A remarkable pseudo-analytic structure

B. Zilber preliminary version

August 10, 2003

1 Spiral structures

We define the (α, ℝ)-canonical two-spiral structure (α ∈ ℝ \ πℚ) to be the structure ℂ_{G,Γ,2} on the complex numbers ℂ in the language

L_{G,Γ,2} = {+, ·, G, G^{†}, Γ, Γ^{†}}, G, G^{†}, Γ, Γ^{†} unitary predicates,

with the interpretation

Γ(ℂ) = exp(αℤ), Γ^{†}(ℂ) = exp(iαℤ),  (1)

G(ℂ) = exp(ϵℝ) · Γ^{†}(ℂ), G^{†}(ℂ) = exp(ϵ̄ℝ) · Γ(ℂ).  (2)

Notice that by easy calculations also

Γ(ℂ) ⊆ G(ℂ) and Γ^{†}(ℂ) ⊆ G^{†}(ℂ)

thus we can add to the above

G(ℂ) = exp(ϵℝ) · Γ(ℂ), G^{†}(ℂ) = exp(ϵ̄ℝ) · Γ^{†}(ℂ).  (3)

Given a L_{G,Γ,2}-definable group H ( written additively, say) and an integer m we also assume that the enriched language L_{G,Γ,2} contains symbols for the (L_{G,Γ,2}-definable) subgoup mH.

1.1

The (α, ℝ)-canonical structure is a model of the first-order theory T_{G,Γ,2}.

Proof See sections 5-6.

1.2

The theory is complete, near model complete, with respect to the enriched L_{G,Γ,2}, and superstable.

Proof See sections 5-6.

1.3

The ’spiral’ subgroups

G_{0}(ℂ) = exp(ϵℝ) and G_{0}^{†}(ℂ) = exp(ϵ̄ℝ)

are type-definable in ℂ_{G,Γ,2}. Also

G_{0} · G_{0}^{†} = ℂ^{∗}

Identifying ϵℝ with G_{0} and ϵ̄ℝ with G_{0}^{†} via exp we can consider the additive group ℂ = ϵℝ + ϵ̄ℝ as definable in the structure, along with the mapping

exp : ϵx + ϵ̄y ↦ exp ϵx · exp ϵ̄y

exp : ℂ → ℂ^{∗}.

Proof Just notice that

z ∈ G_{0} iff ⋀_{m∈ℕ}∃y ∈ G y^{m} = z.  (4)

Similarly with G_{0}^{†}. The rest follows from the facts that

ϵℝ + ϵ̄ℝ = ℂ.

1.4

Notice by straightforward calculations that the type-definable subset

G_{0}(ℂ) ∩ G_{0}^{†}(ℂ) = exp(πϵℤ)

G(ℂ) ∩ G^{†}(ℂ) = exp(πϵℤ + αℤ + αiℤ) = exp πϵℤ) · Γ(ℂ) · Γ^{†}(ℂ).

We define a new sort

V = \frac{G × G^{†}}{Γ · Γ^{†}} ,

a quotient-group with the group operation written additively. As in 1.3 we can idetify V (ℂ) with ℂ and the map

ex : ⟨exp(ϵx + iαm), exp(ϵ̄y + iαn)⟩mod(Γ · Γ^{†}) ↦ exp(ϵx + ϵ̄y)

as exp . In this representation we obviously have

ker ex = \frac{exp(πϵℤ) · Γ(ℂ) · Γ^{†}(ℂ)}{Γ(ℂ) · Γ^{†}(ℂ)} ≅ ℤ.

In the sort V (ℂ) we define in the first-order way

A = \frac{G × \{1\}}{Γ · Γ^{†}}, A^{†} = \frac{\{1\} × G^{†}}{Γ · Γ^{†}}

Δ = \frac{Γ × \{1\}}{Γ · Γ^{†}}, Δ^{†} = \frac{\{1\} × Γ^{†}}{Γ · Γ^{†}}

and define by types

A_{0} = \frac{G_{0} × \{1\}}{Γ · Γ^{†}}, A_{0}^{†} = \frac{\{1\} × G_{0}^{†}}{Γ · Γ^{†}} .

