Fields with pseudo-exponentiation

B. Zilber zilber@maths.ox.ac.uk

10 March 2000

1 Introduction

This research is motivated by the study of model-theoretical properties of classical ’analytic’ structures, i.e. ones having natural analytic representation (see also [Z]). For example, the structure of complex numbers as a field with exponentiation

C_{exp} = (C, +, ·, exp).

One of the questions we can ask is whether C_{exp} is quasi-minimal, i.e. any definable subset of C_{exp} is either countable or the complement to a countable. Another question is about homogeneity of the structure; we do not know any its automorphism except the identity and the complex conjugation. In general we would like to understand the nature of analytic dimension in a context close to model-theoretic stability theory. A slightly weaker analytic structure C_{exp}^{(2)} is a two-sorted structure with both sorts C(1) and C(2) being copies of complex numbers, on both sorts the field structure is given and there is a mapping exp : C(1) → C(2) in the language. The model theory of the both structures, as well as of many others of this kind, seems very hard to study directly. We study here model theory of abstract analogues of C_{exp}^{(2)}. We start by considering the class E_{p} of two-sorted structures of the form (D, ex, R), D, the domain of a mapping ex is a field of characteristic zero, R, a field of characteristic p and ex is a homomorphism of the additive group of D onto the multiplicative group R^{×} of R. Following the ideology of Hrushovski’s construction of non-classical structures with nice dimension notion (see [H]), we consider a notion of a predimension δ(X) for finite subsets X ⊆ D defined as

δ(X) = tr.d.(X) + tr.d.(ex(X)) − dim_{Q}(X),

here dim_{Q}(X) is the dimension of the Q-linear space generated by X. We then define the subclass S_{p} of E_{p}, the weak Schanuel class as the class of structures where δ(X) ≥ 0 for all X and the kernel of ex is standard (a cyclic additive subgroup in case p = 0). Recall that the Schanuel conjecture [L] states that, given Q-linearly independent complex numbers x_{1}, …, x_{n}, the inequality

tr.d.(x_{1}, …, x_{n}, exp(x_{1}), …, exp(x_{n})) − n ≥ 0

holds. We prove that the weak Schanuel class S_{p} is non-empty. We study the predimension δ and the relation A ≤ B following the Hrushovski pattern. We define a notion of exponentially-algebraically closed structures, which are just existentially closed structures with respect to the relation A ≤ B in class S_{p}. Again, by standard model theory as used by Hrushovski, in the class SC_{p} of exponentially-algebraically closed structures δ gives rise to the dimension notion ∂. On the other hand class SC_{p} is shown to be axiomatizable inside S_{p}. More precisely, it is axiomatized by an axiom stating that any normal system of exponential-algebraic equations has a solution in the structure. The definition of a normal system of exponential-algebraic equations is explicit and purely algebraic. It follows A necessary and sufficient condition for the structure of complex numbers with exp to be in SC_{0} is to satisfy the weak Schanuel conjecture and to have solutions to any normal system of exponential-algebraic equations. There are grounds to believe that complex numbers do satisfy the axiom.

Next we proceed by developing analogues of atomic and prime structures in class SC_{p}. A notion of quasi-atomic types and structures as well as quasiprime structures over a given subset are introduced in style of [Sh]. We restrict the study to the case p = 0. Then the usual nice properties for the notions hold. We prove that given a subset C in a structure in SC_{0} the quasi-prime structure E(C) is determined uniquely up to isomorphism over C. Now take C a ∂-independent set of power μ. Then the isomorphism type of E(C) is given by μ, denote the structure E_{μ}. These structures are said to be the canonical weak Schanuel structures of power μ (canonical for short). It is tempting to conjecture that the canonical structure of power 2^{ℵ_{0}} is isomorphic to C_{exp}^{(2)}. This conjecture looks too hard to prove, since the proof would include a proof of the weak Schanuel conjecture. Still it would be interesting to study the properties of the canonical structure compairing these to the known properties of C. It follows from what was stated above that Any bijection of the ∂-base of the canonical structure is extendable to an automorphism and, for μ > ℵ_{0}, the structure is quasi-minimal, i.e. any definable subset of E_{μ} is either of power not greater than ℵ_{0} or of power μ.

In a canonical structure E_{μ} (μ > ℵ_{0}) we study the notion of dimension for definable subsets S ⊆ E^{n}, denoted adim (S), which we call the pseudoanalytic dimension. We define it analogously to the definition of Morley Rank:

adim S = max_{x̄∈S} ∂x̄.

The pseudo-analytic dimension is a good analogue of the analytic dimension in C :

(i) adim S ≥ 0 for any non-empty S; (ii) for non-empty S adim S = 0 iff card (S) ≤ ℵ_{0}; (iii) for non-empty S adim S ≥ m > 0 iff S can be projected to D^{m} so that the complement to the projection in D^{m} is of pseudo-analytic dimension less than m; (iv) For any irreducible algebraic varieties V ⊆ D^{n} and W ⊆ R^{n}

V ∩ ln W ≠ ∅ implies adim (V ∩ ln W) ≥ dim V + dim W − n.

Acknowledgements. The research had been conceived in February 1997 while the author was visiting The Fields Institute in Toronto. The study continued in Kemerovo University, Russia, where I had a permanent position then. Essential progress had been achieved while I was in MSRI, Berkeley, participating in the Model Theory of Fields program. The final version of the paper has been written in University of Oxford. I would like to express my gratitude to all the institutions and people there who supported and helped the research.

2 Definitions and notation

For technical reasons it is more suitable to work with the equivalent class of one sorted structures on D in the language L_{p} which is the version of extended language of fields. On D the following operations and relations are defined: the addition ′+′ and multiplication by all rational scalars; n-ary predicates V (x_{1}, …, x_{n}) for each algebraic subvariety V ⊆ D^{n} defined and irreducible over 0 an equivalence relation E, this allows to consider R = D/E, we think of R as of a field; n-ary predicates EW (y_{1}, …, y_{n}) for each algebraic subvariety W ⊆ R^{n} definable and irreducible over 0.

The interpretation assumes E(x, y) ≡ [ex(x) = ex(y)], V (x_{1}, …, x_{n}) ≡ [⟨x_{1}, …, x_{n}⟩ ∈ V], EW (x_{1}, …, x_{n}) ≡ [⟨ex(x_{1}), …, ex(x_{n})⟩ ∈ W].

Definition E_{p} is the class of structures D in language L_{p} such that D is a field of characteristic zero, E an equivalence relation on D which is congruent with respect to the relations EW (x_{1}, …, x_{n}), D/E when considered with relations EW (x_{1}, …, x_{n}) is embeddable in the group R^{×} where R is a field of characteristic p so that the canonical mapping

ex : D → R^{×}

is a homomorphism of the additive group of D into the multiplicative group R^{×}.

Notation For the sake of simplicity we fix p below and write E instead of E_{p}.

For finite X, X′ ⊆ D, Y, Y′ ⊆ R tr.d.(X) the the transcendence degree of X over 0; dim_{Q}(X) the dimension of the vector space span_{Q}(X) generated by X over Q; tr.d.(Y) the transcendence degree of Y over 0; δ(X) the predimension of finite X ⊆ D : δ(X) = tr.d.(X) + tr.d.(ex(X)) − dim_{Q}(X); δ(X/X′) = δ(XX′) − δ(X′), and XX′ = X ∪ X′; for infinite A ⊆ D δ(X/A) ≥ k, k ∈ Z, means that for any finite Y ⊆ A there is finite Y ⊆ Y′ ⊆ A such that δ(X/Y′) ≥ k, and δ(X/A) = k means δ(X/A) ≥ k and not δ(X/A) = k + 1.

tr.d.(X/X′) = tr.d.(XX′) − tr.d.(X′); tr.d.(Y/Y′) = tr.d.(Y Y′) − tr.d.(Y′); dim_{Q}(X/X′) = dim_{Q}(XX′) − dim_{Q}(X′); ker is the name of a unary predicate of type EW x ∈ ker ≡ ex(x) = 1.

Denote S the subclass of E consisting of all D satisfying the condition:

δ(X) ≥ 0 for all finite X ⊆ D.

Denote subE the class of structures A in language L_{p} satisfying : A is an additive divisible subgroup of the additive group of a field D of characteristic zero and A/E can be embedded into R^{×} for a field R of characteristic p and the canonical mapping ex_{A} : A → R^{×} is a homomorphism.

Denote subS the subclass of subE consisting of A which satisfy δ(X) ≥ 0 for any finite X ⊆ A.

For W an algebraic variety, b̄ = ⟨b_{1}, …, b_{l}⟩ denote

W (b̄) = {⟨x_{l+1}, …, x_{n+l}⟩ : ⟨b_{1}, …, b_{l}, x_{l+1}, …, x_{n+l}⟩ ∈ W }.

Lemma 2.1 If X = {x_{1}, … x_{n+l}} ⊆ D, X′ = {x_{1}, … x_{l}}, x̄ = ⟨x_{1}, … x_{n+l}⟩, x̄′ = ⟨x_{1}, … x_{l}⟩ then: tr.d.(X) = dim V, where V ⊆ D^{n+l} is the minimal algebraic variety over 0 containing x̄; dim_{Q}(X) = dim L where L̄ is the minimal linear subspace of D^{n+l} containing x̄ and given by homogeneous linear equations over Q; tr.d.(exX) is the dimension of the minimal subvariety of R^{n+l} over 0 containing ex(x̄); dim_{Q}(X/X′) = dim L(0^{l}), where 0^{l} is a string of l zeroes.

Proof Immediate from definitions.□

Lemma 2.2 (i) δ(X/X′) = tr.d.(X/X′) + tr.d.(exX/exX′) − dim_{Q}(X/X′); (ii) For any A ⊆ D and finite X ⊆ D there is a finite Y ⊆ A, such that, if Y ⊆ Y′ ⊆ A, then δ(X/Y′) = δ(X/A).

Proof (i) is immediate from definitions. (ii) follows from (i) if we choose Y ⊆ A, such that tr.d.(X/Y) = tr.d.(X/A), tr.d.(ex(X)/ex(Y)) = tr.d.(ex(X)/ex(A)), dim_{Q}(X/Y) = dim_{Q}(X/A). The choice is possible, since tr.d.(X/Y), tr.d.(ex(X)/ex(Y)) and dim_{Q}(X/Y) are non-increasing functions of Y. □

Remark The condition dim_{Q}(X/A) = dim_{Q}(X/Y) for Y ⊆ A is satisfied iff span_{Q}(X) ∩ A ⊆ Y. Correspondingly with the transcendense degree.

Notation For A, B ∈ subE denote A ≤ B the fact that A ⊆ B as structures and δ(X/A) ≥ 0 for all finite X ⊆ B.

Lemma 2.3 For any structure A of the class subE and finite X, Y, Z ⊆ A :

(i) If span_{Q}(X′) = span_{Q}(X) then δ(X′) = δ(X). (ii) If span_{Q}(X′Y) = span_{Q}(XY) then δ(X/Y) = δ(X′/Y). (ii) If span_{Q}(Y) = span_{Q}(Y′) then δ(X/Y) = δ(X/Y′). (iv) δ(XY/Z) = δ(X/Y Z) + δ(Y/Z).

Proof Immediate from the definitions.□

Lemma 2.4 For A, B, C ∈ subE

(i) if A ≤ B and B ≤ C, then A ≤ C; (ii) if A ≤ B, Y ⊆ B, δ(Y/A) = 0, then AY ≤ B.

