Bi-coloured fields on the complex numbers

B.Zilber

August, 2003 ^{∗}

Abstract

We consider two theories of “bad fields” constructed by B.Poizat using Hrushovski’s amalgamation and show that these theories have natural models representable as the field of complex numbers with a distinguished subset given as a union of countably many real analytic curves. One of the two examples is based on the complex exponentiation and the proof assumes Schanuel’s conjecture.

1 Introduction

Following the programme in [Z1] we attempt to find an ’analytic’ interpretation of two-coloured fields of Morley rank ω × 2 with definable multiplicative subgroup [P2] and the similar example with a generic subset from [P1], which have been constructed by B.Poizat using Hrushovski’s method (see also Baldwin and Holland [BH1] and [BH2]). As is usual for Hrushovski’s construction, Poizat produces two sets of axioms: the first axiom scheme states the generalized Schanuel property (in the terminology of [Z1]), the other one is the statement that the structure is existentially closed (in the category of strong embeddings). These axioms along with the other, obvious axioms of the structure determine the complete theory of the bicoloured field. We introduce here two real analytic structures (ℂ, +, ×, P) and (ℂ, +, ×, G), with P and G countable unions of Jordan curves, dense in ℂ, where P is a ’generic’ subset of ℂ and G a multiplicative subgroup. We construct P, the set of ’black’ points, using the Liouville function introduced by Wilkie [W] and also studied by Koiran [K]. We prove that the structure satisfies Poizat’s axioms. Interestingly, we use Wilkie’s theorem on the generalized Schanuel condition for the Liouville function but don’t need Koiran’s theorem which states that the corresponding structure is existentially closed.

∗with later improvements

For G, the subgroup of ’green’ points, we prove the generalized Schanuel condition assuming that the standard Schanuel conjecture is true. By a method very similar to the one for ’black’ points we also prove the existential closedness of the second structure, thus showing that the structures are indeed the same as Poizat’s.

In the notes preceding the present paper I had ’smaller’ versions of P and G both being connected Jordan curves and was able to show the Schanuel conditions for these versions. But Dave Marker pointed out that the second half of the axioms cannot be true with this construction, since a necessary condition for a predicate R ⊆ ℂ containing a Jordan curve to give rise to an ω-stable structure is that R is dense in ℂ. Our present predicates satisfy this condition.

A few comments about the analytic component of the proof are in order. The reader will see that the existential closedness theorem states the solvability of certain systems of analytic equations definable in the structure. For the generic choice of coefficients such a system is solvable for trivial reasons. So, it is enough to find a method to analytically continue the solution from a generic point to a large enough neighborhood. One can do it if the corresponding Jacobian matrix is nondegenerate. Now, crucially, as a consequence of the generalized Schanuel condition, we prove that the set of points where the Jacobian degenerates is very sparse and hence the continuation is possible. Notice similarly the Schanuel conjecture is used in the proof of the solvability of certain systems of exponential equations in [Z4] and indeed for the exponentiation in the reals in [MW]. This presents a good case to treat the general existential closedness condition for analytic and pseudo-analytic structures (see [Z1]) as a dual to the generalized Schanuel condition.

I would also like to mention that the two structures discussed in the paper are very interesting in the real model theory context as well. As a matter of fact, our G contains a connected subgroup G^{0} and is a countable union of cosets of G^{0}, which are Jordan curves. Indeed G^{0} can be identified as the ’spiral’

{e^{t}(cos t + i sin t) : t ∈ ℝ}

studied by C.Miller and P.Speissegger [M] as a subset of ℝ^{2}. The theorem of C. Miller and P. Speissegger states that the real field with the spiral is d-minimal; that is any definable subset of ℝ is a union of an open set and finitely many discrete sets.

This link between real and complex versions of stability seems not to be incidental at all. Another, simpler example is provided by the predicate for the complex roots of unity in the complex field, with or without the reals. In the second case the structure is ω-stable (see [Z3]) while in the first case by a similar proof it has a nice elimination of quantifiers and properties similar to that of the Miller-Speissegger spiral (see [Z5]).

Finally, I would like to mention an important problem regarding other instances of “bad” fields.

Problem. Let T_{P}^{n} and T_{G}^{n} denote the theories of bicoloured fields, with black and green points, of Morley rank ω × n, n > 2, constructed in [P1] and [P2]. Construct explicit predicates P and G, of an ’analytic’ nature, on the complex field ℂ, so that

(ℂ, +, ×, P) ⊨ T_{P}^{n} and (ℂ, +, ×, G) ⊨ T_{G}^{n}.

Notice that if such examples are found, the real dimensions of P and G must be n times less than the real dimension of ℂ, that is equal

to \frac{2}{n}. Thus the theory of subsets of ℝ^{m} of fractional dimension needs to be involved in the construction.

2 Bi-coloured field with “black points”. The generalized Schanuel property

A function

f : ℂ → ℂ

given by

f(x) = Σ_{i≥0}c_{i}x_{i}

is called a Liouville function if c_{i}^{−1} = a_{i} are integers satisfying

for every ℓ ≥ 1, for all sufficiently large i, |a_{i+1}| > |a_{i}|^{i^{ℓ}}

Liouville functions were introduced by A.Wilkie [W] in regard to Problem 2 of [Z1] and the following Schanuel condition has been proved.

Fact 1 [W]. For any distinct nonzero complex numbers z_{1}, …, z_{n}

tr.d.(z_{1}, …, z_{n}, f(z_{1}), …, f(z_{n})) ≥ n,

where tr.d.(X) stands for the transcendence degree of X.

We fix a Liouville function f. Notice that since the coefficients in the definition of f are real, f(x^{c}) = f(x)^{c}, where y^{c} denotes here and below the complex conjugate.

Let ϵ = 1 + i and

P = {f(ϵt + q) : t ∈ ℝ, q ∈ ℚ, (t, q) ≠ (0, 0)}

considered as a unary predicate on ℂ.

Proposition 1 For any distinct p_{1}, …, p_{n} ∈ P,

2tr.d.(p_{1}, …, p_{n}) ≥ n.

That is two times the transcendence degree is not less than the number of points.

Proof Let

T = {ϵt_{1} + q_{1}, …, ϵt_{n} + q_{n}},

with 0 ∉ T, ϵt_{i} + q_{i} ≠ ϵt_{j} + q_{j}, if i ≠ j, t_{i} ∈ ℝ, q_{i} ∈ ℚ, and let f(ϵt_{i} + q_{i}) = p_{i}. We want to prove that

2tr.d.(f(ϵt_{1} + q_{1}), …, f(ϵt_{n} + q_{n})) ≥ n.

