R.

Suppose $\bar {c}'$ is another element of $PrStrat_{\psi ,\varphi ,\theta }(\bar {b},M^{|\bar {z}|},\bar {d})$ . Let $\sigma ' = \sigma (\bar {b},\bar {c}',\bar {d})$ with coordinates $(\sigma _1',\ldots ,\sigma _m')$ . Let $i'$ be the unique index such that $\sigma ^{\prime }_{i'} = \infty $ and let $n_{\sigma '} = \sum _{j \neq i'} \sigma _j$ . Then as before we have

$$\begin{align*}|\varphi(M,\bar{b})| = \frac{G_{i'}(|D_{\bar{d}}|) \cdot (|D_{\bar{d}}| - n_{\sigma'}) + \sum_{j \neq i'} G_j(|D_{\bar{d}}|) \cdot \sigma^{\prime}_j}{n}.\end{align*}$$

Therefore the polynomial $F_{\sigma '}(x) = \frac {G_{i'}(x) \cdot (x - n_{\sigma '}) + \sum _{j \neq i'} G_j(x) \cdot \sigma ^{\prime }_j}{n}$ agrees with the polynomial $F_\sigma (x)$ when applied to the infinite hyper-integer $|D_{\bar {d}}|$ . By Lemma 3.5.1, the polynomials $F_\sigma $ and $F_{\sigma '}$ are equal.

This shows that the choice of $\bar {c} \in PrStrat_{\psi ,\varphi ,\theta }(\bar {b},M^{|\bar {z}|},\bar {d})$ does not affect the polynomial $F_\sigma $ . Formally, we can quotient $\Sigma _{N,m}$ by the equivalence relation $\approx $ where $\sigma \approx \sigma '$ if $F_{\sigma } = F_{\sigma '}$ . Then for each $\approx $ -equivalence class $[\sigma ]_{\approx }$ , we can define the set $\{\bar {b} \in Mor_{\varphi , R}(M^{|\bar {y}|}) : \sigma (\bar {b},\bar {c},\bar {d}) \approx \sigma $ for all $\bar {c} \in PrStrat_{\psi ,\varphi ,\theta }(\bar {b},M^{|\bar {z}|},\bar {d})\}$ by $\pi _{[\sigma ]_\approx }(\bar {y},\bar {d})$ for an L-formula $\pi _{[\sigma ]_\approx }(\bar {y},\bar {w})$ (as $\Sigma _{N,m}$ is finite, we may explicitly list out the tuples which are equivalent to $\sigma $ ). Then Proposition 5.2 holds for R and $\varphi (x,\bar {y})$ , witnessed by the polynomials $F_\sigma (X)$ and formulas $\pi _{[\sigma ]_\approx }(\bar {y},\bar {w})$ as $[\sigma ]_\approx $ ranges over the finitely many equivalence classes.

Proof Let N be the maximum of $Num_T(\xi (v;x\bar {z}\bar {w}))$ and $Num_T(\xi '(v;x \bar {z}\bar {w}))$ (recall Definition 4.5) where $\xi (v;x \bar {z}\bar {w})$ is $\theta (v,\bar {w}) \wedge \psi (x,v,\bar {z})$ and $\xi '(v;x \bar {z}\bar {w})$ is $\theta (v,\bar {w}) \wedge \neg \psi (x,v,\bar {z})$ . Then if $D_{\bar {d}} := \theta (M,\bar {d})$ is strongly minimal, we have that for all $a',\bar {c}' \in M^{1+|\bar {z}|}$ , either $|\psi (a,D_{\bar {d}},\bar {c}')| < N$ or $|D_{\bar {d}} - \psi (a,D_{\bar {d}},\bar {c}')| < N$ .

