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At the end of his paper, Pillay makes this remark: “It is also natural to ask what is the appropriate level of generality of the precise counting result in Theorem 1.1. Firstly there should be no problem obtaining a similar result for pseudofinite ℵ1-categorical theories, where again any definable set will have cardinality an integral polynomial in q where q is the cardinality of a given strongly minimal set.” In this paper, we give a proof of Pillay’s suggested result, with the minor alteration that the polynomial has rational coefficients rather than just integers. We show that this is necessary in Example 5.1. Here we briefly explain our motivations for finding and proving this result. In the paper [4], the authors demonstrate a number of results of the flavor that conditions on pseudofinite dimension in a pseudofinite ultraproduct (pseudofi- nite dimension is information derived from the pseudofinite cardinalities of the definable subsets of a structure) imply stability-theoretic properties of the stuc- ture, such as simplicity and supersimplicity. In one such result, the authors show that if the pseudofinite dimension satisfies a property they refer to as “strong attainability” or “(SA)” for short, then the structure has a supersimple theory. The authors of that paper demonstrate that this result does not reverse, by pro- viding an example of a pseudofinite ultraproduct with supersimple theory which does not satisfy (SA), although there is an elementarily equivalent pseudofinite ultraproduct which does. In unfinished work of ours, we have an example of a supersimple pseudofinite theory T such that no pseudofinite ultraproduct satisfying T has the property (SA). Ultimately we are interested in finding converses to the the results in [4], by which we mean finding conditions on T which imply that conditions such as (SA) must hold, either in some pseudofinite ultraproduct or all pseudofinite ultraproducts satisfying T . In this paper, we obtain one such condition on T – uncountable categoricity – as the conclusion of our Main Theorem implies the condition (SA). In addition to that paper, this work also connects with the notions of multidi- mensional asymptotic class, as developed in [7], and N-dimensional asymptotic class as developed in [3]. As we prove in Proposition 5.5, a class of finite struc- tures with a zero-one law whose limit theory is uncountably categorical gives a multidimensional exact class as well as an N-dimensional asymptotic class, where N is the Morley rank of the limit theory. The author thanks Alf Dolich and Charlie Steinhorn for their conversations and contributions to this paper.
1 Notation
Throughout this paper, L denotes an arbitrary first-order language. For a tuplex ¯, the expression |x¯| denotes the length of the tuple. For an L-formula ϕ(¯x, y¯), an L-structure M, a subset X of M |x¯| and a tuple ¯b ∈ M, the expression ϕ(X, ¯b) denotes the set {a¯ ∈ M |x¯| :a ¯ ∈ X and M |= ϕ(¯a, ¯b)}.
2 |v¯| Formulas ϕ1(¯v),...,ϕn(¯v) are said to partition M when the non-empty |v¯| |v¯| |v¯| sets ϕi(M ) form a partition of M (we allow some of sets the ϕi(M ) to be empty). The word “rank” in this paper will always refer to Morley rank. We will denote the Morley rank of a definable set X as MR(X).
2 Pseudofinite Cardinality
Definition 2.1. A theory T with infinite models is pseudofinite if every sen- tence σ implied by T has a finite model M such that M |= σ. Equivalently, T is pseudofinite if there is some sequence of finite L-structures (Mλ : λ ∈ Λ) and some ultrafilter U on Λ such that the ultraproduct Y Mi is λ→U a model of T . In this paper, the term “pseudofinite ultraproduct” will denote a model of the form Y Mi for some sequence of finite L-structures (Mλ : λ ∈ Λ) and λ→U some ultrafilter U on Λ One nice property of ultraproducts of finite structures is that they come with a notion of subset “cardinality”. Let X be a definable subset of an ultraproduct M = Y Mλ of finite structures. Then X is the ultraproduct Y Xλ, where λ→U λ→U Xλ ⊆ Mλ is defined by the same formula. Each Xλ has a cardinality, being a finite set, and so we can say that X has cardinality (|Xλ|)λ→U , an element of ⋆ the ultrapower R = Y R. If X is finite then |X| will agree with the counting λ→U cardinality of X; if X is infinite then |X| will be an infinite hyperreal in R⋆. We can formalize this notion via the following construction, where we pass from our original language L to a two-sorted expansion, where cardinality takes values in the second sort. Definition 2.2. Let L be a first-order language. We define the expansion L+ to be a two-sorted language. The home sort H has the language L. The second sort OF is an ordered field language (0, 1, −, +, ·,<). L+ also has, for every |y¯| L-formula ϕ(¯x, y¯), a function fϕ(¯x,y¯) : H → OF.
Definition 2.3. Let (Mλ : λ ∈ Λ) be a sequence of finite L-structures and let U be an ultrafilter on Λ. Let M be the ultraproduct Y Mλ. We define the λ→U + + + + + L -expansion M by expanding each Mλ to a L -structure Mλ . In Mλ , we define the ordered field sort OF to be the ordered field of real numbers R. For ¯ ¯ an L-formula ϕ(¯x, y¯) we define fϕ(¯x,y¯) by fϕ(¯x,y¯)(b)= |ϕ(Mλ, b)|, the cardinality ¯ ¯ |y¯| of the set ϕ(Mλ, b), for b ∈ Mλ . + + Having defined each Mλ , we let M be the ultraproduct of the sequence of + + L -structures (Mλ : λ ∈ Λ) with respect to U.
3 For an L-formula ϕ(¯x, y¯), we define the pseudofinite cardinality of ϕ(M, ¯b) ¯ ⋆ + + to be the hyperreal fϕ(¯x,y¯)(b) ∈ R in the L -expansion M . We denote this hyperreal as |ϕ(M, ¯b)|. One benefit of this construction (not used in this paper, but used in e.g. + [4]) is that the ultraproduct M is ω1-saturated not just in L but in the full language L+. Later on in this paper we show that if M is an uncountably categorical structure, then the two-sorted structure M + is no more complex than the multi-sorted disjoint union structure where one sort is M, the other sort is an ultrapower of the reals R⋆, with no model-theoretic interaction between the two (Proposition 5.2). We remark that the pseudofinite cardinality |X| of a pseudofinite ultraprod- uct M depends not just on the L-structure of M, but on the sequence of finite L-structures (Mλ : λ ∈ Λ) and ultrafilter U on Λ such that M = Y Mλ. λ→U The following lemma is the combinatorial core of our main theorem. This lemma lets us express the cardinality of X as a rational expression in terms of the cardinalities of subsets of a set Y and cardinalities of the fibers of a definable relation R(x, y) between X and Y , so long as there are finitely many such cardinalities. Lemma 2.4. Let ϕ(¯x, y,¯ z¯) be a formula. Let c¯ ∈ M |z¯| be parameters. Suppose ⋆ ⋆ ¯ |y¯| that there are finitely many hyperreals A1,...,An such that for all b ∈ M |x¯| ¯ ⋆ ¯ |y¯| there is an i such that ϕ(M , b, c¯) = Ai . For each i, let Zi = {b ∈ M : |x¯| ¯ ⋆ + |ϕ(M , b, c¯)| = Ai } (note that Zi is definable in L ). 1. The equation n |x¯|+|y¯| ⋆ |ϕ(M , c¯)| = X Ai · |Zi| i=1 holds in M +. 2. Suppose in addition that there is a single hyperreal B⋆ such that for all a¯ ∈ M |x¯|, if ϕ(¯a,M |y¯|, c¯) 6= ∅ then |ϕ(¯a,M |y¯|, c¯)| = B⋆. Then the equation
n A⋆ · |Z | |{a¯ ∈ M |x¯| : M |= ∃yϕ¯ (¯a, y,¯ c¯)}| = Pi=1 i i B⋆ holds in M +.
