Abstract

Author: B. Zilber

Title: Raising to powers revisited

email: zilber@maths.ox.ac.uk

Mathematics Subject Classification 2000: 03C98, 11J81

We consider the theory of algebraically closed fields of characteristic zero with raising to powers operations. In an earlier paper we have described complete first-order theories of such a structures, provided that a diophantine conjecture CIT does hold. Here we get rid of this assumption. The theory of complex numbers with raising to real powers satisfies the description if Schanuel’s conjecture holds. In particular, we have proved that a (weaker) version of Schanuel’s conjecture implies that every well-defined system of exponential sums with real exponents has a solution. Recent result by Bays, Kirby and Wilkie states that the required version of Schanuel’s conjecture holds for almost every choice of exponents. It follows that for the corresponding choice of real exponents we have an unconditional description of the first order theory of the complex numbers with raising to these powers.

Raising to powers revisited

Boris Zilber

26 April, 2011

1 Introduction

In this paper we return to the two-sorted structures studied in [6], analogues of the field of complex numbers with the multivalued operations x^{k} = exp(k ln(x)), for k ∈ K ⊆ ℂ. This operation can be better presented in the structure corresponding to the diagram

ℂ ⟶^{exp} ℂ,

where the complex numbers on the left and on the right are seen as two distinct sorts. The right-hand sort is endowed with the usual field structure, whereas the left-hand one is considered a vector space over a subfield K ⊆ ℂ of finite transcendence degree, with one function symbol in the language for each transformation v ↦ k · v, for each k ∈ K. Naturally, for x in the right-hand sort, ln x is defined as exp^{−1}(x), a subset of the left-hand sort. Now x^{k}, k ∈ K, means exp(kv) for some v ∈ ln(x), so ’raising to powers’ is definable in the two-sorted structure.

The abstract structure of raising to powers consists of two sorts V and F^{×}, with F^{×} given a structure of 𝔽\{0}, 𝔽 a field of characteristic zero and V a vector space over a field K of characteristic zero. There is also a map exp which is assumed to be a homomorphism of the additive group V into the multiplicative group of F^{×},

V ⟶^{exp} F^{×}.

In [6] we used a version of Hrushovski’s amalgamation process to construct, for each K, a class of algebraically closed fields with raising to powers K, denoted 𝔽^{K}, satisfying (a corollary of) Schanuel’s conjecture and showed, using CIT, the Conjecture on Intersecting varieties with Tori (see [7]), that any completion of the theory is near model complete and superstable.

Here we refine this result by proving essentially the same statement without any assumptions. The use of CIT is avoided by using the solution of Mordell problem (Falting’s Theorem) about finitely generated subgroups of semi-abelian varieties, in fact for (ℂ^{×})^{n}.

Our main theorem gives, for every 𝔽^{K}, a set of axioms that determines a complete first order theory of 𝔽^{K}, which we also prove to be near model complete and superstable.

The present version of the theorem can be used to study the effect of Schanuel’s conjecture. Let ℂ^{K} denote the field of complex numbers with raising to powers K ⊆ ℂ.

We prove (Theorem 6.2) that in case K ⊆ ℝ, if ℂ^{K} satisfies the corresponding version of Schanuel’s conjecture, it is exponentiallyalgebraically closed, that is any well-defined system of exponential sums equations with exponents in K has a solution in ℂ. The conjecture that ℂ_{exp}, the field of complex numbers with exp, is exponentiallyalgebraically closed was made in [8]. This has been studied by a number of researchers including attempts to refute the conjecture. Theorem 6.2 brings hopes that in general exponential-algebraic closedness follows from Schanuel’s conjecture implies . So far we don’t know if this is true even for ℂ^{K}, when K is not a subfield of the reals.

Another important corollary of the main theorem (Theorem 4.16) states that under Schanuel’s conjecture for ℂ^{K} solutions to an overdetermined system of exponential sums equations lie in a finitely many, modulo 2πi, cosets of proper ℚ-linear subspaces. Moreover, there is a bound on the number of such cosets, uniform in coefficients of the system (but possibly not on the exponents).

Finitely, we invoke a recent result by M.Bays, J.Kirby and A.Wilkie that implies that for ”almost any” tuple λ in ℂ, for K = ℚ(λ), the structure ℂ^{K} satisfies the corresponding version of Schanuel’s conjecture. Thus, the above theorems are applicable to such ℂ^{K} unconditionally. In particular, when also λ ⊆ ℝ, we know the complete theory of ℂ^{K}.

Of course, Faltings’ Theorem has been proven in much greater generality and, as in [6], one can easily replace F^{×} by any semiabelian variety A and carry out the same construction and axiomatisation since also a corresponding analogue of Ax’s Theorem and its corollaries is available. More precisely, one needs the following (weak CIT) to hold for A.

1.1 Theorem (J.Kirby, [5]) Given W (e) ⊆ A^{n}, with W ⊆ A^{n+l} algebraic subvariety defined over k and e ∈ A^{l} there are finitely many codimension 1 EndA-linear subspaces μ(W) = {M_{1}, …, M_{m}} of V^{n} such that for any EndA-linear subspace N ⊆ V^{n}, b ∈ A^{l}, and any positive dimensional atypical component S of the intersection W (e) ∩ exp(N) · b there is M ∈ μ(W) and s ∈ S such that S ⊆ exp(M) · s.

Here, atypical for an irreducible component S of the intersection of algebraic subvarieties W (e) and exp(N)·b of A^{n} (observe that exp(N) is an algebraic subgroup of A^{n}) means that

dim S > dim W (e) + dim exp(N) − dim A^{n}.

To prove our main result in full generality, for semi-abelian A, we would need to consider V as an EndA-module which, for dim A = g > 1, contains divisors of zero and makes the linear structure on V more involved. This would complicate definitions and some argument, without visible advantages for applications. But the proof below goes through practically without changes for any elliptic curve without complex multiplication defined over ℚ.

2 Definitions and notation

2.1 This section along with definitions and notations discusses basic ingredients of Hrushovski’s construction which is standard enough, so the reader can guess the proofs if they seem too short or are absent.

We use here some of the terminology of [7], slightly improved, where we discussed K-linear and affine spaces, tori and their intersections with algebraic varieties.

For technical reasons we find it more convenient to represent the two-sorted structures (V, F^{×}) in the equivalent way as one sorted structures in the language L_{K} which is the extension of the language of vector spaces over ℚ by:

• an equivalence relation E,

• n-ary predicates L(x_{1}, …, x_{n}) for linear subspaces L ⊆ V^{n} given by a set of K-linear equations in x_{1}, …, x_{n},

• n-ary predicates EW for algebraic varieties W ⊆ (F^{×})^{n} definable and irreducible over ℚ.

The interpretation can be explained in the above mentioned terms as follows:

• E(x, y) ≡ [exp(x) = exp(y)],

• L(x_{1}, …, x_{n}) ≡ [⟨x_{1}, …, x_{n}⟩ ∈ L],

• EW (x_{1}, …, x_{n}) ≡ [⟨exp(x_{1}), …, exp(x_{n})⟩ ∈ W].

2.2 Definition E^{K} is the class of structures 𝔽^{K} in language L_{K} with axioms saying that V is an infinite-dimensional vector space over K, E is an equivalence relation on V which is congruent with respect to the relations EW (x_{1}, …, x_{n}), F^{×} = V/E can be identified with the group of the multiplicative group F^{×}, of a field of characteristic zero, and the predicates EW define its algebraic varieties over ℚ. The canonical mapping

exp : V → F^{×}

is a surjective homomorphism of the additive group of V onto the group F^{×}. The underlying set of axioms we denote PK, (powered field with exponents in K).

