Raising to powers in algebraically closed fields

B. Zilber University of Oxford

20 November 2003

Abstract

We study structures on the fields of characteristic zero obtained by introducing (multivalued) operations of raising to power. Using Hrushovski-Fraisse construction we single out among the structures exponentially-algebraically closed once and prove, under certain Diophantine conjecture, that the first order theory of such structures is model complete and every its completion is superstable.

Keywords: Superstable, fields, Schanuel conjecture

2000 Mathematics Subject Classification: 03C45, 03C98

1 Introduction

This paper deals with some of the issues discussed in [Z1] and is part of the program of applying ideas around Hrushovski’s construction of ’new strongly minimal structures’ for understanding classical analytic structures.

We consider here the class of two-sorted structures of the form (D, ex, R) where D (the domain of ex) is an infinite-dimensional vector space over a fixed field K of characteristic zero in the usual language of vector spaces, R (the range) a field of characteristic zero and ex is a homomorphism of the additive group of D onto the multiplicative group R^{×} of the field.

In these structures, for a ∈ K, one can consider the relation

∃z(x = ex(z)&y = ex(a · z))

which in the case of D = R = ℂ and ex = exp is represented locally by a transcendental analytic function y = x^{a}. Also, in the structures where the kernel ker of ex is an infinite cyclic group one may consider definable finitely generated groups of the form a_{1} · ker + … + a_{n} · ker for a_{1}, …, a_{n} ∈ K. At the same time ex(a_{1} · ker + … + a_{n} · ker) is a finitely generated multiplicative subgroup in the field R. Thus the structures carry some interesting Diophantine geometry.

We introduce a predimension δ for finite subsets X ⊆ D :

δ(X) = l.d._{K}(X) + tr.d.(ex(X)) − l.d._{Q}(X)

where l.d._{K}(X) is the dimension of the vector space over K generated by X, l.d._{Q}(X) is the dimension of the vector space over ℚ generated by X, tr.d. the transcendence degree.

Given a non-negative integer d, we consider the subclass E_{d} of the class defined by the condition

δ(X) ≥ −d for any finite X ⊆ D.  (1)

The class is always non-empty. The condition (1) is satisfied for the complex numbers (as D and R) and ex = exp if K is a subfield of ℂ of a finite transcendence degree d and the Schanuel Conjecture holds.

This class proves to have a very nice model theory provided a number-theoretical conjecture on intersections of varieties with tori holds. In the terminology of [Z2] a basic torus is an algebraic subgroup of the multiplicative group (R^{×})^{n} (virtually given by a set of equations of the form y_{1}^{m_{1}} · . . . · y_{n}^{m_{n}} = 1 with integer powers) and a torus is a coset of a basic torus. Notice also that (R^{×})^{n} is a torus itself, so we say that a torus T ⊆ (R^{×})^{n} is a proper subtorus, if T ≠ (R^{×})^{n}.

The conjecture CIT states:

Let W ⊆ R^{n} be a ℚ-definable algebraic variety irreducible over ℚ. Then there is a finite family τ(W) of proper subtori of (R^{×})^{n} such that for any basic torus T ⊆ (R^{×})^{n} and any irreducible component S of the intersection W ∩ T satisfying

dim S > dim W + dim T − n

there is T_{i} ∈ τ(W) with S ⊆ T_{i}.

The Schanuel conjecture states that for any additively independent complex numbers x_{1}, …, x_{n}

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

In [Z2] we formulate and discuss connections between a stronger Uniform Schanuel conjecture and CIT.

Under the assumption that CIT holds we prove that

(i) the class E_{d} is axiomatizable;

(ii) the subclass EC_{d} of E_{d}-existentially closed structures is the model completion of E_{d} in the existential expansion of the language, its theory allows elimination of quantifiers in the expanded language and any completion of the theory is superstable.

This allows us to study the classical structure, the field of the complex numbers with raising to real powers. It corresponds to the case D = R = ℂ, ex = exp and K = ℝ. Using [Z2] (which is based on works of D.Bernstein, A.Kushnirenko, A.Khovanski and B.Kazarnovski) we give a complete set of axioms for the structure and prove that it is superstable and allows elimination of quantifiers to the level of existential formulas, provided the Schanuel Conjecture along with CIT hold. In fact the Uniform Schanuel conjecture is sufficient.

The author is thankful to the referee of the paper for suggesting many useful improvements.

2 Definitions and notation

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 [Z2], 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 (D, R) 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 ⊆ D^{n} given by a set of K-linear equations in x_{1}, …, x_{n},

n-ary predicates EW for algebraic varieties W ⊆ R^{n} definable and irreducible over ℚ.

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

E(x, y) ≡ [ex(x) = ex(y)],

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

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

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

ex : D → R^{×}

is a homomorphism of the additive group of D into the multiplicative group R^{×} of the field. The set of axioms above we denote PF(K) (K-powered field of characteristic zero).

Notation For finite X, X′ ⊆ D, Y, Y′ ⊆ R

l.d._{K}(X) the dimension of the vector space sp_{K}(X) generated by X over K;

l.d._{Q}(X) the dimension of the vector space sp_{ℚ}(X) generated by X over ℚ;

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

δ(X) the predimension of finite X ⊆ D :

δ(X) = l.d._{K}(X) + tr.d.(ex(X)) − l.d._{Q}(X);

δ(X/X′) = δ(X ∪ X′) − δ(X′);

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

We let also

l.d._{K}(X/X′) = l.d._{K}(X ∪ X′) − l.d._{K}(X′);

tr.d.(Y/Y′) = tr.d.(Y ∪ Y′) − tr.d.(Y′);

l.d._{Q}(X/X′) = l.d._{Q}(X ∪ X′) − l.d._{Q}(X′);

ker is the name of a unary predicate of type EW : x ∈ ker ≡ ex(x) = 1. We write ker_{|A} for the realisation of this predicate in A.

Given d ∈ ℤ denote E_{d}(K) the subclass of E(K) consisting of all D satisfying the condition:

δ(X) ≥ −d for all finite X ⊆ D.

Below we fix K and write simply E_{d} instead of E_{d}(K).

Denote SE the class of the substructures of the structures of E in the language L_{K}.

Given an integer d denote SE_{d} the subclass of SE consisting of A which satisfy δ(X) ≥ −d for any finite X ⊆ A.

Remark For any structure A in E and any X ⊆ ker_{|A} in the structure

δ(X) ≤ 0

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

Notation Denote E^{0} (correspondingly SE^{0}) the subclass of E (SE) consisting of the structures A such that

δ(X) = 0

holds for any X ⊆ ker_{|A} .

Remark Evidently E_{0} ⊆ E^{0}.

Notation Denote E_{d}^{0} = E^{0} ∩ E_{d}, SE_{d}^{0} = SE_{d} ∩ SE^{0}.