For $1 \leq i \leq N-1$ , let $\varphi _i(x,\bar {y},\bar {z},\bar {w})$ be the formula such that $\varphi _i(a,\bar {b},\bar {c},\bar {d})$ expresses “ $\varphi (a,\bar {b})$ and $ |\psi (a,D_{\bar {d}},\bar {c})| = i$ ”. Let $\varphi _0(x,\bar {y},\bar {z},\bar {w})$ be the formula such that $\varphi _0(a,\bar {b},\bar {c},\bar {d})$ expresses “ $\varphi _(a,\bar {b})$ and $|\psi (a,D_{\bar {d}},\bar {c})|> N$ ”, and note that $|\psi (a,D_{\bar {d}},\bar {c})|> N$ implies the set $\psi (a,D_{\bar {d}},\bar {c})$ is infinite, in fact cofinite, when $D_{\bar {d}}$ is strongly minimal. For $0 \leq i \leq N-1$ let $\psi _i(x,v,\bar {y},\bar {z},\bar {w})$ be $\psi (x,v,\bar {z}) \wedge \varphi _i(x,\bar {y},\bar {z},\bar {w})$ .

Note that if $D_{\bar {d}}$ is strongly minimal and $\psi (x,w,\bar {c})$ is a stratification of $\varphi (M,\bar {b})$ with respect to $D_{\bar {d}}$ , then the sets $\varphi _0(M,\bar {b},\bar {c},\bar {d}),\ldots , \varphi _N(M,\bar {b},\bar {c},\bar {d})$ partition $\varphi (M,\bar {b})$ .

By induction and by Lemma 5.4, for each i there is an $s_i \in \omega $ and there are polynomials $G_{i,j}(X) \in \mathbb {Q}[X]$ and formulas $\pi _{i,j}(\bar {y},\bar {z},\bar {w})$ for $0 \leq i \leq N-1$ and $1 \leq j \leq s_i$ such that for each i and all $\bar {b},\bar {c},\bar {d}$ , if $D_{\bar {d}}$ is strongly minimal and either $MR(\varphi _i(M,\bar {b},\bar {c})) < R$ or $\varphi _i(M,\bar {b},\bar {c})$ is properly stratified by an instance of $\psi _i(x,v,\bar {y}\bar {z}\bar {w})$ with constant finite $D_{\bar {d}}$ -fibers, then $|\varphi _i(M,\bar {b},\bar {c})| = G_{i,j}(|D_{\bar {d}}|)$ for some j, and for each j the set of all such $(\bar {b},\bar {c}) \in M^{|\bar {y}| + |\bar {z}|}$ is defined by $\pi _{i,j}(\bar {y},\bar {z},\bar {d})$ . To make notation easier, we can let all $s_i = s$ for some large enough s, by including additional (arbitrary) polynomials $G_{i,j}(X)$ and non-realized formulas $\pi _{i,j}(\bar {y},\bar {z},\bar {w})$ (for example, $y_1 \neq y_1)$ for $s_i < j \leq s$ .

Suppose $D_{\bar {d}}$ is strongly minimal and $MR(\varphi (M,\bar {b})) = R$ , and suppose ${M \models PrStrat_{\psi ,\varphi ,\theta }(\bar {b},\bar {c},\bar {d})}$ , so that $\psi (x,v,\bar {c})$ is a stratification of $\varphi (M,\bar {b})$ with respect to $D_{\bar {d}}$ . Then by Lemma 4.12.1, $MR(\varphi _0(M,\bar {b},\bar {c},\bar {d})) < R$ . If $MR(\varphi _i(M,\bar {b},\bar {c},\bar {d})) = R$ , then

$$\begin{align*}\psi_i(x,w,\bar{b},\bar{c},\bar{d}) := \psi(x,w,\bar{c}) \wedge \varphi_i(x,\bar{b},\bar{c},\bar{d})\end{align*}$$

is a proper stratification of $\varphi _i(M,\bar {b},\bar {c},\bar {d})$ with respect to $D_{\bar {d}}$ , and the w-fibers $\psi _i(a,D_{\bar {d}},\bar {b},\bar {c},\bar {d})$ all have finite cardinality i, by the definition of $\varphi _i$ . So in either case, the pseudofinite cardinality $|\varphi _i(M,\bar {b},\bar {c},\bar {d})|$ is $G_{i,j}(|D_{\bar {d}}|)$ , where j is the unique j such that $M \models \pi _{i,j}(\bar {b},\bar {c},\bar {d})$ .