Proof. The equation in Statement 1 is easily seen to hold in the finite case. Each ⋆ ¯ |x¯|+|y¯| ¯ ¯ Ai ·|Zi| is the cardinality of the set {(¯a, b) ∈ M : b ∈ Zi and M |= ϕ(¯a, b)}. To prove Statement 2, note that under our special assumption, Statement 1, with the roles ofx ¯ andy ¯ switched, tells us that |ϕ(M |x¯|+|y¯|, c¯)| = B⋆ · |{a¯ ∈ M |x¯| : |ϕ(¯a,M |y¯|, c¯) = B⋆}| = B⋆ · |{a¯ ∈ M |x¯| : M |= ∃yϕ¯ (¯a, y,¯ c¯)}|. Therefore n ⋆ ⋆ |x¯| ⋆ X Ai · |Zi| = B · |{a¯ ∈ M LM |= ∃yϕ¯ (¯a, y,¯ c¯)}|. Dividing both sides by B i=1 gives the desired result.
4 We also make use of the following easily verified facts.
⋆ Lemma 2.5. Let R = Y R be an ultrapower of the real field R. λ→U 1. Suppose F (X), G(X) ∈ Q[X] are polynomials, and suppose A⋆ ∈ R⋆ is an infinite positive element (i.e. A⋆ > n for every standard natural number n) such that F (A⋆) = G(A⋆). Then F (x) = G(x) for all x ∈ R, i.e. F and G are the same polynomial. 2. Suppose F (X) ∈ Q[X] is a polynomial and suppose A⋆,B⋆ ∈ R⋆ are infinite positive elements such that F (A⋆)= F (B⋆). Then A⋆ = B⋆. 3. Suppose F (X) ∈ Q[X] is a polynomial, and suppose A⋆ ∈ R⋆ is an infinite positive element such that F (A⋆) is positive. Then the leading coefficient of F (X) is positive.
⋆ Proof. 1. Express A as an ultralimit (aλ)λ→U , with each aλ ∈ R. Then ⋆ ⋆ F (A ) = (F (aλ))λ→U and G(A ) = (G(aλ))λ→U . Since these two ultra- ⋆ limits are equal, we have F (aλ) = G(aλ) for almost all λ ∈ Λ. Since A is infinite, for every standard n ∈ ω we have aλ > n for almost all λ. It follows that for every n there is a real aλ > n such that F (aλ) = G(aλ). Therefore the polynomial F (x) − G(x) has arbitrarily large zeroes in R. Hence it must be constant zero, and so F (x)= G(x) for all x. 2. In R, there is an n such that F (x) is strictly increasing or decreasing on (n, ∞) (in R)). In particular, the function x 7→ F (x) is injective on this open ray. This is first-order expressible, hence true in R⋆, and since A⋆ >n and B⋆ >n we get A⋆ = B⋆. 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 = (aλ)λ∈Λ is positive and infinite, the set {λ ∈ Λ: aλ > ⋆ M} is U-large. Then the set {λ ∈ Λ: F (aλ) < 0} = {λ ∈ Λ : (F (A ))λ} is U-large, whence F (A⋆) is negative. Fact 3 follows contrapositively.
3 Uncountably Categorical Theories
Fact 3.1. Let M be an uncountable model of an ℵ1-categorical theory T . Then M is saturated.
Fact 3.2. Let T be an ℵ1-categorical theory. Let M |= T .
1. Let X ⊆ M be definable (possibly with parameters). Then the Morley rank MR(X) exists and is finite. 2. Let ϕ(¯x, y¯) be an L-formula. Then for every n ∈ ω, the set {¯b ∈ M |y¯| : MR(ϕ(M |x¯|, ¯b)) = n} is ∅-definable.
5 We use the following well-known fact about uncountably categorical models (see [2, Lemma 3.1.12]; this fact is also a consequence of Fact 3.2.2)
Fact 3.3. Suppose T is ℵ1-categorical and ϕ(¯x, y¯) is a formula. Then there is a natural number n such that in all models M of T and all ¯b ∈ M |y¯|, either ϕ(M |x¯|, ¯b) is infinite or |ϕ(M |x¯|, ¯b)|≤ n.
Definition 3.4. Let T be an ℵ1-categorical theory (or more generally any theory of which Fact 3.3 holds). Let ϕ(¯x, y¯) be a formula. Define NumT (ϕ(¯x, y¯)) to be the least number N ∈ ω such that in any model M of T and any ¯b ∈ M |y¯|, if ϕ(M |barx|, ¯b) is finite then |ϕ(M |barx|, ¯b)| 1. By induction on S. At S = 0, the statement is trivial. Specifically, let y ∈ Y be any element. Since f −1(y) ⊆ X we have MR(X) ≥ MR(f −1(y)) ≥ R = R + S. Suppose the statement is proven for s If S = 0 then MR(Y ) ≤ S means Y is finite. Let Y = {y1,...,yp}. Then −1 −1 −1 X = f (y1) ∪ ... ∪ f (yp). If each f (yi) has Morley rank ≤ R, then as a finite union of such sets, X also has a Morley rank of ≤ 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 Y is 1, and suppose towards a contradiction that MR(X) > R + S. Then we can find X1,X2,X3,... which are disjoint subsets of X of Morley rank R + S. For each i, let 6 Yi = f(Xi). Then for each i, the map fi which is the restriction of f to Xi is a surjection from Xi onto Yi. The fibers of fi are still of rank ≤ R, as they are subsets of the fibers of f. If MR(Yi) < S then by induction we obtain MR(Xi) ≤ R + MR(Yi) < R + S, a contradiction. Therefore MR(Yi) = S for each i, and since Y has degree 1, we obtain MR(Y \ Yi) Our main tool in proving Theorem 4.1 is an early result of Zil’ber’s found in [8], as well as his book [9]. His result uses the concept of stratification, defined as follows. Definition 3.6. Let A, B be definable unary subsets of a totally transcendental model M. A stratification of A with respect to B is a formula ψ(x, v, c¯) with parameters c¯ such that A = [ ψ(M, b, c¯). b∈B A stratification of rank at most r is a stratification ψ(x, v, c¯) such that MR(ψ(M, b, c¯)) ≤ r for all b ∈ B. A proper stratification is a stratification ψ(x, y, c¯) of rank at most rk(A)−1. 