Proof (i) Let X ⊆ C finite and Z ⊆ A large enough finite so that δ(X/Z) = δ(X/A). We need to prove that δ(X/Z) ≥ 0. Choose Y ⊆ B finite so that span_{Q}(Y Z) = A ∩ span_{Q}(XZ). Then dim_{Q}(X/Y Z) = dim_{Q}(X/B). From the definition of δ it follows that δ(X/Y Z) ≥ δ(X/B) ≥ 0. Hence δ(XY/Z) = δ(X/Y Z)+δ(Y/Z) ≥ 0. At last notice that δ(X/Z) = δ(XY/Z) by Lemma 2.3 (ii).

(ii) For any X ⊆ B we want to show δ(X/AY) ≥ 0. It is given by

δ(X/AY) = δ(XY/A) − δ(Y/A) = δ(XY/A) ≥ 0,

since A ≤ B. □

Notation Let A ∈ subE and X ⊆ A finite. Denote

∂_{A}(X) = min{δ(X′) : X ⊆ X′ ⊆ A}.

Lemma 2.5 Let A ∈ subE and X ⊆ A finite. Choose X′ ⊆ A finite such that

δ(X′) = ∂_{A}(X).

Then X′ ≤ A.

Proof Immediately from the definition.□

Lemma 2.6 Let A, B ∈ subE, A ≤ B and X a finite subset of A. Then

∂_{A}(X) = ∂_{B}(X).

Proof Let a finite Y ⊆ B be such that

δ(XY) = ∂_{B}(X).

Let Y_{0} be a Q-linear base of Y over A and X_{1} ⊆ A a finite superset of X. Then dim_{Q}(Y_{0}/A) = dim_{Q}(Y_{0}/X_{1}), tr.d.(Y_{0}/A) ≤ tr.d.(Y_{0}/X_{1}), and tr.d.(exY_{0}/exA) ≤ tr.d.(exY_{0}/exX_{1}). It follows

δ(Y_{0}/X_{1}) ≥ δ(Y_{0}/A) ≥ 0.

Let X_{1} be a finite subset of A such that X ⊆ X_{1} and

span_{Q}(X_{1}) = span_{Q}(XY) ∩ A.

Then

span_{Q}(XY) = span_{Q}(X_{1}Y_{0}).

Hence

δ(XY) = δ(X_{1}Y_{0}) = δ(X_{1}) + δ(Y_{0}/X_{1}).

By the above proved δ(XY) ≥ δ(X_{1}) and by the definition δ(X_{1}) ≥ ∂_{A}(X). Thus ∂_{B}(X) ≥ ∂_{A}(X), and the converse holds by the definition.□

Lemma 2.7 Suppose A ∈ subS, A′ ∈ subE, A′ = span_{Q}(AX), and δ(X′/A) ≥ 0 for all finite X′ ⊆ span_{Q}X. Then A′ ∈ subS and A ≤ A′.

Proof We may assume that X is Q-linearly independent over A. Let Z ⊆ A′, Z = {z_{1}, … z_{n}}, and z_{i} = x_{i} + y_{i} for some x_{i} ∈ span_{Q}(X), y_{i} ∈ A. Let {x_{1}, … x_{k}} be a Q-linear base of {x_{1}, … x_{n}}. Then, using Lemma 2.3, δ(Z) = δ(x_{1} + y_{1}, … x_{k} + y_{k}, y′_{k+1}, …, y′_{n}) for y′_{k+1}, …, y′_{n} appropriate Qlinear combinations of y_{1}, … y_{n}. Rewrite

δ(Z) = δ({x_{1} + y_{1}, … x_{k} + y_{k}}/{y′_{k+1}, …, y′_{n}}) + δ(y′_{k+1}, …, y′_{n}).

By the assumtions δ(y′_{k+1}, …, y′_{n}) ≥ 0. On the other hand δ({x_{1} + y_{1}, … x_{k} + y_{k}}/{y′_{k+1}, …, y′_{n}}) ≥ δ({x_{1}, … x_{k}}/A) ≥ 0 since tr.d.({x_{1} + y_{1}, … x_{k} + y_{k}}/{y′_{k+1}, …, y′_{n}}) ≥ tr.d.({x_{1} + y_{1}, … x_{k} + y_{k}}/A) ≥ tr.d.({x_{1}, … x_{k}}/A), tr.d.(ex{x_{1} + y_{1}, … x_{k} + y_{k}}/ex{y′_{k+1}, …, y′_{n}}) ≥ tr.d.(ex{x_{1} + y_{1}, … x_{k} + y_{k}}/exA) ≥ tr.d.(ex{x_{1}, … x_{k}}/exA) and dim_{Q}({x_{1} +y_{1}, … x_{k} +y_{k}}/{y′_{k+1}, …, y′_{n}}) = k = dim_{Q}({x_{1}, … x_{k}}/A). Thus

δ(Z) ≥ 0.

The same argument shows that

δ(Z/A) ≥ 0.

□

Definition Define for p > 0

Z[\frac{1}{p}] = \{\frac{m}{p^{n}} : m ∈ Z, n ∈ N\}.

Let A ∈ subE. A is said to be with standard kernel if

Ker = ker_{|A} = \{a ∈ A : ex(a) = 1\} = \{ \begin{matrix}π · Z & if p = 0 \\ π · Z[\frac{1}{p}] & if p > 0 \end{matrix}

for some transcendental π ∈ D. A is said to be with compact kernel if ker_{|A} is an algebraically compact additive group. Denote

Ẑ_{(p)} = ∏_{l≠p, l prime number} Z_{l},

the direct products of additive groups of l-adic integers. These groups are algebraically compact (see [F], Ch. 7). A is said to be with a full kernel if for ker = ker_{|A} = {a ∈ A : ex(a) = 1} the group A/ ker is isomorphic to the multiplicative subgroup of an algebraically closed field containing all the torsion points.

Proposition 1 For any characteristic p

(i) there is an A ∈ subS_{p} with standard full kernel; (ii) there is an A ∈ subS_{p} with compact full kernel.

Proof (i) Take an additive subgroup A = π ·Q ⊆ D for Q the prime subfield of D and π a transcendental element in D. Define H = A/Ker for Ker the standard kernel with generator π defined above. Then H considered as multiplicative group is characterized by the property that it is a torsion group such that any equations of the form x^{n} = h has for any h exactly n solutions when n is prime to p and only one when n = p^{k}. In other words H is isomorphic to the torsion subgroup of an algebraically closed field R of characteristic p. Define ex as the canonical homomorphism A → R^{×} corresponding to this isomorphism. Since π is transcendental, δ(X) = 0 for any finite X ⊆ A. (ii) In a large field D of characteristic 0 consider an additive divisible subgroup A of power continuum and such that the generators of A as a vector space over Q are algebraically independent. Such a group is isomorphic to to the divisible hull of Ẑ(p) and any algebraic dependence between elements of A is a linear dependence over Q. Fix K ⊆ A the corresponding subgroup isomorphic to Ẑ(p). Define H = A/K. Then H is a torsion group such that any equations of the form x^{n} = h has for any h exactly n solutions when n is prime to p and only one when n = p^{k}, i.e. H is isomorphic to the torsion subgroup of an algebraically closed field R of characteristic p. Define ex as the canonical homomorphism A → R^{×} corresponding to this isomorphism. Since the generators of A are algebraically independent, tr.d.(X) = dim_{Q}(X) for any finite X ⊆ A. Thus δ(X) = 0 for any finite X ⊆ A. □

Lemma 2.8 Suppose A ∈ subS and A is with full kernel. Then there is D ∈ S and an embedding of A into D such that A ≤ D and ker_{|D} = ker_{|A} .

Proof Choose a field D of characteristic zero and an algebraically closed field R such that A ⊆ D, A/E ⊆ R^{×} and tr.d.(D/A) = tr.d.(R/exA) ≥ ℵ_{0}. We want to define ex : D → R^{×} extending ex_{A} so that D ∈ S. Denote A_{0} = A, ex_{0} = ex_{A} and H_{0} = ex_{0}(A_{0}). Proceed by induction defining A_{α}, H_{α} and an endomorphism

ex_{α} : A_{α} → H_{α}

by choosing on the even steps: the first element a ∈ D \A_{α} and define ex_{α+1}(a) to be any element in R^{×} \ acl(H_{α}). Put A_{α+1} = A_{α} + Q · a and extend ex_{α+1} to A_{α+1} as a group homomorphism. Put H_{α+1} = ex_{α+1}(A_{α+1}). on the odd steps: the first element h ∈ R^{×} \ H_{α} and define a to be any element in D \ acl(A_{α}) and ex_{α+1}(a) = h. Put A_{α+1} = A_{α} + Q · a and extend ex_{α+1} to A_{α+1} as a group homomorphism. Put H_{α+1} = ex_{α+1}(A_{α+1}). On both even and odd steps it follows from Lemma 2.7 that A_{α+1} ∈ subS and A_{α} ≤ A_{α+1}. Also,

ker_{|A_{α+1}} = ker_{|A_{α}}

since if ex(qa + a′) = 1 for some rational q = \frac{m}{n} and a′ ∈ A_{α} then h^{m} = g^{n} for h = ex(a), g = ex(a′) ∈ H_{α}. Since A_{α} is with full kernel, H_{α} contains a root of degree m of g^{n},and h ∉ H_{α} only q = 0 is possible. □

Notation Let K be a full kernel for some D in S. Denote S(K) the class of structures with kernel K.

3 Exponentially-algebraically closed structures of S(K)

In this section we consider the class of structures with a given full kernel K. We extend the language L_{p} by naming all elements of K, thus when we say that an algebraic variety V is defined over some C ⊆ D, we mean the parameters in the definition of V are from span_{Q}(C) + K. We start by giving some basic definitions and notations.

Definition For C ⊆ D and algebraic varieties V ⊆ D^{n}, W ⊆ R^{n} we say the pair (V, W) is definable over C meaning V is definable over span_{Q}(C) and W over ex(span_{Q}(C)). If the varieties are irreducible over the corresponding sets, then the pair is said to be irreducible over C.

V is said to be free of additive dependencies over C if no ā ∈ V generic over C satisfies m_{1} · a_{1} + … + m_{n} · a_{n} = c for some c ∈ acl(C) and non-zero tuple of integers m_{1}, …, m_{n}. W is said to be free of multiplicative dependencies over C if no b̄ ∈ W generic over ex(C) satisfies a_{1}^{m_{1}} · . . . · a_{n}^{m_{n}} = r for some r ∈ acl(ex(C)). A pair (V, W) is said to be a free pair if both V is free of additive dependencies and W is free of multiplicative dependencies.

Let W ⊆ R^{n} be an algebraic variety defined and irreducible over some ex(C) for some C = span_{Q}(C) ⊆ D. With any such W we associate a sequence \{W^{\frac{1}{l}} : l ∈ N\} of algebraic varieties which are definable and irreducible over ex(C) and satisfy the following: W^{1} = W and for any l, m ∈ N the mapping

[m] : ⟨y_{1}, … y_{n}⟩ ↦ ⟨y_{1}^{m}, … y_{n}^{m}⟩

maps W^{\frac{1}{lm}} onto W^{\frac{1}{l}} . Such a sequence is said to be the sequence associated with W over C. Also with any ⟨y_{1}, … y_{n}⟩ ∈ W as above we associate a sequence

\{⟨y_{1}, … y_{n}⟩^{\frac{1}{l}} : l ∈ N\}

such that for any l, m ∈ N the mapping

[m] : ⟨y_{1}, … y_{n}⟩ ↦ ⟨y_{1}^{m}, … y_{n}^{m}⟩

maps ⟨y_{1}, … y_{n}⟩^{\frac{1}{lm}} onto ⟨y_{1}, … y_{n}⟩^{\frac{1}{l}} . Such a sequence is said to be associated with ⟨y_{1}, … y_{n}⟩.