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

L = {⟨x_{1}, …, x_{n}⟩ ∈ D^{n} : k_{i,1}x_{1} + … + k_{i,n}x_{n} = 0}.

Define dim L to be the corank of the matrix (k_{i,j}), equivalently, the Morley rank of the definable subset L of the vector space D. Let L ⊆ D^{n+l} be a K-linear subspace, ā = ⟨a_{1}, …, a_{l}⟩. Denote

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

An affine subspace V ⊆ D^{n} is said to be K-affine defined over the set C ⊆ D if V = L(c̄) for some K-linear subspace L ⊆ D^{n+l} and c̄ ∈ C^{l}. The same terminology is applied for ℚ instead of K.

For L ⊆ D^{n} K-linear denote L̄ the minimal ℚ-linear subspace of D^{n} containing L. For W ⊆ R^{n+l} an algebraic variety, b̄ = ⟨b_{1}, …, b_{l}⟩ denote

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

Remark A K-affine subspace is defined over a set C iff V = L_{0} + ā for some ā ∈ D^{n} ∩ sp_{K}(C) and L_{0} K-linear.

Lemma 2.1 If X = {x_{1}, … x_{n+l}} ⊆ D, X′ = {x_{n+1}, … x_{n+l}}, x̄ = ⟨x_{1}, … x_{n+l}⟩, x̄′ = ⟨x_{n+1}, … x_{n+l}⟩ then:

l.d._{K}(X) = dim L, for L ⊆ D^{n+l} the minimal K-linear subspace containing x̄;

l.d._{Q}(X) = dim L̄;

tr.d.(ex(X)) is the dimension of the minimal variety over ℚ containing ex(x̄);

δ(X/X′) = l.d._{K}(X/X′) + tr.d.(ex(X)/ex(X′)) − l.d._{Q}(X/X′); l.d._{K}(X/X′) = dim L(0^{l}), where 0^{l} is a string of l zeroes; l.d._{Q}(X/X′) = dim L̄(0^{l}).

Proof Immediate from the definitions.□

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

Lemma 2.2 For any structure A of the class SE and finite X, Y, Z ⊆ A :

(i) If sp_{Q}(X′) = sp_{Q}(X) then δ(X′) = δ(X).

(ii) If sp_{Q}(X′Y) = sp_{Q}(XY) then δ(X/Y) = δ(X′/Y).

(ii) If sp_{Q}(Y) = sp_{Q}(Y′) then δ(X/Y) = δ(X/Y′).

(iv) δ(XY/Z) = δ(X/Y Z) + δ(Y/Z).

Lemma 2.3 For A, B, C ∈ SE (i) if A ≤ B and B ≤ C, then A ≤ C; (ii) if A ≤ B, Y ⊆ B, δ(Y/A) = 0, then AY ≤ B.

Proof Immediate from the definitions.□

Notation Let A ∈ SE_{d} and X ⊆ A finite. Denote

∂_{A}(X) = min{δ(X′) : X ⊆ X′ ⊆ A}.

Lemma 2.4 Let A ∈ SE_{d} and X ⊆ A finite. Choose X′ ⊆ A finite such that

δ(X′) = ∂_{A}(X).

Then X′ ≤ A.

Proof Immediate from the definitions.□

Lemma 2.5 Let A, B ∈ SE_{d}, A ≤ B and X a finite subset of A. Then

∂_{A}(X) = ∂_{B}(X).

Proof Immediate from the definitions.□

Lemma 2.6 Suppose A ∈ SE_{d}, A′ ∈ SE, A′ = sp_{Q}(AX), and δ(X′/A) ≥ 0 for all finite X′ ⊆ sp_{Q}X. Then A′ ∈ SE_{d} and A ≤ A′.

Proof We may assume that X is ℚ-linearly independent over A. Let Z ⊆ A′, Z = {z_{1}, … z_{n}}, and z_{i} = x_{i} + y_{i} for some x_{i} ∈ sp_{Q}(X), y_{i} ∈ A. Let {x_{1}, … x_{k}} be a ℚ-linear base of {x_{1}, … x_{n}}. Then, using Lemma 2.2,

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

for y′_{k+1}, …, y′_{n} appropriate ℚ-linear combinations of y_{1}, … y_{n}. Rewrite

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

By the assumtions δ(y′_{k+1}, …, y′_{n}) ≥ −d. On the other hand δ({x_{1} + y_{1}, … x_{k} + y_{k}}/{y′_{k+1}, …, y′_{n}}) ≥ δ({x_{1}, … x_{k}}/A) ≥ 0

since

l.d._{K}({x_{1}+y_{1}, … x_{k}+y_{k}}/{y′_{k+1}, …, y′_{n}}) ≥ l.d._{K}({x_{1}+y_{1}, … x_{k}+ y_{k}}/A) ≥ l.d._{K}({x_{1}, … x_{k}}/A),

tr.d.(ex{x_{1} + y_{1}, … x_{k} + y_{k}}/ex{y′_{k+1}, …, y′_{n}}) ≥ tr.d.(ex{x_{1} + y_{1}, … x_{k} + y_{k}}/exA) ≥ tr.d.(ex{x_{1}, … x_{k}}/exA) and

l.d._{Q}({x_{1}+y_{1}, … x_{k}+y_{k}}/{y′_{k+1}, …, y′_{n}}) = k = l.d._{Q}({x_{1}, … x_{k}}/A). Thus

δ(Z) ≥ −d.

The same argument shows that

δ(Z/A) ≥ 0.

□

Lemma 2.7 There is an A ∈ SE_{d}.

Proof Take an additive subgroup A = ω · ℚ ⊆ D for ω a non-zero element in D. Define H = A/ωℤ. Then H, considered as a multiplicative group, is characterized by the property that it is a torsion group such that any equations of the form x^{n} = h has for any h exactly n solutions. In other words H is isomorphic to the torsion subgroup of an algebraically closed field R of characteristic 0. Define ex as the canonical homomorphism A → R^{×} corresponding to this isomorphism. Obviously, δ(X) = 0 for any finite X ⊆ A.□

Lemma 2.8 Suppose A ∈ SE_{d} and ex(A) contains the torsion subgroup of the field. Then there is D ∈ E_{d} and an embedding of A into D such that A ≤ D and ker_{|D} = ker_{|A} .

Proof Choose algebraically closed fields D and R of characteristic zero such that A ⊆ D, A/E ⊆ R^{×} and l.d._{K}(D/A) = tr.d.(R/exA) ≥ ℵ_{0}. We want to define ex : D → R^{×} extending ex_{A} so that D ∈ E_{d}.

Denote A_{0} = A, ex_{0} = ex_{A} and H_{0} = ex_{0}(A_{0}).