Let $\Xi _{N,s}$ be the finite set of all tuples $\xi = (\xi _1,\ldots ,\xi _N) \in \{1,\ldots ,s\}^N$ . For ${\xi \in \Xi _{N,s}}$ , let $\pi _\xi (\bar {y},\bar {z},\bar {w})$ be the formula $\bigwedge _{0 \leq i \leq N} \pi _{i, \xi _i}(\bar {y},\bar {z},\bar {w})$ . For $\bar {b} \in P_{\varphi ,R}$ and $\bar {c} \in PrStrat_{\psi ,\varphi ,\theta }(\bar {b},M^{|\bar {z}|},\bar {d})$ , let $\xi (\bar {b},\bar {c},\bar {d})$ be the tuple $\xi = (\xi _1,\ldots , \xi _N)$ such that $M \models \pi _{i,\xi (i)}(\bar {b},\bar {c},\bar {d})$ for each i—i.e., such that $M \models \pi _\xi (\bar {b},\bar {c},\bar {d})$ . Let $F_\xi (X) \in \mathbb {Q}[X]$ be the polynomial $\sum _{i=0}^N G_{i, \xi _i}(X)$ .

When $D_{\bar {d}}$ is strongly minimal, the formulas $\{\pi _\xi (\bar {y},\bar {z},\bar {d}) : \xi \in \Xi _{N,s}\}$ partition

$$\begin{align*}\{(\bar{b},\bar{c}) \in M^{|\bar{y}|+|\bar{z}|} : MR(\varphi(M,\bar{b})) = R \mbox{ and } PrStrat_{\psi,\varphi,\theta}(\bar{b},\bar{c}, \bar{d})\}.\end{align*}$$

If $D_{\bar {d}}$ is strongly minimal and $M \models \pi _\xi (\bar {b},\bar {c},\bar {d})$ then the pseudofinite cardinality $|\varphi (M,\bar {b})|$ is $\sum _{i=0}^N |\varphi _i(M,\bar {b},\bar {c},\bar {d})|$ which equals $F_\xi (|D_{\bar {d}}|)$ , since $|\varphi _i(M,\bar {b},\bar {c},\bar {d})| = G_{i,\xi (i)}(|D_{\bar {d}}|)$ for each i. We note that at least one $\varphi _i(M,\bar {b},\bar {c},\bar {d})$ must have full Morley rank (since this is a finite partition of $\varphi (M,\bar {b})$ ), and so at least one $G_{i,\xi (i)}(X)$ has degree R, hence $G_{\xi }(X)$ does as well.

As in the previous lemma, if $\bar {c}' \in M$ and $M \models PrStrat_{\psi ,\varphi ,\theta }(\bar {b},\bar {c}',\bar {d})$ and $\xi (\bar {b},\bar {c}',\bar {d}) = \xi '$ , then as in the previous lemma we also have $|\varphi (M,\bar {b})| = F_{\xi '}(|D_{\bar {d}}|)$ , so $F_{\xi }(X) = F_{\xi '}(X)$ as polynomials, again by Lemma 3.5.1. Then, as in that proof, we quotient $\Xi _{N,s}$ by the equivalence relation defined as $\xi \approx \xi '$ when $F_\xi = F_{\xi '}$ , and taking our polynomials to be $F_{[\xi ]_\approx }(X)$ and our defining formulas $\pi _{[\xi ]_{\approx }}(\bar {y},\bar {w})$ to be “for all $\bar {z}$ , if $PrStrat_{\psi ,\varphi ,\theta }(\bar {y},\bar {z},\bar {w})$ holds then $\pi _{\xi '}(\bar {y},\bar {z},\bar {w})$ holds for some $\xi ' \approx \xi $ ,” which is definable since there are finitely many $\xi ' \approx \xi $ .

Proof of Proposition 5.2

By Corollary 4.10, there are formulas $\psi _1(x,v,\bar {z}),\ldots ,\psi _s(x,v,\bar {z})$ such that for all $\bar {b} \in M^{|\bar {y}|}$ and all $\bar {d}$ such that $D_{\bar {d}}$ is strongly minimal, there is a unique $i \in \{1,\ldots ,s\}$ such that for some $\bar {c} \in M^{|\bar {z}|}$ , the formula $\psi _i(x,w,\bar {c})$ stratifies $\varphi (M,\bar {b})$ with respect to $D_{\bar {d}}$ . Apply Lemma 5.5 to each $\psi _i$ to obtain $s \in \omega $ and polynomials $G_{i,1}(X),\ldots ,G_{i,s}(X) \in \mathbb {Q}[X]$ and formulas $\pi _{i,1}(\bar {y},\bar {w}),\ldots ,\pi _{i,s}(\bar {y},\bar {w})$ for each i as in the statement of Proposition 5.2. Taking our polynomials to be the polynomials $G_{i,j}(X)$ and our formulas to be “ $(\exists \bar {z} PrStrat_{\psi _i,\varphi ,\theta }(\bar {y},\bar {z},\bar {w}))$ and $\pi _{i,j}(\bar {y},\bar {w})$ ” proves the proposition.