Notice that we may identify

A_{0}(ℂ) = ϵℝ, A_{0}^{†}(ℂ) = ϵ̄ℝ

and

Δ(ℂ) = αℤ, Δ^{†}(ℂ) = iαℤ.

1.5

The involutive transformation of ℂ (as the sort V (ℂ))

I : ϵx + ϵ̄y ↦ ϵx − ϵ̄y (x, y ∈ ℝ)

is type-definable in ℂ_{G,Γ,2}.

Proof Let I(z, ž) be the type

⋀_{m∈ℕ}(∃u, v) u ∈ G & v ∈ G^{†} & u^{m} · v^{m} = z & u^{m} · v^{−m} = ž.  (5)

It is immediate from the definition of the type that I(z, ž) implies z · ž satisfies the type G_{0} and z · ž^{−1} the type G_{0}^{†}. This proves that I restricted to G_{0} × G_{0}^{†} defines I (with the translation of · into +.)

1.6

The field ℝ ⊆ ℂ of reals along with all C^{∞} functions are L_{ω_{1},ω}-definable in L_{G,Γ,2}.

Proof By section 4.

1.7

If ℂ_{G,Γ,2} ⪹ M in the language with the definable ’metric’ relation |z_{1}| < |z_{2}| on ℂ and the exponentiation exp on the reals then there exists 1 ≠ γ_{0} ∈ Γ(M) such that for every positive integer m the equation

γ^{m} = γ_{0}

is solvable in Γ(M), in other words there is no group embedding of Γ(M) into the additive group ℤ̂.

Proof It is well known and easy to see that when |z_{1}| < |z_{2}| is in the language the reals are definable in ℂ, ℂ = ℝ + iℝ and the same holds in M with corresponding formal reals.

We also have a definable embedding

f_{α} : Γ(M) → ℝ(M) the reals of M

corresponding to

f_{α} : exp(αm) ↦ m

on ℂ_{G,Γ,2}. Hence f_{α}(G_{0}(M)) = ℤ(M), the ring of integers of M, which must be a proper extension of standard integers, since the reals of M have been properly extended. If η ∈ ℤ(M) is a non-standard (infinite) integer then obviously η! ∈ ℤ(M) is divisible by any standard integer and hence the element γ_{0} ∈ Γ(M) corresponding to η! has the required property.□

Question. Would the same be true without assuming the exponentiation definable?

1.8

We say that a model M of T_{G,Γ,2} is proper if both groups Γ(M) and Γ^{†}(M) are embeddable into ℤ̂.

For proper M the statements above about type-definable sets and relations hold.

1.9

Assuming the model M is proper define a type-definable bijection ·^{i} : Γ(M) ↦ Γ^{†}(M) as a relation between z and z^{i} :

z ∈ Γ & z^{i} ∈ Γ^{†} & z · z^{i} ∈ G^{0}.

This is a group isomorphism. Also, by construction z ↦ z · z^{i} is a map from Γ(M) to G_{0}(M) and similarly we have a definable embedding into G_{0}^{†}(M).

The same relation written in terms of sort V gives a type-definable map

i· : Δ → Δ^{†}.

2 The (e^{α}, ℤ̂)-canonical structures

A weakly (e^{α}, ℤ̂)-canonical structure is a model M of T_{G,Γ,2} with both Γ(M) and Γ^{†}(M) isomorphic to the additive group Ẑ.

2.1

There exists a weakly (e^{α}, ℤ̂)-canonical structure with any fixed cardinality κ = card G(M) ≥ 2^{ℵ_{0}}.