Proceed by induction defining A_{α}, H_{α} and an endomorphism

ex_{α} : A_{α} → H_{α}

by choosing:

On the even steps: the first element a ∈ D\A_{α} and define ex_{α+1}(a) to be any element in R^{×} \ acl(H_{α}). Put A_{α+1} = A_{α} + ℚ · a and extend ex_{α+1} to A_{α+1} as a group homomorphism. Put H_{α+1} = ex_{α+1}(A_{α+1}).

On the odd steps: the first element h ∈ R^{×} \H_{α} and define a to be any element in D \ sp_{K}(A_{α}) and ex_{α+1}(a) = h. Put A_{α+1} = A_{α} + ℚ · a and extend ex_{α+1} to A_{α+1} as a group homomorphism. Put H_{α+1} = ex_{α+1}(A_{α+1}).

On both even and odd steps it follows from Lemma 2.6 that A_{α+1} ∈ SE_{d} and A_{α} ≤ A_{α+1}.

Also,

ker_{|A_{α+1}} = ker_{|A_{α}}

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

3 Exponentially-algebraically closed structures

Definition A structure D in E_{d}^{0} is said to be E_{d}^{0}-exponentially-algebraically closed (e.a.c.) if for any D′ ∈ E_{d}^{0}, such that D ≤ D′, any finite quantifier-free type over D which is realized in D′ has a realization in D.

Denote EC_{d}^{0} the class of E_{d}^{0}-exponentially-algebraically closed structures, or, in the shorter form, EC.

It follows from the transitivity of ≤-embedding and the inductiveness of the class E_{d} in the standard way

Proposition 1 For any D in E_{d}^{0} there exists E_{d}^{0}-e.a.c. structure containing D.

Below D is always an E_{d}^{0}-exponentially-algebraically closed structure.

By Lemma 2.5 we may omit D when writing ∂_{D}.

Notation cl(A) = {b : ∂(Ab) = ∂(A)}

Lemma 3.1 The operator A ↦ cl in D is a closure operator, i.e. it satisfies

(i) A ⊆ B implies A ⊆ cl(A) ⊆ cl(B);

(ii) cl(cl(A)) = cl(A);

(ii) For any b, c ∈ D : b ∈ cl(A, c) \ cl(A) → c ∈ cl(A, b).

Proof Standard.□

We want to find out now what are the systems of equations and inequalities that have solutions in any e.a.c.-structure.

Definition For C ⊆ D, an K-affine variety V ⊆ D^{n} and an algebraic variety W ⊆ R^{n} it is said that the pair (V, W) is definable over C if V is definable over sp_{Q}(C) and the variety W is definable over the field ℚ(ex(sp_{Q}(C))) (we often say ’defined over ex(sp_{Q}(C))).

If W is irreducible over the corresponding set, then the pair is said to be irreducible over C. V is said to be free of additive dependencies over C if there is no proper ℚ-affine subspace of D^{n} containing V. W is said to be free of multiplicative dependencies over C if no connected component of W lies in a proper subtorus of (R^{×})^{n}. A pair (V, W) is said to be a free pair if both V is free of additive dependencies and W is free of multiplicative dependencies.

Let W ⊆ R^{n} be an algebraic variety defined and irreducible over some ex(C) for some C = sp_{Q}(C) ⊆ D. A pair (V, W) is said to be a normal pair over C if in some extension of D there are ⟨a_{1}, …, a_{n}⟩ ∈ V and ⟨b_{1}, …, b_{n}⟩ ∈ W such that for any k ≤ n independent integer vectors m_{i} = ⟨m_{i,1}, … m_{i,n}⟩, i = 1, …, k, and

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

we have

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

Lemma 3.2 Let C, A ∈ SE finite, C ≤ A, c̄ be the string of all elements of C and ā be the string of elements of A. Let L be the minimal K-linear space containing āc̄ and W the minimal algebraic variety over ℚ containing ex(āc̄). Then the pair (L(c̄), W (ex(c̄))) is normal.

Proof Take ā for ⟨a_{1}, …, a_{n}⟩ and ex(ā) for ⟨b_{1}, …, b_{n}⟩ in the definition of normality. Then C ≤ A implies the inequalities required in the definition. □

To formulate an equivalent definition of normality we introduce the following:

Notation Let V ⊆ D^{n} be an affine K-space defined with parameters c̄. Choose a generic n-tuple ā in the space. Given a matrix m̄ of integer vectors m_{i} = ⟨m_{i,1}, … m_{i,n}⟩, i = 1, …, k, consider a′_{i} = m_{i,1}a_{1} + … + m_{i,n}a_{n} and denote m̄V the minimal K-affine subspace over c̄ containing ⟨a′_{1}, …, a′_{k}⟩, (the K-locus over c̄).

Similarly, for an algebraic variety W ⊆ R^{n} defined over d̄ and the same m̄ choose a generic n-tuple b̄ in W, consider b′_{i} = b_{1}^{m_{i,1}} · . . . · b_{n}^{m_{i,n}} and denote W m̄ the algebraic locus over d̄ of ⟨b′_{1}, …, b′_{k}⟩.

Evidently, the definitions do not depend on the choice of the generic tuples.

Lemma 3.3 The pair (V, W) is normal if and only if for any independent integer vectors m_{i} = ⟨m_{i,1}, … m_{i,n}⟩, i = 1, …, k

dim(m̄V) + dim(W m̄) ≥ k.

Proof Immediate from the definitions.□

Lemma 3.4 Let C ⊆ D and (V, W) a normal free irreducible pair over C. Let V′ ⊆ V be a finite union of proper K-affine subspaces definable over C and W′ ⊆ W a proper algebraic ex(C)-definable subvariety. Then there is ā in D such that ā ∈ V \V′ and ex(ā) ∈ W \W′. Moreover in some extension D′ ≥ D ā can be chosen generic in V over C and ex(ā) generic in W over ex(C).

Proof Take ā in some extension of D to be generic in V over D and b̄ in some extension of R generic in W over R. Choose a sequence \{b̄^{\frac{1}{l}} : l ∈ ℕ\} associated with b̄ in the following sense:

b̄^{1} = b̄ and (b̄^{\frac{1}{ml}})^{m} = b̄^{\frac{1}{l}} for any m, l ∈ ℕ, and (x̄)^{m} is understood coordinatewise.

Define ex on A = D + sp_{Q}(a_{1}, … a_{n}) as:

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

for any integers m_{i}, l ≠ 0 and element d ∈ D. The definition is consistent since V is free of additive dependencies. Evidently the formula defines a homomorphism. The kernel of the homomorphism coincides with that of ex on D, since W has no multiplicative dependencies. Thus A ∈ SE. Notice that by the normality for any k independent integer vectors m_{i} = ⟨m_{i,1}, … m_{i,n}⟩, i = 1, …, k, it holds δ({m_{1}ā, …, m_{k}ā}/D) ≥ 0.