Let (V, W), (V′, W′) be pairs over C, (V, W) irreducible, V′ ⊆ V, W′ ⊆ W, \{W^{\frac{1}{l}} : l ∈ N\} a sequence associated with W. Then the triple

τ = (V \ V′, W \ W′, \{W^{\frac{1}{l}} : l ∈ N\})

is said to be an almost finite [quantifier-free] type over C. A pair of the form (V \ V′, W \ W′) is said to be a finite type over C. A tuple ⟨a_{1}, … a_{n}⟩ ∈ D^{n} is said to be realizing type τ if ⟨a_{1}, … a_{n}⟩ ∈ V \ V′, ex⟨a_{1}, … a_{n}⟩ ∈ W \ W′ and ex\frac{1}{l}⟨a_{1}, … a_{n}⟩ ∈ W^{\frac{1}{l}} for all l ∈ N.

A pair (V, W) is said to be a normal pair over C if in some extensions of the fields there are ⟨a_{1}, …, a_{n}⟩ ∈ V and ⟨b_{1}, …, b_{n}⟩ ∈ W such that for any k ≤ n independent integer vectors m_{i} = ⟨m_{i,1}, … m_{i,n}⟩, i = 1, …, k, and

a′_{i} = m_{i,1}a_{1} + … + m_{i,n}a_{n}, b′_{i} = b_{1}^{m_{i,1}} · . . . · b_{n}^{m_{i,n}}

it holds

tr.d.(⟨a′_{1}, …, a′_{k}⟩/C) + tr.d.(⟨b′_{1}, …, b′_{k}⟩/ex(C)) ≥ k.

Equivalently, for the varieties

V′_{1,...,k} = locus_{C}(a′_{1}, …, a′_{k}) and W′_{1,...,k} = locus_{ex(C)}(b′_{1}, …, b′_{k})

it must hold

dim V′_{1,...,k} + dim W′_{1,...,k} ≥ k.

Notice that in case

⟨a′_{1}, …, a′_{k}⟩ = ⟨a_{i_{1}}, …, a_{i_{k}}⟩ and ⟨b′_{1}, …, b′_{k}⟩ = ⟨b_{i_{1}}, …, b_{i_{k}}⟩

the varieties V′_{1,...,k} and W′_{1,...,k} are just the projections V_{i_{1},...,i_{k}} and W_{i_{1},...,i_{k}} of the initial varieties on the (i_{1}, …, i_{k})-subspaces.

Definition A structure D in S(K) is said to be S(K)-exponentially-algebraically closed (e.a.c.) if for any D′ ∈ S(K), such that D ≤ D′, any almost finite quantifier-free type over D which is realized in D′ has a realization in D. The class of S(K)-exponentially-algebraically closed structures is denoted SC(K). We omit K when the kernel is fixed.

We continue by studying properties of associated sequences. A sequence associated with ȳ is not uniquely determined; for ȳ^{\frac{1}{l}} there are l^{n} possible values. Evidentely, one can get all the values multiplying a value ȳ^{\frac{1}{l}} by all ξ̄ = ⟨ξ_{1}, …, ξ_{n}⟩, where ξ_{i}’s are roots of unity of degree l. It follows

Lemma 3.1 Let ȳ ∈ W and \{ȳ^{\frac{1}{l}} ∈ l ∈ N\}, \{W^{\frac{1}{l}} : l ∈ N\} be sequences associated with ȳ and W correspondingly. Then there is a sequence {ξ̄(l) : l ∈ N} of roots of unity of power l, such that

ξ̄(l) · ȳ^{\frac{1}{l}} ∈ W^{\frac{1}{l}}

for all l ∈ N and ξ(l · m)^{m} = ξ(l) for all m ∈ N.

Lemma 3.2 Assume D ∈ subS is with a compact kernel, W a nonempty algebraic subvariety of R^{n} and \{W^{\frac{1}{l}} : l ∈ N\} a sequence associated with W. Then there is x̄ ∈ D^{n} such that

ex(\frac{1}{l} · x̄) ∈ W^{\frac{1}{l}}

for all l ∈ N.

Proof Take any ȳ ∈ W and v̄ ∈ D^{n} such that ex(v̄) = ȳ. Evidentely, \{ex(\frac{1}{l} · v̄) : l ∈ N\} is a sequence associated with ȳ. By Lemma 3.1 there is a sequence \{ξ̄(l) : l ∈ N\} such that ξ̄(l) · ex(\frac{1}{l} · v̄) ∈ W^{\frac{1}{l}} for all l ∈ N. Let ᾱ(l) ∈ D^{n} be such that ex(ᾱ(l)) = ξ̄(l) for all l ∈ N. Consider a system of equations in the additive language in variables χ̄, z̄(l) (l ∈ N) running in ker^{n} :

{χ̄ = l · ᾱ(l) + l · z̄(l) : l ∈ N}.

Any finite subsystem

{χ̄ = l · ᾱ(l) + l · z̄(l) : l ≤ m}

of the system has a solution χ̄ = m!ᾱ(m!). Indeed,

ex(\frac{1}{l} · χ̄) = ξ̄(m!)^{\frac{m!}{l}} = ξ̄(l) = ex(α(l)),

hence χ̄ − l · ᾱ(l) ∈ l · ker . Thus the whole system has a solution χ̄ ∈ ker^{n} . Put x̄ = χ̄ + v̄.□

If the number of the varieties W^{\frac{1}{l}} for a given W grows with l, then by the symmetry coming from the action of Galois group (multiplication by roots of unity of degree l), it follows that the number of the sequences associated with W is 2^{ℵ_{0}}. This is the case when W is not free of multiplicative dependence. Indeed, if y_{1}^{m_{i,1}} · . . . · y_{1}^{m_{i,n}} = c holds on W then for W^{\frac{1}{l}} holds y^{m_{i,1}} · . . . · y^{m_{i,n}} = c^{\frac{1}{l}} with exactly l different choices for c^{\frac{1}{l}} . Nevertheless, in case W is free of multiplicative dependencies F.Voloch answered our question by proving

Proposition 2 [F.Voloch] Let W be an irredicible algebraic variety such that W is not contained in a coset of any proper torus. Then there exist a number c such that the number of irreducible components of W^{\frac{1}{l}} is bounded by c independently on l.

Proof First we reduce the proof to the case when W is a curve by cutting W by a generic linear subspace of appropriate dimension. Denote K = R(W) the field of rational functions on the curve W defined over R. Let x_{1}, …, x_{n} ∈ K^{×} be the coordinate functions, which are multiplicatively independent over R^{×} by assumptions. Claim. Let H be the group generated by x_{1}, …, x_{n} in K^{∗}. Then there is d > 0, such that #H/H ∩ (K^{∗})^{l} ≥ dl^{n} for any l ≥ 1. Indeed, by hypothesis x_{1}, …, x_{n} generate a free abelian group of rank n in K^{∗}/R^{∗} isomorphic to H, and so #H/H^{l} = l^{n}. So we need to show that #(H ∩ K^{l}/H^{l}) is bounded independent of l. Let D be the group of divisors of K/R and D_{S} the subgroup generated by S = {v|∃i v(x_{i}) ≠ 0}. So D_{S} is a f.g. abelian group and H embeds in D_{S}. Let

G = D_{S} ∩ (H ⊗ Q) ⊆ D_{S} ⊗ Q

which is also a lattice and H ⊆ G with G/H finite, #G/H = c. Now, if x ∈ H ∩ (K^{∗})^{l} then x = y^{l}, y ∈ K^{∗}. But (y) ∈ D_{S} and also H ⊗ Q, so (y) ∈ G, so y^{c} ∈ H, so x^{c} = y^{cl} ∈ H^{l}, hence H ∩K^{l}/H^{l} is an abelian group of exponent c generated by ≤ n elements, so #H ∩ K^{l}/H^{l} ≤ c^{n} and d = 1/c^{n}. Claim proved. Let W^{\frac{1}{l}} be an irreducible curve such that

[l](W^{\frac{1}{l}}) = W.

Then for the field R(W^{\frac{1}{l}}) of rational functions of the curve by Galois theory

|R(W^{\frac{1}{l}}) : K| = #H/H ∩ (K^{∗})^{l}.

The latter is bounded from below by the estimate dl^{n}, by the claim. On the other hand the mapping [l] on the full inverse [l]^{−1}(W) by Bezout theorem is of degree l^{n}. It follows that the number of irreducible components in the inverse image is at most 1/d = c. □

Corollary 1 If W is an algebraic variety free of multiplicative dependencies then any irreducible component of W satisfies assumptions of Proposition 2 and thus there is an l_{0} ∈ N such that sequences \{W^{\frac{1}{l}} : l ∈ N\} and \{ȳ^{\frac{1}{l}} : l ∈ N} satisfy

ȳ^{\frac{1}{l}} ∈ W^{\frac{1}{l}} for all l ∈ N iff ȳ^{\frac{1}{l}} ∈ W^{\frac{1}{l}} for all l ≤ l_{0}.

In other words, the sequence \{W^{\frac{1}{l}} : i ∈ N\} is determined by its cut of length l_{0}.

Lemma 3.3 Let W ⊆ R^{n+m} be an algebraic variety definable and irreducible over some subfield F and free of multiplicative dependencies. Let A be a torsion-free divisible subgroup of the additive group of D of rank m, ā ∈ A^{m}, W_{a} = W (ex(ā)). Then over ex(A) · F any associated sequence \{Wa^{\frac{1}{l}} : l ∈ N\} is determined by its finite cut.

Proof We may assume ā generates A over Q. Starting the proof suppose the contrary holds. Then there are infinitely many sequences associated with W_{a}. Let \{b̄^{\frac{1}{l}} : l ∈ N\} be a realization of an associated sequence \{Wa^{\frac{1}{l}} : l ∈ N\}. Then the algebraic locus of b̄^{\frac{1}{l}} for an l ∈ N over ex(A) is of the form W′(ex(ā′)) for some ā′ ∈ A^{m}, W′ ⊆ R^{n+m} over F. Since ā generates A for some k ∈ N ex(ā′) is a rational combination of ex(\frac{1}{lk} ā). Hence we may assume ā′ = \frac{1}{lk} ā in the expression above. On the other hand the locus of (n + m)- tuple b̄^{\frac{1}{lk}} ex(\frac{1}{lk} ā) over F is given by a member W^{\frac{1}{lk}} of a sequence associated with W. In other words W′(ex(ā′)) can be represented as (W^{\frac{1}{lk}} (ex(\frac{1}{lk} ā))^{k} for some k ∈ N. Suppose l ≥ l_{0}. Then W^{\frac{1}{lk}} is determined uniquely ones W^{\frac{1}{l}} and k is given. It follows

(W^{\frac{1}{lk}} (ex(\frac{1}{lk} ā))^{k} = W^{\frac{1}{l}} (ex(\frac{1}{l}ā).