Proof Start with the class of two sorted structures (V, F), with F a field of characteristic zero, V an additive group, ex : V → F^{∗} a homomorphism, A, A^{†} ≤ V subgroups,

A = A_{0} ˙+Δ^{†}, A^{†} = A_{0}^{†} ˙+Δ,

where

Δ ≅ Δ^{†} ≅ Ẑ ≅ ker ex ,

Δ is pseudo-generated by some ex (α) and Δ^{†} by ex (iα),

A ∩ A^{†} = Δ + Δ^{†}, Δ ∩ Δ^{†} = 0

and

A ∩ ker ex = A^{†} ∩ ker ex = 0.

We also assume the property

δ(X) ≥ 0,

for the (A, A^{†})-predimension δ, all finite subsets X in the class.

Show that the class has the amalgamation over strong substructures property. Thus there is an universal structure in the class satisfying usual Hrushovskistyle axioms which are exactly T_{G,Γ,2} as far as the sort F with subgroups

G = ex (A), G^{†} = ex (A^{†}), Γ = ex (Δ) and Γ^{†} = ex (Δ^{†}).

Since ex is bijective on G and G^{†} and G_{0}, G_{0}^{†} are type-definable, by the argument 1.3, the sort V and its subgroups A and A^{†} in the constructed model is determined uniquely by the description of G and G^{†}. See sections 5-6 for details. □

2.2

By elementary equivalence and the fact that for a canonical M

⋂_{n∈ℕ}Δ(M) = 0 = ⋂_{n∈ℕ}Δ^{†}(M)

all the facts of 1.4 hold in M with ℤ replaced by Ẑ.

2.3

The predimension δ of 2.1 generates a dimension notion under which

dim G(M) = 1 = dim A(M), dim Γ(M) = 0 = dim Δ(M)

and moreover G_{0}(M) (and A_{0}(M)) is quasi-minimal, i.e. the set can not be divided into two dimension 1 parts of dimension 1. We thus can consider a pregeometry on G_{0}(M) (which is probably locally modular) and hence define a basis and thereafter the pseudo-linear dimension of G_{0}(M) as the cardinality of the (pseudo-linear) basis. As usual, any two pseudo-linear bases of G_{0}(M) are isomorphic by an elementary bijection over Γ · Γ^{†}. Using an argument from [ZPE] (which is actually Shelah’s, but in a weaker form than excellency) we can show that

Given a pseudo-linear basis B of cardinality κ there is a unique, up to isomorphism over Γ·Γ^{†}, prime over Γ·Γ^{†} and B weakly (e^{α}, ℤ̂)-canonical model of T_{G,Γ,2}.

Definition A weakly (e^{α}, ℤ̂)-canonical model which is prime over Γ·Γ^{†} and a pseudo-linear basis B of cardinality 2_{0}^{ℵ} we will call a (e^{α}, ℤ̂)-canonical model.

We fix notation 𝔽 for an (e^{α}, ℤ̂)-canonical model of T_{G,Γ,2}.We also abbreviate, unless we mean otherwise, strongly (e^{α}, ℤ̂)-canonical model of T_{G,Γ,2} to just canonical model.

3 A canonical model with topology

3.1

We now introduce a topology on a definable extension of the canonical model 𝔽. We denote 𝔽̄ the structure with

the universe F̄ = F ∪ {∞}, where ∞ is an extra point named by a new constant, in fact set-wise F̄ can be naturally identufied as the projective line P(F); the basic predicates on F̄ are going to be all the Zariski-closed subsets of F̄^{n}, all n along with predicates G, Γ G^{†} and Γ^{†} are being declared closed in 𝔽̄ Fand interpreted in 𝔽̄ as follows

G(F̄) = G(F) ∪ {0, ∞}, G^{†}(F̄) = G^{†}(F) ∪ {0, ∞}
Γ(F̄) = Γ(F) ∪ {0, ∞} and Γ^{†}(F̄) = Γ^{†}(F).