Thus D ⊆ A satisfy the assumptions of Lemma 2.6 and hence A ∈ SE_{d}, D ≤ A. By the choice ā ∈ V \ V′ and ex(ā) ∈ W \ W′. Since D ∈ EC there is a realization of the type in D. □

Proposition 2 A structure D in E_{d}^{0} is e.a.c. iff given any C ⊆ D, a normal free pair (V, W) over C and a finite union V′ ⊆ V of proper K-affine subspaces definable over C, there is ā ∈ V ∩ ln W such that ā ∉ V′.

4 Definability of normality and freeness conditions

This technical section heavily relies on [Z2] where in particular a theorem of J.Ax is used. Later, while preliminary versions of the present paper were put on my web-page, B.Poizat [P] and K.Holland [H] used a preliminary version of [Z2] and the theorem of Ax to prove technical results very similar to the main result in this section, for their own purposes (however, linked with a Hrushovski style construction). So it would be difficult to resolve the priority question if such one happens to arise.

Let V (a) ⊆ D^{n} be a K-affine subspace defined over some finite tuple a from D, V′(a) a finite union of K-affine subspaces defined over a, W (b) an algebraic variety defined over b, a tuple from R. In fact, we may assume a ∈ D^{n} is a vector such that V (a) = L + a and L is K-linear. Thus dim V (a) = dim L does not depend on a. Also, V (a) is free of additive dependencies iff L is. It is evident that the set of a for which V′(a) is a proper subset of V (a) is quantifier-free definable in the K-vector space language. Also, by basic algebraic geometry the set of b satisfying for a given l the statement:

W (b) is irreducible and dim W (b) ≥ l

is quantifier-free definable in the language of fields.

Our further arguments use the following statement (Corollary 3 of [Z2]).

Fact 1 Given an algebraic variety W (ā) ⊆ ℂ^{n} there is a finite collection μ(W) of non-zero integer vectors such that for any torus T ⊆ ℂ^{∗}^{n} and an infinite atypical component S ⊆ W (ā) ∩ T of the intersection there is m̄ ∈ μ(W) and a constant c (depending on a and T) such that all (a_{1}, …, a_{n}) in the component satisfy a_{1}^{m_{1}} · . . . · a_{n}^{m_{n}} = c.

Lemma 4.1 The set of b such that W (b) is of dimension l, irreducible and free of multiplicative dependencies is quantifier-free definable in the language of fields.

Proof Given W (b) which is not free of multiplicative dependencies, W (b) ⊆ T for some proper torus. This implies that W (b) is an atypical component of W (b) ∩ T. By Fact 1 this is equivalent to the statement

⋁_{m̄∈μ(W)} ∀ȳ ∈ W (b) ȳm̄ = const.

□

Notation Given a basic torus T ⊆ (R^{×})^{n} there is a uniquely determined algebraic (group) variety (R^{×})/T and the corresponding regular homomorphism

[T] : (R^{×})^{n} → (R^{×})^{n}/T.

We write W/T for the image of W under the homomorphism instead of [T](W). Also, since T is uniquely determined by any of it cosets, we use the notation also when T is a non-basic torus.

Let T ⊆ P be tori, W ⊆ P. We say that W/T is an atypical image with respect to P if

dim W/T < min{dim P/T, dim W }.

Easy dimension calculations show for irreducible W ⊆ P, W/T atypical image, that for any generic w ∈ W it holds

dim W ∩ T w > 0  (2)

and

dim W ∩ T w > dim W − dim P/T.  (3)

The following Fact has been proved in [Z2]: The statement of the Proposition 1 of [Z2] (or rather its reformulation in the proof) is stronger than the Fact. The proof assumes CIT but the Corollary 3 of [Z2] (Fact 1 above, the function field case of CIT) in an obvious way replaces CIT in this proof to yield:

Fact 2 Let W ⊆ R^{N} be an algebraic variety, a ∈ R^{r}, some r < N, P ⊆ R^{N} a torus and

{y ∈ P : y^{⌢}a ∈ W } = W (a) ⊆ P.

Then there is a finite collection π_{P} (W) of basic subtori of P depending on W only, such that given a torus T ⊆ P, for any connected infinite atypical component X of W (a)∩T, there exists Q ∈ π_{P} (W) and c ∈ P such that X ⊆ Q · c and X is typical in W (a) ∩ T with respect to Q · c.

Proposition 3 ^{1} Given W (a) ⊆ P = (F^{∗})^{n} an irreducible algebraic variety, for any basic torus T ⊆ P with atypical image W (a)/T with respect to P, there is Q ∈ π_{P} (W) such that

dim W (a)/Q = dim W (a)/T − dim Q/(Q ∩ T)

and

dim W (a)/T = dim W (a)/(Q ∩ T).

Proof Let w ∈ W (a) be generic and X ⊆ W (a) ∩ T · w be a component of the intersection of maximal dimension. Then by the additive formula

dim W (a)/T = dim W (a) − dim X  (4)

and dim X = dim W (a) ∩ T · w > 0. We may assume w ∈ X. By Fact 2 there is Q ∈ π_{P} (W) such that (i) X ⊆ Q · w and (ii) X is a typical component of the intersection (W (a) ∩ Qw) ∩ T w with respect to Qw. By (i) and the maximality of dim X, we have dim W (a)/T = dim W (a)/(Q∩T). And (ii) means that, given a connected component Y ⊇ X of the variety W (a) ∩ Qw, we have

dim X = dim Y + dim Q ∩ T − dim Q.  (5)

But Y is a component of a generic fiber of the mapping W (a) → W (a)/Q, and by the classical theorem on dimension of fibers ([S], Chapter 1, s.6, Thm 7)

dim Y = dim W (a) ∩ Qw = dim W (a) − dim W (a)/Q.  (6)

Combining (4), (5) and (6) we get the requred equality on dim W (a)/Q.□

In case P = (R^{×})^{n} we write π(W) instead of π_{P} (W).

Lemma 4.2 If a pair (V, W (a)) in n-spaces is not normal then either dim V +dim W (a) < n, or there is Q ∈ π(W) defined by a matrix q on l = codim Q independent integer n-rows as Q = {y ∈ (F^{×})^{n} : y^{q} = 1} such that

dim qV + dim W (a)^{q} < l.

1I am grateful to Kitty Holland for detecting a serious error in the formulation of the Proposition in the previous version of the paper. The present version is quite similar to an unpublushed result of her’s, and her proof is based on the similar results from Section 5 of [Z2]

Proof Suppose dim V + dim W (a) ≥ n, and the pair is not normal, which is witnessed by m̄, a set of k < n independent integer n-vectors, as

dim m̄V + dim W (a)m̄ < k.

By definitions the mapping x → m̄x is is a linear surjective mapping D^{n} → D^{k} and y → ym̄ is a surjective homomorphism (R^{×})^{n} → (R^{×})^{k}. Denote the kernel of the second one T, thus the latter mapping in notations above is P → P/T, and W (a)m̄ = W (a)/T.