Now consider the system

T^{∗} = T ∪ T^{c} = {ϵt_{1} + q_{1}, …, ϵt_{n} + q_{n}, ϵ^{c}t_{1} + q_{1}, …, ϵ^{c}t_{n} + q_{n}}, (1)

where ϵ^{c} = 1 − i. We want to calculate the number of points in T^{∗}.

The only form of equality that may happen between elements of T^{∗} is

ϵt_{i} + q_{i} = ϵ^{c}t_{j} + q_{j}, for some i, j ≤ n.  (2)

Since the sum ϵℝ + ℝ is direct the equality is equivalent to

t_{i} = −t_{j} & t_{i} = \frac{1}{2}(q_{j} − q_{i})

which implies that for each i there is at most one j such that (2) holds and that in this case all the elements involved are algebraic. Thus, if k ≤ n is the number of equalities of the form (2), then

|T^{∗}| = 2n − k, and there is a subset S ⊆ T ∩ aclℚ of size at least k.

Thus,

tr.d.T^{∗} = tr.d.T ≤ n − k.

It follows, that

|T^{∗}| − tr.d.T^{∗} ≥ n.

By properties of Liouville function f

tr.d.(T^{∗} ∪ f(T^{∗})) ≥ |T^{∗}|.

Hence

tr.d.f(T^{∗}) ≥ |T^{∗}| − tr.d.T^{∗} ≥ n.

Recalling that f(ϵt_{i} + q_{i}) = p_{i} and f(ϵ^{c}t_{i} + q_{i}) = p_{i}^{c}, the latter can be rewritten as

tr.d.(p_{1}, …, p_{n}, p_{1}^{c}, …, p_{n}^{c}) ≥ n.

Since complex conjugation is a field isomorphism we have by symmetry

2tr.d.(p_{1}, …, p_{n}) ≥ tr.d.(p_{1}, …, p_{n}, p_{1}^{c}, …, p_{n}^{c}) ≥ n.

□

3 Existential Closedness Theorem for “black points”

In order to prove that (ℂ, +, ×, P) is a model of Poizat’s theory it remains to prove the following

Existential Closedness Theorem Let V ⊆ ℂ^{k} be an irreducible algebraic variety defined over a finitely generated subfield ℚ(C) which has a point ⟨a_{1}, …, a_{k}⟩ ∈ V satisfying:

a_{i} ≠ a_{j} and a_{i} ∉ acl(C), for all i ≠ j, i ≤ k (P -freeness)

and

tr.d.(a_{i_{1}}, …, a_{i_{m}}) ≥ \frac{m}{2}, for any i_{1} < ⋯ < i_{m} ≤ k (P -normality).

Then there is ⟨a_{1}, …, a_{k}⟩ ∈ V ∩ P^{k} generic in V over C.

The rest of the section is devoted to the proof of the theorem.

We assume that V satisfies the assumptions of the theorem. We may assume that k is even, for if k = 2n − 1 we can replace V by V × ℂ ⊆ ℂ^{2n} which is both P -free and P -normal. We also assume dim V = n by the following

Lemma 3.1 Let V ⊆ ℂ^{2n} be a P -free and P -normal irreducible variety over some C_{0} such that dim V = d > n. Then, for any c_{1}, …, c_{2n} ∈ ℂ algebraically independent over C_{0}, an irreducible component V′ of the variety V ∩ H_{c}, for H_{c} = {x ∈ ℂ^{2n} : c_{1} · x_{1} + ⋯ + c_{2n} · x_{2n} = 1}, is P -normal and P -free, and dim V′ = d − 1.

Proof V′ is defined over some acl(C_{0} ∪ {c_{1}, …, c_{2n}}). dim V′ = d − 1, as V ⊈ H_{c}. Let c = ⟨c_{1}, …, c_{2n}⟩ and let a = ⟨a_{1}, …, a_{2n}⟩ be a generic, over C_{0}c, point in V′. Then a is generic in V over C_{0} and tr.d.(c/C_{0}a) = 2n − 1, by elementary calculation.

Claim. Let b = ⟨b_{1}, …, b_{m}⟩ ∈ ℂ^{m} with b_{i} ∈ acl(C_{0}a), all i ≤ m. Then

tr.d.(b/C_{0}) > tr.d.(b/C_{0}c) implies tr.d.(b/C_{0}) = tr.d.(a/C_{0}) = d. Proof. Let U ⊆ ℂ^{2n} be the minimal algebraic variety over acl(C_{0}b) containing c coordinatewise. Suppose the inequality holds. Then tr.d.(c/C_{0}b) < tr.d.(c/C_{0}) and so dim U < 2n. U must contain the hyperplane

H_{a} = {x ∈ ℂ^{2n} : a_{1} · x_{1} + ⋯ + a_{2n} · x_{2n} = 1},

for otherwise dim U ∩H_{a} < dim U contradicting tr.d.(c/C_{0}a) = 2n−1. Hence H_{a} is a component of U and so U = H_{a}.

Notice also that H_{a} must contain a point b′ with coordinates in F_{b} = acl(C_{0}b), a field of definition of H_{a}. This follows from the elimination of quantifiers in the theory of algebraically closed fields.

But then a is the unique vector satisfying (b′, a) = 1 and the equations (z − b′, a) = 0 for all z ∈ H_{a}. Thus a ∈ acl(C_{0}b) coordinatewise. This proves the claim.

Now we can prove that V′ is P -free and P -normal.

Freeness. If a_{i} ∈ acl(C_{0}c) then tr.d.(a_{i}/C_{0}c) < tr.d.(a_{i}/C_{0}) = 1, since V is free. So, by the Claim, with m = 1, we will have tr.d.(a_{i}/C_{0}) = d, which contradicts the condition d > 1.

Normality. Assuming, towards a contradiction, that tr.d.(a_{i_{1}}, …, a_{i_{m}}/C_{0}c) < m/2, we get by normality of V, tr.d.(a_{i_{1}}, …, a_{i_{m}}/C_{0}) > tr.d.(a_{i_{1}}, …, a_{i_{m}}/C_{0}c). Hence, by the claim, tr.d.(a_{i_{1}}, …, a_{i_{m}}/C_{0}c) = tr.d.(a_{1}, …, a_{2n}/C_{0}c) = d − 1 ≥ n ≥ m/2, the contradiction. □

Let N = 2n and V̄ ⊆ ℂ^{N} be an irreducible variety over some C̄ such that, given generic ⟨y_{1}, … y_{N}⟩ ∈ V and a generic ⟨ȳ_{1}, … ȳ_{N}⟩ ∈ V̄ , for any selection of indices 1 ≤ i_{1} < … i_{m} ≤ N,

tr.d._{C}(y_{i_{1}}, … y_{i_{m}}) = tr.d.C̄(ȳ_{i_{1}}, … ȳ_{i_{m}}).