Using identification of 2.1 we have also

A(F̄) = A(F) ∪ {□, ◇}, A^{†}(F̄) = A(F) ∪ {□, ◇}

where

ex (□) = ∞ and ex (◇) = 0.

Also

Δ(F̄) = Δ(F) ∪ {□, ◇} and Δ^{†}(F̄) = Δ^{†}(F).

This leads us to

V (F̄) = V (F) ∪ {□, ◇}

with the addition becoming a ternary relation SUM(x, y, z) on V (F̄) as in [Z?] which is the union of semigroups V (F) ∪ {□} and V (F) ∪ {◇} with □ + ◇ = V (F̄).

Finally, we postulate ex in the extended 𝔽̄ to be continuos. In particular

ker ex (F̄) = ker ex (F)

is closed.

3.2

A, A^{†}, Δ and Δ^{†} are closed as images of closed sets under continuous map ex .

A_{0}(F̄) = ⋃_{m∈ℕ}mA(F̄) = A_{0}(F) ∪ {□, ◇}

and

A_{0}^{†}(F̄) = ⋃_{m∈ℕ}mA^{†}(F̄) = A_{0}^{†}(F) ∪ {□, ◇}

are closed as intersection of closed sets. The closure of the map i· of 1.9 becomes the binary relation K ⊆ Δ × Δ^{†} K(x, y) iff (∃z ∈ A_{0}) SUM(x, y, z) & x ∈ Δ & y ∈ Δ^{†} iff y = i·x ∨ x = □ ∨ x = ◇, In particular i· is not a continuos mapping. Similarly, the involutive transformation I (see 1.5) is not continuos, since by

(5) the closure of its graph is I(F̄) = I(F) ∪ {□, ◇} × V (F̄) ∪ V (F̄) × {□, ◇}.

Define a closed subset Z of A corresponding to 2αℤ

z ∈ Z iff (∃x, y) x ∈ Δ & y ∈ Δ^{†} & SUM(x, y, z)

it follows

Z(F̄) = Z(F) ∪ {□, ◇} where Z(F) ≅ Ẑ.

On the other hand iZ(F̄), the closure of the subset corresponding to 2iαℤ is defined by

z ∈ iZ iff (∃x, y) x ∈ Δ & y ∈ Δ^{†} & SUM(z, x, y)

and

iZ(F̄) = iZ(F) ≅ Ẑ.

Thus, one can say that the topology considers the ’real direction’ to be compactified with extra end points □ and ◇, while the ’imaginary direction’ is compact in 𝔽 already. So, there is no continuous map between the two, though there is the type-definable involutive transformation I exchanging the directions.

3.3 Questions and conjectures

Prove that the canonical structure with the topology is analytic Zariski.

Prove that the topology is compatible with toric geometry.

Prove that the involutive transformation I of the canonical structure transforms the topology X ↦ X^{†}.

Appendix 1

4 Jordan curve on the complex plane

Let C ⊆ ℂ be a Jordan curve (without endpoints) on the complex plane, that is the image of a continuous differentiable map f : ℝ → ℂ = ℝ + iℝ which has a local differentiable inverse almost everywhere on ℝ. We also assume that C is not dense in ℂ. Then

4.1

By applying a conformal transformation

z ↦ \frac{az − b}{z − c}, a, b, c ∈ ℂ, |c − C| > ϵ,

we may assume that C is a subset of a bounded disk in ℂ.

4.2

There is a point on C, which we assume to be 0 (by a choice of a and b), where C is well approximated by a straight line ℝ, by a choice of a, that is for any small enough positive real ϵ, for each t ∈ (−ϵ, +ϵ)

f(t) = t + ig(t), g(t) ∈ ℝ, |t^{−1}g(t)| < ϵ.

It follows that the Jacobian matrix of the map

f + if : ℝ^{2} → ℂ^{2}

is non-degenerate around (0, 0).