7 Remark 3.7. For formulas ϕ(x, y¯), θ(v, y¯) and ψ(x, v, z¯), the notion ψ(x, v, c¯) is a stratification of ϕ(M, ¯b) with respect to θ(v, d¯) is first-order ∅-definable in ¯b, c,¯ d¯. If Morley Rank is definable as in Fact 3.2.2, the notions “ψ(x, w, d¯) is a stratification of rank at most r of ϕ(M, ¯b) with respect to θ(v, d¯)” and “ψ(x, v, c¯) is a proper stratification of ϕ(M, ¯b) with respect to θ(v, d¯)” are ∅-definable in ¯b, c,¯ d¯. Zil’ber’s result tells us that in a ℵ1-categorical theory, any infinite definable subset has a proper stratification over any strongly minimal set. Fact 3.8 ([8], Lemma 1). Let T be an uncountably categorical theory and let M |= T be uncountable. Let A, B ⊂ M be infinite definable sets, with B strongly minimal. Then there exists a proper stratification of A with respect to B. We need a slightly stronger statement for our main proof, which follows quickly from Zil’ber’s result and from the saturation of any uncountable model of T . Corollary 3.9. Let T be an uncountably categorical theory. Let ϕ(x, y¯) and θ(v, w¯) be formulas. Then there are formulas ψ1(x, v, z¯),...,ψn(x, v, z¯) such that for all uncountable M |= T and ¯b, d¯ ∈ M such that θ(M, d¯) is strongly minimal, there is an i such that for some c¯ ∈ M, the formula ψi(x, y, c¯) is a proper stratification of ϕ(M, ¯b) with respect to θ(M, ¯b′). Furthermore, we may choose formulas ψi so that in the above paragraph, there is a unique i such that for some c¯ ∈ M, the formula ψi(x, v, c¯) is a proper stratification. Proof. Consider the partial type Π(¯z, z¯′) which contains formulas saying ∀w¯[(MR(θ(M, z¯) ∧ ϕ(M, w¯)) = 0) ∨ (MR(θ(M, z¯) ∧ ¬ϕ(M, w¯)) = 0)] and ∀w¯[ψ(x, y, w¯) is not a proper stratification of ϕ(M, z¯) w.r.t. θ(M, z¯′)] for all L-formulas φ(x, w¯) and ψ(x, y, w¯) (using Fact 3.2 and Remark 3.7). Suppose the first paragraph of the corollary were false. Then for any finite collection of L formulas ψ1(x, y, w¯),...,ψn(x, y, w¯), there is an M |= T and ¯b, ¯b′ ∈ M such that θ(M, ¯b′) is strongly minimal and such that for each i, there is noc ¯ ∈ M such that ψi(x, y, c¯) is a proper stratification of ϕ(M, ¯b). Then Π(¯z, z¯′) is consistent: we can realize any finite subset of Π(¯z, z¯′) with such a ¯b, ¯b′. Choose an uncountable M which is a model of T . Then M is a saturated model, by categoricity. Therefore Π is realized by some ¯b, ¯b′ ∈ M. Let A = ϕ(M, ¯b) and B = θ(M, ¯b′). Then B is strongly minimal, and there is no proper stratification of A with respect to B, contradicting Fact 3.8. 8 Therefore the first paragraph of the corollary holds. We can force the index i to be unique by modifying our formulas so that, for instance, ψ2(x, y, c¯) implies ′ that ψ1(x, y, c¯) does not properly stratify ϕ(M, ¯b) with respect to θ(M, b¯) (which is a first-order definable property of ¯b, b¯′, c¯, as per Remark 3.7). The two statements in the following Lemma are noted and used in proofs by Zil’ber in [9]. We use them to obtain the fact that the degree of the polynomial giving the cardinality of a definable set is exactly the Morley rank of the set. First we need another fact from [8] (also found in [9]). Fact 3.10 ([8], Theorem 3). Let ∆(x), Φ(x) be L-formulas over A ⊂ M with MR(∆(M)) = m and let ψ(v0, v1) be a stratification of Φ(M) with respect to ∆(M) of rank at most n. Then MR(Φ(M)) ≤ m + n. Lemma 3.11. Let M be a structure. Let X,D ⊆ M be definable subsets with D strongly minimal. Let ψ(x, y) be a proper stratification of X with respect to D (possibly with parameters). 1. There is an N ∈ ω such that rk({a ∈ X : |ψ(a,D)| >N}) < rk(X). 2. The set {d ∈ D : rk(ψ(X, d)) = rk(X) − 1} is cofinite in D. Proof. 1. As D is strongly minimal, there is an N ∈ ω such that for all a ∈ M we have that either |ϕ(a,D)| ≤ N or |¬ϕ(a,D)| ≤ N. Let d1,...,dN , dN+1 ∈ D be distinct elements. For each i = 1,...,N +1 let Xi = ψ(X, di). As ψ is a proper stratification, we have MR(Xi) < MR(X) for each i. Therefore MR(X1 ∪ ... ∪ XN+1) < MR(X). If a ∈ X and |ψ(a,D)| >N then ψ(a,D) is cofinite in D with |D − ψ(a,D)|≤ N. Therefore at least one of d1,...,dN+1 must be an element of ψ(a,D), that is, M |= ψ(a, di) for some i. Then a ∈ ϕ(X, di) = Xi. Hence {a ∈ X : |ψ(a,D)| >N}⊆ X1 ∪ ... ∪ XN+1, and therefore MR({a ∈ X : |ψ(a,D)| >N}) ≤ MR(X1 ∪ ...XN+1) < MR(X). 2. Suppose otherwise. Then {d ∈ D : MR(ψ(M, d)) = MR(X) − 1} is finite and {d ∈ D : MR(ψ(M, d)) ≤ MR(X) − 2} is cofinite. Let {d ∈ D : MR(ψ(M, d)) = MR(X) − 1} = {d1,...,dn}. For i = 1,...,n let Xi = ′ ′ ψ(M, di) and let X = [ ψ(M, d). Then X = X1∪...∪Xn∪X . d∈D−{d1,...,dn} ′ Each Xi has Morley rank MR(X) − 1. X is stratified by ψ(x, y) over D −{d1,...,dn}, and this stratification has rank at most MR(X) − 2. ′ Therefore MR(X ) ≤ MR(X)−2+MR(D −{d1,...,dn})= 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 ∈ D : rk(ψ(X, d)) = rk(X) − 1} must be cofinite in D. 9 4 Main Theorem Theorem 4.1. Let T be an uncountably categorical theory in the language L. Let θ(v, w¯) be an L-formula. Then for every L-formula ϕ(x1,...,xn, y¯), there are finitely many polynomials F1(X),...,Fr(X) ∈ Q[X] and L-formulas π1(¯y, w¯),...,πr(¯y, w¯) such that for all pseudofinite ultraproducts M and all d¯∈ M such that D = θ(M, d¯) is strongly minimal, we have that for all ¯b ∈ M |y¯|, n the pseudofinite cardinality |ϕ(M , ¯b)| is Fi(|D|) for some i, and furthermore for each i the set n {¯b ∈ M : |ϕ(M , ¯b)| = Fi(|D|)} is definable over X by πϕ,i(¯y, d¯). Additionally, if ¯b satisfies πϕ,i(¯y, d¯) , then the degree of the polynomial Fi(X) is the Morley rank of the set ϕ(M n, ¯b). We first prove Theorem 4.1 for the case n = 1 (Proposition 4.2), which takes up most of the proof. The full theorem will follow from a relatively fast inductive fiber-decomposition argument. First we give an overview of our proof of the single-variable case, leaving aside the fine details to the forthcoming sequence of lemmas. Let X ⊆ M be a definable subset of rank R of which we will find the pseudofinite cardinality. Let ψ(x, y) be a stratification of X with respect to D, with parameters hidden for the moment. Partition X into X1,X2,...,Xs so that the pseudofinite cardinality ⋆ of the fiber ψ(a,D) is a constant Aj over a ∈ Xj . By strong minimality of D, s there are only finitely many such cardinalities. Then |X| = X |Xi|, so it sufices i=1 to show that each Xi has cardinality F (|D|) for some