Notation For finite X, X′ ⊆ V, Y, Y′ ⊆ F^{×}

lin.dim_{K}(X) the dimension of the vector space span_{K}(X) generated by X over K;

lin.dim_{ℚ}(X) the dimension of the vector space span_{ℚ}(X) generated by X over ℚ;

tr.deg(Y) the transcendence degree of Y;

δ^{K}(X) the predimension of finite X ⊆ V :

δ^{K}(X) = lin.dim_{K}(X) + tr.deg(exp(X)) − lin.dim_{ℚ}(X);

δ^{K}(X/X′) = δ^{K}(X ∪ X′) − δ^{K}(X′);

For infinite Z ⊆ V and k ∈ ℤ δ^{K}(X/Z) ≥ k by definition means that for any Y ⊆_{fin} Z there is Y ⊆_{fin} Y′ ⊆ Z such that δ^{K}(X/Y′) ≥ k, and δ^{K}(X/Z) = k means δ^{K}(X/Z) ≥ k and not δ^{K}(X/Z) ≥ k + 1.

We recall that A ⊂ V is said to be self-sufficient in 𝔽^{K} if δ^{K}(X/A) ≥ 0 for all finite X ⊆ V. This is written as

A ≤ 𝔽^{K}.

2.3 We let also lin.dim_{K}(X/X′) = lin.dim_{K}(X ∪ X′) − lin.dim_{K}(X′); tr.deg(Y/Y′) = tr.deg(Y ∪ Y′) − tr.deg(Y′); lin.dim_{ℚ}(X/X′) = lin.dim_{ℚ}(X ∪ X′) − lin.dim_{ℚ}(X′); ker is the name of a unary predicate of type EW : x ∈ ker ≡

exp(x) = 1. We write ker_{|A} for the realisation of this predicate in A.

Given d ∈ ℤ, let E_{d}^{K} be the subclass of E^{K} consisting of all 𝔽^{K} satisfying the condition:

δ^{K}(X/ ker) ≥ −d for all finite X ⊆ V,

where ker = ker_{|𝔽} .

2.4 Denote subE^{K} the class of the substructures of the structures of E^{K} in the language L_{K}.

Given an integer d, let subE_{d}^{K} be the subclass of subE^{K} consisting of A which satisfy δ^{K}(X) ≥ −d for any finite X ⊆ A.

Remark For any structure A in subE^{K} and any X ⊆ ker_{|A} in the structure

δ^{K}(X) = 0

and thus E_{d}^{K} is empty for d < 0.

On the other hand, for any K we have by Lemmas 2.7 and 2.8 of [6]

E_{0}^{K} ≠ ∅.

2.5 Assuming F^{×} = ℂ^{×}, the algebraic torus, and the Schanuel conjecture we can make a better estimates for the minimal d such that ℂ^{K}, the complex numbers with raising to powers K ⊆ ℂ, belongs to E_{d}^{K}.

By Schanuel’s conjecture, for any finite X ⊆ ℂ

tr.deg(X, exp(X)) ≥ lin.dim_{ℚ}(X).

Recall that we assumed that tr.deg(K) is finite. Obviously,

lin.dim_{K}(X) + tr.deg(exp X) + tr.deg(K) ≥ tr.deg(X, exp(X)).

Hence,

δ^{K}(X) ≥ tr.deg(X, exp(X))−lin.dim_{ℚ}(X)−tr.deg(K) ≥ −tr.deg(K).

Since lin.dim_{K}(ker) = lin.dim_{ℚ}(ker) = 1 for ker = 2πiℤ, we have

δ^{K}(X/ ker) ≥ −(tr.deg(K) + 1).

Thus, under the conjecture,

ℂ^{K} ∈ E_{d}^{K}, for d = tr.deg(K) + 1.

Remark. In case of an elliptic curve one can produce similarly the estimate d = tr.deg(K) + 2 assuming the corresponding analogue of Schanuel’s conjecture (see Be).

2.6 Given 𝔽^{K} ∈ E_{d}^{K} \ E_{d−1}^{K} one can find a finite A ⊆ 𝔽^{K} with

δ^{K}(A/ ker) = −d.  (1)

By minimality

A ∪ ker ≤ 𝔽^{K}.

The equality (1) does not change if we extend A by elements of ker . In case δ^{K}(X) is bounded from below by, say −d′, for all finite X ⊂ ker, in particular, if ker is a finite rank group, then

δ^{K}(Y) ≤ −(d + d′) for all finite Y ⊆ 𝔽.

It follows that there is a finite A_{0} ⊆ ker such that if A_{0} ⊆ A and (1) holds then the value of δ^{K}(A) reaches minimum and so

A ≤ 𝔽^{K}  (2)

that is

δ^{K}(X/A) = δ^{K}(X/A ∪ ker) ≥ 0 for every X ⊆ 𝔽.

2.7 We will assume throughout that A with the property (1) and (2) does exist, as is the case for ℂ^{K} under Schanuel’s conjecture.

In fact, this is a form of Schanuel’s conjecture for ℂ^{K} for a given K ⊆ ℂ.

2.8 Definition A structure 𝔽^{K} in E_{d}^{K} is said to be E_{d}^{K}-exponentially-algebraically closed (e.a.c.) if for any 𝔽_{1}^{K} ∈ E_{d}^{K}, such that 𝔽^{K} ≤ 𝔽_{1}^{K}, any finite quantifier-free type over 𝔽^{K} which is realized in 𝔽_{1}^{K} has a realization in 𝔽^{K}. Denote EC_{d}^{K} the class of E_{d}^{K}-exponentially-algebraically closed structures, or, in the shorter form, EC^{K}.

Using the standard Fraisse construction in the class E_{d}^{K} relative to the strong embedding ≤ one can prove:

2.9 Propositon [Proposition 1 of [6]] For any 𝔽^{K} in E_{d}^{K} there exists an E_{d}^{K}-e.a.c. structure containing 𝔽^{K}.

More difficult is the following result, based on Theorem 1.1. Its proof using Ax’s Theorem is given in [6].

2.10 Proposition [Corollaries 1 and 2, section 4 of [6]] There exists a set EC of first order ∀∃ axioms such that, for any 𝔽^{K} ∈ E_{d}^{K},

𝔽^{K} ⊧ EC iff 𝔽^{K} is exponentially-algebraically closed.

2.11 For W ⊆ (F^{×})^{n+l} an algebraic variety, b = ⟨b_{1}, …, b_{l}⟩ denote

W (b) = {⟨x_{1}, …, x_{n}⟩ ∈ (F^{×})^{n} : ⟨x_{1}, …, x_{n}, b_{1}, …, b_{l}⟩ ∈ W }.

A subspace L ⊆ V^{n} is said to be K-linear if there are k_{ij} ∈ K (i ≤ r, j ≤ n) such that

L = {⟨x_{1}, …, x_{n}⟩ ∈ V^{n} : k_{i1}x_{1} + ⋯ + k_{in}x_{n} = 0}.

Define dim L = co-rank(k_{ij}), the co-rank of the matrix (k_{ij}).

Let L ⊆ V^{n+l} be a K-linear subspace, ā = ⟨a_{1}, …, a_{l}⟩. Let

L(a) = {⟨x_{1}, …, x_{n}⟩ ∈ V^{n} : ⟨x_{1}, …, x_{n}, a_{1}, …, a_{l}⟩ ∈ L}.

We call such an L(a) a K-affine subspace defined over a. The same terminology is applied for ℚ instead of K.

2.12 Lemma A K-affine subspace L(a) ⊆ V^{n} can be represented equivalently and uniformly on a as

L(a) = L(0) + r(a), r(a) ∈ V^{n}, r is a K-linear map, V^{l} → V^{n}

and L(0) is a K-linear subspace.

Moreover, if r′ is any K-linear map such that r′(a) ∈ L(a) for all a ∈ pr_{n+1...n+l}L, then also

L(a) = L(0) + r′(a).