We call such a V̄ a formal conjugate of V.

An example of a formal conjugate V̄ would be V^{c}, the image of V under complex conjugation.

Let W_{s} ⊆ ℂ^{5N} be the algebraic variety in variables x_{1}, … x_{N}, x̄_{1}, … x̄_{N}, y_{1}, … y_{N}, ȳ_{1}, … ȳ_{N}, t_{1}, … t_{N} and parameters C and s_{1}, … s_{N} ∈ ℂ

given by the system of algebraic equations (3) and (4):

y = ⟨y_{1}, … y_{N}⟩ ∈ V, ȳ = ⟨ȳ_{1}, … ȳ_{N}⟩ ∈ V̄ ,  (3)
x_{i} = ϵt_{i} + s_{i}, x̄_{i} = ϵ^{c}t_{i} + s_{i}, i = 1, …, N.  (4)

We are going to consider W_{s} in conjunction with

y_{i} = f(x_{i}), ȳ_{i} = f(x̄_{i}), i = 1, …, N.  (5)

We say that an irreducible variety Z ⊆ ℂ^{2m} over C in variables x_{1}, … x_{m}, y_{1}, … y_{m} is f-free if there is a point ⟨a_{1}, … a_{m}, b_{1}, … b_{m}⟩ ∈ Z such that a_{i} ≠ a_{j} for i ≠ j and a_{i} ∉ acl(C) for any i ≤ m.

We say that Z is also f-normal if for the point above, for any 1 ≤ i_{1} < ⋯ < i_{k} ≤ m,

tr.d._{C}(a_{i_{1}}, …, a_{i_{k}}, b_{i_{1}}, …, b_{i_{k}}) − k ≥ 0.  (6)

Notice that any solution

⟨t_{1}, …, t_{N}, x_{1}, …, x_{N}, x̄_{1}, …, x̄_{N}, y_{1}, …, y_{N}, ȳ_{1}, …, ȳ_{N}⟩

of (3)-(5) is uniquely and explicitly determined by the choice of t_{1}, …, t_{N}. We thus will often refer to t = ⟨t_{1}, …, t_{N}⟩ as a solution of (3)-(5). We call a solution t semi-generic if in the full solution corresponding to t, ⟨y_{1}, … y_{N}⟩ is generic in V over C and ⟨ȳ_{1}, … ȳ_{N}⟩ generic in V̄ over C̄.

Lemma 3.2 Given s_{1}, …, s_{N} ∈ ℂ, let for some U and any u ∈ U t_{i}(u) ∈ ℂ, x_{i}(u) = ϵt_{i}(u)+s_{i}, x̄_{i}(u) = ϵ^{c}t_{i}(u)+s_{i}, for i = 1, …, N, x(u) = ⟨x_{1}(u), … x_{N}(u)⟩, x̄(u) = ⟨x̄_{1}(u), … x̄_{N}(u)⟩. Assume that f(x(u)) ∈ V, f(x̄(u)) ∈ V̄ , t_{i}(u) ≠ t_{j}(u′) if i ≠ j or u ≠ u′, for any u, u′ ∈ U and i, j ≤ N,

and suppose also that t_{1}(u) ∈ acl(ℚ), for all u ∈ U.

Then U is at most finite.

Proof Assume towards a contradiction that U is infinite, say U = ℕ.

We also may assume that ϵt_{i}(u) + s_{i} ≠ ϵ^{c}t_{j}(u′) + s_{j} for all u ≠ u′ and i, j ≤ N, since, given u, i and j, such an equality happens only for at most finitely many values of u′.

Recall that Hrushovski predimension for structure (ℂ, +, ×, f) is

δ(X) = tr.d.(X, f(X)) − |X|, δ(X/C) = δ(X ∪ C) − δ(C).

We can extend C and assume s_{1}, …, s_{N} ∈ C, C̄ = C. Then

tr.d.(f(x(u))/C) ≤ n, tr.d.(f(x̄(u))/C) ≤ n

and

acl(x(u), x̄(u)) ⊆ acl(t_{2}(u), …, t_{N}(u), C),

hence

δ(x(u), x̄(u)/C) ≤ N − 1 + 2n − 2N < 0

for all u. Moreover, by the same argument,

δ(x(u + 1), x̄(u + 1)/C ∪ {x(1), x̄(1), …, x(u), x̄(u)}) < 0.

Hence inductively

δ(C ∪ {x(1), x̄(1), …, x(u), x̄(u)}) ≤ δ(C) − u.

For large enough u this contradicts the Schanuel property of the Liouville function f.□

Corollary 1 There is no continuos function u ↦ t(u) from an open subset of ℂ^{p}, p > 0, non-constant in all coordinates t_{i}(u), such that

⟨f(ϵt_{1}(u) + s_{1}), …, f(ϵt_{N}(u) + s_{N})⟩ ∈ V \ ⋃_{i≠j}{y_{i} = y_{j}}

and

⟨f(ϵ^{c}t_{1}(u) + s_{1}), …, f(ϵ^{c}t_{N}(u) + s_{N})⟩ ∈ V̄ \ ⋃_{i≠j}{y_{i} = y_{j}}.

Proof Otherwise, for every finite A ⊆ ℂ,

⋂_{a∈A, i≤N}{u ∈ U : t_{i}(u) ≠ a}

is open in U. Thus we can have infinitely many instances of u with t_{1}(u) ∈ acl(ℚ) and distinct coordinates, contradicting Lemma 3.2.□

Lemma 3.3 Given s_{1}, …, s_{N}, let t^{0} = ⟨t_{1}^{0}, …, t_{N}^{0}⟩ be a semi-generic solution of (3)-(5). Then there is a neighborhood of t^{0} in ℂ^{N} such that t^{0} is the only solution in the neighborhood.

Proof We use induction on n, 2n = N.