Finally notice that the sequence \{W^{\frac{1}{l}} (ex(\frac{1}{l} ā) : l ∈ N\} is determined by its l_{0}-cut.□

Notation For subsets A ⊆ B ⊆ D with D ∈ S(K) the notion A ≤ B is applied in the same sense as for substructures.

Now we come to study the rank notion ∂_{D} for D e.a.c..

Lemma 3.4 For D ∈ SC, given finite A ⊆ D and D′ ∈ S(K) such that D ≤ D′,

∂_{D}(A) = ∂_{D′}(A).

Proof Immediate from the definitions.□

By Lemma above we may omit D when writing ∂_{D}.

Lemma 3.5 For any finite A ⊆ D and any a, b ∈ D

(i) ∂(A) ≤ ∂(aA) ≤ ∂(A) + 1; (ii) ∂(abA) = ∂(aA) = ∂(A) implies ∂(bA) = ∂(A); (iii) ∂(abA) = ∂(aA)&∂(A) < ∂(bA) implies ∂(abA) = ∂(bA); (iv) ∂(aA) = ∂(A) = ∂(bA) implies ∂(abA) = ∂(A);

(v) ∂(aA) = ∂(A) implies ∂(bA) = ∂(abA).

Proof (i) follows immediately from the definitions of δ and d. (ii) and (iii) are immediate from (i).

(iv) Let B′ ⊇ aA, B′′ ⊇ bA be such that δ(B′) = ∂(aA) and δ(B′′) = ∂(bA). Denote B = B′ ∩ B′′. Notice that δ(B′ ∪ B′′) ≤ δ(B′′). Indeed by Lemma 2.3(iii) δ(B′ ∪ B′′) = δ(B′/B′′) + δ(B′′) = = [tr.d.(B′/B′′) + tr.d.(ex(B′)/ex(B′′)) − dim_{Q}(B′/B′′)] + δ(B′′). By the modularity of linear dimension dim_{Q}(B′/B′′) = dim_{Q}(B′/B). Also, by properties of algebraic dependence tr.d.(B′/B′′) ≤ tr.d.(B′/B) and tr.d.(ex(B′)/ex(B′′)) ≤ tr.d.(ex(B′)/ex(B)) Hence δ(B′/B′′) ≤ δ(B′/B). The latter is less or equal to zero by the choice of A, B′ and B. Now, since abA ⊆ B′∪B′′ and δ(B′∪B′′) ≤ δ(B′′) = ∂(A), we have ∂(abA) = ∂(A).

(v) is immediate from (iv).□

Corollary 2 The operator

A ↦ cl(A) = {b : ∂(Ab) = ∂(A)}

in D is a closure operator. cl(A) is said to be the ∂-closure of A.

Definition Let C ⊆ D and ā ∈ D^{n} be given. The ex-locus of ā over C is given as

(V, W, \{W^{\frac{1}{l}} : l ∈ N\}),

where V ⊆ D^{n} is the minimal algebraic variety over C containing ā and W^{\frac{1}{l}} ⊆ R^{n} the minimal algebraic variety over ex(C) containing ex(\frac{1}{l} ā). We will call (V, W) the [incomplete] ex-locus.

Lemma 3.6 Let C, A ⊆ D finite, C ≤ A, c̄ be the string of all elements of C and ā be a string of elements of A forming a Q-linear base of A over C. Then the incomplete ex-locus (V, W) of Ā over C is normal over C.

Proof Let ⟨b_{1}, …, b_{n}⟩ in the definition of normality be ex(ā). Then the inequality in the definition follows from C ≤ A.□

Lemma 3.7 Let C contain ker, τ = (V \ V′, W \ W′, \{W^{\frac{1}{l}} : l ∈ N\}) be an almost finite type over C and assume the pair (V, W) is normal over C and free. Then there is ā in D realizing τ. Moreover in some extension D′ ≥ D ā can be chosen generic in V (of maximal tr.d.) over C and ex(ā) generic in W over ex(C).

Proof Under the assumptions we may assume that τ = (V \ V′, W \ W′). Take ā in some extension of field D to be generic in V over D and b̄ in some extension of R generic in W over R. Choose a sequence \{b̄^{\frac{1}{l}} : l ∈ N\} associated with b̄. Define ex on A = D + span_{Q}(a_{1}, … a_{n}) as:

ex(∑_{i} \frac{m_{i}}{l} a_{i} + d) = ∏_{i}(b_{i}^{\frac{1}{l}})^{m_{i}} · ex(d)

for any integers m_{i}, l ≠ 0 and element d ∈ D. The definition is consistent since V is free of additive dependencies. Evidently the formula defines a homomorphism. The kernel of the homomorphism coincides with that of ex on D, since W has no multiplicative dependencies. Thus A ∈ subE. Notice that by the normality for any independent integer vectors m_{i} = ⟨m_{i,1}, … m_{i,n}⟩, i = 1, …, k, it holds δ(m_{1}ā, …, m_{k}ā/D) ≥ 0. Thus D ⊆ A satisfy the assumptions of Lemma 2.7 and hence A ∈ subS, D ≤ A. By the choice ā realizes τ. Since D ∈ SC there is a realization of the type in D. □

Theorem 1 A structure D ∈ S(K) is in SC(K) iff for any normal free pair (V, W) over D there is a realization of the type (V, W) in D.

First we prove

Lemma 3.8 For any free normal pair (V, W) (V ⊆ D^{n}, W ⊆ R^{n}) and V′ ⊆ V, W′ ⊆ W there is a free normal pair (V^{∗}, W^{∗}) (V^{∗} ⊆ D^{n+m}, W ⊆ R^{n+m}) such that ⟨a_{1}, …, a_{n}⟩ ∈ D^{n} realizes (V \ V′, W \ W′) iff there is a_{n+1}, …, a_{n+m} ∈ D^{m} such that ⟨a_{1}, …, a_{n}, a_{n+1}, …, a_{n+m}⟩ realizes (V^{∗}, W^{∗}).

Let f(x_{1}, …, x_{n}) be a polynomial in AnnV′\AnnV. Choose a natural number k such that there is no rational function r such that r^{k} ≡ f on a component of V. Add new variable x_{n+1} together with new identity f(x_{1}, …, x_{n}) = x_{n+1}^{k}, which defines together with AnnV new algebraic variety V^{f}. Notice that the projection of V^{f} onto the first n coordinates is V \ V_{f} where V_{f} is the subvariety of V defined by f = 0. Also, V^{∗} is free of additive dependencies since otherwise ^{k} \sqrt{f} = x_{n+1} is a rational function on V. Using this method finitely many times we come to V^{∗} with the projection V \ V′. Let g(y_{1}, …, y_{n}) be a polynomial in AnnW′ \ AnnW. We may assume that there is no integer k and m_{1}, …, m_{n} not all zero such that

g(y_{1}, …, y_{n})^{k} · y_{1}^{m_{1}} · . . . · y_{n}^{m_{n}}

is constant on a component of W since otherwise g(y_{1}, …, y_{n}) is nonvanishing on the component. Add new variable y_{n+1} together with new identity g(x_{1}, …, x_{n}) = g_{n+1}, and the resulting variety W^{g} projects onto the first n coordinates as W \ W_{g} where W_{g} is the subvariety of W defined by g = 0. By our assumption W^{g} is free of multiplicative dependencies and we can come to W^{∗} as above. The normality of the pair is evident from the construction.□

Proof of Theorem. The left-to-right implication follows from Lemma 3.7. To get the inverse assume ā is in some D′ ≥ D and we need to realize an almost finite type (V \ V′, W \ W′, \{W^{\frac{1}{l}} : l ∈ N\}) where τ = (V, W, \{W^{\frac{1}{l}} : l ∈ N\}) is the ex-locus of ā over D. It is enough to solve the problem for Q-linear base ā_{0} of ā over D, so we may assume ā is Q-linearly independent over D. Thus (V, W) is a free normal pair, and so we may assume τ is a finite type. By Lemma 3.8 we reduce the type (V \ V′, W \ W′) to a type of the form (V, W) and (V, W) normal and free. By the assumptions of Theorem the type is realized in D. □

Corollary 3 The structure (C, exp, C) on complex numbers is in SC iff it satisfies the weak Schanuel conjecture and for any normal free pair (V, W) over C there is a realization of the type (V, W) in C.

4 Atomic and prime structures

In this section D is always exponentially-algebraically closed with a fixed kernel. The cardinality of the kernel ker is denoted κ. Definition Let C_{0} ⊆ C ≤ D and V ⊆ D^{n} algebraic variety over C_{0}, W ⊆ R^{n} variety over ex(C_{0}), V and W irreducible over corresponding sets. The type τ = (V, W, \{W^{\frac{1}{l}} : l ∈ N\}) is said to be a q-atom over C defined over C_{0}, if there is a realization ā of τ in some D′ ≥ D and for every ā realizing τ in a D′ ≥ D, ā is generic over C realization of τ, i.e. ā generic in V over C and ex(ā) generic in W over ex(C). Any realization ā in D of a type as above is said to be q-atomic over C and C_{0} is said to be isolating ā over C.

Notation For C ⊆ D denote ecl(C) the field obtained by the following process in ω steps: C^{0} = acl(C), C^{(2n+1)} = ln(acl(ex(C^{(2n)}))), C^{(2n+2)} = acl(C^{(2n+1)}),

ecl(C) = ⋃_{n<ω} C^{(n)}.

Remark. If X ⊆ C^{(m)}, then by the definition

δ(X/C^{(m−1)}) ≤ 0

(here C^{(−1)} = C). If C^{(m−1)} ≤ D, then the equality holds.

Lemma 4.1 (i) For any n ∈ N, C ≤ D, X ⊆ C^{(n)} there is X ⊆ X′ ⊆ C^{(n)}, such that δ(X′/C) = 0; (ii) If C ≤ D, then C^{(n)} ≤ D for all n ∈ N and ecl(C) ≤ D; (iii) If X ⊆ ecl(C) and C ≤ CX ≤ D, then δ(X/C) = 0.

Proof (i) The statement is obvious for n = 0. So, we assume n > 0. Let X ⊆ C^{(n)} and Y ⊆ C^{(n−1)} be such that

δ(X/C^{(n−1)}) = δ(X/CY) = 0.

By induction hypothesis we may assume δ(Y/C) = 0. Since

δ(XY/C) = δ(X/CY) + δ(Y/C)

we get the required by putting X′ = XY. (ii) Let X ⊆ D be finite. We want to show that δ(X/C^{(n)}) ≥ 0. By definition δ(X/C^{(n−1)}) = δ(X/CY) for some finite Y ⊆ C^{(n)}. By (i) we may choose Y so that δ(Y/C) = 0. Then δ(X/CY) = δ(XY/C) ≥ 0, since C ≤ D. (iii) follows from (i). □

Lemma 4.2 Let C ≤ D, |ā| ⊆ ecl(C) and Cā ≤ D. Then δ(ā/C) = 0 and ā is q-atomic over C.

Proof δ(ā/C) = 0 by Lemma 4.1(iii). To prove the rest of Lemma we first prove Claim.