Proof Let L be determined by the system of linear equations

∑_{j=1}^{n} q_{ij}v_{j} + ∑_{s=1}^{l} k_{is}w_{s} = 0, q_{ij}, k_{is} ∈ K.  (3)

Then the system

∑_{j=1}^{n} q_{ij}v_{j} + ∑_{s=1}^{l} k_{is}a_{s} = 0,

determines L(a). It follows that

∑_{j=1}^{n} q_{ij}v_{j} = 0  (4)

determines L(0).

By linear algebra there is an n-tuple {r_{j}(w_{1}, …, w_{l}) : j = 1, …, n} of K-linear functions V^{l} → V such that, if for a given (w_{1}, …, w_{l}) the system (3) is consistent, then v_{j} = r_{j}(w_{1}, …, w_{l}) gives a solution to the system. Hence, v − r(a) is a solution to the homogeneous system (4) iff v is a solution of the system (3) with w = a.

The ’moreover’ statement follows immediately from the fact that r′(a) − r(a) ∈ L(0). □

2.13 Lemma Given a K-linear L ⊆ V^{n+l} and 0 the zero of V^{l}

(i) there exists a unique maximal ℚ-linear subspace N_{L} ⊆ L; (ii) N_{L(0)} = N_{L}(0); (iii) given a ∈ pr_{n+1...n+l}L and q(a) ∈ L(a) ∩ span_{Q}(a) there exists

a maximal ℚ-affine subspace N_{L,q(a)}(a) ⊆ L(a) over span_{Q}(a) containing q(a), and in this case N_{L,q(a)}(a) = N_{L}(0) + q(a).

Proof (i) N_{L}(0) exists since the sum of two ℚ-linear subspaces of L(0) is again a ℚ-linear subspace of L(0).

(ii) Obviously, N_{L}(0) is a ℚ-linear subspace of L(0), so N_{L(0)} ⊇ N_{L}(0).

N_{L(0)} is a ℚ-linear subspaces of L(0), so N_{L(0)} × {0} is a ℚ-linear subspaces of L, hence N_{L(0)} × {0} ⊆ N_{L} and N_{L(0)} ⊆ N_{L}(0).

(iii) N_{L}(0) + q(a) is a ℚ-affine subspace of L(0) + q(a) = L(a). If M + q(a) is another ℚ-affine subspace of L(a), containing q(a) then M = (M + q(a)) − q(a) ⊆ L(0) and hence M ⊆ N_{L}(0), M + q(a) ⊆ N_{L}(0) + q(a). □

3 Intersections in semi-abelian varieties

3.1 Lemma. In the statement of Theorem 1.1 we can assume that s ∈ acl(b, e).

Proof S is an irreducible component of the set W (e) ∩ exp(N) · b definable over (e, b) hence it is definable over acl(b, e). Thus it contains points from the algebraically closed field acl(b, e).□

3.2 Proposition. Given W ⊆ (F^{×})^{n+l}, with W algebraic subvariety defined over ℚ there are finitely many proper ℚ-linear subspaces π(W) = {M_{1}, …, M_{m}} of V^{n} such that for any e, b ∈ (F^{×})^{l} and a ℚ-linear subspaces N ⊆ V^{n}, for any positive dimensional atypical component S of the intersection W (e)∩exp(N)·b there is a M ∈ π(W) and s ∈ S ∩ acl(e, b) such that S ⊆ exp(M) · s and S is a typical component of exp(N) · b ∩ exp(M) · s ∩ W (e) with respect to the group variety exp(M) · s.

Proof Notice first that by obvious transformations of W we can assume that the family {W (e) : e ∈ (F^{×})^{l}} is invariant with respect to shifts by elements of (F^{×})^{l}, that is, for every b, e ∈ (F^{×})^{l} there is e′ ∈ (F^{×})^{l} such that

W (e) · b = W (e′).

By induction on the dimension of a proper algebraic subgroup P of (F^{×})^{n}, for any algebraic subvariety W_{P} of P ⊆ (F^{×})^{l} over ℚ we construct a collection of proper algebraic subgroups π_{P} (W_{P}) of P such that the statement of the lemma holds for exp(N) · b ∩ W_{P} (e) ⊆ P.

For dim P = 1 the statement is trivially true for there is no atypical components in any intersection.

Consider the general case. Assume by induction that π_{P} (W_{P}) has been constructed for dim P < dim(F^{×})^{n}. Notice that by invariance π_{P} (W_{P}) will be the same if we replace P by P · b, for b ∈ (F^{×})^{l} a parameter.

Assume that for all P ⊂ (F^{×})^{n} proper, π_{P} (W_{P}) exists.

Given W ⊆ (F^{×})^{n+l}, we let

π(W) = ⋃_{Q=exp(M), M∈μ(W)} π_{Q}(W_{Q}) ∪ {M},

where W_{Q} is W ∩ (Q × (F^{×})^{l}).

Now, if S ⊆ exp(N) · b ∩ W (e) is atypical, then by Theorem 1.1 S ⊆ exp(M) · s for some M ∈ μ(W). Hence S ⊆ exp(N) · b ∩ W_{Q}(e), for Q = exp(M) · s, and is a component of the intersection. In other words, S is a component of the intersection exp(N ∩M)·b′ ∩W_{Q}(e) for some b′ ∈ (F^{×})^{l}. Either S is typical in this intersection with respect to Q = exp(M), and hence the statement of the proposition holds for the chosen M belonging to π_{Q}(W_{Q}) by definitions, or S is atypical but we can find by induction M′ ∈ π_{Q}(W_{Q}) ⊆ π(W) such that S ⊆ exp(M′)·s and is a typical component in the intersection with respect to exp(M′).

□

We want to show now that under certain conditions we can factor out M in the previous proposition.

3.3 Let M ⊆ V^{n} be a ℚ-linear subspace. We see V^{n} as a subspace of V^{n+l}, equivalently V^{n+l} = V^{n} ˙+V^{l}, with a ∈ V^{l}, e = exp(a) ∈ (F^{×})^{l}.

By definitions M is definable by c = codim M independent Qlinear equations

m_{i1}v_{1} + ⋯ + m_{in}v_{n} = 0, i = 1, … c,

where (v_{1}, …, v_{n}) ∈ V^{n}. The same in matrix notation

m̄v̄ = ¯0.

We now choose m̄^{⊥}, a (n − c) × n-matrix consisting of vectors (m_{j1}, …, m_{jn}) ∈ ℚ^{n}, j = c + 1, …, n, which extend m̄ to the basis of the ℚ-vector space ℚ^{n}. We let M^{⊥} to be the set of solutions to the system

m̄^{⊥}v̄ = ¯0.

This determines the definable decomposition

V^{n} = M ˙+M^{⊥} ≅ M ˙×V^{n}/M.

Applying exp we correspondingly have the decomposition

(F^{×})^{n} = Q · Q^{⊥} ≅ Q × (F^{×})^{n}/Q,

where Q = exp(M) and Q^{⊥} = exp(M^{⊥}).

Note that the structure (M^{⊥}, exp, Q^{⊥}) is by construction isomorphic to (V^{c}, exp, (F^{×})^{c}) in language L_{K}.

We denote the natural mappings

V^{n} → M^{⊥} and (F^{×})^{n} → Q^{⊥}

associated with the above decomposition as

v ↦ v + M and x ↦ x · Q,

correspondingly.

It is easy to see that x ↦ x · Q is a proper mapping on (F^{×})^{n} (and (F^{×})^{n+l}) hence the images W/ exp M (that is W/Q) and W (e)/ exp M are algebraic subvarieties of exp M^{⊥} ×(F^{×})^{l} and exp M^{⊥}, correspondingly.

The same algebraicity holds for S/ exp M and b · exp(N)/ exp M).