F ∈ Q[x] (the details of definability in this argument are in Lemma 4.5). If rk(Xj ) < rk(X) then this is true by induction. If rk(Xj ) = rk(X) then this is shown in Lemma 4.4. In that proof, we use Lemma 2.4 with respect to the relation ψ(x, y) between Xj and D to express |Xi| as a polynomial in the sizes of subsets of D and fibers ψ(Xi, d) for d ∈ D. Each of these cardinalities is itself a polynomial in |D| – subsets of D are by strong minimality, and fibers ψ(Xi, d) are by induction as ψ is a proper stratification. We now prove Proposition 4.2 rigorously, and in a particular form which lends itself to our inductive argument. Proposition 4.2. For every R ∈ ω and every L-formula ϕ(x, y¯), there are polynomials F1(x),...,Fn(x) of degree R and L-formulas π1(¯y, w¯),...,πn(¯y, w¯) such that for all ¯b ∈ M |y¯| and all d¯ ∈ M |w¯| such that D = θ(M, d¯) is strongly minimal, • If MR(ϕ(x, ¯b)) = R then |ϕ(M, ¯b)| = Fi(|D|) for some i, and • For all i, the set |y¯| {¯b ∈ M : MR(ϕ(M, ¯b)) = R and |ϕ(M, ¯b)| = Fi(|D|)} is definable by πi(¯y, d¯). 10 We will prove this proposition by induction on R, through a sequence of lemmas. First we handle the base case R = 0. For this sequence of lemmas, recall (definition 3.4) that NumT (ϕ(¯x, y¯) is the least number N ∈ ω such that whenever M |= T and ¯b ∈ M |y¯|, if ϕ(M |x¯|, ¯b) is finite then |ϕ(M |x¯|, ¯b)| Proof. Let n = NumT (ϕ(x, y¯)). Then for M |= T and ¯b ∈ M , if RK(ϕ(M, ¯b)) = 0 – which means ϕ(M, ¯b) is finite – then |ϕ(M, ¯b)| For formulas ϕ(x, y¯), ψ(x, v, z¯) and θ(v, w¯), let PrStratψ,ϕ,θ(¯y, z,¯ w¯) be a first-order formula such that M |= PrStratψ,ϕ,D(¯b, c,¯ d¯) exactly when ψ(x, t, c¯) is a proper stratification of ϕ(M, ¯b) with respect to θ(M, d¯). Lemma 4.4. Let ϕ(x, y¯) be an L-formula. Suppose there is a formula ψ(x, v, z¯) and a natural number n such that for all M |= T , all ¯b ∈ M |y¯|, and all d¯∈ M |z¯| ¯ ¯ ¯ such that Dd¯ = θ(M, d) is strongly minimal, if MR(ϕ(M, b)) ≤ R and ϕ(M, b) is infinite then there is a c¯ ∈ M |z¯| such that ¯ 1. ψ(x, v, c¯) is a proper stratification of ϕ(M, b) with respect to Dd¯ ¯ 2. For every a ∈ ϕ(M, b), the cardinality of the set ψ(a,Dd¯, c¯) is n. Then Proposition 4.2 holds of R and ϕ(x, y¯). Proof. By the inductive hypothesis, there are polynomials G1,...,Gm ∈ Q[x] of degree < R and formulas ρ1(v, z,¯ w¯),...,ρm(v, z,¯ w¯) such that for all models ¯ |w¯| ¯ M |= T , all d ∈ M such that Dd¯ = θ(M, d) is strongly minimal, and all (e, c¯) ∈ M 1+|z¯|, • if MR(ψ(M, e, c¯)) < R then |ψ(M, e, c¯)| = Gi(|Dd¯|) for some i, and • each ρi defines the set {(e, c,¯ d¯): MR(ψ(M, e, c¯)) < R and |ψ(M, e, c¯)| = Gi(|Dd¯|)}, and • if M |= ρi(e, c,¯ d¯) then deg Gi = MR(ψ(M, e, c¯)). Let N be the maximum of the 2m numbers given by NumT (ξi(v;¯zw¯)) and ′ NumT (ξi(v;¯z)) for i = 1,...,m, where ξi(v;¯zw¯) is θ(v, w¯) ∧ ρi(v, z,¯ w¯) and ′ ¯ ξi(v;¯zw¯) is θ(v, w¯) ∧ ¬ρi(v, z,¯ w¯). Then if θ(M, d) is strongly minimal, we have 11 ′ |z¯| ′ that for each i = 1,...,m and for allc ¯ ∈ M , either |ρi(D, c¯ )| < N or ′ |D − ρi(D, c¯ )| Gi(|Dd¯|) · (|Dd¯|− nσ)+ Gj (|Dd¯|) · σj |ϕ(M, ¯b)| = Pj6=i . n ¯ Therefore |ϕ(M, b)| = Fσ(|Dd¯|), where Gi(X) · (X − nσ)+ Gj (X) · σj F (X)= Pj6=i ∈ Q[X]. σ n By induction, each Gj (X) has degree < R. Moreover, Gi(X) has degree exactly R−1, which follows from Lemma 3.11.2, since i is the unique i such that {b ∈ Dd¯ : |ψ(M, b, c¯)| = Gi(Dd¯)} is cofinite. Therefore Fσ(X) is a polynomial of degree R. ′ |z¯| ′ ′ Supposec ¯ is another element of PrStratψ,ϕ,θ(¯b,M , d¯). Let σ = σ(¯b, c¯ , d¯) ′ ′ ′ ′ with coordinates (σ1,...,σm). Let i be the unique index such that σi′ = ∞ and let nσ′ = X σj . Then as before we have j6=i′ ′ ′ ′ Gi (|Dd¯|) · (|Dd¯|− nσ )+ ′ Gj (|Dd¯|) · σ |ϕ(M, ¯b)| = Pj6=i j . n 12 ′ ′ ′ Gi (x) · (x − nσ )+ Pj6=i′ Gj (x) · σj Therefore the polynomial F ′ (x)= agrees σ n with the polynomial Fσ(x) when applied to the infinite hyper-integer |Dd¯|. By Lemma 2.5.1, the polynomials Fσ and Fσ′ are equal. |z¯| This shows that the choice ofc ¯ ∈ PrStratψ,ϕ,θ(¯b,M , d¯) does not affect the polynomial Fσ. Formally, we can quotient ΣN,m by the equivalence relation ′ ≈ where σ ≈ σ if Fσ = Fσ′ . Then for each ≈-equivalence class [σ]≈, we can |z¯| define the set {¯b ∈ P (ϕ, R): σ(¯b, c,¯ d¯) ≈ σ for allc ¯ ∈ PrStratψ,ϕ,θ(¯b,M , d¯)} ¯ by π[σ]≈ (¯y, d) for an L-formula π[σ]≈ (¯y, w¯) (as ΣN,m is finite, we may explicitly list out the tuples which are equivalent to σ). The polynomials Fσ(X) and formulas π[σ]≈ (¯y, w¯) as [σ]≈ ranges over the finitely many equivalence classes give Proposition 4.2 for R and ϕ(x, y¯). Now we can remove the restriction that the v-fibers of ψ(x, v, z¯) have a constant finite cardinality. Lemma 4.5. Suppose there is a formula ψ(x, v, w¯) such that for all M |= T , ¯ |y¯| ¯ |w¯| ¯ all b ∈ M and all d ∈ M such that Dd¯ = θ(M, d) is strongly minimal, if MR(ϕ(x, ¯b)) = R then there is a c¯ ∈ M |w¯| such that ψ(x, v, c¯) is a proper ¯ stratification of ϕ(M, b) with respect to Dd¯. Then Proposition 4.2 holds for ϕ(x, y¯). ′ Proof. Let N be the maximum of NumT (ξ(v; xz¯w¯) and NumT (ξ (v; xz¯w¯) (re- call Definition 3.4) where ξ(v; xz¯w¯) is θ(v, w¯) ∧ ψ(x, v, z¯) and ξ′(v; xz¯w¯) is ¯ θ(v, w¯) ∧ ¬ψ(x, v, z¯). Then if Dd¯ := θ(M, d) is strongly minimal, we have that ′ ′ 1+|z¯| ′ ′ for all a , c¯ ∈ M , either |ψ(a,Dd¯, c¯ )| 13 R, then ψi(x, w, ¯b, c,¯ d¯) := ψ(x, w, c¯) ∧ ϕi(x, ¯b, c,¯ d¯) ¯ ¯ is a proper stratification of ϕi(M, b, c,¯ d) with respect to Dd¯, and the w-fibers ¯ ¯ ψi(a,Dd¯, b, c,¯ d) all have finite cardinality i, by the definition of ϕi. So in either ¯ ¯ case, the pseudofinite