(i) If |ā| ⊆ C^{(2n+2)}, then there is b̄ in C^{(2n+1)} ∩ span_{Q}(Cā), such that ā ∈ acl(Cb̄) and δ(b̄/C) = 0. (ii) If |ā| ⊆ C^{(2n+1)}, then there is b̄ in C^{(2n)} ∩ span_{Q}(Cā), such that ex(ā) ∈ acl(ex(Cb̄)) and δ(b̄/C) = 0. Proof of (i). Choose finite b̄ to be a Q-basis of C^{(2n+1)} ∩ span_{Q}(Cā) Then dim_{Q}(ā/Cb̄) = dim_{Q}(ā/C^{(2n+1)}), since if a Q-linear combination u of ā belongs to C^{(2n+1)}, then u ∈ C^{(2n+1)} ∩ span_{Q}(Cā). It follows that δ(ā/Cb̄) = tr.d.(ā/Cb̄) ≥ 0, since tr.d.(ex(ā)/ex(Cb̄)) ≤ dim_{Q}(ā/Cb̄) and tr.d.(ex(ā)/ex(Cb̄)) ≥ tr.d.(ex(ā)/ex(C^{(2n+1)})) = dim_{Q}(ā/C^{(2n+1)}) = dim_{Q}(ā/Cb̄). On the other hand, δ(ā/Cb̄) = δ(āb̄/C) − δ(b̄/C) = 0 = δ(b̄/C), since δ(āb̄/C) = δ(ā/C) = 0 and δ(b̄/C) ≥ 0 by C ≤ D. Hence tr.d.(ā/Cb̄) = 0, which means |ā| ⊆ acl(Cb̄). Proof of (ii) is symmetric. Now we continue the proof of Lemma. By induction on m we prove that if |ā| ⊆ C^{(m)}, then ā is q-atomic over C. For m = 0 ā ∈ acl(C) and the statement is evident. If |ā| ⊆ C^{(m)} for m > 1, then by the Claim there is b̄ in C^{(m−1)} ∩ span_{Q}(Cā) such that ā ∈ acl(Cb̄) or ex(ā) ∈ acl(ex(Cb̄)). Let ā ∈ acl(Cb̄) ⊆ (Cb̄)^{(0)}, more precisely for some C_{0} ⊆ C finite ā is isolated by C_{0}b̄ over Cb̄, b̄ ∈ span_{Q}(C_{0}ā), Cb̄ ≤ D and C_{0} isolates b̄ over C. Let τ = (V, W, \{W^{\frac{1}{l}} : l ∈ N\}) be a quantifier-free type over C_{0} satisfied by āb̄, such that

τ(b̄) = (V (b̄), W (ex(b̄)), \{W^{\frac{1}{l}} (ex(b̄)) : l ∈ N\})

is a q-atom over Cb̄ and τ implies the q-atomic type τ_{0} of b̄ over C. Then x̄ realizes τ(b̄) implies x̄ ∈ acl(C_{0}b̄). Since b̄ ∈ span_{Q}(C_{0}ā), there is a Q-linear combination with parameters in C_{0} q̄· x̄, such that b̄ = q̄· ā. Then the type saying that x̄ȳ satisfies τ and ȳ = q̄ · x̄ is a q-atom of ā over C. Indeed, if ā′ satisfies the type and b̄′ = q̄· ā′, then ā′b̄′ satisfies τ. Then b̄′ realizes τ_{0}, thus b̄′ is generic in τ_{0} over C. It follows that Cb̄′ is isomorphic to Cb̄. Then τ(b̄′) is a q-atom over Cb̄′ satisfied by ā′. It implies that ā′ is generic in τ(b̄′) over Cb̄′, hence ā′b̄′ is generic in τ over C. □

Lemma 4.3 If C ≤ D, then For any X ⊆ C^{(2n+2)}

tr.d.(ex(X)/ex(C^{(2n+1)})) = dim_{Q}(X/C^{(2n+1)});

For any X ⊆ C^{(2n+1)}

tr.d.(X/C^{(2n)}) = dim_{Q}(X/C^{(2n)}).

Proof By definition of δ, using Lemma 4.1(ii).□

Proposition 3 Assume C ≤ A ≤ D and C ≤ B ≤ D are isomorphic as substructures of D over C, ecl(C) = C, by an isomorphism φ : A → B. Then φ can be extended to an isomorphism φ : ecl(A) → ecl(B), provided

(i) the kernel in D is compact or (ii) A \ C is finite, C countable.

Proof First we show that Claim 1. If φ extends to A → B then it extends to acl(A) → acl(B). For this take any φ preserving the field structure and notice that for any Qlinearly independent over A elements a_{1}, …, a_{n} ∈ acl(A) the algebraic type of ex(a_{1}), …, ex(a_{n}) over ex(A) is uniquely determined by the requirement of algebraic independence. Claim 2. If A \ C is finite then φ extends to Ax̄ for any x̄ in ln(acl(ex(A))). Indeed, let ex(x̄)) be in acl(ex(A)). Let WA^{\frac{1}{l}} for l ∈ N be the algebraic loci of ex(\frac{1}{l} ·v̄) over ex(A). W_{A} can be represented as W (ex(ā)) for some finite string ā from A and W an irreducible variety over ex(C), which is algebraically closed. By Lemma 3.3 the sequence \{WA^{\frac{1}{l}} : l ∈ N\} is determined by WA^{\frac{1}{l}} for some l ∈ N. Take ȳ in D such that

ex(\frac{1}{l} · ȳ) ∈ WB^{\frac{1}{l}}

where WB^{\frac{1}{l}} is obtained by applying φ to WA^{\frac{1}{l}} . Since A, B ≤ D and

tr.d.(ex(x̄)/ex(A)) = 0 = tr.d.(ex(ȳ)/ex(B)),

necessarily

tr.d.(x̄/A) = n = tr.d.(ȳ/B),

i.e. both strings are algebraically independent over the corresponding sets. Hence one can extend φ(x̄) = ȳ. Claim 3. If A \ C is finite and φ : A → B is an isomorphism over C, x ∈ ecl(A), then there is x̄ in D extending x, such that Aā ≤ D and there is an isomorphism Ax̄ → ecl(B) extending φ. Proof of Claim 3. Take for x̄ a finite string from ecl(A) extending x such that Ax̄ ≤ D. By Lemma 4.2 x̄ up to the ordering can be represented as x̄_{0} … x̄_{k} such that x̄_{i+1} is in acl(Ax̄_{0} … x̄_{i}) or ex(x̄_{i+1}) is in acl(ex(Ax̄_{0} … x̄_{i})) depending on i odd or even. Use Claims 1 and 2 and induction on k to extend φ to Ax̄. If A \ C is finite and C countable, then ecl(A) and ecl(B) can be enumerated and by back-and-force arguments using Claim 3 one gets the isomorphism. Claim 4. If the kernel is compact then φ extends to ln(acl(ex(A))) → ln(acl(ex(B)). Indeed, let ex(x̄) be in acl(ex(A)), WA^{\frac{1}{l}} for l ∈ N be the algebraic loci of ex(\frac{1}{l} · x̄) over ex(A). By Lemma 3.2 there is ȳ such that

ex(\frac{1}{l} · x̄) ∈ WB^{\frac{1}{l}}

for all l ∈ N, where WB^{\frac{1}{l}} are obtained by applying φ to WA^{\frac{1}{l}} . Since A, B ≤ D and

tr.d.(ex(x̄)/ex(A)) = 0 = tr.d.(ex(ȳ)/ex(B)),

necessarily

tr.d.(x̄/A) = n = tr.d.(ȳ/B),

i.e. both strings are algebraically independent over the corresponding sets. Hence one can extend φ(x̄) = ȳ. Claims 3 and 4 combined with Claim 1 prove the both cases of Proposition.□

Corollary 4 Suppose D is with a standard kernel, A ≤ D and B ≤ D are ∂-independent sets, and φ : A → B is a bijection. Then φ can be extended to an isomorphism φ : ecl(A) → ecl(B).

Lemma 4.4 Let (V, W) be a free normal pair in D^{n} over some C_{0} ≤ D such that dim V +dim W −n = d > 0 and dim V > 1. Then for any c_{1}, …, c_{n} ∈ D algebraically independent over C_{0} there is V′ ⊆ V defined over C_{0}c̄ (c̄ = ⟨c_{1}, …, c_{n}⟩) such that (V′, W) is a normal free pair and dim V′+dim W −n = d − 1. If ā realizes (V′, W) then δ(c̄/C_{0}ā) < n.

Proof Let V′ be an irreducible over C_{0}c̄ component of the variety

V ∩ {x̄ : c_{1} · x_{1} + … + c_{n} · x_{n} = 1}.

Let ā be a generic over C_{0}c̄ point in V′. Then ā is generic in V over C_{0} and tr.d.(ā/C_{0}c̄) = dim V − 1, tr.d.(c̄/C_{0}ā) = n − 1. In particular, it follows that δ(c̄/C_{0}ā) < n. Claim 1. For any k ≤ n and any distinct i_{1}, …, i_{k} ∈ {1, …, n} tr.d.(a_{i_{1}}, …, a_{i_{k}}/C_{0}) > tr.d.(a_{i_{1}}, …, a_{i_{k}}/C_{0}c̄) implies tr.d.(a_{i_{1}}, …, a_{i_{k}}/C_{0}) = tr.d.(ā/C_{0}). Proof Let U ⊆ R^{n} be the minimal algebraic variety over C_{0} ∪ {a_{i_{1}}, …, a_{i_{k}}} containing c̄. Suppose the inequality holds. Then dim U < n. Also, U must contain the hyperplane H_{a} defined by z̄ā = 1, for otherwise tr.d.(c̄/C_{0}ā) < n − 1. Hence H_{a} is a component of U. But then ā ∈ acl(C_{0} ∪ {a_{i_{1}}, …, a_{i_{k}}}), since ā is the only vector satisfying z̄ā = 1 for all z̄ ∈ H_{a}. It follows that tr.d.(a_{i_{1}}, …, a_{i_{k}}/C_{0}) = tr.d.(ā/C_{0}). Claim 2. For any non-zero integer n-tuple m̄ = ⟨m_{1}, …, m_{n}⟩ the element m̄ · ā = m_{1} · a_{1} + … + m_{n} · a_{n} is not in acl(C_{0}c̄). Proof Notice first that a ∉ acl(C_{0}), since V is free of additive dependencies. Thus, if a ∈ acl(C_{0}c̄), then tr.d.(c̄/C_{0}a) < n. By the argument from the proof of Claim 1 it would follow that |ā| ⊆ acl(C_{0}a). This contradicts the fact that dim V > 1.

Proof of Lemma. It follows from Claim 2, that V′ is free of additive dependencies. Thus the pair (V′, W) is free. To prove the normality of the pair consider for any distinct i_{1}, …, i_{k} ∈ {1, …, n} the number

dim V′_{i_{1},...,i_{k}} + dim W_{i_{1},...,i_{k}} − k.

Suppose towards the contradiction that the number is negative. Then

dim V_{i_{1},...,i_{k}} + dim W_{i_{1},...,i_{k}} − k ≤ 0,

dim V′_{i_{1},...,i_{k}} ≥ dim V_{i_{1},...,i_{k}} − 1. It follows that dim V′_{i_{1},...,i_{k}} < dim V, since dim W ≤ dim W_{i_{1},...,i_{k}} + (n − k) and d > 0. Thus

tr.d.(a_{i_{1}}, …, a_{i_{k}}/C_{0}) < tr.d.(ā/C_{0}).

Then by Claim 1

tr.d.(a_{i_{1}}, …, a_{i_{k}}/C_{0}) = tr.d.(a_{i_{1}}, …, a_{i_{k}}/C_{0}c̄).