3.4 On the other hand, exp can be naturally extended to the quotient spaces

exp : V^{n}/M → (F^{×})^{n}/Q.

So, (V^{n}/M, exp, (F^{×})^{n}/Q) and (V^{n+l}/M, exp, (F^{×})^{n+ℓ}/Q) are canonically isomorphic to (V^{c}, exp, (F^{×})^{c}) and (V^{c+l}, exp, (F^{×})^{c+l}) correspondingly and hence, for (V, exp, F^{×}) ∈ E_{d}^{K}, u ∈ V^{n} and \overrightarrow{A} ∈ V^{l}, A ≤ 𝔽^{K},

δ^{K}(u+M) = lin.dim_{K}(u+M)+tr.deg(exp(u+M))−lin.dim_{ℚ}(u+M) ≥ d,

and

δ^{K}(u+M/A) = lin.dim_{K}(u+M/A)+tr.deg(exp(u+M)/ exp A)−lin.dim_{ℚ}(u+M/A) ≥ 0.

3.5 We have also the decomposition

L(a) = L(¯0) ∩ M ˙+L/M(a)  (5)

where

L/M = L/(L∩(M×{¯0})) ⊆ V^{n+ℓ}/(M×{¯0}), ¯0 ∈ V^{ℓ}, L/M(a) ⊆ M^{⊥}.

Thus we can naturally identify L(a)/M with L/M(a).

4 Axiomatizing E_{d}^{K}.

Fix A ⊆ V and consider pairs (L(a), W (exp a)), where L is a K-affine subspace of V^{n} over a = \overrightarrow{A} ∈ V^{l} and W an algebraic subvariety of (F^{×})^{n} over ℚ.

Definition A pair (L(a), W (exp a) is said to be special if L is not contained in any proper ℚ-linear subspace of V^{n+ℓ} and

dim L(a) + dim W (exp a) < n.  (6)

(This corresponds to 0-special in the terminology of [6].)

4.1 Let, for a ℚ-subspace M of V^{n}

d(W (exp a), exp(M)) = min\{dim(W̄ (exp a)∩w·\overline{exp M} : w ∈ W (exp a)\},

where W̄ and \overline{exp M} is the closure in the ambient projective space.

Let

W^{exp M}(exp a) = {w ∈ W (exp a) : dim(W (exp a)∩w·exp M) > d(W (exp a), exp M)}.

Since minimal dimension fibers are located over an open subset W^{exp M}(exp a) is a proper closed subset of W (exp a), maybe empty, if d(W (exp a), exp M) = dim(W (exp a)∩exp(x) exp M).

4.2 Suppose that (L(a), W exp a)) is special. Suppose M ⊆ L(0). Consider the quotients M^{⊥} = V^{n}/M, exp(M)^{⊥} = (F^{×})^{n}/ exp(M) and subsets L(a)/M and W (exp a)/ exp(M).

Then, dim W (exp a)/ exp(M) < n − dim M, that is

W (exp, a)/ exp(M) is a proper subvariety of (F^{×})^{n}/ exp(M).

Indeed, by addition formula,

dim W (exp a)/ exp(M) = dim W (exp a) − d(W (exp a), exp M)

and

d(W (exp a), exp M) ≥ 0 > dim W (exp a) + dim M − n,

since W (exp a), L(a) is special.

4.3 Assume A ≤ 𝔽^{K} is finite and let x ∈ V^{n}. We analyse first order consequences of this assumption. We aim to show also that the analysis yields the same conclusions and formulas when we replace A by B with qftp(A) = qftp(B).

Suppose (L(a), W exp a)) is special, Suppose x ∈ L(a) and exp(x) ∈ W (exp a), in 𝔽^{K} and A ≤ 𝔽^{K}.

Since

lin.dim_{K}(x/A)+tr.deg(exp(x)/ exp(A))−n ≤ dim L(a)+dim W (exp a)−n < 0

and, by A ≤ 𝔽^{K},

δ^{K}(x/A) ≥ 0,

we have lin.dim_{ℚ}(x/A) < n, so x^{⌢}a ∈ N for some proper ℚ-linear subspace of V^{n+l}, We assume N is minimal for x^{⌢}a.

We have exp(x) ∈ S_{x} ⊆ exp(N(a)) ∩ W (exp a), where S_{x} is a component of exp(N(a)) ∩ W (exp a).

Case 1. The component S_{x} is of dimension 0.

Then tr.deg(exp(x)/ exp(A)) = 0, which implies lin.dim_{K}(x/a) = lin.dim_{ℚ}(x/a), that is dim N(a) = dim N(a) ∩ L(a) and so N(a) ⊆ L(a) is a ℚ-affine subspace over a, thus N(a) = N_{L}(0) + q(a), for some q(a) ∈ L(a) ∩ span_{Q}(a) (Lemma 2.13).

So

x ∈ N_{L}(0) + q(a) for some q(a) ∈ L(a) ∩ span_{Q}(a).

Subcase 1.1 d(W (exp a), exp(N_{L}(0))) < dim(W (exp a)∩exp(x+ N_{L}(0))).

Under this assumption W^{exp(N_{L}(0))}(exp a) is a proper closed subset of W (exp a) containing exp(x), by 4.1.

Otherwise we have

Subcase 1.2. d(W (exp a), exp(N_{L}(0))) = dim(W (exp a)∩exp(x+ N_{L}(0))).

We have W^{exp(N_{L}(0))}(exp a) = ∅ in this case.

Consider the quotients N_{L}^{⊥}(0) = V^{n}/N_{L}(0), exp(N_{L}(0))^{⊥} = (F^{×})^{n}/ exp(N_{L}(0)) and subsets L/N_{L}(a) and W (exp, a)/ exp(N_{L}(0)). By 4.2 W (exp a)/ exp(N_{L}(0)) ⊊ (F^{×})^{n}/ exp(N_{L}(0)).

Obviously, for the x above, exp(x/N_{L}(0)) is a singleton in W (exp a)/ exp(N_{L}(0) and is also equal to exp(q(a)/N_{L}(0)).

Let

Γ_{a} = {exp(q(a)^{⌢}a) : q(a) ∈ span_{Q}(a)}

This is a coset Γ^{0} · s(exp a) of a finite rank subgroup Γ^{0} of (F^{×})^{n+l} (depending on the choice of exp a).

By Faltings’ Theorem there are finitely many, say k_{a}, cosets T_{i}(exp a) ⊆ (F^{×})^{n+l} of group subvarieties (tori) T_{i} ⊆ (F^{×})^{n+l}, T_{i}(exp a) ⊆ W (exp a) (note the notation, T_{i}(exp a) as defined in 2.11) such that

Γ_{a} ∩ W (exp a) = ∪_{i≤k_{a}}Γ_{a} ∩ T_{i}(exp a).  (7)

Hence, in this case

exp(x/N_{L}(0)) ∈ ⋃_{i≤k_{a}} T_{i}(exp a)  (8)

and T_{i}(exp a) ⊆ W (exp a)/ exp N_{L}(0) ⊊ (F^{×})^{n}/ exp(N_{L}(0).

Note that (7) and (8) continue to hold with the same T_{i} and k_{a} if we replace a by b with exp \frac{a}{n} ≡ exp \frac{b}{n}, all n, in the field language, that is Galois conjugated.

Case 2. dim S_{x} > 0. Then, by 3.2, S_{x} ⊆ c · exp(M), for some ℚ-linear subspace M ∈ π_{W} of V^{n+ℓ}, c ∈ acl(exp a), and S_{x} is typical in the intersection W (exp a) ∩ c · exp(M) with respect to c · exp(M). The latter gives us, for a′ = ln c,

dim S_{x} = dim(exp(N(a))∩exp(M+a′))+dim(W (exp a)∩exp(M+a′))−dim exp(M+a′).