Claim. W (a)/T is an atypical image.

Suppose not. Then, in case dim P/T ≤ dim W (a), we get dim W (a)/T = dim P/T = k, a contradiction. In case dim W (a) < dim P/T we get dim W (a)/T = dim W (a). It follows dim m̄V + dim W (a)^{m} = dim mV + dim W (a) ≥ dim V − dim T + dim W (a) ≥ n − dim T = dim P/T, which contradicts the assumptions again. Claim proved.

By Proposition 3 there is Q ∈ π(W) with dim W (a)/Q = dim W (a)/T − dim Q/(Q ∩ T) and dim W (a)/(Q ∩ T) = dim W (a)/T.

Claim 2. W.l.o.g. we may assume that Q ⊇ T.

Indeed, the basic torus Q∩T is given by a sytem of k′ = codim Q∩ T ≥ k independent equations y^{m′} = 1.

By definition m′ defines a linear surjective mapping m′ : D^{n} → D^{k′}, with ker m′ ⊆ ker m, so m can be obtained as the composition of m′ with another linear mapping with fibers of dimension k′ − k. Thus,

dim m′V ≤ dim mV + k′ − k.

On the other hand

dim W (a)^{m′} = dim W (a)/(Q ∩ T) = dim W (a)/T = dim W (a)^{m}.

Thus

dim m′V + dim W (a)^{m′} ≤ dim mV + dim W (a)^{m} + k′ − k < k′.

In other words, we can replace T by Q ∩ T, and so m by m′, and still witness the failure of normality. Claim proved.

Let now the above basic torus Q ⊇ T be given by l = codim Q ≤ k equations of the form y^{q} = 1, and the matrix q induce the surjective mappings

D^{n} → D^{l} and (R^{∗})^{n} → (R^{∗})^{l}.

Since Q ⊇ T we have dim qV ≤ dim mV, while on R we have dim W (a)^{q} = dim W (a)^{m} − (k − l), by the definition of Q.

The two last formulas yield

dim qV + dim W (a)^{q} ≤ dim mV + dim W (a)^{m} + l − k.

It follows

dim qV + dim W (a)^{q} < l.

□

Corollary 1 Given V (a), V′(a), W (ex(a)), as defined in the beginning of the section, the statement about parameters a :

(V (a), W (ex(a))) is a free normal pair and V′(a) is a proper subset of V (a)

is quantifier-free definable in L_{K}.

Denote the formula from the corollary NF_{V,V′,W} (a). Denote EC the set of axioms of the form

∀x[NF_{V,V′,W} (x) → ∃y((y ∈ V (x))&(y ∉ V′(x))&(ex(y) ∈ W (ex(x))))]

It follows from Proposition 2

Corollary 2 For any D ∈ E_{d}^{0}

D ⊧ EC iff D is exponentially-algebraically closed.

5 Axiomatizing E_{d}

Notation Denote E_{d/ ker} the subclass of E for which δ(X/ ker) ≥ −d holds for any finite X ⊆ D. Denote E_{d/ ker}^{0} = E^{0} ∩ E_{d/ ker}.

Lemma 5.1 For {x_{1}, …, x_{n}} ⊆ ker (i) δ(x_{1}, …, x_{n}) = 0 iff l.d._{K}(x_{1}, …, x_{n}) = l.d._{Q}(x_{1}, …, x_{n}); (ii) the condition δ(X) = 0 for all X ⊆ ker is equivalent to:

x_{n} = k_{1}x_{1} + … + k_{n−1}x_{n−1} with k_{1}, …, k_{n−1} ∈ K and 0 ≠ x_{i} ∈ ker for any i ≤ n implies k_{1}, …, k_{n−1} ∈ ℚ.

Proof (i) is immediate from the definitions. To see (ii) assume first that the condition on ker holds and x_{n} = k_{1}x_{1}+…+k_{n−1}x_{n−1} is a minimal counterexample. By (i) then one gets x_{n} = q_{1}x_{1} + … + q_{n−1}x_{n−1} for some q_{1}, …, q_{n−1} ∈ ℚ. Combining the two linear combinations one comes to k_{i} = q_{i} for all i < n. The converse is obvious.□

Lemma 5.2 Let A ∈ SE^{0}, B ∈ SE and A ≤ B. Then B ∈ SE^{0}.

Proof Let X ⊆ B and X ⊆ ker . Then, since tr.d.(X/A) = 0,

0 ≤ δ(X/A) = l.d._{K}(X/A) − l.d._{Q}(X/A)

and hence l.d._{K}(X/A) = l.d._{Q}(X/A).

We want to prove that if x_{n} = k_{1}x_{1}+…+k_{n−1}x_{n−1} for x_{1}, …, x_{n} ∈ X then all k_{i} ∈ ℚ. Suppose x_{1}, …, x_{n} is a counterexample with k_{n−1} ∈ K\ℚ, x_{1}, …, x_{l} ∈ A and x_{l+1}, …, x_{n} ∈ B \ A with n − l minimal possible.

Notice that then l.d._{K}(x_{l+1}, …, x_{n}/A) = n − l − 1 and hence l.d._{Q}(x_{l+1}, …, x_{n}/A) = n − l − 1. Thus there are non-trivial integer coefficients m_{l+1}, …, m_{n} such that m_{l+1}x_{l+1} + … + m_{n}x_{n} = y ∈ A. It follows y ∈ ker . Combining this with the initial combination one contradicts minimality.□

Lemma 5.3 E_{d/ ker}^{0} = E_{d}^{0}

Proof We assume d ≥ 0. Let D ∈ E_{d/ ker}^{0}, X ⊆ D. Then δ(X/ ker) ≥ −d which means that δ(X ∪ Y) ≥ −d for appropriate finite Y ⊆ ker . Claim δ(X ∪ Y) ≤ δ(X). It is enough to prove that δ(Xy) ≤ δ(X) for any y ∈ ker . If y ∈ sp_{Q}(X) then the equality holds. If y ∉ sp_{Q}(X) then l.d._{Q}Xy = l.d._{Q}X + 1, l.d._{K}Xy ≤ l.d._{K}X + 1, tr.d.ex(Xy) = tr.d.ex(X). Claim proved. It follows from the Claim that D ∈ E_{d}^{0} and thus E_{d/ ker}^{0} ⊆ E_{d}^{0}. Assume now D ∈ E_{d}^{0}. Then for any finite X ⊆ D any Y ⊆ ker one has δ(X ∪ Y) ≥ −d. It follows δ(X/ ker) ≥ −d. Thus D ∈ E_{d/ ker}^{0}. □

Notation Let AK be the following set of axioms: for any k_{1}, …, k_{n} ∈ K such that 1, k_{1}, …, k_{n} are ℚ-linearly independent there is an axiom stating:

∀x̄(x_{1}, …, x_{n} ∈ ker \{0} → k_{1} · x_{1} + … + k_{n} · x_{n} ∉ ker).