Assume towards a contradiction that t^{0} is not isolated. Notice that full solutions of (3)-(5) form an analytic set A in ℂ^{5N}. It follows from our assumption that the set is of a positive dimension p at the point a^{0} ∈ A corresponding to t^{0}. Representing A as a local analytic covering of a unit disc in ℂ^{p} (see e.g. Theorem 3.7 of [C]) we get a local biholomorphic isomorphism ϕ : U → A between an open subset U of ℂ^{p} and an open subset ϕ(U) of A such that the closure of ϕ(U) contains a^{0}. We may assume that 0 is in the closure of U and a^{0} ∈ lim_{u→0} φ(u), that is a^{0} belongs to the set of limit points. We write t(u), x(u), x̄(u), y(u), ȳ(u) for the corresponding coordinates of the solution ϕ(u), u ∈ U. In particular, with some abuse of notation we right t^{0} = t(0) and a^{0} = ϕ(0) is a semi-generic solution.

By Corollary 1 at least one of t_{i}(u) is constant, equivalently, t′_{i} = 0. It corresponds to the fact that y_{i} is constant in Corollary 1. Notice that this is not possible if n = 1, for then both y_{1} and y_{2} are constant because of algebraic interdependence. This would imply both t_{1}(u) and t_{2}(u) constant, contradicting the assumptions. Thus we assume below that n > 1 (and so N > 2).

Let r be the minimal even number such that

tr.d._{C}(y_{i_{1}}(0), …, y_{i_{r}}(0)) = \frac{r}{2}.  (7)

We may assume {i_{1}, …, i_{r}} = {1, …, r}. Let V^{1} ⊆ ℂ^{r} be the locus of ⟨y_{1}(0), …, y_{r}(0)⟩ over C and V̄^{1} the locus of ⟨ȳ_{1}(0), …, ȳ_{r}(0)⟩ over C̄. It is obvious from assumptions and definitions that dim V^{1} = r/2, half the number of variables, and that V^{1} is P -free and P -normal. Thus V^{1} satisfies the condition of the Lemma. So, by induction, either r = N or t_{1}(u), …, t_{r}(u) are constant and r ≤ N − 2.

Consider the first case, r = N.

By Corollary 1 we assume t_{1} and y_{1} are constants. We also may assume that t_{N−1} and y_{N−1} are not constants.

Notice that

tr.d._{C}(y_{2}(0), …, y_{N−1}(0)) = n = tr.d._{C}(y_{1}(0), y_{2}(0), …, y_{N−1}(0))

(8)

by our assumption and P -normality.

Let V′ ⊆ ℂ^{N−2} be the locus of ⟨y_{2}(0), …, y_{N−1}(0)⟩ over acl(C ∪ {y_{1}(0)}) and let C′ be a finite subset of parameters defining the variety.

Claim 1. dim V′ = n − 1 and V′ is P -normal and P -free.

Proof. By the calculations in (8) y_{1}(0) ∈ acl_{C}(y_{2}(0), …, y_{N−1}(0)). Hence

tr.d._{C,y_{1}(0)}(y_{2}(0), …, y_{N−1}(0)) = tr.d._{C}(y_{1}(0), y_{2}(0), …, y_{N−1}(0))−1 = n−1.

Freeness. Suppose towards a contradiction that for some non-zero 1 < i ≤ N − 2, y_{i}(0) ∈ acl(C, y_{1}(0)). Then tr.d._{C}(y_{1}(0), y_{i}(0)) = 1, contradicting r = N > 2. This contradicts the assumption of the claim and proves that V′ is P -free.

For normality we need to show that for any 1 < i_{1} < ⋯ < i_{ℓ} ≤ N − 2

tr.d._{C′}(y_{i_{1}}(0), …, y_{i_{ℓ}}(0)) ≥ \frac{ℓ}{2}.

For any odd ℓ < N − 2 we have by assumption,

tr.d._{C}(y_{1}(0), y_{i_{1}}(0), …, y_{i_{ℓ}}(0)) ≥ \frac{ℓ + 1}{2} + 1 ≥ \frac{ℓ}{2} + 1.

For an even ℓ < N − 2, letting ℓ′ = ℓ + 1 and considering above an extra term y_{i_{ℓ+1}} we then have

tr.d._{C}(y_{1}(0), y_{i_{1}}(0), …, y_{i_{ℓ}}(0)) ≥ tr.d._{C}(y_{1}(0), y_{i_{1}}(0), …, y_{i_{ℓ}}(0), y_{i_{ℓ+1}}(0))−1 ≥

≥ \frac{ℓ′ + 1}{2} = \frac{ℓ}{2} + 1.

Hence in either case

tr.d._{C′}(y_{i_{1}}(0), …, y_{i_{ℓ}}(0)) ≥ \frac{ℓ}{2},

which proves the claim.

Since t_{2}(u), …, t_{N−1}(u), y_{2}(u), …, y_{N−1}(u), ȳ_{2}(u), …, ȳ_{N−1}(u) is a solution of the system corresponding to V′, we get a contradiction with the induction hypothesis. Hence the case r = N is eliminated and we are left with r ≤ N − 2 and t_{1}(u), …, t_{r}(u) all constant.

Let w(u) = {w_{1}(u), …, w_{2m}(u)⟩ be y_{r+1}(u), …, y_{N}(u), 2m = N − r, at least one of them non-constant by assumptions. Let V′′ ⊆ ℂ^{2m} be the locus of ⟨w_{1}(0), …, w_{2m}(0)⟩ over acl(C∪{y_{1}(0), …, y_{r}(0)}) and C′′ the finite set of parameters for V′′.

Claim 2. dim V′′ = m, V′′ is P -free and P -normal.

Proof. We have

dim V′′ = tr.d.(y_{r+1}(0), …, y_{N}(0)/C ∪ {y_{1}(0), …, y_{r}(0)) =

= tr.d._{C}(y_{1}(0), …, y_{N}(0))−tr.d._{C}(y_{1}(0), …, y_{r}(0)) = N/2−r/2 = m.

Freeness. We need to show that for any i ≤ 2m, w_{i}(0) ∉ acl(C′′). Assuming w_{i}(0) = y_{j}(0) ∈ acl(C′′), some j ≥ r + 1, we will get

tr.d._{C}(y_{1}(0), …, y_{r}(0), y_{j}(0)) = \frac{r}{2},

contradicting P -normality of V.

Normality. For i_{1} < ⋯ < i_{l} ≤ 2m by P -normality of V

tr.d._{C}(y_{1}(0) …, y_{r}(0), w_{i_{1}}(0), …, w_{i_{l}}(0)) ≥ \frac{l + r}{2} ,

hence

tr.d._{C′′}(w_{i_{1}}(0), …, w_{i_{l}}(0)) ≥ \frac{l + r}{2} − tr.d._{C}C′′ ≥ \frac{l + r}{2} − \frac{r}{2} = \frac{l}{2}.

Claim proved.