4.3

Now the definable set B = C + iC is inside a disk, say of radius R, and contains an open disk around 0 of radius, say r.

4.4

We now define a binary relation u << w on ℂ :

u << w iff B ⊆ \frac{w}{u}B.

Obviously,

|\frac{w}{u}| > \frac{R}{r} ⇒ u << w  (6)

and

u << w ⇒ |u| < |w|.  (7)

4.5

Now we introduce an L_{ω_{1},ω}-definable relation

u ≺ w ⇔ ⋁_{n∈ℕ}u^{n} << w^{n}.

We claim that u ≺ w is equivalent to |u| < |w|. Indeed, if |u| < |w| then for some n

|\frac{w}{u}|^{n} > \frac{R}{r}

and hence, by (6), u^{n} << w^{n}, and so u ≺ w. Conversely if, for some n, u^{n} << w^{n} then by (7) |u^{n}| < |w^{n}| and hence |u| < |w|.

4.6

The reals are L_{ω_{1},ω}-definable in (ℂ, +, ·, C). Indeed, we have from above that the relation |u| ≤ |w| is L_{ω_{1},ω}-definable in the structure. We now show how to define in the first order way, without parameters, the reals. It is enough to notice that

{u ∈ ℂ : ∀v |v| = |u| → |u + 1| ≥ |v + 1|} = {u ∈ ℝ : u > 0}. (8)

□

4.7

Suppose now that

(ℂ, +, ·, C) ≼_{ω_{1},ω} (F, +, ·, C(F)).

Then, the L_{ω_{1},ω}-formula in (8) still defines the positive formal reals, and F is the complexes over the formal reals. But also it is an L_{ω_{1},ω}-property that the formal reals are Archimedean. Hence, F = ℂ. There is no proper L_{ω_{1},ω}-elementary extensions.

We also have, since the order of length continuum is definable in the structure

Corollary 4.1 The structure (ℂ, +, ·, C) has no uncountably categorical L_{ω_{1},ω}- description, moreover it is not a member of an excellent class.

Appendix 2

5 A superstable field with two subgroups

Let C be the class of two-sorted structures V = (V, ex , F) with F a field and V a ℚ-vector space, with two distinguished subgroups A, A^{†} ≤ V, and a homomorphism ex : V → F^{∗} between the sorts satisfying the following axioms

A and A^{†} are subgroups of V of the form A_{0} + Δ, A_{0}^{†} + Δ^{†}, A_{0}, A_{0}^{†}divisible and Δ, Δ^{†}elementarily equivalent to the abelian group ℤ

(9)

\begin{matrix}A + A^{†} = V, \\ Δ + Δ^{†} = A ∩ A^{†} \end{matrix}  (10)
ker ex ∩ A = {0} = ker ex ∩ A^{†};  (11)

the predimension condition, for every finite X ⊆ V containing d ∈ Δ and d^{†} ∈ Δ^{†}, δ(X) = 2tr.d.(ex X) − (lin.d.(X ∩ A) + lin.d.(X ∩ A^{†})) ≥ −2

(12)

It follows from (12) that at some finite D ⊆ V containing d, d^{†} the minimum δ(D) ≥ −2 of δ(X) is reached and then

δ(X/D) ≥ 0 for all finiteX ⊆ V,  (13)

where δ(X/D) = δ(X ∪ D) − δ(D). Obviously, the theory of V depends on the type of D.

We also are going to consider the following conditions.

Definition Given V ∈ C and an irreducible algebraic variety W ⊆ F^{2n} in variables x_{1}, …, x_{n}, y_{1}, …, y_{n} defined over a finite C we say that W is GHfree if, for G = ex (A), H = ex (A^{†}), and a generic point (g_{1}, …, g_{n}, h_{1}, …, h_{n}) in W

gp(g_{1}, …, g_{n}) ∩ acl(C) = {1} = gp(h_{1}, …, h_{n}) ∩ acl(C),

where gp(X) stands for a multiplicative subgroup of F generated by X.