Remark If ker is a cyclic subgroup then AK holds in the structure.

Lemma 5.4 The subclass of E axiomatized by AK is exactly E^{0}.

Proof By Lemma 5.1 AK holds for any D ∈ E^{0}. To prove the converse, by the same Lemma, we need to prove that for any k_{1}, …, k_{n} ∈ K and a_{1}, …, a_{n}, b ∈ ker if k_{1} · a_{1} + … + k_{n} · a_{n} = b then there are q_{1}, …, q_{n} ∈ ℚ such that q_{1} · a_{1} + … + q_{n} · a_{n} = b.

Suppose w.l.o.g. that a_{1}, …, a_{n} are ℚ-linearly independent and k_{1} · a_{1} + … + k_{n} · a_{n} = b ∈ ker . By the axioms there are integers m_{1}, …, m_{n+1} such that k_{1} · m_{1} + … + k_{n} · m_{n} + m_{n+1} = 0 with, say, m_{1} ≠ 0. It follows that

m_{1} · b = k_{2} · (m_{1} · a_{2} − m_{2} · a_{1}) + … + k_{n} · (m_{1} · a_{n} − m_{n} · a_{1}) − m_{n+1} · a_{1}.

Since m_{1} ·a_{i} −m_{i} ·a_{1} ∈ ker for i = 2, …, n and m_{1} ·b+m_{n+1} ·a_{1} ∈ ker, by induction hypothesis m_{1} · b + m_{n+1} · a_{1} is a ℚ-linear combination of m_{1} · a_{i} − m_{i} · a_{1} for i = 2, …, n and thus b is a ℚ-linear combination of a_{1}, …, a_{n}. □

From now on we have to use the conjecture CIT formulated in the introduction and discussed in [Z2]. Recall that τ(W) is the finite set of basic tori stipulated in the conjecture.

Notation Let, for a definable K-linear L ⊆ D^{n+l}, a natural number l and an algebraic variety W ⊆ R^{n} defined and irreducible over ℚ, the formula

A_{L,W} (x̄) := (∀z̄ ∈ ker^{l})[(x̄z̄ ∈ L)&(ex(x̄) ∈ W) → ⋁_{T ∈τ(W)} ex(x̄) ∈ T].

Definition The pair (L, W) as above is said to be m-special if the minimal torus T(W) containing W is ex(L̄(0^{l})) and

dim L(0^{l}) + dim W < m + dim L̄(0^{l}).

Equivalently, for an ā generic in L and a b̄ generic in W,

l.d._{K}(ā/ ker) + tr.d.(b̄) − l.d._{Q}(ā/ ker) < m.

Lemma 5.5 For a structure D in E and c̄ ∈ D^{n}, given m ∈ ℤ

δ(c̄/ ker) ≥ m iff ⋀_{(L,W) is m-special} A_{L,W} (c̄)

Proof Suppose, given m and c̄, all the formulae hold in D.

Let l, ā ∈ ker^{l} and L be chosen so that c̄ā ∈ L and dim L(ā) = l.d._{K}(c̄/ ker), dim L̄(ā) = l.d._{Q}(c̄/ ker) (see Lemma 2.1.) Remember that dim L(ā) = dim L(0^{l}) and dim L̄(ā) = dim L̄(0^{l}). Let W be the minimal algebraic variety over ℚ containing ex(c̄). We claim that

δ(c̄/ ker) = dim L(0^{l}) + dim W − dim L̄(0^{l}) ≥ m.

Suppose the opposite is true. Then (L, W) is m-special and

D ⊧ A_{L,W} (c̄).

Hence, by the choice of L, ā and W, necessarily ex(c̄) ∈ T for a proper torus T ∈ τ(W). This contradicts the minimality of W. The right-to-left implication in the statement is proved.

To prove the converse suppose that

δ(c̄/ ker) ≥ m and for some m-special (L, W) D ⊧ ¬A_{L,W} (c̄).

Then for some ā ∈ ker^{l}

c̄ā ∈ L & ex(c̄) ∈ W \ ⋃_{T ∈τ(W)} T.

Let b̄ ∈ ker^{r} and N be a ℚ-linear subspace of D^{n+r} such that c̄b̄ ∈ N and

dim N(b̄) = dim N(0^{r}) = l.d._{Q}(c̄/ ker).

Notice that ex(N(0^{l})) is then equal to the minimal torus T_{c} containing ex(c̄) and dim T_{c} = dim N(0^{l}). Also notice that

l.d._{K}(c̄/ ker) ≤ dim L(0^{l}),
tr.d.(c̄/ex(ker)) = tr.d.(c̄) ≤ dim W ∩ T_{c}.

Since δ(c̄/ ker) ≥ m,

dim L(0^{l}) + dim W ∩ T_{c} − dim T_{c} ≥ m.

By our assumptions T_{c} is not a subtorus of any T ∈ τ(W), thus by CIT

dim W ∩ T_{c} = dim W + dim T_{c} − dim T(W).

Notice that by assumptions T(W) = ex(L̄(0^{l})) and dim T(W) = dim L̄(0^{l}). Hence

dim L(0^{l}) + dim W − dim L̄(0^{l}) ≥ m,

which contradicts the fact that (L, W) is m-special. □

Corollary 3 The subclass E_{d/ ker} of E is axiomatized by the set of axioms:

AS_{d}  (∀x̄)A_{L,W} (x̄)  (L, W) is (−d)-special.

Corollary 3 and Lemma 5.4 immediately imply

Theorem 1 The subclass of E axiomatized by AS_{d} and AK is exactly E_{d}^{0}. The class of structures axiomatized by PF(K), AS_{d}, AK and EC is exactly EC_{d}^{0}.

Notation Denote PCF_{d}(K) the theory of EC_{d}^{0}(K). In what follows we omit K.

6 The theory of algebraically closed K-powered fields of characteristic zero

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

E_{P} (x̄) ≡ ∃ȳP (x̄, ȳ),

where P is a quantifier-free formula, is denoted L_{K}^{E}, and these predicates are called E-predicates. Notice that negations of A_{L,W} (x̄) are equivalent to E-predicates.

Lemma 6.1 For M, N ∈ E^{0}, if M ⊆ N in the language L_{K}^{E}, then M ≤ N.

Proof Given a finite X ⊆ N, its L_{K}-quantifier-free type obviously tells the value of δ(X).

Also, the statement ’∂_{M}(X) = δ(X)’ follows from the L_{K}^{E}-quantifier-free type of X. Indeed, using the Claim from the proof of Lemma 5.3, one easily sees that, if m = δ(X), the statement is equivalent to

δ(X) = m & ∀Z δ(XZ/ ker) ≥ m.

By Lemma 5.5 the second part of the expression is given by negations of E-predicates.