Proof of Proposition 5.1

The case $n = 1$ is proven in Proposition 5.2.

Let us assume the theorem is true for n and let $\varphi (x_1,\ldots ,x_n,x_{n+1},\bar {z})$ be a formula. By Proposition 5.2, there are $G_1(X),\ldots ,G_s(X) \in \mathbb {Q}[X]$ of degree at most N such that for all $\bar {a} \in M^n$ , all $\bar {b} \in M^{|\bar {y}|}$ and all $\bar {d} \in M^{|\bar {w}|}$ such that $D_{\bar {d}}$ is strongly minimal, we have $|\varphi (M,\bar {a},\bar {b})| = G_i(|D_{\bar {d}}|)$ for some i, and for each i the set of all $\bar {a}\bar {b}$ such that $|\varphi (M,\bar {a},\bar {b})| = G_i(|D_{\bar {d}}|)$ is definable by the formula $\pi _i(x_2,\ldots ,x_{n+1},\bar {y},\bar {d})$ .

By induction, for each i there are polynomials $H_{i,1}(X),\ldots ,H_{i,m}(X) \in \mathbb {Q}[X]$ and formulas $\tau _{i,1}(\bar {y},\bar {w}), \ldots , \tau _{i,m}(\bar {y},\bar {w})$ such that for all $\bar {b} \in M^{|\bar {y}|}$ and $\bar {d} \in M^{|\bar {w}|}$ with $D_{\bar {d}}$ strongly minimal, there is a $j \in \{1,\ldots ,m\}$ such that $|\pi _i(M^n,\bar {b},\bar {d})| = H_{i,j}(|D_{\bar {d}}|)$ , and $deg H_{i,j}(X) = MR(\pi (\bar {x},\bar {b},\bar {d}))$ , and for all j the formula $\tau _{i,j}(\bar {y},\bar {d})$ defines the set of all $\bar {b}$ such that $|\pi _i(M^n,\bar {b},\bar {d})| = H_{i,j}(|D_{\bar {d}}|)$ . (As in the proofs above, we may not have the same number of polynomials for each $\pi _i$ , but we may add additional unused polynomials to get a uniform number.)

For every tuple $\sigma = (\sigma _1,\ldots ,\sigma _s) \in \{1,\ldots ,m\}^s$ , let $\tau _\sigma (\bar {z},\bar {w})$ be the formula $\bigwedge _i \tau _{i,\sigma _i}(\bar {z},\bar {w})$ . Note that the formulas when $D_{\bar {d}}$ is strongly minimal the formulas $\tau _\sigma (\bar {z},\bar {d})$ partition $M^{|\bar {z}|}$ as $\sigma $ ranges over $\{1,\ldots ,m\}^s$ (with some empty sets in this partition, if any of the polynomials $H_{i,j}$ is unattained).

If $\sigma = (\sigma _1, \ldots , \sigma _s)$ , $M \models \tau _\sigma (\bar {c})$ , and $D_{\bar {d}}$ is strongly minimal, then $|\pi _i(M^{|\bar {y}|}, \bar {b},\bar {d})| = H_{i,\sigma _i}(|D_{\bar {d}}|)$ for each i. By Lemma 3.4.1, we have that

$$\begin{align*}|\varphi(M^{n+1},\bar{b})| = \sum_{i=1}^s F_i(|D_{\bar{d}}|) \cdot H_{i,\sigma_i}(|D_{\bar{d}}|) = F_\sigma(|D_{\bar{d}}|),\end{align*}$$