cardinality |ϕi(M, b, c,¯ d)| is Gi,j (|Dd¯|), where j is the unique j such that M |= πi,j (¯b, c,¯ d¯). N Let ΞN,s be the finite set of all tuples ξ = (ξ1,...,ξN ) ∈ {1,...,s} . For ¯ ξ ∈ ΞN,s, let πξ(¯y, z,¯ w¯) be the formula ^ πi,ξi (¯y, z,¯ w¯). For b ∈ Pϕ,R and 0≤i≤N |z¯| c¯ ∈ PrStratψ,ϕ,θ(¯b,M , d¯), let ξ(¯b, c,¯ d¯) be the tuple ξ = (ξ1,...,ξN ) such that ¯ ¯ ¯ ¯ M |= πi,ξ(i)(b, c,¯ d) for each i – i.e., such that M |= πξ(b, c,¯ d). Let Fξ(X) ∈ Q[X] N be the polynomial X Gi,ξi (X). i=0 ¯ When Dd¯ is strongly minimal, the formulas {πξ(¯y, z,¯ d): ξ ∈ ΞN,s} partition |y¯|+|z¯| {(¯b, c¯) ∈ M : MR(ϕ(M, ¯b)) = R and PrStratψ,ϕ,θ(¯b, c,¯ d¯)}. ¯ ¯ If Dd¯ is strongly minimal and M |= πξ(b, c,¯ d) then the pseudofinite cardinal- N ¯ ¯ ¯ ¯ ¯ ity |ϕ(M, b)| is X |ϕi(M, b, c,¯ d)| which equals Fξ(|Dd¯|), since |ϕi(M, b, c,¯ d)| = i=0 ¯ ¯ Gi,ξ(i)(|Dd¯|) for each i. We note that at least one ϕi(M, b, c,¯ d) must have full Morley rank (since this is a finite partition of ϕ(M, ¯b), and so at least one Gi,ξ(i)(X) has degree R, hence Gξ(X) does as well. ′ ′ As in the previous lemma, ifc ¯ ∈ M and M |= PrStratψ,ϕ,θ(¯b, c¯ , d¯) and ¯ ′ ¯ ′ ¯ ′ ξ(b, c¯ , d)= ξ , then as in the previous lemma we also have |ϕ(M, b)| = Fξ (|Dd¯|), so Fξ(x)= Fξ′ (x) as polynomials, again by Lemma 2.5.1. Then, as in that proof, ′ we quotient ΞN,s by the equivalence relation defined as ξ ≈ ξ when Fξ = Fξ′ , and taking our polynomials to be F[ξ]≈ (X) and our defining formulas π[ξ]≈ (¯y, w¯) to be “for allz ¯, if PrStratψ,ϕ,θ(¯y, z,¯ w¯) holds then πξ′ (¯y, z,¯ w¯) holds for some ξ′ ≈ ξ,” which is definable since there are finitely many ξ′ ≈ ξ. Now we can prove the full one-variable case. Proof of Proposition 4.2. By Corollary 3.9, there are formulas ¯ |y¯| ¯ ψ1(x, v, z¯),...,ψs(x, v, z¯) such that for all b ∈ M and all d such that Dd¯ is strongly minimal, there is a unique i ∈{1,...,s} such that for somec ¯ ∈ M |z¯|, ¯ the formula ψi(x, w, c¯) stratifies ϕ(M, b) with respect to Dd¯. Apply Lemma 4.5 to each ψi to obtain s ∈ ω and polynomials Gi,1(X),...,Gi,s(X) ∈ Q[X] and formulas πi,1(¯y, w¯),...,πi,s(¯y, w¯) for each i as in the statement of Proposition 4.2. Taking our polynomials to be the polynomials Gi,j (X) and our formulas to be “(∃zStrat¯ ψi,ϕ,θ(¯y, z,¯ w¯)) and πi,j (¯y, w¯)” proves the proposition. We can now the proof of the full main theorem. 14 Proof of Proposition 4.1. The case n = 1 is proven in Proposition 4.2. Let us assume the theorem is true for n and let ϕ(x1,...,xn, xn+1, z¯) be a formula. By Proposition 4.2, there are G1(X),...,Gs(X) ∈ Q[X] of degree at n ¯ |y¯| ¯ |w¯| most N such that for alla ¯ ∈ M , all b ∈ M and all d ∈ M such that Dd¯ ¯ is strongly minimal, we have |ϕ(M, a,¯ b)| = Gi(|Dd¯|) for some i, and for each i ¯ ¯ the set of alla ¯b such that |ϕ(M, a,¯ b)| = Gi(|Dd¯|) is definable by the formula πi(x2,...,xn+1, y,¯ d¯). By induction, for each i there are polynomials Hi,1(X),...,Hi,m(X) ∈ Q[X] |y¯| |w¯| and formulas τi,1(¯y, w¯),...,τi,m(¯y, w¯) such that for all ¯b ∈ M and d¯ ∈ M n ¯ ¯ with Dd¯ strongly minimal, there is a j ∈{1,...,m} such that |πi(M , b, d)| = ¯ ¯ ¯ Hi,j (|Dd¯|), and degHi,j (X)= MR(π(¯x, b, d)), and for all j the formula τi,j (¯y, d) ¯ n ¯ ¯ defines the set of all b such that |πi(M , b, d)| = Hi,j (|Dd¯|). (As in the proofs above, we may not have the same number of polynomials for each πi, but we may add additional unused polynomials to get a uniform number). s For every tuple σ = (σ1,...,σs) ∈ {1,...,m} , let τσ(¯z, w¯) be the formula ^ τi,σi (¯z, w¯). Note that the formulas when Dd¯ is strongly minimal the formulas i |z¯| s τσ(¯z, d¯) partition M as σ ranges over {1,...,m} (with some empty sets in this partition, if any of the polynomials Hi,j is unattained). If σ = (σ1,...,σs), M |= τσ(¯c), and Dd¯ is strongly minimal, then |y¯| ¯ ¯ |πi(M , b, d)| = Hi,σi (|Dd¯|) for each i. By Lemma 2.4.1, we have that s n+1 ¯ |ϕ(M , b)| = X Fi(|Dd¯|) · Hi,σi (|Dd¯|)= Fσ(|Dd¯|), i=1 s where Fσ(X) is the polynomial X Gi(X) · Hi,σi (X). We note that this equa- i=1 tion comes from the fact that ϕ(M n+1, ¯b) is the disjoint union of the sets Zi := {(c, ¯a, ¯b): M |= πi(¯a, ¯b) and M |= ϕ(c, a,¯ ¯b)} for i = 1,...,s, each of which has cardinality Gi(|Dd¯|) · Hi,σi (|Dd¯|). The fibers {c ∈ M : (c, a¯) ∈ Zi} all have pseudofinite cardinality Gi(|Dd¯|) by the definition of Zi, hence they all have the same Morley rank which is deg Gi(X). Therefore by Lemma 3.5, taking the definable surjection f to be the projection of Zi onto the coordinates n (x2,...,xn), the Morley rank of Zi is deg Gi +MR(πi(M , ¯b, d¯)) = deg Gi(X)+ deg Hi,σi (X), which is the degree of the polynomial Gi(X) · Hi,σi (X). So deg Fσ(X) = max(deg Gi(X) · Hi,σi (X): i ≤ s) = max(MR(Zi): i ≤ s) = MR(ϕ(M n+1, ¯b)). 5 Additional Results We illustrate that it our counting polynomials may need to have strictly rational coefficients, as opposed to the strongly minimal case, where the coefficients are integer-valued. Example 5.1. Let L contain unary predicates P0, P1 and a binary relation R. 15 Let T be the complete theory axiomatized by • P0 and P1 partition the universe M into infinite disjoint sets • For all x, y ∈ M if R(x, y) holds then x ∈ P0 and y ∈ P1. • For all x ∈ P0 there are exactly three y ∈ P1 such that R(x, y) holds. • For all y ∈ P1 there are exactly two x ∈ P0 such that R(x, y) holds. ′ ′ ′ ′ • For all x, x ∈ P0 and y,y ∈ P1, if R(x, y) and R(x ,y) and R(x, y ) hold then R(x′,y′) holds. The reduct of any model M of T to {R} looks like an infinite disjoint union of complete two-to-three bipartite digraph, directed versions of the complete bi- partite