Thus dim V′_{i_{1},...,i_{k}} = dim V_{i_{1},...,i_{k}} and hence

dim V_{i_{1},...,i_{k}} + dim W_{i_{1},...,i_{k}} − k = dim V′_{i_{1},...,i_{k}} + dim W_{i_{1},...,i_{k}} − k < 0,

contradicting the normality of (V, W). To finish the proof of normality we need to consider also

a′_{i} = m_{i,1}a_{1} + … + m_{i,n}a_{n}, and b′_{i} = b_{1}^{m_{i,1}} · . . . · b_{n}^{m_{i,n}} i = 1, … n

where ⟨m_{i,1}, …, m_{i,n}⟩ are independent integer vectors and ⟨b_{1}, …, b_{n}⟩ a generic element of W over ex(C_{0}c̄). The corresponding inequalities for the loci over C_{0}c̄ and ex(C_{0}c̄) of the ā′ and b̄′ are proved by the same argument. □

Lemma 4.5 Let C ≤ D and V ⊆ D^{n} an algebraic variety over a finite C_{0} ⊆ C, W ⊆ R^{n} a variety over ex(C_{0}). Let ā in D realize the type τ = (V, W, \{W^{\frac{1}{l}} : l ∈ N\}) and satisfy the following minimality condition: for any ā′ ∈ D realizing τ either dim_{Q}(ā′/C) > dim_{Q}(ā/C) or dim_{Q}(ā′/C) = dim_{Q}(ā/C) and δ(ā′/C) ≥ δ(ā/C). Suppose the ex-locus τ_{0} = (V_{0}, W_{0}, \{W0^{\frac{1}{l}} : l ∈ N\}) of ā over C is defined over C_{0}. Then τ_{0} is a q-atom over C defined over C_{0}. If C = ecl(C), then δ(ā/C) = 0.

Proof Let ā′ be any realization of τ_{0} and assume towards a contradiction that ā′ is not generic over C. Then

tr.d.(ā′/C) + tr.d.(ex(ā′)/ex(C)) < tr.d.(ā/C) + tr.d.(ex(ā)/ex(C)).

By definition dim_{Q}(ā′/C) ≥ dim_{Q}(ā/C), so δ(ā′/C) < δ(ā/C), the contradiction. This proves the first part of the statement. Assume C = ecl(C) and assume towards a contradiction that δ(ā/C) = d > 0. We can reduce the problem to a Q-linear basis of ā over C, so we assume that ā = ⟨a_{1}, …, a_{n}⟩ is Q-linearly independent over C. It follows that (V_{0}, W_{0}) is free and normal. By the assumptions dim V_{0} + dim W_{0} − n = δ(ā/C) > 0. Take c_{1}, …, c_{n} ∈ C algebraically independent over C_{0} (such ones exist in C_{0}^{(1)}). By Lemma 4.4 either dim V_{0} = 1, or there is V′_{0} ⊆ V_{0} over C̄_{0}c̄ such that (V′_{0}, W_{0}) is free and normal. In the second case by Lemma 3.7 there is a generic over C_{0}c̄ realization ā′ of

(V′_{0}, W_{0}, \{W0^{\frac{1}{l}} : l ∈ N\}),

which is also generic realization of τ_{0} over C_{0}, such that

δ(ā′/C_{0}c̄) = dim V′_{0} + dim W_{0} − n < d.

The contradiction. In the first case, since d > 0, necessarily dim W_{0} = n, i.e. W_{0} = R^{n}. Choose c ∈ C \ C_{0}^{(1)} and let

W′_{0} = {⟨y_{1}, …, y_{n}⟩ : y_{1} + … + y_{n} = c}.

Since ex(c) ∉ acl(ex(C_{0})), any generic over ex(C_{0}c) realization ⟨b_{1}, …, b_{n}⟩ of W′ is generic in W over ex(C_{0}). Also, if n > 1, for any k < n and distinct i_{1}, …, i_{k} ∈ {1, …, n}

tr.d.(b_{i_{1}}, …, b_{i_{k}}/ex(C_{0}c)) = tr.d.(b_{i_{1}}, …, b_{i_{k}}/ex(C_{0})),

which yields the normality of (V_{0}, W′_{0}). The same argument shows that b_{1}^{m_{1}} · . . . · b_{n}^{m_{n}} ∉ acl(ex(C_{0}c)) for any non-zero integer n-tuple ⟨m_{1}, …, m_{n}⟩, which shows that W′_{0} is free of multiplicative dependencies. Thus (V_{0}, W′_{0}) is free. Again it gives by Lemma 3.7 as above a generic over C_{0} realization of τ_{0} over C_{0}, such that

δ(ā′/C_{0}c̄) = dim V_{0} + dim W′_{0} − n < d.

The final contradiction. □

Lemma 4.6 Let ecl(∅) ⊆ C ≤ D, C \ ecl(∅) be finite, āb̄ a tuple in D′ ≥ D, τ = (V, W, \{W^{\frac{1}{l}} : l ∈ N\}) be the ex-locus of āb̄, σ be the ex-locus of ā over C and assume δ(ā/C) = 0. Then for any ā′ realizing σ there is b̄′ in some D′′ ≥ D′ such that ā′b̄′ realize τ.

Proof We prove the statement by induction on |b̄| = n ≥ 1 assuming also that āb̄ are Q-linearly independent over C. It follows from the assumptions that (V, W) is a free and normal pair. Let n = 1. If both dim V (ā) = 1 and dim W (ex(ā)) = 1, then, by the elementary algebraic geometry, for any ā′ realizing σ dim V (ā′) = 1 and dim W (ex(ā′)) = 1. Hence V (ā′) and W (ex(ā′)) are just the whole affine lines and we can take for b′ any point in D. If dim V (ā) = 0, then necessarily dim W (ex(ā)) = 1 and again dim V (ā′) ≥ 0 and dim W (ex(ā′)) = 1. Take for b′ any point in V (ā′). If dim W (ex(ā)) = 0, then dim V (ā) = 1 and dim W (ex(ā′)) ≥ 0, dim V (ā′) = 1. Let

τ_{a} = (V (āc̄), W (ex(āc̄)), \{(W (ex(āc̄)))^{\frac{1}{l}} : l ∈ N\})

be the quantifier-free type of b̄ over Cā, where c̄ is a finite string from C and V, W are irreducible varieties over ecl(∅), ex(ecl(∅)), correspondingly. tau_{a′} is the type obtained by replacing all occurencies of ā by ā′. By Lemma 3.3 the sequence \{(W (ex(ā′c̄)))^{\frac{1}{l}} : l ∈ N\}) is determined by its cut of a length l_{0}, thus there is y in R and an associated sequence such that

y^{\frac{1}{l}} ∈ (W (ex(ā′c̄)))^{\frac{1}{l}} for all l ∈ N\}.

Take b′ ∈ D such that

ex(\frac{1}{l_{0}} · b′) = y^{\frac{1}{l_{0}}} .

Then

ex(\frac{1}{l} · b′) ∈ (W (ex(ā′c̄)))^{\frac{1}{l}} for all l ∈ N.

Since V (ā′) = D, b′ satisfies τ_{a′} and ā′b′ satisfies τ.

For n > 1 notice first that by the assumptions Cā ≤ D, thus (V (ā), W (ex(ā))) is a normal pair. To prove the statement of Lemma we consider two alternative cases: Case 1. There are distinct i_{1}, … i_{k} in {1, …, n−1} such that the projections V_{i_{1},...i_{k}}(ā) and W_{i_{1},...i_{k}}(ex(ā)) onto corresponding coordinates satisfy

dim V_{i_{1},...i_{k}}(ā) + dim W_{i_{1},...i_{k}}(ex(ā)) = k.

Let for simplicity ⟨i_{1}, … i_{k}⟩ = ⟨1, … k⟩. Then δ(⟨b_{1}, … b_{k}⟩/Cā) = 0 and one can consider ā_{+} = ā⟨b_{1}, … b_{k}⟩ in place of ā and b̄_{−} = ⟨b_{k+1}, … b_{n}⟩ in place of b̄. By induction hypothesis find type σ_{+} over C which says about any ā′_{+} that there is b̄′_{−} such that ā′_{+}b̄′_{−} satisfy τ. Case 2.

dim V_{i_{1},...i_{k}}(ā) + dim W_{i_{1},...i_{k}}(ex(ā)) > k

for any distinct i_{1}, … i_{k} ∈ {1, …, n − 1}. If the pair (V (ā), W (ex(ā)) is free then the statement follows from Lemma 3.7. Suppose W (ex(ā)) has a multiplicative dependence:

y_{1}^{m_{1}} · . . . · y_{n}^{m_{n}} = r ∈ acl(Cā)

for some m_{1}, … m_{n} ∈ Z, r ∈ R. Notice that by definitions

ex(b_{1})^{m_{1}} · . . . · ex(b_{n})^{m_{n}} = r

holds, and so

m_{1}b_{1} + . . . · m_{n}b_{n} = d

holds for some d ∈ D such that ex(d) = r. Assume m_{n} ≠ 0. Apply the induction hypothesis considering ād in place of ā and ⟨b_{1}, …, b_{n−1}⟩ in place of b̄. We have then that whenever ā′d′ satisfy the same type as ād, there is ⟨b′_{1}, …, b′_{n−1}⟩ such that ā′⟨d′, b′_{1}, …, b′_{n−1}⟩ is of the same type as ā⟨d, b_{1}, …, b_{n−1}⟩. On the other hand, considering n = 1, we proved that whenever ā′ satisfies the type of ā, there is d′ such that ā′d′ satisfy the type of ād, which completes the proof in the case. If V (ā) has an additive dependence we act symmetrycally. □

Lemma 4.7 Let ecl(∅) ⊆ C ≤ D, ā, b̄ in D, δ(ā/C) = 0 and ā q-atomic over C, b̄ q-atomic over Cā. Then āb̄ is q-atomic over C.

Proof Choose finite C_{0} ⊆ C such that C_{0} isolates ā over C and C_{0}ā isolates b̄ over Cā. Let τ = (V, W, \{W^{\frac{1}{l}} : l ∈ N\}) be the ex-locus of āb̄ over C_{0} and σ the type obtained by projecting the varieties onto ā-coordinates. Then σ is a q-atom isolating ā over C. Also, the type of b̄ over Cā is determined over C_{0}ā by τ(ā). We claim that whenever ā′b̄′ in some D′ ≥ D realizes τ, ā′b̄′ is generic over C. This would prove Lemma. Assume ā′b̄′ realizes τ. Then ā′ satisfies σ and, since σ is a q-atom over C, the quantifier free types of ā and ā′ over C coincide. Suppose towards a contradiction that b̄′ is not generic in τ(ā′) over C. Then b̄′ satisfies τ(ā′) and a pair (V′(ā′), W′(ex(ā′))) with

dim V′(ā′) < dim V (ā′) or dim W′(ex(ā′)) < dim W (ex(ā′)).

By Lemma 4.6, since τ and (V′, W′) are defined over some finite set C_{1}, C_{0} ⊆ C_{1} ⊆ C, there is b̄′′ which satisfies τ(ā) and (V′(ā), W′(ex(ā)). This contradicts the assumption that τ(ā) is a q-atom over C.