where $F_\sigma (X)$ is the polynomial $\sum _{i=1}^s G_i(X) \cdot H_{i,\sigma _i}(X)$ . We note that this equation comes from the fact that $\varphi (M^{n+1},\bar {b})$ is the disjoint union of the sets ${Z_i := \{(c,\bar {a},\bar {b}) : M \models \pi _i(\bar {a},\bar {b})}$ and $M \models \varphi (c,\bar {a},\bar {b})\}$ for $i = 1, \ldots , s$ , each of which has cardinality $G_i(|D_{\bar {d}}|) \cdot H_{i, \sigma _i}(|D_{\bar {d}}|)$ . The fibers $\{c \in M : (c,\bar {a}) \in Z_i\}$ all have pseudofinite cardinality $G_i(|D_{\bar {d}}|)$ by the definition of $Z_i$ , hence they all have the same Morley rank which is $\deg G_i(X)$ . Therefore by Lemma 4.6, taking the definable surjection f to be the projection of $Z_i$ onto the coordinates $(x_2,\ldots ,x_n)$ , the Morley rank of $Z_i$ is $\deg G_i + MR(\pi _i(M^n,\bar {b},\bar {d})) = \deg G_i(X) + \deg H_{i, \sigma _i}(X)$ , which is the degree of the polynomial $G_i(X) \cdot H_{i,\sigma _i}(X)$ . So $\deg F_\sigma (X) = \max (\deg G_i (X) \cdot H_{i, \sigma _i}(X) : i \leq s) = \max (MR(Z_i) : i \leq s) = MR(\varphi (M^{n+1},\bar {b}))$ .

Proof Let $D = \theta (M,\bar {d})$ be a definable strongly minimal set. Then $|X| = F(|D|)$ for some polynomial $F(X) \in \mathbb {Q}[x]$ , by Theorem 5.1. Since $|X|$ and $|D|$ are both non-standard integers, the hyperreal $|D|$ is the unique positive hyperreal z such that $F(z) = |X|$ , by Lemma 3.5.2. So $|D|$ is definable over $|X|$ in $\mathbb {R}^{\mathcal {U}}$ . Therefore to prove our theorem, it suffices to let $c = |D|$ , and show that $M^+$ is definable in the disjoint union $L^+$ -structure $M \cup \mathbb {R}^{\mathcal {U}}$ over $c \in \mathbb {R}^\star $ and $\bar {d} \in M$ .

To show that $M^+$ is definable in the disjoint union, we show that for every L-formula $\varphi (\bar {x},\bar {y})$ , the function $f_{\varphi (\bar {x},\bar {y})}$ is definable over $c = |D| \in \mathbb {R}^\star $ and $\bar {d} \in M$ . Let $\varphi (\bar {x},\bar {y})$ be an L-formula. By Theorem 5.1, there are finitely many polynomials $F_1(X),\ldots ,F_r(X) \in \mathbb {Q}[X]$ and formulas $\pi _1(\bar {y},\bar {w}),\ldots ,\pi _r(\bar {y},\bar {w})$ such that the formulas $\pi _i(\bar {y},\bar {d})$ partition $M^{|\bar {y}|}$ and for every $\bar {b}$ we have $|\varphi (M,\bar {b})| = F_i(|D|)$ iff $M \models \pi _i(\bar {b})$ . Then for any $\bar {b} \in M^{|\bar {y}|}$ and any $e \in \mathbb {R}^{\mathcal {U}}$ ,

$$\begin{align*}M^+ \models f_{\varphi(\bar{x},\bar{y})}(\bar{b}) = e\end{align*}$$

if and only if

$$\begin{align*}(M,\mathbb{R}^{\mathcal{U}}) \models \bigvee_{i=1}^r [\pi_i(\bar{b},\bar{d}) \wedge e = F_i(c)]\end{align*}$$

which is definable in the disjoint union.

Proof Let $\theta (v,\bar {w})$ be an L-formula such that some/every (saturated) model M of the theory T has a $\bar {d} \in M$ such that $\theta (M,\bar {d})$ is strongly minimal.

Let $\varphi (\bar {x},\bar {y})$ be an L-formula. Let $F_1(X),\ldots ,F_r(X) \in \mathbb {Q}[X]$ be the polynomials and $\pi _1(\bar {y},\bar {w}),\ldots ,\pi _r(\bar {y},\bar {w})$ be the formulas obtained in Theorem 5.1 as applied to $\varphi (\bar {x},\bar {y})$ and $\theta (v,\bar {w})$ .