The claim contradicts the induction hypothesis. □

Lemma 3.4 Given s^{0} ∈ ℂ^{N} and a corresponding semi-generic solution t^{0} ∈ ℂ^{N} of the system (3)-(5) there are neighborhoods S of s^{0} and T of t^{0} such that for every s ∈ S there is a unique t(s) ∈ T, satisfying (3)-(5). Moreover s ↦ t(s) is a holomorphic function on S.

Proof Let W ⊆ ℂ^{6N} be the algebraic variety in variables s_{i}, t_{i}, x_{i}, x̄_{i}, y_{i}, ȳ_{i}, i = 1, …, N, defined by

(w, s) ∈ W iff w ∈ W_{s}.

Let

W = W ∩ {f(x) = y & f(x̄) = ȳ}.

We assume C = C̄. By assumption a connected component W^{0} of W containing (w^{0}, s^{0}) with w^{0} given by t^{0} is not in any algebraic subvariety of W of the form W ∩ {y ∈ Y ∨ ȳ ∈ Ȳ } for Y, Ȳ formally conjugated proper algebraic subvarieties of V and V̄ over C.

Let W_{s}^{0} ⊆ ℂ^{5N} × {s} be the fiber over s of the projection map (w, s) ↦ s on W^{0} and let (w_{0}, s) ∈ W_{s}^{0} be an isolated generic point of W_{s}^{0}. By Lemma 3.3 the set

W_{s}^{0} \ ⋃{y ∈ Y ∨ ȳ ∈ Ȳ : Y, Ȳ proper, defined over C}

consists of isolated points, hence is open in W_{s}^{0}. Thus, for an open polydisc D ⊆ ℂ^{5N} containing w_{0}, for any proper algebraic C-definable subvarieties W_{Y} = {y ∈ Y ∨ ȳ ∈ Ȳ } of W

(D × {s}) ∩ W^{0} ∩ W_{Y} = ∅, but (D × {s}) ∩ W^{0} ≠ ∅.

By continuity, for some open S, for any W_{Y} ,

(D × S) ∩ W^{0} ∩ W_{Y} = ∅.

Hence (D × {s}) ∩ W^{0}, for each s ∈ S, consists of isolated points of W_{s}^{0} and so the projection (D × S) ∩ W^{0} → S has 0-dimensional fibers. For D and S small enough we get singleton fibers and the projection becomes bijective. By [C], 1.3.3, Proposition 3, the map s ↦ w(s), giving the unique point of the fiber, is holomorphic. Finally recall that any point (w, s) ∈ W_{s}^{0} is determined by its projection on t-coordinates. □

Proof of the Existential Closedness Theorem

Now let V̄ be V^{c}. Let y^{0} be a generic point of the algebraic variety V and x^{0} ∈ ℂ^{N}, f(x^{0}) = y^{0}.

Since ℂ = ϵℝ+ℝ, we can represent x_{i}^{0} = ϵt_{i}^{0}+s_{i}^{0} for some t_{i}^{0}, s_{i}^{0} ∈ ℝ, i = 1, …, N. It follows that t^{0}, x^{0}, x̄^{0}, y^{0}, ȳ^{0}, where x̄^{0} and ȳ^{0} are the complex conjugates of x^{0} and y^{0} correspondingly, satisfy (3)-(5) and form a semi-generic solution.

For S and T of Lemma 3.4 consider s ∈ ℝ^{N} ∩ S and t = t(s). We claim that t has to be real as well. Indeed, consider the full solution (t, x, x̄, y, ȳ, z) of (3)-(5) with parameters s. Then, by the form of the equation, (t^{c}, x̄^{c}, x^{c}, ȳ^{c}, y^{c}, z^{c}) is a solution of the same system.

For small S the t is close to t^{0}, a real number, hence t^{c} is close to t^{0} as well. Thus t^{c} ∈ T and is a solution of (3)-(5) corresponding to s. We then have by uniqueness t^{c} = t(s), which proves that t is real.

It remains now to choose a rational point q ∈ S and consider the solution t(q) of the system (3)-(5). Since t_{i}(q) are real and q_{i} ∈ ℚ we have that y_{i}(q) = f(ϵt_{i} + q_{i}) ∈ P and ⟨y_{1}(q), …, y_{N}(q)⟩ ∈ V. □

Corollary 2 The complete theory of (ℂ, +, ×, P) is axiomatised by Poizat’s axioms. The theory is ω-stable with MR(ℂ) = ω × 2 and MR(G) = ω.

Proof Poizat’s axiomatisation (Proposition 2.1 of [P1]) consists of an obvious description of an algebraically closed field of characteristic zero with a subset P along with two lists of non-obvious axioms, ∀- axioms and ∀∃-axioms. ∀-axioms are equivalent to the statement of our Proposition 1. ∀∃-axioms are equivalent to the statement that for any V satisfying the assumptions of the Existential Closedness Theorem and any algebraic subvariety V′ ⊂ V of smaller dimension there is a point a ∈ (V \V′)∩G^{k}. Our Existential Closedness Theorem is formally stronger as a it provides is even generic over any given C, in fact, it is equivalent to ∀∃-axioms when applied to an ω-saturated model.

The statements on Morley rank is the property of the first order theory shown to be ω2 by Poizat. □

4 The field with “green points”

Let

G = {exp(ϵt + q) : t ∈ ℝ, q ∈ ℚ}

considered as a unary predicate on ℂ. This is a multiplicative torsionfree subgroup of ℂ^{∗}. It has a subgroup

G^{0} = {exp(ϵt) : t ∈ ℝ}

which can be naturally considered as the “connected component” of G.

For a finite subset X of a torsion-free abelian group, the subgroup generated by X is obviously embeddable into a vector space over ℚ of a large enough dimension. The group rank grk(X) is defined as the size of a maximal independent subset of X in the vector space, equivalently, the dimension of the divisible hull of X as a vector space.

We recall that the standard Schanuel conjecture states that for any additively independent complex numbers z_{1}, …, z_{n}

tr.d.(z_{1}, …, z_{n}, exp(z_{1}), …, exp(z_{n})) ≥ n.

Proposition 2 Assume Schanuel’s conjecture. Then, for any h_{1}, …, h_{n} ∈ G,

2tr.d.(h_{1}, …, h_{n}) ≥ grk(h_{1}, …, h_{n}).

Proof We may assume that h_{1}, …, h_{n} are multiplicatively independent, h_{i} = exp(ϵt_{i} + q_{i}), thus ϵt_{1} + q_{1}, …, ϵt_{n} + q_{n} are additively independent. We want to prove that

2tr.d.(h_{1}, …, h_{n}) ≥ n.