It is easy to see that δ^{K}(x/Aa′) ≥ 0 and hence we obtain

dim(L(a) ∩ N(a) ∩ (M + a′)) + dim S_{x} − dim(N(a) ∩ (M + a′)) ≥ 0.

Combining with the above we get

dim(L(a)∩N(a)∩(M+a′))+dim(W (exp a)∩exp(M+a′))−dim exp(M+a′) ≥ 0.

And so

dim(L(0) ∩ M) + dim(W (exp a) ∩ exp(M + a′)) − dim M ≥ 0. (9)

Subcase 2.1 d(W (exp a), exp(M)) < dim(W (exp a)∩c·exp(M)).

Under this assumption W^{exp(M)}(exp a) is a proper closed subset of W (exp a) containing exp(x).

Otherwise, we have

Subcase 2.2. d(W (exp a), exp(M)) = dim(W (exp a)∩c·exp(M)). So,

W^{exp(M)}(exp a) = ∅.  (10)

We now apply the factorisation of 3.3-3.4.

Obviously, for our x,

exp(x/M) ∈ S_{x}/ exp(M) ∩ exp(L(a)/M)

and S_{x}/ exp(M) is a singleton in (F^{×})^{n−dim M} defined over the same parameters as S_{x}, that is over acl(exp(A)).

Now we notice that

dim L/M(a) + dim W (exp a)/ exp M =

= dim L(a)−dim L(0)∩M +dim W (exp a)−d(W (exp a), exp(M)) =

= [dim L(a)+dim W (exp a)]−[dim L(0)∩M +d(W (exp a), exp(M))].

The sum in the first bracket is less than n by assumptions, and the sum in the second bracket is not less than dim M by (9). Hence

dim L/M(a) + dim W (exp a)/ exp M < n − dim M,

that is the pair is special.

This means that after factorisation by M we are in case 1 again. Hence either, as in subcase 1.1,

4.4 W^{exp(N_{L/M} (0))}(exp a) is a proper closed subset of W (exp a)/ exp M containing exp(x + M).

or, as in subcase 1.2,

4.5

exp(x + N_{L/M}(0) + M) ∈ ⋃_{i≤k_{a}} T_{i}(exp a),

for group subvarieties T_{i},

T_{i}(exp a) ⊆ W (exp a)/ exp(N_{L/M}(0)+M) ⊊ (F^{×})^{n}/ exp(N_{L/M}(0)+M).

Note again as in subcase 1.2 that the latter continues to hold with the same T_{i} if we replace a by b with exp \frac{a}{n} ≡ exp \frac{b}{n}, all n, in the field language, that is Galois conjugate.

4.6 For L and W as above let, for a of the length equal to the size of A,

Φ_{L,W,A}(a, x) := x^{⌢}a ∈ L & exp(x^{⌢}a) ∈ W → ⋁_{M∈π_{W} ∨M=N_{L}(0)∨M={0}}

[exp(x) ∈ W^{exp M}(exp a) ∨ exp(x + M) ∈ W^{exp N_{L/M}} (exp a) ∨ exp(x + M) ∈ ⋃_{i≤k_{a}} exp(N_{i,a,M})]

Ths is a quantifier-free formula and, by the analysis above

A ≤ 𝔽^{K} ⇒ 𝔽^{K} ⊧ ∀xΦ_{L,W,A}(\overrightarrow{A}, x).

4.7 Consequence of the analysis. Given A ≤ 𝔽^{K} and a special pair L, W, the ingredients of formula Φ_{L,W,A}(a, x), except for exp(N_{i,a,M}), depend on A as parameters only. And exp(N_{i,a,M}), as noted in cases 1.2 and 2.2 where these have been introduced, are of the form T_{i}(exp \frac{a}{m}), m ∈ ℕ, T_{i} are group subvarieties (subtori) of (F^{×})^{n+ℓ} and T_{i} depend on the Galois type of exp(span_{Q}A) ⊆ F^{×} only.

In particular, if

B ≤ 𝔽^{K} and exp(span_{Q}A) ≡_{fields} exp(span_{Q}B)

then

𝔽^{K} ⊧ ∀xΦ_{L,W,A}(\overrightarrow{B}, x).

4.8 We define a strong embedding type with variables Y, |Y | = |A|, \overrightarrow{Y} = y,

sttp_{A}(y) := qftp_{A}(y) ∪ {∀x Φ_{L,W,A}(y, x) : (L(y), W (exp y)) special},

where qftp_{A}(y) denotes the quantifier-free type of A in variables Y. This is a type consisting of universal formulas.

Now we can reformulate 4.7

4.9 B ≤ 𝔽^{K} and 𝔽^{K} ⊧ qftp_{A}(B) ⇒ 𝔽^{K} ⊧ sttp_{A}(B).

We assume below that 𝔽^{K} ∈ E_{d}^{K} and A has been chosen so that A ≤ 𝔽^{K} as well as A ∪ ker ≤ 𝔽^{K} as discussed in 2.7.

4.10 Lemma.

𝔽^{K} ⊧ sttp_{A}(B) ⇒ B ≤ 𝔽^{K}.

Proof Assume w.l.o.g. that x ∈ V^{n} is ℚ-linearly independent over B, a = \overrightarrow{B}, L(a) is the minimal K-affine subspace over A containing x, and W (exp a) the minimal algebraic variety over exp(B) containing exp(x). Notice that under this choice exp x is multiplicatively independent over exp B.

We show that (L(a), W (exp a) can not be special, thus proving δ^{K}(x/B) ≥ 0.

Indeed, if the pair were special, ∀xΦ_{L,W,A}(a, x) would imply that x satisifes one of the conditions on the second or third line of the definition of Φ_{L,W,A}(a, x).

Any of the conditions on the second line contradicts the assumptions that W (exp a) is the algebraic locus of exp x over exp B, since W^{Q}(exp a), for Q a group subvariety, is a proper subvariety of W (exp a) by 4.1 and 4.4.

The condition on the third line can not hold because by 4.5 it would contradict the fact that exp x is multiplicatively independent over exp B. □

4.11 Proposition. The following two conditions are equivalent:

𝔽^{K} ⊧ sttp_{A}(B)  (11)

and

B ≤ 𝔽^{K} & 𝔽^{K} ⊧ qftp_{A}(B)  (12)

Proof Lemma 4.9 proves (12) ⇒ (11). The converse follows from Lemma 4.10 and the definition of sttp.□

4.12 Let 𝔽^{K} be a member of E_{d}^{K} and A ⊆ 𝔽^{K} be a finite subset containing generators of ker(exp) such that

δ^{K}(A) = −d.

It follows that A ≤ 𝔽^{K}.

Let

SCH_{A} = {∃X ⋀ S(X) : S ⊂ sttp_{A}(X), finite, |X| = n},

be the set of ∃∀-sentences stating the consistency of type sttp_{A}.

Recall the notation PK for the axioms of powered fields with powers in K.

4.13 Lemma. Let 𝔽^{K} be a model of PK + SCH_{A} which realises the type sttp_{A}. Then 𝔽^{K} ∈ E_{d}^{K} \ E_{d−1}^{K}.

Proof By assumption we have a finite subset B ⊆ 𝔽 such that 𝔽^{K} ⊧ sttp_{A}(B). By Proposition 4.11

B ≤ 𝔽^{K}.

As a consequence of qftp_{A} we have δ^{K}(B) = −d. It follows that 𝔽^{K} ∉ E_{d−1}^{K}.

To see that 𝔽^{K} ∈ E_{d}^{K} we need to prove that δ^{K}(Z) ≥ −d for any finite Z ⊆ 𝔽^{K}.