Let $\Psi (\bar {w})$ be an $L^+$ -formula such that $\Psi (\bar {d})$ expresses the conjunction of the two conditions:

1. 1. $\pi _1(\bar {y},\bar {d}),\ldots ,\pi _r(\bar {y},\bar {d})$ partition $M^{|\bar {y}|}$ , and

2. 2. for all $\bar {b} \in M^{|\bar {y}|}$ we have $|\varphi (M^{|\bar {x}|},\bar {b})| = F_i(|\theta (M,\bar {d})|)$ if and only if ${M \models \pi _i(\bar {b},\bar {d})}$ .

Then for every $M \models T$ and $\bar {d} \in M$ , if $\theta (M,\bar {d})$ is strongly minimal then ${M^+ \models \Psi (\bar {d})}$ . Since every uncountable model of T is saturated, every uncountable model of T contains a tuple $\bar {d}$ such that $\theta (M,\bar {d})$ is strongly minimal. Therefore whenever M is a pseudofinite ultraproduct which satisfies T, the expanded structure $M^+$ satisfies the $L^+$ -sentence $\exists \bar {w} \Psi (\bar {w})$ .

By assumption, every nonprincipal ultraproduct of the family $(M_\lambda : \lambda \in \Lambda )$ satisfies the theory T. Therefore every infinite ultraproduct of the expanded finite structures $M^+_\lambda $ satisfies the sentence $\exists \bar {w} \Psi (\bar {w})$ . Hence this sentence is satisfied in all but finitely many of the $L^+$ -expansions $M^+_\lambda $ . If $M^+_\lambda $ satisfies $\exists \bar {w} \Psi (\bar {w})$ then, taking $\bar {d}_\lambda $ to be a witness, the structure $M^+_\lambda $ satisfies the conclusion of the proposition.

Let $M_1, \ldots , M_k$ be the finitely many structures in our family whose expansions $M_i^+$ do not satisfy the $L^+$ -formula $\exists \bar {w} \Psi (\bar {w})$ . Let N be a number greater than $\max \{ |M_1|, \ldots , |M_k|\}$ . Let us add polynomials $F_{r+1}(X),\ldots ,F_{r+N}(X)$ where $F_{r+k}(X)$ is the constant polynomial $k-1$ , and add new formulas $\pi _{r+1}(\bar {y},\bar {w}),\ldots , \pi _{r+N}(\bar {y},\bar {w}),$ where $\pi _{r+k}(\bar {b},\bar {d})$ expresses “ $|\varphi (M,\bar {b})| = k-1$ ”. Finally, let us modify the formulas $\pi _1(\bar {y},\bar {w}), \ldots , \pi _r(\bar {y},\bar {w})$ by conjoining them each with a formula such that $\pi _i(\bar {b},\bar {d})$ implies “ $|\varphi (M,\bar {b})| \geq N$ ”. With this alteration, the conclusion of the proposition holds for every structure $M_\lambda $ .

Fact 6.7. Let T be an uncountably categorical theory. Then every model of T contains a strongly minimal set defined by a formula whose parameters have an isolated type. That is, there exist formulas $\kappa (\bar {w})$ and $\theta (v,\bar {w})$ where $\kappa (\bar {w})$ isolates a type p such that for a model $M \models T$ and $\bar {d} \in M$ , if $tp(\bar {d}) = p$ then $\theta (M,\bar {d})$ is strongly minimal.

Proof We slightly modify the proof of Proposition 6.5. We conjoin to $\Psi (\bar {w})$ the formula $\kappa (\bar {w})$ . Then it remains true that all-but-finitely many of the finite expansions $M^{+}_\lambda $ satisfy $\exists \bar {w} \Psi (\bar {w})$ , and we may explicitly encode out the finitely many counterexamples as in the proof of Proposition 6.5.

Proof We compare $F(X)$ to polynomials of the form

$$\begin{align*}a_n(x + \alpha)^n = a_n x^n + a_n \cdot n \alpha x^{n-1} + \dots + a_n \cdot n \alpha^{n-1} x + a_n \alpha^n.\end{align*}$$

Note that the inverse of $a_n(x + \alpha )^n$ is $(x / a_n)^{1/n} - \alpha $ .