Notice that any ℤ-linear transformations of the system

T = {ϵt_{1} + q_{1}, …, ϵt_{n} + q_{n}}  (9)

corresponds to a multiplicative transformation of h_{1}, …, h_{n}. Thus, a ℤ-linear non-degenerate transformation does not change the inequality in the statement of the proposition. By applying such a transformation we can assume that q_{1} = … q_{n−1} = 0 in (9). It follows, in particular, that t_{1}, …, t_{n−1} are ℚ-linearly independent.

Now consider the system

T^{∗} = {ϵt_{1}, …, ϵt_{n−1}, ϵt_{n} + q_{n}, ϵ^{c}t_{1}, …, ϵ^{c}t_{n−1}, ϵ^{c}t_{n} + q_{n}}  (10)

where ϵ^{c} = 1 − i, the result of complex conjugation. We want to calculate the ℚ-linear dimension lin.d. T^{∗} as well as the transcendence degree of the system T^{∗}.

Since ϵ and ϵ^{c} are ℝ-linearly independent,

span_{Q}{ϵt_{1}, …, ϵt_{n−1}, ϵ^{c}t_{1}, …, ϵ^{c}t_{n−1}} = span_{Q}{t_{1}, …, t_{n−1}, it_{1}, …, it_{n−1}}

is of ℚ-linear dimension 2n − 2 and

lin.d.T^{∗} = 2n − 2 iff q_{n}, t_{n} ∈ span_{Q}(t_{1}, …, t_{n−1}).  (11)

Either of conditions in (11) imply that tr.d.T^{∗} ≤ n − 2.

Notice further that

tr.d.T^{∗} = tr.d.(t_{1}, …, t_{n})

hence, if lin.d. T^{∗} < 2n we get tr.d.T^{∗} < n. It follows, that under all circumstances

lin.d.T^{∗} − tr.d.T^{∗} ≥ n.

By the Schanuel conjecture

tr.d.(T^{∗} ∪ exp T^{∗}) ≥ lin.d.T^{∗}.

Hence

tr.d. exp T^{∗} ≥ lin.d. T^{∗} − tr.d.T^{∗} ≥ n.

Remembering that exp(ϵt_{i} +q_{i}) = h_{i} and exp(ϵ^{c}t_{i} +q_{i}) = h_{i}^{c}, the latter can be rewritten as

tr.d.(h_{1}, …, h_{n}, h_{1}^{c}, …, h_{n}^{c}) ≥ n.

Since complex conjugation is a field isomorphism we have by symmetry

2tr.d.(h_{1}, …, h_{n}) ≥ n.

□

5 Existential Closedness Theorem for “green points”

Let V ⊆ ℂ^{k} be an irreducible algebraic variety defined over a finitely generated subfield ℚ(C) and a = ⟨a_{1}, …, a_{k}⟩ ∈ V is a generic point of V over C.

We will say that V is G-free if

a_{1}^{m_{1}} · . . . a_{k}^{m_{k}} ∉ acl(C), for any ⟨m_{1}, …, m_{k}⟩ ∈ ℤ^{k} \ {0}.

We will say that V is weakly G-normal if for any 1 ≤ i_{1} < ⋯ < i_{r} ≤ k

tr.d._{C}(a_{i_{1}}, …, a_{i_{r}}) ≥ \frac{r}{2}.

Given a non-degenerate (k × k)-matrix M over ℤ (that is with linearly independent rows m_{1}, …, m_{k} ∈ ℤ^{k} ) we consider a transformation

ϕ_{M} : (ℂ^{∗})^{k} → (ℂ^{∗})^{k} : y ↦ y^{M}

where

y = ⟨y_{1}, …, y_{k}⟩, y^{M} = ⟨y^{m_{1}}, …, y^{m_{k}}⟩ and y^{m_{i}} = y_{1}^{m_{i1}} · · · · · y_{k}^{m_{ik}}.

ϕ_{M} is a local analytic automorphism of the torus. The image ϕ_{M}(V) is a dense subset of an irreducible algebraic variety defined over C, which we denote V^{M}. We shall call a transformation of the form ϕ_{M} an admissible transformation.

A generic point a of V corresponds to a^{M}, a generic point of V^{M}.

Remark It follows immediately from definitions that V is G-free if and only if V^{M} is.

We will say that V is G-normal if V^{M} is weakly G-normal for every non-degenerate integer matrix M.

In order to prove that (ℂ, +, ×, G) is a model of Poizat’s theory we prove the following

Existential Closedness Theorem Let V ⊆ ℂ^{k} be an irreducible algebraic variety defined over a finitely generated subfield ℚ(C). Assume that V is G-normal and G-free. Then there is ⟨a_{1}, …, a_{k}⟩ ∈ V ∩ G^{k} generic in V over C.

Below in this section V is a a variety satisfying the assumptions of the theorem. We may also assume as in the previous section that k = 2n. We also assume dim V = n by the following analog of Lemma 3.1

Lemma 5.1 Let V ⊆ ℂ^{2n} be a G-free and G-normal irreducible variety over some C_{0} such that dim V = d > n. Then for any c_{1}, …, c_{2n} ∈ ℂ algebraically independent over C_{0} an irreducible component V′ of the variety V ∩ H_{c}, for H_{c} = {y ∈ ℂ^{2n} : c_{1} · y_{1} + ⋯ + c_{2n} · y_{2n} = 1}, is G-normal and G-free and dim V′ = d − 1.

Proof V′ is defined over some C_{1} ⊆ acl(C_{0} ∪ {c_{1}, …, c_{2n}}). Then dim V′ = k − 1, as V ⊈ H_{c}. Let c = ⟨c_{1}, …, c_{2n}⟩ and a be a generic over C_{0}c point in V′. Then a is generic in V over C_{0} and tr.d.(c/C_{0}a) = 2n − 1.

Let M_{1}, …, M_{k} ∈ ℤ^{2n} be linearly independent.

Claim. tr.d.(a^{M_{1}}, …, a^{M_{k}}/C_{0}) > tr.d.(a^{M_{1}}, …, a^{M_{k}}/C_{0}c) implies tr.d.(a^{M_{1}}, …, a^{M_{k}}/C_{0}) = tr.d.(a/C_{0}) = d. This is a special case of the claim in the proof of Lemma 3.1. Now, to see that V′ is G-free consider a non-zero M ∈ ℤ^{2n}. If

a^{M} ∈ acl(C_{0}c) then tr.d.(a^{M}/C_{0}c) < tr.d.(a^{M}/C), since V is free, and so by the claim above, tr.d.(a^{M}/C_{0}) = d, which contradicts the condition d > 1.