Lemma 4.8 Let ecl(∅) ⊆ C ≤ Cā ≤ D and āb̄ be q-atomic over C. Then ā is q-atomic over C and b̄ is q-atomic over Cā.

Proof Let

τ = (V, W, \{W^{\frac{1}{l}} : l ∈ N\})

be the q-atomic type of āb̄ over C defined over some finite C_{0}. Let V_{1}, W_{1} be the projections of V, W onto the coordinates corresponding to ā, ex(ā) and

σ = (V_{1}, W_{1}, \{W1^{\frac{1}{l}} : l ∈ N\}).

Suppose ā′ realizes type σ. Then, by Lemma 4.6, there is b̄′ such that ā′b̄′ realizes τ. Since τ is a q-atom over C, ā′b̄′ is generic in (V, W) over C. It follows ā′ is generic in (V_{1}, W_{1}) and b̄′ generic in (V (ā′), W (ex(ā′))). The first fact yields that ā is q-atomic over C, the second fact, after assuming ā′ = ā, yields b̄ is q-atomic over Cā.□

Definition A structure A ∈ subS is said to be q-atomic over C ≤ A if for any ā in A such that Cā ≤ A the tuple ā is q-atomic over C.

Lemma 4.9 Assume A is with full kernel, ecl(∅) ⊆ C ≤ A.

(i) If B ⊆ A finite, δ(B/C) = 0 and A is q-atomic over C, then A is q-atomic over CB. (ii) If ecl(C) ⊆ A and A is q-atomic over ecl(C), then A is q-atomic over C and for any ā in A, such that Cā ≤ A, it holds δ(ā/C) = 0.

Proof (i) Suppose CBā ≤ A. By assumtions, Bā is q-atomic over C. Then by Lemma 4.8 ā is q-atomic over CB. (ii) Suppose Cā ≤ A. Extend ā to ā′ so, that ecl(C)ā′ ≤ A. Then, by assumtions, ā′ is q-atomic over ecl(C) and by 4.5 δ(ā′/ecl(C)) = 0. Choose finite c̄ in ecl(C) such that ā′ is q-atomic over Cc̄, δ(ā′/Cc̄) = 0 and Cc̄ ≤ A. By Lemmas 4.2 and 4.7, c̄ā′ is q-atomic over C. Also

δ(c̄ā′/C) = δ(ā′/Cc̄) + δ(c̄/C) = 0.

Since c̄ā′ extends ā and Cā ≤ A, δ(ā/C) = 0. Finally, by Lemma 4.8, ā is q-atomic over C. □

Lemma 4.10 Over any C ∈ subS such that C = ecl(C) there is a q-atomic structure E(C) ∈ SC such that for any embedding C ≤ D ∈ SC there is an extension of the embedding E(C) ≤ D.

Proof E(C) will be represented as ⋃_{n} C_{n} for some

C_{0} ≤ … ≤ C_{n} ≤ … ⊆ D.

Put C_{0} = C and assume C_{n} is constructed. Let {μ_{n,α} : α < λ_{n}} be the set of all quantifier-free almost finite types over C_{n}, λ_{n} a cardinal. Put C_{n,0} = C_{n} and, when C_{n,α} is constructed, use Lemma 4.5 to find in some normal extension of C_{n,α} a q-atomic over C_{n,α} tuple ā_{n,α} realizing μ_{n,α}. The q-atomic type is realizable in every q-e.a.c. structure D, so we assume the tuple is in D. Put

C_{n,α+1} = C_{n,α} ∪ |ā_{n,α}|.

For limit ordinals σ

C_{n,σ} = ⋃_{α<σ} C_{n,α}.

Finally, C_{n+1} = C_{n,λ_{n}}. Claim. For any n and α < λ_{n}

(i) C_{n,α} is q-atomic over C_{0}; (ii) C_{n,α} ≤ D; (iii) C_{n,α} is q-atomic over C; Proof of Claim. By induction on lexicographically ordered pairs ⟨n, α⟩. For ⟨0, 0⟩ the statements follow from properties of ecl. The inductive step for limit ordinals is trivial, so we assume the Claim is true for C_{n,α} and want to prove it for C_{n,α+1}. Let ā be a tuple from C_{n,α+1}, such that C_{0}ā ≤ D. By the construction there is b̄ in C_{n,α} which isolates ā_{n,α} over C_{n,α}. When we extend b̄ the property is preserved, so we assume C_{0}b̄ ≤ D and |ā| ⊆ |b̄ā_{n,α}|. Then, by induction, b̄ is q-atomic over C_{0} and by Lemma 4.5 δ(b̄/C_{0}) = 0. Now by Lemma 4.7 b̄ā_{n,α} is q-atomic over C_{0}. Since |b̄ā_{n,α}| = |āb̄′| for some b̄′, by Lemma 4.8 ā is q-atomic over C_{0}, which proves (i). To prove (ii) we must show that δ(X/C_{n,α+1}) ≥ 0 for any finite X ⊆ D. Notice that δ(X/C_{n,α+1}) = δ(X/C_{0}Y) for large enough finite Y ⊆ C_{n,α+1}. We can choose Y = |b̄ā_{n,α}| with b̄ as above. Then Y is q-atomic over C_{0} and by Lemma 4.5 δ(Y/C_{0}) = 0. Hence δ(X/C_{0}Y) ≥ 0, by Lemma 2.4. Which finishes the proof of (ii). (iii) follows from (i) and Lemma 4.9. It follows from the claim that E(C) is q-atomic over C and E(C) is q-e.a.c. Also, E(C) ≤ D for chosen D, while the construction and the isomorphism type of E(C) is independent on D.□

Definition A structure E(C) ∈ SC is said to be [constructible] q-prime over C if

(i) C ≤ E(C), (ii) E(C) is q-atomic over ecl(C), (iii) there is a sequence {ā_{j} : j < λ} of tuples of E(C) for some limit ordinal λ, such that E(C) = ⋃{|ā_{j}| : j < λ} and for any j ≤ λ ā_{j} is a realization of a q-atom τ_{j} over ecl(C) ∪ A_{j}, where A_{j} = ⋃_{i<j} |ā_{i}|. The corresponding sequence {(ā_{j}, τ_{j}) : j < λ} is said to be a q-construction. We assume |ā_{j}| ∩ (ecl(C) ∪ A_{j}) = ∅.

Proposition 4 Over any C ∈ subS, with C = ecl(C), there is a q-prime structure E(C) such that for any embedding C ≤ D ∈ SC there is an extension of the embedding E(C) ≤ D. The cardinality of E(C) is not greater than card (C)+κ. Any two constructible q-prime structures over C are isomorphic over C.

Proof The first statement is proved in Lemma 4.10, where the sequence {ā_{j} : j < λ} is just {ā_{n,α} : n < ω, α < λ_{n}} getting ordinal enumeration and τ_{j} are the corresponding types. The second statement follows from the fact that λ_{n} ≤ card (C)+κ, since ecl(C) ≤ card (C)+κ and the number of types over a finite set is not greater than 2^{ℵ_{0}} ≤ κ. To prove the rest we use the classical argument following [B] and [Sh]. A subset B of a constructible q-prime structure over C is said to be closed if for all ā_{j}, if |ā_{j}| ∩ B ≠ ∅, then |ā_{j}| ⊆ CB, CB contains all parameters of type τ_{j} and CB ≤ E(C). Notice that each point b ∈ B can be included in some finite string b̄ from B such that Cb̄ ≤ E(C). Since E(C) is q-atomic, by Lemma 4.5 δ(b̄/C) = 0. In particular, if B is finite δ(B/C) = 0. Claim 1. If B is a closed subset of E(C), j < λ, then

(i) CBA_{j} ≤ E(C); (ii) δ(ā_{j}/CBA_{j}) = 0; (iii) ā_{j} is q-atomic over CBA_{j} and the corresponding q-atom is τ_{j}. Proof of Claim. For any j < λ we now prove (i) - (iii) for B^{i} = B ∩ A_{i}, in place of B for all i ≤ λ by induction on i. For i ≤ j this is true, since then B^{i} ⊆ A_{j}, (i) and (iii) are true by the definition and (ii) follows by the following argument, which works for any i, provided (i) and (iii) hold. Choose finite A ⊆ B^{i}A_{j}, such that CA ≤ E(C) and A contains the parameters of τ_{j}. Then A is q-atomic over C and ā_{j} is q-atomic over CA. By Lemma 4.5 Aā_{j} is q-atomic over C, hence by the same Lemma δ(Aā_{j}/C) = 0, δ(A/C) = 0. It follows δ(ā_{j}/CA) = 0, which in turn implies δ(ā_{j}/CB^{i}A_{j}) = 0. Assume now that (i)-(iii) hold for some i, j ≤ i < λ, we want to prove it for i + 1. Assume B^{i+1} ≠ B^{i}. Then B^{i+1} = |ā_{i}| ∪ B^{i}. Since B^{i}A_{j} ⊆ A_{i} and the parameters of τ_{i} are in B^{i}, ā_{i} is q-atomic over CB^{i}A_{j}ā_{j}. By the induction hypothesis ā_{j} is q-atomic over CB^{i}A_{j} by τ_{j} and δ(ā_{j}/CB^{i}A_{j}) = 0. Then by 4.7 ā_{j}ā_{i} is q-atomic over CB^{i}A_{j}. More precisely, the type of ā_{j}ā_{i} over CB^{i}A_{j} is determined by τ_{j}(x̄)&τ_{i}(ȳ) (where x̄, ȳ correspond to ā_{j}, ā_{i}). Since ā_{i} is q-atomic over CB^{i}A_{j} and CB^{i}A_{j} ≤ E(C), by the argument above δ(ā_{i}/CB^{i}A_{j}) = 0. Then CB^{i}A_{j}ā_{i} ≤ E(C). Hence, by Lemma 4.8 and the above proved, ā_{i} is q-atomic over CB^{i}A_{j} and ā_{j} is q-atomic over CB^{i}A_{j}ā_{i}, more presicely the q-atom is again τ_{j}. Since CB^{i}A_{j}ā_{i} = CB^{i+1}A_{j}, we proved (i) and (iii) for B^{i+1}. Once again (ii) is proved by the argument above. For limit ordinals i (i) - (iii) is evident. This finishes the proof of Claim. Claim 2. Suppose B is closed. Then for any finite X ⊆ E(C) there is finite B_{X} ⊆ E(C), such that BB_{X} is closed and X ⊆ B_{X}. Proof of Claim. We prove by induction on j that for X ⊆ A_{j} there is B_{X} ⊆ A_{j}, such that BB_{X} is closed. If j = 0, X = ∅ = B_{X}. Suppose |ā_{j}| ⊆ X ⊆ A_{j+1}. Choose finite Y ⊆ A_{j}, such that X ∩ BA_{j} ⊆ Y, CBB_{Y} is closed and CB_{Y} contains all the parameters of τ_{j}. Then there is finite Z ⊆ B such that CZB_{Y} ≤ CBB_{Y} ≤ E(C) and ā_{j} is q-atomic over CZB_{Y} . Since E(C) is q-atomic over C, ZB_{Y} is q-atomic and by 4.5, 4.7 ZB_{y}ā_{j} is q-atomic over C. Hence δ(ZB_{y}ā_{j}/C) = 0, thus CZB_{Y} ā_{j} ≤ E(C). Since Z can be chosen as large as we want, it follows CBB_{Y} ≤ E(C). We state that BB_{Y} ā_{j} is closed. It remains to prove that if there is a ∈ |ā_{i}| such that a ∈ BB_{Y} |ā_{j}| then |ā_{i}| ⊆ CBB_{Y} |ā_{j}| and the parameters of τ_{i} are in the same set. By the construction it is true if a ∈ BB_{Y} . If a ∈ |ā_{j}|, then i = j and both |ā_{j} and the set of all parameters of τ_{j} are subsets of CBB_{Y} |ā_{j}|. Thus BB_{Y} |ā_{j}| is a closed set which contains X. Claim proved. Claim 3. E(C) is q-atomic over CB for any closed B and if CBā ≤ E(C), then δ(ā/CB) = 0. Proof of Claim. It is enough to prove that any ā in CBA_{j}, such that CBā ≤ E(C), E(C) is q-atomic over CB and δ(ā/CB) = 0 CB for all j < λ. We prove it by induction on j. The statement is trivial for j = 0. Suppose it holds for j and ā is a tuple in CBA_{j+1} such that CBā ≤ E(C). Choose finite tuple b̄ in CBA_{j} such that CBb̄ ≤ E(C), |ā| ⊆ |b̄ā_{j}| and ā_{j} is q-atomic over Cb̄ (Claim 1). By Lemma 4.7 b̄ā_{j} is q-atomic over CB. By Lemma 4.8 ā is q-atomic over CB. Since |b̄ā_{j}| = |āb̄′| for some b̄′, by Lemma 4.8 ā is q-atomic over CB. For large enough b̄

δ(b̄ā_{j}/CB) = δ(ā_{j}/CBb̄) + δ(b̄/CB) = δ(ā_{j}/CBb̄) = δ(ā_{j}/CBA_{j}) = 0.