It follows from general properties of ≤ that for any Y ⊆ N, given X ⊆ M such that ∂_{M}(X) = δ(X), one has δ(Y/X) ≥ 0. Thus M ≤ N.□

Lemma 6.2 Assume M_{1}, M_{2} ∈ E_{d}^{0} and both satisfy EC.

(i) Suppose A ⊆ M_{1}, A ⊆ M_{2} and b̄_{i} ∈ M_{i}^{n} are such that Ab̄_{i} ≤ M_{i} for i = 1, 2 and the L_{K}-quantifier-free types of b̄_{1} and b̄_{2} over A coincide. Then the L_{K}^{E}-quantifier-free types of b̄_{1} and b̄_{2} over A coincide.

(ii) Suppose M_{2} ≤ M_{1}. Then M_{2} ⊆ M_{1} in the language L_{K}^{E}.

Proof (i) Let ∃x̄P_{b_{1}}(x̄) be an E-predicate with parameters in Ab̄_{1} which holds in M_{1}, with P_{b_{1}} quantifier-free. Let d̄ be a string in M_{1} for which P_{b_{1}}(d̄) holds. Then P_{b_{1}}(x̄) is a consequence of a formula P_{b_{1}}^{0}(x̄) of the form

x̄āb̄_{1} ∈ L \ (L^{0} ∪ … ∪ L^{k}) & (ex(x̄āb̄_{1}) ∈ W \ W^{0}),

where ā is the string of all elements of A, L is the minimal K-linear space containing d̄āb̄_{1} and W is the minimal algebraic variety over ℚ containing ex(d̄āb̄_{1}), L^{i} ⊆ L are K-linear subspaces and W^{0} ⊆ W is an algebraic subvariety over ℚ. By Lemma 3.2 it follows that (L, W) is normal over āb̄_{1}. Moreover, since normality is expressible quantifierfreely in L_{K}, the pair is normal over āb̄_{2}. It follows from axioms EC that ∃x̄P̄_{b_{2}}^{0}(x̄) holds in M_{2} and hence ∃x̄P_{b_{2}}(x̄) holds. Thus b̄_{1} and b̄_{2} satisfy the same E-predicates over A.

(ii) Let A = M_{2} ≤ M_{1} and b_{1}, b_{2} ∈ A be of the same L_{K}-quantifier-free type. Then we have the assumptions of (i) satisfied, and the argument above proves that every E-predicate with parameters in M_{2} which holds in M_{1} must also hold in M_{2}. The converse is obvious, thus M_{2} ⊆ M_{1} as an L_{K}^{E}-substructure. □

Corollary 4 For D_{1}, D_{2} ∈ EC_{d}^{0}

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

Notation Define ID to be the set of axioms of the form

∃x_{1}, …, x_{m}∀y_{1}, …, y_{n}A_{L,W} (x_{1}, …, x_{m}, y_{1}, …, y_{n})

for positive integers m and (L, W) ranging over all the m-special pairs.

Remark If for any m there is X ⊆ D such that ∂(X) ≥ m then D ⊧ ID. Thus any e.a.c. infinite dimensional structure satisfies ID. It is probable that any D ∈ EC_{d}^{0} satisfies ID, so this set of axioms is redundant.

Proposition 4 Suppose D ⊧ PCF_{d}+ID. Then any finite L_{K}^{E}-quantifier-free type which is realized in an E_{d}^{0}-extension of D is realized in D itself.

Proof Consider an elementary extension D^{∗} of D. We prove that for any finite A ⊆ D^{∗} and c̄ in an extension D′, with D^{∗} ≤ D′, the L_{K}^{E}- quantifier-free type q of c̄ over A is realized in D^{∗}. W.l.o.g. we assume that A ≤ D^{∗}.

Consider first the case ∂(c̄/A) = 0. Choose c̄′ in D′ extending c̄ such that ∂(c̄/A) = δ(c̄′/A) = 0. Since A ≤ D′ by Lemma 3.2 and axioms EC the L_{k}-quantifier-free type of c̄′ over A is realized in D^{∗}. Let b̄′ be the realization. Since ∂(c̄′/A) = δ(c̄′/A) we have Ac̄′ ≤ D′. Since δ(b̄′/A) = 0 and A ≤ D^{∗} we have Ab̄′ ≤ D^{∗}. Hence by Lemma 6.2 the L_{K}^{E}-quantifier-free types of b̄′ and c̄′ over A coincide. Hence the corresponding substring of b̄′ is a realization we sought for.

Let now ∂(c̄/A) = k > 0 and {c_{1}, …, c_{k}} be a ∂-base of c̄ over A. It follows from ID and the saturatedness that there are {b_{1}, …, b_{k}} in D^{∗} which are ∂-independent over A. Evidently the L_{K}-quantifier-free types of ⟨b_{1}, …, b_{k}⟩ and ⟨c_{1}, …, c_{k}⟩ over A coincide. Also A ∪ {b_{1}, …, b_{k}} ≤ D^{∗} and A∪{c_{1}, …, c_{k}} ≤ D′. Hence the L_{K}^{E}-quantifier-free types of the strings coincide too. Thus we may identify the strings and since ∂(c̄/A ∪ {c_{1}, …, c_{k}}) = 0 by the case considered above the L_{K}-quantifier-free type of c̄ over A is realized in D^{∗}. □

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

D_{1} ⊧ S(a) iff D_{2} ⊧ S(φ(a)).

Lemma 6.3 Let D_{1} and D_{2} satisfy PCF_{d} + ID, and A_{1} ≤ D_{1}, A_{2} ≤ D_{2} such that there is an L_{K}^{E}-monomorphism

φ : A_{1} → A_{2}.

or A_{1} = A_{2} = ∅.

Let D_{1}^{A} and D_{2}^{A} be the expansions of D_{1}, D_{2} by the set of constants naming elements of A_{1} and A_{2} in correspondence with φ. Then

D_{1}^{A} ≡ D_{2}^{A}.

Proof We prove that given ω-saturated elementary extensions D_{1}^{∗} of D_{1} and D_{2}^{∗} of D_{2}, given finite B ⊆ D_{1}^{∗}, c ∈ D_{1}^{∗} and a L_{K}^{E}- monomorphism φ of A_{1}∪B into D_{2}^{∗} 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 A_{1} ∪ B = A = φ(A). It is enough to show that under the assumption for any c ∈ D_{1}^{∗} we can extend φ to some A′ ⊇ Ac as an L_{K}^{E}-monomorphism and A′ ≤ D_{1}^{∗}, φ(A′) ≤ D_{2}^{∗}.