graph K2,3, with arrows going from the side with two elements to the side with three elements. The predicate P0 picks out the source nodes and the predicate P1 picks out the target nodes. This theory is totally categorical. In any model M of T , the sets P0(M) and P1(M) are strongly minimal. T is a pseudofinite theory which may be satisfied by any ultraproduct of M1,M2,M3,... where Mn is the disjoint union of n copies of the complete two-to-three bipartite digraph. Then |Mn| = 5n and |P0(Mn)| = 2n and |P1(Mn)| = 3n. It follows 2 that in any infinite ultraproduct M = M we have |P (M)| = |P (M)|, Y n 0 3 1 n→U 3 and |P (M)| = |P (M)|. This shows that the polynomials in the theorem may 1 2 0 be required to have strictly rational coefficients. In our next result, we show that when a pseudofinite ultraproduct M has uncountably categorical theory, then the two-sorted structure M + is uncompli- cated in a model-theoretic sense. To formulate this result, we need the notion of a disjoint union structure. Suppoe L1 and L2 are disjoint languages, and let M1 be an L1-structure and M2 an L2-structure. The disjoint union of M1 and M2 is a two-sorted structure. One sort is for M1 and has the language L1, and the other sort is for M2 in the language L2. Each sort inherits the full structure of the Mi, and there is no defined interaction between the two sorts. The disjoint union is, in a sense, the least model-theoretically complicated way of joining the structures M1 and M2. In this next proposition, we show that the counting structure M + is definable in the disjoint union of M and the real closed field R⋆. Proposition 5.2. Let (Mλ : λ ∈ Λ) be a sequence of finite L-structures and let + U be an ultrafilter on Λ such that M := Y Mλ is ℵ1-categorical. Let M be the λ→U + + L -expansion of M with respect to (Mλ : λ ∈ Λ) and U. Then M is definable over a singleton c ∈ OF and a tuple d¯ ∈ M in the two-sorted disjoint union structure of M and RU , where d¯ is such that θ(M, d¯) is strongly minimal for an L-formula θ(v, w¯) and where c is the pseudofinite cardinality of any infinite definable set X ⊆ M m (for any arity m). 16 Proof. Let D = θ(M, d¯) be a definable strongly minimal set. Then |X| = F (|D|) for some polynomial F (X) ∈ Q[x], by Theorem 4.1. 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 2.5.2. So |D| is definable over |X| in RU . Therefore to prove our theorem, it suffices to let c = |D|, and show that M + is definable in the disjoint union L+-structure M ∪ RU over c ∈ R⋆ and d¯∈ M. To show that M + is definable in the disjoint union, we show that for every ⋆ ¯ L-formula ϕ(¯x, y¯), the function fϕ(¯x,y¯) is definble over c = |D| ∈ R and d ∈ M. Let ϕ(¯x, y¯) be an L-formula. By Theorem 4.1, there are finitely many polynomials F1(X),...,Fr(X) ∈ Q[X] and formulas π1(¯y, w¯),...,πr(¯y, w¯) such |y¯| that the formulas πi(¯y, d¯) partition M and for every ¯b we have |ϕ(M, ¯b)| = |y¯| U Fi(|D|) iff M |= πi(¯b). Then for any ¯b ∈ M and any e ∈ R , + ¯ M |= fϕ(¯x,y¯)(b)= e if and only if r U ¯ ¯ (M, R ) |= _[πi(b, d) ∧ e = Fi(c)] i=1 which is definable in the disjoint union. In our final two propositions, we will demonstrate how Theorem 4.1 passes down to give information about cardinalities of sets in finite structures. These propositions will be about particular kinds of families of sets defined as follows. Definition 5.3. Let L be a language and let (Mλ : λ ∈ Λ) be a sequence of L- structures. We say that (Mλ : λ ∈ Λ) has a zero-one law if for every L-sentence φ, either Mλ |= φ for all-but-finitely many λ or Mλ |= ¬φ for all-but-finitely many λ. Equivalently, (Mλ : λ ∈ Λ) has a zero-one law if any two non-principal ultraproducts of the family are elementarily equivalent. We also say that (Mλ : λ ∈ Λ) is a zero-one class. If (Mλ : λ ∈ Λ) is a zero-one clas, we call the theory of any/all non-principal ultraproducts of the family the limit theory. Let us also recall the little-o notation “f = o(g)”, which is an abbreviation for the statement f(x) lim =0. x→∞ g(x) First we show that a zero-one class with an uncountably categorical limit theory is an R-mec, a notion devised by Anscombe, Macpherson, Steinhorn and Wolf to appear in their upcoming paper [1], and explored in detail in Wolf’s thesis [7]. Definition 5.4. Let C be a class of finite L-structures. Let R be a set of functions from C to R≥0. Then C is an R-mec if for every L-formula ϕ(¯x, y¯) there are finitely many h1(X),...,hn(X) ∈ R and L-formulas ψ1(¯y),...,ψn(¯y) such that for each M ∈C, 17 |y¯| • for each ¯b ∈ M , there is an i such that |ϕ(M, ¯b)| = hi(M), and |y¯| • for each i the formula ψi(¯y) defines the set {¯b ∈ M : |ϕ(M, ¯b)| = hi(M)}. The word “mec” is short for “multidimensional exact class”. It is a special case of the more general notion of multidimensional asymptotic class, introduced in [1] and explored in [7] as R-mecs are, the definition of which is similar to the above definition except instead of stipulating that |ϕ(M, ¯b)| = hi(M), we stipulate that the two are “asymptotically” equal, in the sense that the quotient |ϕ(M, ¯b)|− hi(M) tends to zero as hi(M) goes to infinity. That is, |ϕ(M, ¯b)|− hi(M) y¯| hi(M) = o(hi(M)), as M ranges over C and ¯b ranges over ψi(M ). We note that hi takes as input the structure M, not the number |M|. In this proposition, we prove that a class of finite structures whose ultra- product theory is uncountably categorical is an R-mec for a particularly simple class of functions R. Proposition 5.5. Let T be an uncountably categorical pseudofinite theory. Sup- pose (Mλ : λ ∈ Λ) is a sequence of finite L-structures with