Let Y be a ℚ-linear basis of span_{Q}(Z) ∩ span_{Q}(B). We have then lin.dim_{ℚ}(Z/B) = lin.dim_{ℚ}(Z/Y) and thus δ^{K}(Z/Y) ≥ δ^{K}(Z/B) ≥ 0. But δ^{K}(Z) = δ^{K}(Z/Y) + δ^{K}(Y), so δ^{K}(Z) ≥ δ^{K}(Y) ≥ −d.□

4.14 Theorem. Assume δ^{K}(A) = −d. The following two conditions are equivalent for a structure 𝔽^{K} :

(i) 𝔽^{K} ⊧ PK+SCH_{A};

(ii) 𝔽^{K} ∈ E_{d}^{K} \ E_{d−1}^{K} and qftp_{A} is realised in some ^{∗}𝔽^{K} ≻ 𝔽^{K}.

If in (i) 𝔽^{K} also satisfies EC then in (ii) E_{d}^{K} should be replaced by EC_{d}^{K}.

Proof Assume (i). By the definition of SCH_{A} there is ^{∗}𝔽^{K} ≻ 𝔽^{K} which realises sttp_{A}, say by B. By Lemma 4.13, ^{∗}𝔽^{K} ∈ E_{d}^{K} \ E_{d−1}^{K}, so ∗𝔽^{K} ∈ E_{d}^{K} \ E_{d−1}^{K}.

It follows that 𝔽^{K} ∈ E_{d}^{K}, since δ^{K}(X/ ker) ≥ −d for all X ⊆ ^{∗}V. It remains to see that 𝔽^{K} ∉ E_{d−1}^{K}. Indeed, if it were in E_{d−1}^{K}, we

would have A′ ≤ 𝔽^{K} with δ^{K}(A′) = −d′, d′ ≤ d − 1, and by the analysis in 4.3 arrive at the fact that 𝔽^{K} and ^{∗}𝔽^{K} realise sttp_{A′}, hence using again 4.13, ^{∗}𝔽^{K} ∈ E_{d′}^{K}, a contradiction. This proves (ii).

Now, conversely, assume (ii).

We claim that^{∗}𝔽^{K} ∈ E_{d}^{K}. Indeed, since 𝔽^{K} ∈ E_{d}^{K} by 4.3 we find a B with δ^{K}(B/ ker) = −d, so B ≤ 𝔽^{K} and 𝔽^{K} ⊧ SCH_{B}. So^{∗}𝔽^{K} ⊧ SCH_{B} and as shown in the first part of the proof, it follows ^{∗}𝔽^{K} ∈ E_{d}^{K}.

By assumptions, up to isomorphism, A ⊆ ^{∗}𝔽^{K}. Since δ^{K}(A) = −d, we have A ≤ ^{∗}𝔽^{K}. It follows ^{∗}𝔽^{K} ⊧ SCH_{A}, so 𝔽^{K} ⊧ SCH_{A} and (i) proved.

The last statement of the theorem follows from Proposition 2.10. □

4.15 Let L ⊆ V^{n} be a K-linear subspace of dimension l and W ⊆ (𝔽^{×})^{n+m} be a variety over ℚ. Define the set of special parameters

P_{l}(W) = {p ∈ 𝔽^{m} : dim W (p) < n − l}.

This set is constructible (quantifier-free definable in the field language).

Theorem.Assume 𝔽^{K} ⊧ PK+SCH_{A}. There are a number N and codimension 1 ℚ-linear subspaces M_{1}, …, M_{N} ⊊ V^{n} depending on L and W such that for every p ∈ P_{l}(W) for some a_{1}, … a_{N} ∈ V^{n}

{z ∈ V^{n} : z ∈ L & exp(z) ∈ W (p)} ⊆ ⋃_{i≤N}(M_{i} + a_{i} + ker^{n}).

Proof First we consider the case of a single p ∈ P_{l}. Choose a finite set B ⊂ V so that p ⊆ exp(B) and B ∪ ker ≤ 𝔽^{K}. Let z = ⟨z_{1}, …, z_{n}⟩ ∈ L such that exp(z) ∈ W (p). By the choice of B

δ^{K}(z/B ∪ ker) ≥ 0.

But lin.dim_{K}(z/B ∪ ker) + tr.deg(exp(z)/ exp(B) < n. It follows, lin.dim_{ℚ}(z/B ∪ ker) < n. In other words,

m_{1}z_{1} + … m_{n}z_{n} − b ∈ ker,

for some m_{1}, …, m_{n} ∈ ℤ, not all zero, and b ∈ span_{ℚ}(B). Denote

M = {⟨x_{1}, …, x_{n}⟩ ∈ V^{n} : m_{1}x_{1} + … m_{n}x_{n} = 0}.

We have proved that

z ∈ L & exp(z) ∈ W (p) ⇒ z ∈ M+a+ker^{n} for some a ∈ span_{ℚ}(B)^{n} and M

(13)

Claim. For a given p there is finitely many M and a such that (13).

Indeed, if not then the type saying that z ∈ L & exp(z) ∈ W (p) and z ∉ M + a + ker^{n}, for M running through all ℚ-linear subspaces of codimension 1 and a ∈ span_{ℚ}(B)^{n} is consistent. This type would be realised in some ^{∗}𝔽^{K} ≻ 𝔽^{K} contradicting (13).

The proved claim implies the existence of the bound N_{p} on the number of cosets M + a + ker^{n} satisfying (13). We need to show that there is an N that bounds all the N_{p}. Assuming such a bound does not exist we can find a p ∈ P_{l}(W) in some ^{∗}𝔽^{K} ≻ 𝔽^{K} for which no finite bound N_{p} does exist, contradicting the Claim.□

4.16 Corollary. Suppose ℂ^{K} the structure on complex numbers, for some K ⊆ ℂ satifies the assumptions 2.7 for some A ⊆ ℂ, that is

ℂ^{K} ⊧ PK+SCH_{A}.

Then there are codimension-1 ℚ-linear subspaces M_{1}, …, M_{N} ⊊ ℂ^{n} such that for every p ∈ P_{l}(W) there are a_{1}, … a_{N} ∈ ℂ^{n} with the property that for every irreducible component S of the analytic set {z ∈ ℂ^{n} : z ∈ L & exp(z) ∈ W (p)} there is j ∈ {1, …, N} and k ∈ ℤ^{n} such that S ⊆ M_{j} + a_{j} + 2πik.

The corollary is immediate from the theorem once one takes into account that irreducible components of the analytic set M_{i} + a_{i} + ker^{n} are of the form M_{i} + a_{i} + 2πik.

5 Completeness, near model completeness and superstability

5.1 Definition The extension of the initial language L_{K} by existential predicates

E_{P} (x̄) ≡ ∃ȳP (x̄, ȳ),

where P is a quantifier-free formula, is denoted L_{K}^{E}.

We assume throughout that A is a finite subset of subE_{d}^{K}, δ^{K}(A) = −d

5.2 Lemma. Assuming 𝔽_{1}^{K} ⊆ 𝔽^{K} as L_{K}^{E}-structures and 𝔽^{K} ⊧ PK+SCH_{A}, we have 𝔽_{1}^{K} ⊧ PK+SCH_{A} and 𝔽_{1}^{K} ≤ 𝔽^{K}.

Proof 𝔽_{1}^{K} ∈ E_{d}^{K} for every L_{K}-substructure of 𝔽^{K}, since facts of the form δ^{K}(X) = m are fixed by quantifier-free types.

To see that 𝔽_{1}^{K} ≤ 𝔽^{K} it is enough to show that for a finite B

B ≤ 𝔽_{1}^{K} ⇒ B ≤ 𝔽^{K}.

This follows from Proposition 4.11 if we take into account that sttp_{B} is L_{K}^{E}-quantifier-free.