It is easily seen that $a_n(x+\alpha )^n> F(x)$ for sufficiently large x if $a_n \cdot n \alpha> a_{n-1}$ —that is, if $\alpha> a_{n-1} / (n a_n)$ —and that $a_n(x+\beta )^n < F(x)$ for sufficiently large x if $\beta < a_{n-1} / (n a_n)$ .

In general, if $f(x),g(x) :\mathbb {R} \to \mathbb {R}$ are increasing functions such that $f(x)> g(x)$ for all (sufficiently large) x, then $f^{-1}(x) < g^{-1}(x)$ for all (sufficiently large) x. Therefore if $\alpha> a_{n-1} / (n a_n)$ we have that $(x/a_n)^{1/n} - \alpha < F^{-1}(x)$ eventually, and if ${\beta < a_{n-1} / (n a_n)}$ then $(x/a_n)^{1/n} - \beta> F^{-1}(x)$ eventually. Rearranging these inequalities, we obtain that $\beta < (x/a_n)^{1/n} - F^{-1}(x) < \alpha $ eventually for all ${\alpha> a_{n-1} / (n a_n) > \beta }$ , which proves that

$$\begin{align*}\lim_{x \to \infty} (x/a_n)^{1/n} - F^{-1}(x) = a_{n-1} / (n a_n).\end{align*}$$

To prove that

$$\begin{align*}(b_m/ a_n^{m/n}) x^{m/n} - G(F^{-1}(x)) = o(x^{m/n})\end{align*}$$

it suffices to prove the asymptotic equations

$$\begin{align*}b_m(x/a_n)^{m/n} - G((x / a_n)^{1/n}) = o(x^{m/n})\end{align*}$$

and

$$\begin{align*}G((x / a_n)^{1/n}) - G(F^{-1}(x))= o(x^{m/n}).\end{align*}$$

For the first, rewrite

$$\begin{align*}\lim_{x \to \infty} \frac{b_m(x/a_n)^{m/n} - G((x / a_n)^{1/n})}{x^{m/n}}\end{align*}$$

as

$$\begin{align*}\lim_{z \to \infty} \frac{b_m z^m - G(z)}{a_n^{m/n} z^m}\end{align*}$$

after making the variable substitution $z = (x / a_n)^{1/n}$ . The numerator is a polynomial of degree less than m, therefore the limit is 0 as required.

To verify the second asymptotic equation, we observe that the first part of this lemma implies the existence of a B such that ${F^{-1}(x) \in ((x/a_n)^{1/n} - B, (x/ a_n)^{1/n} + B)}$ for all x. Then

$$\begin{align*}|G(F^{-1}(x)) - G((x/a_n)^{1/n})| < |G((x/a_n)^{1/n} + B) - G((x/a_n)^{1/n} - B)|\end{align*}$$

for sufficiently large x. Therefore it suffices to show

$$\begin{align*}\lim_{x \to \infty} \frac{{G((x/a_n)^{1/n} + B) - G((x/a_n)^{1/n} - B)}}{x^{m/n}} = 0,\end{align*}$$

which is equivalent to saying

$$\begin{align*}\lim_{z \to \infty} \frac{G(z + B) - G(z - B)}{a_n^{m/n} z^m} = 0\end{align*}$$

as in the previous calculation. The polynomial $G(z + B) - G(z - B)$ simplifies to a polynomial of degree less than m, and so the limit is correct, and the conclusion of the lemma is true.

Proof Let $\kappa (\bar {w})$ , $\theta (v,\bar {w})$ and p be as in Fact 6.7.