Normality: Assuming, towards a contradiction, that for some linearly independent M_{1}, …, M_{k} ∈ ℤ^{2n}, tr.d.(a^{M_{1}}, …, a^{M_{k}}/C_{0}c) < k/2 we get by normality of V tr.d.(a^{M_{1}}, …, a^{M_{k}}/C_{0}) > tr.d.(a^{M_{1}}, …, a^{M_{k}}/C_{0}c). Hence by the claim tr.d.(a^{M_{1}}, …, a^{M_{k}}/C_{0}c) = tr.d.(a_{1}, …, a_{2n}/C_{0}c) = d − 1 ≥ n, the contradiction. □

In full analogy to section 2 we use notation N = 2n and V̄ for a variety over some C̄ with the property that given generic ⟨y_{1}, … y_{N}⟩ ∈ V, a generic ⟨ȳ_{1}, … ȳ_{N}⟩ ∈ V̄ , for any selection of integer vectors M_{1}, … M_{l} ∈ ℤ^{N},

tr.d._{C}(y^{M_{1}}, … y^{M_{l}}) = tr.d.C̄(ȳ^{M_{1}}, … ȳ^{M_{l}}).

Again, an example of such a V̄ would be V^{c}, the image of V under complex conjugation.

We also consider the variety W_{s} ⊆ ℂ^{5N} over parameters C and s ∈ ℂ^{N} given by equations (3) and (4) (with the new meaning of V and V̄) and a system of equations

y_{i} = exp(x_{i}), ȳ_{i} = exp(x̄_{i}), i = 1, …, N.  (12)

Lemma 5.2 Given s_{1}, …, s_{N} ∈ ℂ, let for an index set U, for each u ∈ U, x_{i}(u) = ϵt_{i}(u) + s_{i}, x̄_{i}(u) = ϵ^{c}t_{i}(u) + s_{i}, for i = 1, …, N,

x(u) = ⟨x_{1}(u), … x_{N}(u)⟩, x̄(u) = ⟨x̄_{1}(u), … x̄_{N}(u)⟩.

Assume that for all u

exp(x(u)) ∈ V, exp(x̄(u)) ∈ V̄ , {t_{1}(u), …, t_{N}(u)} is ℚ-linearly independent over C ∪ C̄ ∪ {s_{1}, …, s_{N}} and x(u) ≠ x(u′) if u ≠ u′, for any u, u′ ∈ U.

Then U is at most countable.

Proof Let C′ = C ∪ C̄ ∪ {s_{1}, …, s_{N}}. Obviously, for each u

lin.d.(x(u), x̄(u)/C′) = lin.d.({ϵt_{1}(u), …, ϵt_{N}(u), ϵ^{c}t_{1}(u), …, ϵ^{c}t_{N}(u)}/C′) =

= lin.d.({t_{1}, …, t_{N}, it_{1}, …, it_{N}}/C′) = 2lin.d.({t_{1}, …, t_{N}}/C′) = 2N

by assumptions.

We also have

(exp(x(u)), exp(x̄(u)) ∈ V ×V̄ , acl(x(u), x̄(u)) ⊆ acl(t_{1}(u), …, t_{N}(u), C′).

Thus,

tr.d.(exp(x(u)), exp(x̄(u)/C′) ≤ 2n = N, tr.d.(x(u), x̄(u)/C′) ≤ N.

Define a predimension for the structure ℂ_{exp} = (ℂ, +, ·, exp) as in [Z2]

δ(X) = tr.d.(X ∪ exp(X)) − lin.d.X, δ(X/C) = δ(X ∪ C) − δ(C).

By above we have

δ(x(u), x̄(u)/C′) ≤ 0

for all u. It follows that x(u) ∈ cl_{F} (C) as defined in [Z2], section 4, for some exponentially-algebraically closed field F with pseudoexponentiation, extending ℂ. By Lemma 5.7 of [Z2] (countable closure property) which only needs the Schanuel conjecture for ℂ, the set U has to be at most countable. □

Lemma 5.3 For a continuous function u ↦ t(u) from an open subset U ⊆ ℂ^{p}, non-constant in all coordinates t_{i}(u), such that

x(u) = ⟨exp(ϵt_{1}(u) + s_{1}), …, exp(ϵt_{N}(u) + s_{N})⟩ ∈ V

and

x̄(u) = ⟨exp(ϵ^{c}t_{1}(u) + s_{1}), …, exp(ϵ^{c}t_{N}(u) + s_{N})⟩ ∈ V̄ ,

there is a proper ℚ-affine subspace

L_{m}(d) = {t ∈ ℂ^{N} : m_{1}t_{1} + ⋯ + m_{N}t_{N} = d},

some d ∈ span_{Q}C′ (C′ = C ∪ C̄ ∪ {s_{1}, …, s_{N}})) such that

J_{m}(d) = {u ∈ U : ⟨t_{1}(u), …, t_{N}(u)⟩ ∈ L_{m}(d)}

contains an open subset.

Proof Suppose that for any m ∈ ℤ^{N} \ 0 and d ∈ span_{Q}C′ the set J_{m}(d) is nowhere dense in U. Then the set

U′ = U \ ⋃{J_{m}(d) : m ∈ ℤ^{N} \ 0, d ∈ span_{Q}C′}

is the complement of the union of countably many nowhere-dense subsets, so is of cardinality continuum. For any u ∈ U′ by construction the coordinates of t(u) are linearly independent over C′. This contradicts Lemma 5.2.□

Lemma 5.4 Given s_{1}, …, s_{N}, let t^{0} = ⟨t_{1}^{0}, …, t_{N}^{0}⟩ be a semi-generic solution of (3),(4) and (12). Then there is a neighborhood of t^{0} in ℂ^{N} such that t^{0} is the only solution in the neighborhood.

Proof Assume towards a contradiction that t^{0} is not isolated. By repeating the argument in the proof of Lemma 3.3 we then are given an open disc U, the point 0 belonging to its closure, and an analytic map from U into the set of solutions, such that t^{0} = t(0) and (t(0), x(0), x̄(0), y(0), ȳ(0)) is a semi-generic solution of (3),(4) and (12).

We are going to show by induction on n, 2n = N, that this is not possible.

By Lemma 5.3 there is a non-zero integer tuple m, a complex number d and an open U_{m}(d) ⊆ U such that at least one of

∑_{1≤i≤N} m_{i}t_{i}(u) = d for all u ∈ U_{m}(d),

or equivalently

∑_{1≤i≤N} m_{i}t′ = 0 on the open set.