Since

0 = δ(āb̄′/CB) = δ(b̄′/CBā) + δ(ā/CB)

and both summands on the right are non-negative by CB ≤ CBā ≤ E(C), we get δ(ā/CB) = 0. Claim proved.

Now we come to the proof of Proposition. We consider two constructible qprime structures A and A′ over C with q-constructions {(ā_{j}, τ_{j}) : j < λ} and {(ā′_{j}, τ′_{j}) : j < λ′} correspondingly. Assume λ ≤ λ′. We define by induction ascending chains of isomorphic closed sets C_{j} ≤ A and C_{j} ≤ A′ for j < λ and the isomorphisms f_{j} : C_{j} → C_{j}. We also satisfy the condition that if j = η + (2n + 2) η limit, n natural, then ā_{η+n} lives in C_{j}; if j = η + (2n + 1) then ā′_{η+n} lives in C_{j}. Consider a typical case of the construction: j = η + (2n + 2). By Claim 2 there is a finite set B_{0} containing |ā_{η+n}|, such that C_{j−1}B_{0} is closed. By Claim 3 A is q-atomic over C_{j−1} and δ(B_{0}/C_{j−1}) = 0. Since A is q-atomic over C_{j−1} there is an isomorphism g_{0} extending f_{j−1} and taking B_{0} onto some B′_{0} ⊆ A′. By Claim 2 B′_{0} can be included in some B′_{1} ⊆ A′ such that C′_{j−1}∪B′_{1} is closed. By the isomorphism δ(B′_{0}/C′_{j−1}) = 0, hence by Lemma 4.9 A′ is q-atomic over C′_{j−1}B′_{0}. Thus B′_{1} satisfies a q-atom over C′_{j−1}B′_{0}. It follows there is an isomorphism g_{1} extending g_{0} and taking C_{j−1} ∪ B_{1} for some B_{1} onto C′_{j−1} ∪ B′_{1}. Continuing in this way we get a chain of finite sets B_{m} ⊆ A (m ∈ N), B′_{m} ⊆ A′ and corresponding isomorphisms g_{m} from C_{j−1}B_{m} onto C′_{j−1}B′_{m}, such that C_{j−1}B_{m} is closed if m is even and C′_{j−1}B′_{m} is closed if m is odd. Finally the sets

C_{j} = C_{j−1} ∪ ⋃_{m∈N} B_{m}, C′_{j} = C′_{j−1} ∪ ⋃_{m∈N} B′_{m} and f_{j} = ⋃_{m∈N} g_{m}

satisfy the conditions required for j. For odd j the procedure is symmetrically from A′ to A. In the end the construction yields an isomorphism f = f_{λ} from A onto the closed subset C_{λ} of A′. Since A′ is q-atomic over C_{λ} and by the isomorphism any q-atom over C′_{λ} is realized in C′_{λ} we have necessarily C′_{λ} = A′.□

Theorem 2 Let C ≤ D and C be a ∂-independent set of power λ. Suppose that D is q-prime constructible over C. Then

(i) given any C′ maximal ∂-independent subset of A, card (C) = card (C′; ) (ii) any bijection f : C → C′ can be extended to an automorphism of D; (iii) for finite S ⊆ D card (cl(S)) ≤ κ; (iv) if λ > ℵ_{0} then any definable subset of D is either of power λ or at most κ.

Proof (i) By Lemma 4.5 C′ is in cl(C), the ∂-closure of C. By Lemma 3.5 card (C′) ≤ card (C). On the other hand cl(C′) must contain C for otherwise C′ is not maximal. Hence card (C′) = card (C). (ii) Notice that C′ ≤ D so we can consider E(C′). C′ is isomorphic to C since any two ∂-independent tuple c̄ and c̄′ of the same length n realize the same quantifier-free type (V, W, \{W^{\frac{1}{l}} : l ∈ N\}), namely, V and W are trivial (the n-spaces itself). By Corollary 4 and Proposition 4 E(C′) is isomorphic to E(C) = D. (iii) By the definitions and the above proved there is a finite subset C_{0} ⊆ C, such that S ⊆ cl(C_{0}). Consider E(C_{0}), which may be assumed embedded in E(C). Now construct inside E(C) structure A = E(E(C_{0}) ∪ C). Notice, that since E(C_{0}) ⊆ cl_{A}(C_{0}), the set C \ C_{0} is independent over E(C_{0}). Let a ∈ cl_{A}(C_{0}). Since A is q-atomic over E(C_{0}) ∪ C, there is ā extending a, such that ā is q-atomic over E(C_{0}) ∪ C. Let τ be the corresponding almost finite type determining the q-atom of ā over the set. More precisely, τ is defined over finite Bc̄ where B ⊆ E(C_{0}) and c̄ is a string from C \ C_{0} and

τ = (V (c̄), W (ex(c̄)), \{W^{\frac{1}{l}} (ex(\frac{1}{l}c̄)) : l ∈ N\})

with (V, W) defined over B minimal containing āc̄. Since c̄ is ∂-independent over Bā, by the argument in the proof of (ii), V (c̄) = V_{0}, for a B-definable variety V_{0}, and W (ex(c̄)) = W_{0}, for a ex(B)-definable variety W_{0}. Thus, τ can be chosen to be defined over B. Since E(C_{0}) is existentially-algebraically closed, τ is realized in E(C_{0}) by a tuple ā′. Since τ is a q-atom over E(C_{0})∪C, we get ā = ā′. Thus we have proved that ā ∈ E(C_{0}). It follows cl_{A}(C_{0}) ⊆ E(C_{0}) and card (cl_{A}(C_{0}) ≤ card (E(C_{0})) ≤ κ. On the other hand E(C) is embeddable in A and by this embedding cl(C_{0}) goes into cl_{A}(C_{0}). It finishes the proof of (iii). (iv) If a set X definable over finite S ≤ D is of power greater than κ, then it contains an element c ∉ cl(S). Then by (ii) there is an automorphism over S sending c to any point in C \ C_{0}. It follows X is of power λ. □

5 Pseudo-analytic dimension in canonical structures

Throughout this section D is a canonical structure over a ∂-independent set of power λ > ℵ_{0} and the standard kernel. We expand the language by naming all elements of the standard kernel.

Definition For a subset S ⊆ D^{n} definable over some finite C ≤ D the pseudo-analytic dimension of S, denoted adim S, is given as

adim S = max_{ā∈S} ∂(ā/C).

Lemma 5.1 (i) The definition of adim does not depend on the choice of parameters C ≤ D. (ii) adim S ≥ 0 for any non-empty S. (iii) For non-empty S adim S = 0 iff card S ≤ κ. (iv) For non-empty S adim S ≥ m > 0 iff S can be projected to D^{m} so that the complement to the project ion in D^{m} is of pseudo-analytic dimension less than m or empty.

(v) For algebraic varieties V ⊆ D^{n}, W ⊆ R^{n}

adim V = dim V, adim ln W = dim W.

Proof Immediate.□

Proposition 5 Suppose (V, W) is a normal free pair over C ≤ D. Then

adim (V ∩ ln W) ≥ dim V + dim W − n.

Proof By induction on d = dim V + dim W − n ≥ 0. If dim W = n, then V ∩ ln W = V. Thus we may assume dim V > 1. Choose c_{1}, …, c_{n} ∈ D such that ∂(⟨c_{1}, …, c_{n}⟩/C) = n. By Lemma 4.4 we get then algebraic subvariety V′ ⊆ V defined over C ∪ {c_{1}, …, c_{n}} such that pair (V′, W) is normal, free and

dim V′ + dim W′ − n = d − 1.

By induction there is ā ∈ V ∩ ln W, ∂(ā/Cc̄) = d − 1. But, again by Lemma 4.4, ∂(c̄/Cā) < n = ∂(c̄/C), hence ∂(ā/C) > ∂(ā/Cc̄) = d − 1. □

Lemma 5.2 Let (V, W) be a normal pair over C ≤ D and suppose

V ∩ ln W ≠ ∅.

Then

adim (V ∩ ln W) ≥ dim V + dim W − n.

Proof We use induction on n. The case n = 1 is trivial. Consider the general case. Choose ⟨a_{1}, …, a_{n}⟩ ∈ V ∩ ln W. By the induction hypothesis there is ⟨a′_{1}, …, a′_{n−1}⟩ ∈ V (a_{n}) ∩ ln W (ex(a_{n})) with

∂(⟨a′_{1}, …, a′_{n−1}⟩/Ca_{n}) ≥ dim V (a_{n}) + dim W (ex(a_{n})) − (n − 1).

If dim V (a_{n}) = dim V or dim W (ex(a_{n})) = dim W then we would have the desired estimate

∂(⟨a′_{1}, …, a′_{n−1}⟩/Ca_{n}) ≥ dim V + dim W − n.

Hence we assume dim V (a_{n}) = dim V − 1 and dim W (ex(a_{n})) = dim W − 1 which yields that the estimates are true for generic fibers and thus the projections V_{n} of V and W_{n} of W to the last coordinate are dense in D and R correspondingly. Choose a′_{n} ∈ D ∂-independent over C. By the induction hypothesis there is ⟨a′_{1}, …, a′_{n−1}⟩ ∈ V (a′_{n}) ∩ ln W (ex(a′_{n})) with

∂(⟨a′_{1}, …, a′_{n−1}⟩/Ca′_{n}) ≥ dim V (a_{n})+dim W (ex(a_{n}))−(n−1) = dim V +dim W −n−1.

By our choice

∂(⟨a′_{1}, …, a′_{n}⟩/C) = ∂(⟨a′_{1}, …, a′_{n−1}⟩/Ca′_{n}) + 1 ≥ dim V + dim W − n

and ⟨a′_{1}, …, a′_{n}⟩ ∈ V ∩ ln W. □

References

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