It remains to see that an elementary extension ^{∗}𝔽_{1}^{K} of 𝔽_{1}^{K} contains a copy of A. This is immediate by the fact that the condition on consistency of qftp_{A} is given by existential L_{K}-formulas, so that is by L_{K}^{E}-quantifier-free ones. □

5.3 Lemma. Assume 𝔽_{1}^{K}, 𝔽_{2}^{K} ∈ E_{d}^{K} and 𝔽_{1}^{K} ⊧ EC. Suppose 𝔽_{2}^{K} ≤ 𝔽_{1}^{K}. Then 𝔽_{2}^{K} ⊆ 𝔽_{1}^{K} in the language L_{K}^{E}.

Proof Recall that by Proposition 2.10 𝔽_{1}^{K} ∈ EC^{K}. Let a ⊆ 𝔽_{1}^{K} be finite and suppose 𝔽_{2}^{K} ⊧ ∃y P (a, y), where P (x, y) is quantifier-free. By the definition of EC^{K} we get then 𝔽_{1}^{K} ⊧ ∃y P (a, y). □

5.4 Corollary. For 𝔽_{1}^{K}, 𝔽_{2}^{K} ∈ EC_{d}^{K}

𝔽_{1}^{K} ⊆ 𝔽_{2}^{K} as L_{K}^{E}-structures iff 𝔽_{1}^{K} ≤ 𝔽_{2}^{K}.

5.5 We can always extend 𝔽^{K} to contain also an infinite cl-independent over A subset I, so suppose A ∪ I ≤ 𝔽^{K} ∈ E_{d}^{K}.

We will denote SCH_{A} + ID the set of axioms stating that the type sttp_{A∪I} is consistent.

Notice that given a finite I_{0} ⊆ I we have an embedding A∪I_{0} ≤ 𝔽^{K} for every ω-saturated model 𝔽^{K} of SCH_{A} + ID.

We then have by Proposition 4.11.

5.6 Corollary. PK + SCH_{A}+EC + ID axiomatises the subclass of EC_{A}^{K} whose ω-saturated structures are infinite dimensional.

We say that a (partial) map φ : 𝔽_{1}^{K} → 𝔽_{2}^{K} is an L_{K}^{E}-monomorphism, if it is injective and for any k-ary L_{K}^{E}-predicate S and any k-tuple a from the domain of φ

𝔽_{1}^{K} ⊧ S(a) iff 𝔽_{2}^{K} ⊧ S(φ(a)).

5.7 Lemma. Let 𝔽_{1}^{K} and 𝔽_{2}^{K} satisfy PK + SCH_{A} + EC + ID, and B_{1} ≤ 𝔽_{1}^{K}, B_{2} ≤ 𝔽_{2}^{K} such that there is an L_{K}-monomorphism

φ : B_{1} → B_{2}.

Let 𝔽_{B_{1}}^{K} and 𝔽_{B_{2}}^{K} be the expansions of 𝔽_{1}^{K}, 𝔽_{2}^{K} by constants naming elements of B_{1} and B_{2} in correspondence with φ. Then

𝔽_{B_{1}}^{K} ≡ 𝔽_{B_{2}}^{K}.

Proof We prove that given ω-saturated elementary extensions^{∗}𝔽_{1}^{K} of 𝔽_{1}^{K} and ^{∗}𝔽_{2}^{K} of 𝔽_{2}^{K}, given finite C ⊆ ^{∗}𝔽_{1}^{K}, c ∈ ^{∗}𝔽_{1}^{K} and a L_{K}^{E}- monomorphism φ of B_{1} ∪ C into ^{∗}𝔽_{2}^{K} one can extend the monomorphism to c. By symmetry, this yields a winning strategy for the Ehrenfeucht-Fraisse game, and we are done.

We may assume that φ is the identity and B_{1} ∪ C = B = φ(B). It is enough to show that under the assumption for any c ∈ ^{∗}𝔽_{1}^{K} we can extend φ to some B′ ⊇ Bc as an L_{K}-monomorphism and B′ ≤ ^{∗}𝔽_{1}^{K}, φ(B′) ≤ ^{∗}𝔽_{2}^{K}.

If ∂(c/B) = 1 then define B′ = Bc and φ(c) to be any element from ^{∗}𝔽_{2}^{K} which is not in the ∂-closure of A in ^{∗}𝔽_{2}^{K} (use ID). Then B′ and φ(B′) are as required.

If ∂(c/B) = 0 then extend c to a finite string c̄ from ^{∗}𝔽_{1}^{K} so that δ^{K}(c̄/B) = 0. The quantifier free type of c̄ over B is consistent with 𝔽_{1}^{K}, by 5.3, and so is realised in ^{∗}𝔽_{1}^{K}, by b̄ say. Since δ^{K}(b̄/B) = 0, we have Ab̄ ≤ ^{∗}𝔽_{2}^{K}. So, we can define B′ = Bc̄ and φ(c̄) = b̄. □

5.8 Lemma. Let 𝔽_{1}^{K}, 𝔽_{2}^{K} be ω-saturated models of PK+SCH_{A} +ID, B_{1}, B_{2} finite subsets of 𝔽_{1}^{K}, 𝔽_{2}^{K}, correspondingly, and φ : B_{1} → B_{2} is a L_{K}^{E}-monomorphism. Then, there exists a finite subset B̃_{1} such that B̃_{1} ≤ 𝔽_{1}^{K} and φ can be extended to B̃_{1} in such a way that

φ(B̃_{1}) = B̃_{2} ≤ 𝔽_{2}^{K}.

Proof Let b̄_{1} be the string of all elements of B_{1} and c̄ in 𝔽_{1}^{K} such that δ^{K}(b̄_{1}c̄) = ∂(b̄_{1}). It follows B_{1}c̄ ≤ 𝔽_{1}^{K}. Let m = ∂(b̄_{1}).

Let q^{0}(x̄ȳ) be the L_{K}-quantifier-free type of b̄_{1}c̄. Let b̄_{2} be a string in 𝔽_{2}^{K} which corresponds to b̄_{1}. Then the L_{K}^{E}-monomorphism guarantees that q^{0}(b̄_{2}ȳ) is consistent and thus there is d̄ in 𝔽_{2}^{K} realising the type, in particular ∂(b̄_{2}) ≤ δ^{K}(b̄_{1}c̄) = m = ∂(b̄_{1}). By symmetry ∂(b̄_{2}) = m = ∂(b̄_{1}). Since δ^{K}(b̄_{2}d̄) = ∂(b̄_{2}), we have b̄_{2}d̄ ≤ 𝔽_{2}^{K}. Now Lemma 5.3 says that b̄_{2}d̄ is of the same L_{K}^{E}-quantifier-free type as b̄_{1}c̄. □

5.9 Main Theorem. Given 𝔽^{K} ∈ EC_{d}^{K}, let finite A ≤ 𝔽^{K}. Then the following hold:

(i) The axioms PK + SCH_{A} + ID + EC determine the complete theory Th(𝔽^{K}) of 𝔽^{K}.

(ii) The theory Th(𝔽^{K}) has quantifier elimination in language L_{K}^{E}. (iii) Further on, Th(𝔽^{K}) is superstable. (iv) Moreover, the group structure on ker is embedded in 𝔽^{K} con-

servatively, that is no new relations are induced (using parameters) on ker from 𝔽^{K}.

Proof (i) and (ii) It follows from Lemmas 5.7 (with B_{1} ≅ A ≅ B_{2}) and 5.8 that the theory is complete and submodel complete. The latter implies elimination of quantifiers (see e.g. Theorem 13.1 of S).

(iii) To prove superstability consider 𝔽^{K} ∈ EC_{A}^{K} of cardinality λ. We want to establish the cardinality of the set S(𝔽^{K}) of complete 1-types over 𝔽^{K}. Let ^{∗}𝔽^{K} be an elementary extension of 𝔽^{K} which realises all n-types over 𝔽^{K} for all n. Let S^{#}(𝔽^{K}) the set of all complete n-types over 𝔽^{K} which are realised in ^{∗}𝔽^{K} by n-tuples b̄ = ⟨b_{1}, …, b_{n}⟩ such that δ^{K}(b̄/𝔽^{K}) = ∂(b_{1}/𝔽^{K}). It follows that card S(𝔽^{K}) ≤ card S^{#}(𝔽^{K}).