We also say that W is GH-normal if for every multiplicatively independent ĝ_{i_{1}}, …, ĝ_{i_{m}} ∈ gp(g_{1}, …, g_{n}) and multiplicatively independent ĥ_{i_{1}}, …, ĥ_{i_{l}} ∈ gp(h_{1}, …, h_{n})

tr.d._{C}(ĝ_{i_{1}}, …, ĝ_{i_{m}}, ĥ_{i_{1}}, …, ĥ_{i_{l}}) ≥ \frac{m + l}{2}.

Any GH-free and GH-normal W ⊆ F^{2n} has a generic point (g_{1}, …, g_{n}, h_{1}, …, h_{n}) ∈ W such that g_{1}, …, g_{n} ∈ G and h_{1}, …, h_{n} ∈ H

(14)

We denote Γ) = ex (Δ) and Γ^{†} = ex (Δ^{†}).

As usual we write, for X ⊆ Y ⊆ V, V ∈ C, δ(Y/X) = δ(Y) − δ(X) and

X ≤ Y iff δ(X′/X) ≥ 0 for every X ⊆ X′ ⊆ Y.

Theorem 1 There exists an existentially closed, with resepect to ≤, structure V in the class C. The class of existentially closed structures C^{∗} is first order axiomatizable by a theory T (C^{∗}) and any its subsclass with a fixed quantifier-free type tpD of the minimal D is completely axiomatizable by the theory T (C^{∗}) ∪ tpD, superstable with

U(V) = U(F) = ω×2, U(A) = U(A^{†}) = ω, U(Δ) = U(Δ^{†}) = 1, U(Γ) = U(Γ^{†}) = 1.

More precisely, the first order theory T (C^{∗}) is given by axioms (9)-(14).

Proof The construction of an existentially closed structure is given by the usual Hrushovski-Fráisse amalgamation very similar to that in [P]. It then satisfies (9)-(14). The fact that (12)-(14) is first order follows from the same argument as in [P] (uses Ax’s theorem in interpretation of [2]. The superstability and rank computation is straightforward.□

6 Interpretation in the complexes

Let α ∈ ℝ such that α/π ∉ ℚ, ϵ = 1 + i, ϵ̄ = 1 − i,

A = ϵℝ + iαℤ, A^{†} = ϵ̄ℝ + iαℤ,
Δ = αℤ, Δ^{†} = iαℤ.

Obviously, A+A^{†} = ℂ and Δ+Δ^{†} = A∩A^{†} = ϵℤ+ϵ̄ℤ and (9)-(11) is satisfied.

Lemma 6.1 Assume Schanuel’s conjecture. Then, for any finite S, T ⊆ ℝ

2tr.d. exp(ϵS ∪ ϵ̄T) ≥ lin.d.S + lin.d.T.

Proof We assume that T ∩ S = U, T = T′ ∪ U, S = S′ ∪ U and S′, T′ and U are mutually linearly independent. Let

C = ϵU ∪ ϵ̄U, D = (ϵS ∪ ϵ̄T) \ C.

Then, we have the following properties of C, D and their complex conjugates

C = C̄;  (15)
lin.d.(CDD̄) = |C| + 2|D| = 2|U| + 2|D|;  (16)
acl(C) = acl(U); acl(CD) = acl(CD̄) = acl(U ∪ D);  (17)
tr.d. exp(CD) = tr.d. exp(CD̄).  (18)

We use Schanuel’s conjecture in the form

∂(X) = tr.d.(X ∪ exp X) − lin.d.X ≥ 0 for any finite X ⊆ ℂ.

Notice that

∂(X/Y) = ∂(XY) − ∂(Y) = tr.d.(X exp X/Y exp Y) − lin.d.(X/Y),

where XY here and below stands for X ∪ Y.

Claim 1.