First we show Definition 6.6 holds for the formula $\theta (x,\bar {y}) \wedge \kappa (\bar {w})$ (with variables as in that definition partitioned as $x,\bar {y}\bar {w})$ . We apply Corollary 6.8 to the formula “ $x = x$ ” to obtain polynomials

$$\begin{align*}F_1(X),\ldots,F_r(X) \in \mathbb{Q}[X]\end{align*}$$

of the Morley rank of the formula (which is N) and formulas $\pi _1(\bar {w}),\ldots ,\pi _r(\bar {w})$ . Let i be such that $p \vdash \pi _i(\bar {w})$ . Then $M_\lambda \models (\kappa (\bar {w}) \rightarrow \pi _i(\bar {w}))$ for cofinitely many $M_\lambda $ . For such a $M_\lambda $ , if $M_\lambda \models \kappa (\bar {d})$ then $|M_\lambda | = F_i(|\theta (M_\lambda ,\bar {d})|)$ . If $M_\lambda \models \kappa (\bar {d}')$ as well then $|M_\lambda | = F_i(|\theta (M_\lambda , \bar {d}')|)$ . By injectivity of polynomials on a tail, this implies the existence of a C such that for $|M_\lambda |> C$ , if $\bar {d}, \bar {d}' \in \kappa (M_\lambda ^{|\bar {w}|})$ then $|\theta (M_\lambda , \bar {d})| = |\theta (M_\lambda , \bar {d}')|$ , and $|M_\lambda | = F_i(|\theta (M_\lambda , \bar {d})|)$ . Let ${F(X) = F_i(X) = a_N X^N + \dots + a_1 X + a_0}$ . It follows from Lemma 6.9 that

$$\begin{align*}\lim_{|M_\lambda| \to \infty} \frac{|\theta(M_\lambda, \bar{d})| - (1/a_N)^N |M_\lambda|^{1/N}}{|M_\lambda|^{1/N}} = 0\end{align*}$$

if we take $\bar {d} \in \kappa (M_\lambda ^{|\bar {w}|})$ .

Now let $\varphi (x_1,\ldots ,x_n,\bar {y})$ be an arbitrary L-formula. Apply Corollary 6.8 to $\varphi (x_1,\ldots ,x_n,\bar {y})$ to get polynomials $G_1(X),\ldots ,G_s(X)$ and formulas $\pi _1(\bar {y},\bar {w}), \ldots , \pi _s(\bar {y},\bar {w})$ such that the conclusion of that proposition holds. Then if ${\bar {b},\bar {d} \in M_\lambda }$ with $|M_\lambda |$ sufficiently large and $M_\lambda \models \kappa (\bar {d}) \wedge \pi _i(\bar {b},\bar {d})$ then $|\varphi (M_\lambda ^n,\bar {b})| = G_i(|\theta (M_\lambda ,\bar {d})|) = G_i(F^{-1}(|M_\lambda |))$ . For each $i = 1,\ldots ,s$ let $G_i(X) = a_{i,N_i} X^{N_i} + \dots + a_{i,1} X + a_{i,0}$ . By Lemma 6.9, we have

$$\begin{align*}\lim_{x \to \infty} \frac{G_i(F^{-1}(x)) - (a_{i,N_i} / a_N^{1/N}) x^{N_i / N}}{x^{N_i / N}} = 0.\end{align*}$$

Therefore we have

$$\begin{align*}\lim_{x \to \infty} \frac{|\varphi(M_\lambda^n, \bar{b})| - (a_{i,N_i} / a_N^{1/N}) |M_\lambda|^{N_i / N}}{|M_\lambda|^{N_i / N}} = 0\end{align*}$$

for all $\bar {b},\bar {d} \in M_\lambda $ with $M_\lambda \models \pi _i(\bar {b},\bar {d}) \wedge \kappa (\bar {d})$ . That is,

$$\begin{align*}|\varphi(M_\lambda^n,\bar{b})| - (a_{i,N_i} / a_N^{1/N}) |M_\lambda|^{N_i/N} = o(|M_\lambda|^{N_i/N}),\end{align*}$$

whence the definition of N-dimensional asymptotic classes almost applies to the formula $\varphi (\bar {x},\bar {y})$ , with $(\mu _i, d_i)$ being $((a_{i,N_i} / a_N^{1/N}), N_i)$ and defining formulas being “ $\exists \bar {w}[ \kappa (\bar {w}) \wedge \pi _i(\bar {y},\bar {w})]$ .” The only difference from the definition is that the defining formulas do not necessarily partition every $M_\lambda $ . However they do partition all $M_\lambda $ with $|M_\lambda |$ sufficiently large. This is sufficient for the family to be an N-dimensional asymptotic class—in general, an N-dimensional asymptotic class remains so after adding arbitrarily many structures of size at most K, for any fixed K.