a zero-one law and limit theory T . Let θ(v, w¯) be a formula such that θ(M, d¯) is strongly minimal for some M |= T and d¯ ∈ M. Then for every formula ϕ(¯x, y¯), there are poly- nomials F1,...,Fr ∈ Q[x] and formulas π1(¯y, w¯),...,πr(¯y, w¯) such that in all ¯ |w¯| ¯ |y¯| finite structures Mλ there is a tuple dλ ∈ Mλ such that for all b ∈ Mλ , there |x¯| ¯ ¯ is i ∈ {1,...,r} such that |ϕ(Mλ , b)| = Fi(|θ(Mλ, dλ|), and for each i the set |y¯| ¯ ¯ |y¯| πi(Mλ , dλ) is the set of all b ∈ Mn for which this equation holds. In particular, the class (Mλ : λ ∈ Λ) is an R-mec, where R is the set of functions defined by f(Mλ)= F (|θ(Mλ, d¯λ)|) as F (x) ranges over Q[x] and d¯λ ranges over elements of Mλ. Proof. Let θ(v, w¯) be an L-formula such that some/every (saturated) model M of the theory T has a d¯∈ M such that θ(M, d¯) is strongly minimal. Let ϕ(¯x, y¯) be an L-formula. Let F1(X),...,Fr(X) ∈ Q[X] be the poly- nomials and π1(¯y, w¯),...,πr(¯y, w¯) be the formulas obtained in Theorem 4.1 as applied to ϕ(¯x, y¯) and θ(v, w¯). Let Ψ(¯w) be an L+-formula such that Ψ(d¯) expresses the conjunction of the two conditions |y¯| 1. π1(¯y, d¯),...,πr(¯y, d¯) partition M , and |y¯| |x¯| 2. for all ¯b ∈ M we have |ϕ(M , ¯b)| = Fi(|θ(M, d¯)|) if and only if M |= πi(¯b, d¯). Then for every M |= T and d¯∈ M, if θ(M, d¯) is strongly minimal then M + |= Ψ(d¯). Since every uncountable model of T is saturated, every uncountable model of T contains a tuple d¯ such that θ(M, d¯) is strongly minimal. Therefore 18 whenever M is a pseudofinite ultraproduct which satisfies T , the expanded structure M + satisfies the L+-sentence ∃w¯Ψ(¯w). By assumption, every nonprincipal ultraproduct of the family (Mλ : λ ∈ Λ) satisfies the theory T . Therefore every infinite ultraproduct of the expanded + finite structures Mλ satisfies the sentence ∃w¯Ψ(¯w). Hence this sentence is sat- + + + isfied in all but finitely many of the L -expansions Mλ . If Mλ satisfies ∃w¯Ψ(¯w) ¯ + then, taking dλ to be a witness, the structure Mλ satisfies the conclusion of the proposition. Let M1,...,Mk be the finitely many structures in our family whose expan- + + sions Mi do not satisfy the L -formula ∃w¯Ψ(¯w). Let N be a number greater than max{|M1|,..., |Mk|}. Let us add polynomials Fr+1(X),...,Fr+N (X) where Fr+k(X) is the constant polynomial k−1, and add new formulas πr+1(¯y, w¯),...,πr+N (¯y, w¯) where πr+k(¯b, d¯) expresses “|ϕ(M, ¯b)| = k − 1”. Finally, let us modify the for- mulas π1(¯y, w¯),...,πr(¯y, w¯) by conjoining them each with a formula such that πi(¯b, d¯) implies “|ϕ(M, ¯b)| ≥ N”. With this alteration, the conclusion of the proposition holds for every structure Mλ. The following notion was introduced by Elwes in [3]. Definition 5.6. A family of finite L-structures (Mλ : λ ∈ Λ) is a N-dimensional asymptotic class if for every L-formula ϕ(¯x, y¯), there exist finitely many pairs ≥0 (µ1, d1),..., (µs, ds) ∈ R × ω and L-formulas π1(¯y),...πs(¯y) over ∅ such that for every λ ∈ Mλ, the following two conditions hold: • |y¯| The formulas π1(¯y),...,πs(¯y) partition Mλ , • |x¯| ¯ di/N di/N For i =1,...,s, we have that |ϕ(Mλ , b)|− µi|Mλ| = o(|Mλ| ) as ¯ |y¯| |Mλ| → ∞, for b ∈ πi(Mλ ). Clause 2 may be restated as: for every ǫ> 0 there is a number C such that ¯ |y¯| whenever |Mλ| > C and b ∈ πi(Mλ ) we have |x¯| ¯ di/N di/N |ϕ(Mλ , b) − µi|Mλ| <ǫ|Mλ| . We prove that a zero-one family of finite structures whose limit theory is uncountably categorical is an N-dimensional asymptotic class, where N is the Morley rank of the limit theory. As the definition of an N-dimensional asymp- totic class requires that the formulas πi be definable without parameters, while the conclusion of Proposition 5.5 allows for parameters, we shall need the fol- lowing fact, which for example may be found as a consequence of [5, 6.1.16]. Fact 5.7. 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 κ(¯w) and θ(v, w¯) where κ(¯w) isolates a type p such that for a model M |= T and d¯∈ M, if tp(d¯)= p then θ(M, d¯) is strongly minimal. We shall need the following corollary of Proposition 5.5. 19 Corollary 5.8. Assume that (Mλ : λ ∈ Λ) is a zero-one class with uncountably categorical limit theory T . Let κ(¯w) and θ(v, w¯) be as in Fact 5.7. Then in the conclusion of Proposition 5.5, the tuple d¯λ may be taken to be any tuple in |w¯| κ(Mλ ), provided that this set is nonempty, which is the case in cofinitely many Mλ. Proof. We slightly modify the proof of Proposition 5.5. We conjoin to Ψ(w ¯) the formula κ(¯w). Then it remains true that all-but-finitely many of the finite + expansions Mλ satisfy ∃w¯Ψ(¯w), and we may explicitly encode out the finitely many counterexamples as in the proof of Proposition 5.5. For the counting clause of the definition of N-dimensional asymptotic class, we shall apply the following lemma about the asymptotics of the inverse function of a polynomial. n n−1 Lemma 5.9. Let F (X)= anx + an−1x ... + a1x + a0 ∈ R[X], with an > 0. Let C ∈ R be such that the function F (x) is increasing on [C, ∞), and let F −1 :[F (C), ∞) → [C, ∞) be the inverse function. Then 1/n −1 an−1 lim [(x/an) − F (x)] = . x→∞ nan m m−1 Furthermore, let G(x)= bmx + bm−1x + ... + b1x + b0. Then m/n m/n −1 m/n (bm/an )x − G(F (x)) = o(x ). Proof. We compare F (X) to polynomials of the form n n n−1 n−1 n an(x + α) = anx + an · nαx + ... + an · nα x + anα . n 1/n Note that the inverse of an(x + α) is (x/an) − α. n It is easily seen that an(x+α) > F (x) for sufficiently large x if an·nα>an−1 n – that is, if α>an−1/(nan) – and that an(x + β) < F (x) for sufficiently large x if β 1/n −1 lim (x/an) − F (x)= an−1/(nan). x→∞ To prove that m/n m/n −1 m/n (bm/an )x − G(F (x)) = o(x ) it suffices to prove the asymptotic equations m/n 1/n m/n bm(x/an) − G((x/an) )= o(x ) 20 and 1/n −1 m/n G((x/an) ) − G(F (x)) = o(x ). For the first, rewrite b (x/a )m/n − G((x/a )1/n) lim m n n x→∞ xm/n as b zm − G(z) lim m z→∞ m/n m an z 1/n after making the variable substitution z = (x/an) . 