2∂(CD) ≥ ∂(CDD̄) + ∂(C) + tr.d.(CD exp CD) − tr.d.(C exp(CD)).

Indeed,

2∂(CD) − ∂(CDD̄) − ∂(C) =

= 2tr.d.(CD exp CD)−2lin.d.(CD)−tr.d.(CDD̄ exp(CDD̄))+lin.d.(CDD̄)−∂(C) = = 2tr.d.(CD exp CD) − tr.d.(CDD̄ exp(CDD̄)) − |C| − ∂(C) = = 2tr.d.(CD exp CD) − tr.d.(CD exp(CDD̄)) − tr.d.(C exp C) =

= tr.d.(CD exp CD) − tr.d.(exp D̄/CD exp(CD)) − tr.d.(C exp C) ≥

≥ tr.d.(CD exp CD) − tr.d.(C exp(CD)) ≥ 0

Claim proved. Claim 2.

2tr.d. exp(CD) − lin.d.(CD) ≥ 0.

Indeed,

2tr.d. exp(CD) + 2tr.d.(CD/ exp(CD)) − 2lin.d.(CD) ≥ 2∂(CD) thus 2tr.d. exp(CD)−lin.d.(CD) ≥ 2∂(CD)+lin.d.(CD)−2tr.d.(CD/ exp(CD)) and by Claim 1

2∂(CD) ≥ tr.d.(CD exp CD) − tr.d.(C exp(CD)),

hence

2tr.d. exp(CD) − lin.d.(CD) ≥

≥ lin.d.(CD)−2tr.d.(CD/ exp(CD))+tr.d.(CD exp CD)−tr.d.(C exp(CD)) ≥ ≥ lin.d.(CD) − tr.d.(CD/ exp(CD)) + tr.d.(exp CD) − tr.d.(C exp(CD)) ≥ ≥ lin.d.(CD)−(tr.d.(CD/ exp(CD))+tr.d.(C/ exp(CD))) ≥ |C|+|D|−(|U|+|D|+|U|) = 0. Lemma 6.2 Assume Schanuel’s conjecture. Then, for any finite W ⊆ A∪A^{†} such that {α, iα} ⊆ W, δ(W) = 2tr.d. exp(W) − (lin.d.span_{ℚ}(W) ∩ A + lin.d.span_{ℚ}(W) ∩ A^{†}) ≥ −2.

Proof We may assume that span_{ℚ}(W) ∩ A is generated by a finite set of the form Y ∪ {iα}, ϵα ∈ Y ⊆ ϵℝ, and span_{ℚ}(W) ∩ A^{†} by Z ∪ {iα}, with ϵ̄α ∈ Z ⊆ ϵℝ. Notice that by definitions

{α, iα} ⊆ Y ∪ {iα} and {α, iα} ⊆ Z ∪ {iα}.

By Lemma 6.1

2tr.d.(exp(Y ∪ Z)) ≥ lin.d.Y + lin.d.Z.  (19)

By definitions

tr.d.(exp(Y ∪ Z)) = tr.d.(exp(Y ∪ Z ∪ {α, iα}) = tr.d. exp W

and

lin.d.span_{ℚ}(W) ∩ A = lin.d.Y + 1 and lin.d.span_{ℚ}(W) ∩ A^{†} = lin.d.Z + 1.

Hence the inequality of Lemma follows.□

Corollary 6.1 The structure V(ℂ, ℂ) with V = ℂ, F = ℂ, ex = exp and A, A^{†} as above satisfies axioms (9) - (12) with a D dependimg on the choice of α.

We will denote V_{α} the structure above.

Proposition 6.1 V_{α} satisfies axiom (14).

Proof This is basically the same proof as the proof of the existential closedness axiom in [ZBF].□

Thus we get

Theorem 2 V_{α} is a model of the superstable theory T_{α}.

Problem Prove that some model V of T_{α} can be compactified to analytic Zariski.