From general properties of ≤ we get 𝔽b̄ ≤ ^{∗}𝔽^{K}, and by Lemma 5.3 the L_{K}^{E}-quantifier-free type of b̄ over 𝔽^{K} is determined by the L_{K}- quantifier-free type of that. Thus card S(𝔽^{K}) is less or equal to the cardinality of QF S(𝔽^{K}), the set of all L_{K}-quantifier-free complete types over 𝔽^{K}.

We claim that QF S(𝔽^{K}) ≤ λ + 2^{ω}. Indeed, each quantifier-free L_{K}-type of b̄ over 𝔽^{K} is uniquely determined by the minimal K-affine subspace L over 𝔽^{K} containing b̄ and, for each l ∈ ℕ, the minimal algebraic variety W^{\frac{1}{l}} containing exp(\frac{b̄}{l}). Notice that, once W = W^{1} is known, for each l there is at most l^{n} choices of W^{\frac{1}{l}} (n = |b̄|), all conjugated by torsion elements of (𝔽^{×})^{n} of order l. This branches into at most 2^{ω} types for each of λ-many varieties W.

(iv) Consider again a saturated model 𝔽^{K} of the theory, let C ≤ 𝔽^{K} be an arbitrary finite self-sufficient set and let B = ker ∪C. Clearly, B ≤ 𝔽^{K}. We claim first that for every finite tuple b̄ in B the complete L_{K}-type of C ∪ b̄ is determined by the quantifier-free L_{K}-type of the tuple. This is again a direct consequence of C∪b̄ ≤ 𝔽^{K}, by Lemma 5.7. Now, since any type of a tuple in the definable B is equivalent to a L_{K}-quantifier-free type, any definable subset of B^{n} is quantifier-free definable, by compactness. We deduce that any C-definable subset of ker^{n} is L_{K}(C)-quantifier-free definable, hence any subset of ker^{n} definable with parameters is L_{K}-quantifier-free definable.

More specifically, let b̄ be ℚ-linearly independent over C. We claim that then it is K-linearly independent over C, which follows from the assumption that δ^{K}(b̄/C) ≥ 0.

It follows that quantifier-free L_{K}(C)-formulas without parameters restricted to ker are Boolean combinations of formulas of the form m_{1}x_{1} + … + m_{n}x_{n} = k_{1}c_{1} + … + k_{p}c_{p}, for some m_{1}, …, m_{n} ∈ ℤ, k_{1}, …, k_{p} ∈ K, and of the form \frac{x}{m} ∈ ker (equivalently, exp(\frac{x}{m}) = 1). The latter can be equivalently rewritten as ∃y ∈ ker x = my. This is the standard form for core formulas in the theory of the ℤ-group (ker, +, 0). Which proves that the subsets of ker^{n} definable in 𝔽^{K} are the same as ones definable in (ker, +, 0). □

6 Raising to powers in the complex numbers

6.1 Consider the structure ℂ^{K} for K ⊆ ℂ. Assume Schanuel’s conjecture or, more specifically, its form derived in 2.5:

ℂ^{K} ∈ E_{d}^{K} and A ≤ ℂ^{K}.

6.2 Theorem. Assume the corollary of Schanuel’s conjecture in the form 6.1. Suppose K ⊆ ℝ. Then

ℂ^{K} ⊧ PK+SCH_{A}+EC + ID.

In particular, the axioms define the complete theory of the structure which also has the properties described in the Main Theorem 5.9 .

Proof PK and SCH_{A} are immediate by assumptions. The axiom ID is a corollary of the Countable Closure property that has been proved for ℂ_{exp} in [8] under Schanuel’s conjecture. The same proof works for ℂ^{K} under the version 6.1 of Schanuel’s conjecture. It remains to establish EC, the exponential-algebraic closedness.

This property was proved in [7], Theorem 3, under the extra assumption that Schanuel’s conjecture holds uniformly. The latter is used in the proof just ones, as a condition for the statement of Theorem 2. But the statement of Theorem 2 is exactly Corollary 4.16, proved here using only assumptions of the present theorem.□

Note that the theorem states in particular that the corresponding form of Schanuel’s conjecture (for K ⊆ ℝ) implies EC, exponentialalgebraic closedness.

6.3 We recall the following result by A.Wilkie, J.Kirby and M.Bays.

Theorem. ([4] 1.3) Let 𝔽_{exp} be any exponential field, let ker be the kernel of its exponential map, let C be an ecl-closed subfield of 𝔽_{exp}, and let λ be an m-tuple which is exponentially algebraically independent over C, K = ℚ(λ). Then for any tuple z from 𝔽 :

tr.deg(exp(z)/C(λ)) + lin.dim_{K}(z/ ker) − lin.dim_{ℚ}(z/ ker) ≥ 0. (14)

In particular, this holds for the exponential field ℂ_{exp} of complex numbers and C = ecl(∅).

Here, an exponential field 𝔽_{exp} is a (𝔽, +, ·, exp) a field structure with a homomorphism exp : 𝔽 → 𝔽^{×}. An ecl-closed subfield is an exponential subfield C ⊆ 𝔽 that is exponentially-algebraically closed inside 𝔽_{exp} (see [4] for details). In the exponential field ℂ_{exp} the eclclosure ecl(X) of a countable subset X is countable, by Lemma 5.12 of [8]. In particular, all but countably many complex numbers are exponentially algebraically independent over ecl(∅).

6.4 Corollary of (14).

Since

tr.deg(exp(z)/C, λ) = tr.deg(exp(z), λ/C)−tr.deg(λ/C) ≤ tr.deg(exp z/C)

we have as a corollary

tr.deg(exp(z)/C) + lin.dim_{K}(z/ ker) − lin.dim_{ℚ}(z/ ker) ≥ 0,

and a weaker version, which is of interest to us here,

δ^{K}(z/ ker) = tr.deg(exp z) + lin.dim_{K}(z/ ker) − lin.dim_{ℚ}(z/ ker) ≥ 0,

which amounts to say that

𝔽^{K} ∈ E_{0}.

6.5 Corollary. Let a finite subset λ ⊆ ℂ be exponentially-algebraically independent over ecl(∅) and let K = ℚ(λ). Then ℂ^{K} satisfies PK+SCH_{0}+ID, where SCH_{0} denotes SCH_{A} with A = {2πi}.

In particular, the statement of Corollary 4.16 holds for ℂ^{K}.

References

[1] J.Ax, On Schanuel Conjectures, Annals of Mathematics, 93 (1971), 252 - 258

[2] C.Bertolin, Périodes de 1-motifs et transcendance, J. Number Theory, 97(2), 2002, pp.204-221,

[3] W.D.Brownawell and K.K.Kubota, Algebraic independence of Weierstrass functions, Acta Arithmetica, 33 (1977), 113-148

[4] M.Bays, J.Kirby and A. J. Wilkie A Schanuel property for exponentially transcendental powers to appear in the Bull. London Math. S.

[5] J.Kirby, The theory of the exponential differential equations of semiabelian varieties, Selecta Mathematica, 15, (2009) no. 3, 445– 486

[6] B.Zilber, Raising to powers in algebraically closed fields. JML, v.3, no.2, 2003, 217-238

[7] B.Zilber, Exponential sums equations and the Schanuel conjecture, J. London Math. Soc. 65(1) (2002), pp.27-44

[8] B.Zilber, Algebraically closed field with pseudo-exponentiation, Annals of Pure and Applied Logic, 132 (2004) 1, pp. 67-95