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 −1 1/n of this lemma implies the existence of a B such that F (x) ∈ ((x/an) − 1/n B, (x/an) + B) for all x. Then −1 1/n 1/n 1/n |G(F (x)) − G((x/an) )| < |G((x/an) + B) − G((x/an) − B) for sufficiently large x. Therefore it suffices to show G((x/a )1/n + B) − G((x/a )1/n − B) lim n n =0 x→∞ xm/n which is equivalent to saying G(z + B) − G(z − B) lim =0 z→∞ zm 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. With this lemma, we are able to prove that any class of finite structures with a zero-one law (as defined earlier) and uncountably categorical limit theory is an N-dimensional asymptotic class, where N is the Morley rank of the limit theory. Proposition 5.10. Let T be an uncountably categorical pseudofinite theory. Suppose (Mλ : λ ∈ Λ) is a sequence of finite L-structures such that Y Mλ |= λ→U T for every non-principal ultrafilter U on ω. Then (Mλ : λ ∈ Λ) is an N- dimensional asymptotic class, where N is the Morley rank of T . Proof. Let κ(¯w), θ(v, w¯) and p be as in Fact 5.7. First we show Definition 5.6 holds for the formula θ(x, y¯) ∧ κ(¯w) (with vari- ables as in that definition partitioned as x, y¯w¯). We apply Corollary 5.8 to the formula “x = x” to obtain polynomials F1(X),...,Fr(X) ∈ Q[X] 21 of the Morley rank of the formula (which is N) and formulas π1(¯w),...,πr(¯w). Let i be such that p ⊢ πi(¯w). Then Mλ |= (κ(¯w) → πi(¯w)) for cofinitely many ′ Mλ. For such a Mλ, if Mλ |= κ(d¯) then |Mλ| = Fi(|θ(Mλ, d¯)|). If Mλ |= κ(d¯) ′ as well then |Mλ| = Fi(|θ(Mλ, d¯)|). By injectivity of polynomials on a tail, ¯ ¯′ |w¯| this implies the existence of a C such that for |Mλ| > C, if d, d ∈ κ(Mλ ) ′ then |θ(Mλ, d¯)| = |θ(Mλ, d¯)|, and |Mλ| = Fi(|θ(Mλ, d¯)|). Let F (X)= Fi(X)= N aN X + ... + a1X + a0. It follows from Lemma 5.9 that N 1/N |θ(Mλ, d¯)|− (1/aN ) |Mλ| lim 1/N =0 |Mλ|→∞ |Mλ| ¯ |w¯| if we take d ∈ κ(Mλ ). Now let ϕ(x1,...,xn, y¯) be an arbitrary L-formula. Apply Corollary 5.8 to ϕ(x1,...,xn, y¯) to get polynomials G1(X),...,Gs(X) and formulas π1(¯y, w¯),...,πs(¯y, w¯) such that the conclusion of that proposition holds. Then if ¯b, d¯∈ Mλ with |Mλ| ¯ ¯ ¯ n ¯ ¯ sufficiently large and Mλ |= κ(d) ∧ πi(b, d) then |ϕ(Mλ , b)| = Gi(|θ(Mλ, d)|) = −1 Ni Gi(F (|Mλ|)). For each i =1,...,s let Gi(X)= ai,Ni X + ... + ai,1X + ai,0. By Lemma 5.9, we have −1 1/N Ni/N G (F (x)) − (a i /a )x lim i i,N N =0. x→∞ xNi/N Therefore we have n ¯ 1/N Ni/N |ϕ(M , b)|− (a i /a )|M | lim λ i,N N λ =0 Ni/N x→∞ |Mλ| for all ¯b, d¯∈ Mλ with Mλ |= πi(¯b, d¯) ∧ κ(d¯). That is, n ¯ 1/N Ni/N Ni/N |ϕ(Mλ , b)|− (ai,Ni /aN )|Mλ| = o(|Mλ| ), whence the definition of N-dimensional asymptotic classes almost applies to 1/N the formula ϕ(¯x, y¯), with (µi, di) being ((ai,Ni /aN ),Ni) and defining formulas being “∃w¯[κ(¯w) ∧ πi(¯y, w¯)].” The only difference from the definition is that the defining formulas do not necessarily partition every Mλ. However they do partition all Mλ with |Mλ| 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. References [1] Anscombe, S., Macpherson, D.,Steinhorn, C., Wolf, D. “Multidimensional asymptotic classes and generalised measurable structures” In preparaion. (2016) [2] Buechler, S. Essential Stability Theory Springer, Perspectives in mathemat- ical logic series (2002) 22 [3] Elwes, R “Asymptotic classes of finite structures” The Journal of Symbolic Logic, vol. 72, no. 2 (2007) [4] Garcia, D., Macpherson, D., Steinhorn, C. “Pseudofinite structures and sim- plicity” Journal of Mathematical Logic, vol. 15, no. 1 (2014) [5] Marker, D. Model Theory: An Introduction Springer (2002) [6] Pillay, A. “Strongly Minimal Pseudofinite Structures” http://arxiv.org/abs/1411.5008 (2014) [7] Wolf, D. “Model theory of multidimensional asymptotic classes” PhD Thesis, University of Oxford (2016) [8] Zil’ber, B. “The Transcendental Rank of the Formulas of an ℵ1-Categorical Theory”, Mathematical Notes of the Academy of Sciences of the USSR, vol. 15, pp 182-186 (1974) [9] Zil’ber, B. Uncountably Categorical Theories American Mathematical Soci- ety (1993) 23 0 then − the fibers of fi restricted to Xi have Morley rank ≤ R−1, so by induction, − + MR(Xi ) ≤ R − 1+ S. So in either case, MR(Xi )= MR(Xi)= R + S + + and MR(X \ Xi ) < R + S. If MR(Yi ) < S then as in the previous + + paragraph, since fi surjects Xi onto Yi with fibers of rank R, we would + + have MR(Xi ) < R+S, which is false. Therefore MR(Yi )= S for each i, + and since Y has degree 1, we have MR(Y \ Yi ) type π = {y ∈ Yi : i ∈ ω}, which is definable with parameters from the set of parameters which define the sets X1,X2,.... + + Since Y1 ∩ ... ∩ Yn has Morley rank S and is in particular non-empty for every n, the type π is consistent. Because M is ω1-saturated, π is −1 realized by some y ∈ Y . Then MR(fi (y)) = R for all i ∈ ω. But then −1 −1 −1 f1 (y),f2 (y),... are disjoint subsets of f (y) of Morley rank R, hence MR(f −1(y)) > R, a contradiction. This proves the case where the Morley degree of Y is 1. If the degree of Y is D, then Y is the disjoint union of definable sets Y1,...,YD ⊆ Y , each −1 of Morley rank S and Morley degree 1. Letting Xi = f (Yi) we get by the D = 1 case that MR(Xi) ≤ R + S, and since X = X1 ∪ ... ∪ Xi this completes the proof.