It corresponds to the fact that

∏_{1≤i≤N} y_{i}^{m_{i}}(u) = exp ∑_{1≤i≤N} m_{i}(ϵt_{i}(u) + s^{0})

is constant, where the string y(u) is in V.

We can now replace variables in a non-degenerate way by introducing, say

t̂_{1} = ∑_{1≤i≤N} m_{i}t_{i} and ŷ_{1} = ∏_{1≤i≤N} y_{i}^{m_{i}}

and leaving the remaining unchanged. This corresponds to an admissible transformation

ϕ : ⟨y_{1}, …, y_{n−1}, y_{n}, y_{n+1}, …, y_{N}⟩ ↦ ⟨ ∏_{1≤i≤N} y_{i}^{m_{i}}, y_{2}, …, y_{N}⟩.

Thus we assume w.l.o.g. that ŷ_{1} = y_{1} is constant for u in the open set, and so is t_{1}. Notice that this is not possible if n = 1 by the same argument as in the proof of Lemma 3.3. Thus we assume below that N > 2.

Let r be the minimal even number such that for some M_{1}, …, M_{r} ∈ ℤ^{N} linearly independent

tr.d._{C}(y^{M_{1}}(0), …, y^{M_{r}}(0)) = \frac{r}{2}.  (13)

The equality is satisfied by r = N, hence the minimal r exists.

By an admissible transformation of variables we may assume that ⟨y^{M_{1}}, …, y^{M_{r}}⟩ = ⟨y_{1}, …, y_{r}⟩.

Let V^{1} ⊆ ℂ^{r} be the locus of ⟨y_{1}(0), …, y_{r}(0)⟩ over C and V̄^{1} the locus of ⟨ȳ_{1}(0), …, ȳ_{r}(0)⟩ over C̄. It is obvious from assumptions and definitions that dim V^{1} = r/2, half the number of variables, and that V^{1} is G-free and G-normal. Thus V^{1} satisfies the condition of the Lemma. So, by induction, either r = N or t_{1}(u), …, t_{r}(u) are constant and r ≤ N − 2.

Consider the first case, r = N.

We assume t_{1} and y_{1} are constant and t_{N−1} and y_{N−1} are not constant.

Similarly to (8) we notice that

tr.d._{C}(y_{2}(0), …, y_{N−1}(0)) = n = tr.d._{C}(y_{1}(0), y_{2}(0), …, y_{N−1}(0)).

(14)

Let V′ ⊆ ℂ^{N−2} be the locus of ⟨y_{2}(0), …, y_{N−1}(0)⟩ over acl(C ∪ {y_{1}(0)}) and let C′ be a finite subset of parameters defining the variety.

Claim 1. dim V′ = n − 1 and V′ is G-normal and G-free.

Proof. dim V′ = n − 1 by the same reasons as in the proof of Lemma 3.3.

Freeness. Suppose towards a contradiction that for some non-zero M ∈ ℤ^{N}, m_{1} = 0, y^{M}(0) ∈ acl(C, y_{1}(0)). Then tr.d._{C}(y_{1}(0), y_{M}(0)) = 1 and r = 2. This contradicts the assumption of the claim.

For normality we need to show that for any M_{1}, …, M_{ℓ} ∈ ℤ^{N−2} linearly independent ( ℓ ≤ N − 2) and z(0) = ⟨y_{2}(0), …, y_{k}(0)⟩

tr.d._{C′}(z^{N_{1}}(0), …, z^{N_{ℓ}}(0)) ≥ \frac{ℓ}{2},

where C′ ⊆ acl(C ∪ {y_{1}(0)}) is a finite set of parameters needed to define V′.

We use the same calculations as in the proof of Claim 1 of Lemma 3.3 to show this and so prove the claim.

The Claim eliminates the case r = N and we assume again that r ≤ N − 2 and t_{1}(u), …, t_{r}(u) are all constant.

Let w(u) = {w_{1}(u), …, w_{2m}(u)⟩ be y_{r+1}(u), …, y_{N}(u), 2m = N − r, at least one of them non-constant by assumptions. Let V′′ ⊆ ℂ^{2m} be the locus of ⟨w_{1}(0), …, w_{2m}(0)⟩ over acl(C∪{y_{1}(0), …, y_{r}(0)}) and C′′ the finite set of parameters for V′′.

Claim 2. dim V′′ = m and V′′ is G-free and G-normal.

Proof. dim V′′ = m by the same calculation as in the proof of Claim 2 of 3.3.

Freeness. Up to transformation of variables we need to show that w_{i}(0) ∉ acl(C′′). Assuming w_{i}(0) = y_{j}(0) ∈ acl(C′′), some j ≥ r + 1, we will get

tr.d._{C}(y_{1}(0), …, y_{r}(0), y_{j}(0)) = \frac{r}{2},

contradicting G-normality of V.

Normality. For linearly independent M_{1}, …, M_{ℓ} ∈ ℤ^{2m} by Gnormality of V

tr.d._{C}(y_{1}(0) …, y_{r}(0), w^{M_{1}}(0), …, w^{M_{ℓ}}(0)) ≥ \frac{ℓ + r}{2} ,

hence

tr.d._{C′′}(w^{M_{1}}(0), …, w^{M_{ℓ}}(0)) ≥ \frac{ℓ + r}{2} − tr.d._{C}C′′ ≥ \frac{ℓ + r}{2} − \frac{r}{2} = \frac{ℓ}{2}.

Claim proved.

The Claim contradicts the induction hypothesis and finishes the proof of the lemma. □

Proof of the Existential Closedness Theorem

We first recall the statement of Lemma 3.4 with f replaced by exp, with the same proof.

In the same way the final part of the proof of the Existential Closedness Theorem for P is valid for G. □

Corollary 3 The complete theory of (ℂ, +, ×, G) is axiomatised by Poizat’s axioms. The theory is ω-stable with MR(ℂ) = ω × 2 and MR(G) = ω.

Proof Poizat’s axiomatisation consists of the obvious description of an algebraically closed field of characteristic zero with a torsion-free subgroup G along with two lists of non-obvious axioms, ∀-axioms and ∀∃-axioms (section 3 of [P2]). ∀-axioms, as proved in his paper, are equivalent to the statement of our Proposition 2. ∀∃-axioms are equivalent to the statement that for any V satisfying the assumptions of the Existential Closedness Theorem and any algebraic subvariety V′ ⊂ V of smaller dimension there is a point a ∈ (V \V′)∩G. This is provided by our Existential Closedness Theorem. □

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