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

If ∂(c/A) = 0 then extend c to a finite string c̄ from D_{1}^{∗} so that δ(c̄/A) = 0. Again, as in the proof of Proposition 4, there is b̄ in D_{2}^{∗} which realizes the L_{K}-quatifier-free type of c̄ over A. Since δ(b̄/A) = 0, Ab̄ ≤ D_{2}^{∗} holds. Thus by Lemma 6.2 the L_{K}^{E}-quatifier-free types of b̄ and c̄ over A coincide. Put φ(c̄) = b̄, A′ = Ac̄.□

To apply the Lemma we need A_{1}, A_{2} satisfying the assumption. In particular, we can not start with A_{1} = A_{2} = ∅ if d > 0. The next lemma solves the problem for D_{1} and D_{2} satisfying the same L_{K}-existential sentences.

Lemma 6.4 For any finite subset A_{1} of a structure D_{1}, model of PCF_{d} + ID, there is a finite subset Ã_{1} such that

(i) Ã_{1} ≤ D_{1};

(ii) if A_{2} is a subset of an ω-saturated model D_{2} of PCF_{d} +ID and there is a L_{K}^{E}-monomorphism φ : A_{1} → A_{2}, then φ can be extended to Ã_{1} and

φ(Ã_{1}) = Ã_{2} ≤ D_{2}.

Proof Let ā_{1} be the string of all elements of A_{1} and c̄ in D_{1} such that δ(ā_{1}c̄) = ∂(ā_{1}). It follows A_{1}c̄ ≤ D_{1}. Let m = ∂(ā_{1}).

Let q^{0}(x̄ȳ) be the L_{K}-quantifier-free type of ā_{1}c̄. Let ā_{2} be a string in D_{2} which is L_{K}^{E}-monomorphic to ā_{1}. Then the E-predicates guarantee that q^{0}(ā_{2}ȳ) is consistent and thus ∂(ā_{2}) ≤ m = ∂(ā_{1}). By symmetry ∂(ā_{2}) = m = ∂(ā_{1}). Let ȳ = d̄ be the realization of q^{0}(ā_{2}ȳ) in D_{2}. Since δ(ā_{2}d̄) = ∂(ā_{2}), we have ā_{2}d̄≤ D_{2}. Now Lemma 6.2 says that ā_{2}d̄ is of the same L_{K}^{E}-quantifier-free type as ā_{1}c̄. □

Theorem 2 The theory PCF_{d} + ID is a model completion of PF + AK + AS_{d} + ID in the language L_{K}^{E}. The theory has quantifier elimination in this language.

Proof It follows from Lemmas 6.3 and 6.4 that the theory is submodel complete. Thus (see e.g. Theorem 13.1 of [S]) it has elimination of quantifiers. □

Remark In fact, given a model D of PCF_{d} + ID we may assume that there is a finite A ⊆ D with δ(A) = −d (otherwise D is a model of PCF_{d′} for some d′ < d ) and thus A ≤ D. The fact that there exists X ≅ A in the basic language can be expressed by the formula ∃XA_{L,W} (X) for some pair (L, W) witnessing the fact that δ(A) = δ(A/ker) = −d (see Lemma 5.5). Then, by Lemmas 6.3 and 6.2,

PCF_{d} + ID + ∃XA_{L,W} (X)

is a complete theory.

Theorem 3 Any completion of PCF_{d} + ID is superstable.

Proof Let D ∈ EC_{d}^{0} satisfy ID and card D = λ. We want to establish the cardinality of the set S(D) of complete 1-types over D. Let D^{∗} be an elementary extension of D which realizes all n-types over D for all natural n. Let S^{#}(D) the set of all complete n-types over D which are realized in D^{∗} by n-tuples b̄ = ⟨b_{1}, …, b_{n}⟩ such that δ(b̄/D) = ∂(b_{1}/D). It follows that card S(D) ≤ card S^{#}(D). From general properties of ≤ we get Db̄ ≤ D^{∗}, and by Lemma 6.2 the L_{K}^{E}- quantifier-free type of b̄ over D is determined by the L_{K}-quantifier-free type of that. By quantifier elemination the complete type of b̄ over D is determined by the L_{K}-quantifier-free subtype. Thus card S(D) is less or equal to the cardinality of QS(D), the set of all L_{K}-quantifier-free complete types over D, which is of power λ + 2^{ω}, since each such type is uniquely determined by (V, W, \{W^{\frac{1}{l}} : l ∈ ℕ\}) for some Kaffine space V, an algebraic variety W and an associated sequence of varieties \{W^{\frac{1}{l}} : l ∈ ℕ\}. □

7 Raising to real powers in the complex field

Let K ⊆ ℂ be of finite transcendence degree d. Notice that

l.d._{K}(X) ≥ tr.d.(X/K) ≥ tr.d.(X) − tr.d.(K)

in this case. Thus

l.d._{K}(X)+tr.d.(exp(X))−l.d._{Q}(X) ≥ [tr.d.(X)+tr.d.(exp(X))−l.d._{Q}(X)]−d.

Assuming the Schanuel conjecture, the expression in the brackets is non-negative. Thus one gets

δ(X) ≥ −d

for

δ(X) = l.d._{K}(X) + tr.d.(exp(X)) − l.d._{Q}(X).

Assume now K is a subfield of the reals ℝ and has transcendence degree d, D = R = ℂ, ex = exp, and let ℂ^{(K)} = (ℂ, exp, ℂ) be the corresponding two-sorted structure on the complex numbers in the language L_{K}.

Lemma 7.1 (i) Assume SchC. Then ℂ^{(K)} satisfies PF+AK; (ii) Assume also CIT. Then ℂ^{(K)} satisfies AS_{d}.

Proof (i) Follows from the remarks above.

(ii) Again, Schanuel’s conjecture implies ℂ^{(K)} ∈ E_{d/ ker}, so the statement follows from Corollary 3.□

Theorem 4 Assuming SchC+CIT, for any field K ⊆ ℝ of finite transcendence degree d, the structure ℂ^{(K)} satisfies PCF_{d}+ ID. Thus the theory of the structure allows quantifier elimination in the language L_{K}^{E} and is superstable.

Proof The main result of [Z2], Theorem 5 followed by a Remark, state under the assumtions of the theorem under the proof.

Fact 3 Let L ⊆ ℂ^{n} be an ℝ-linear subspace and W a family of algebraic varieties such that (L, W (a)) is normal and free for any a in a definable set of parameters C(W). Then there is a positive real constant R(L, W) such that, given a ball B ⊆ Re(L) of radius R(L, W), there is a point

x ∈ (Re(L) + ıB) ∩ ln W (a) (notice that (Re(L) + ıB) ⊆ L). (7)

Moreover, for any number l we can choose a real constant R(L, W, l) such that, given any ℝ-affine hyperplanes H_{i} ⊆ ℂ^{n}, (i = 1, …, l) and a ball B ⊆ Re(L), there is an x satisfying (7) with

x ∉ ⋃_{i=1}^{l} H_{i}.

The Fact yields condition EC. Thus ℂ^{(K)} satisfies PCF_{d}(K), so it is a structure from EC_{d}^{0}(K).

Claim. Given countable A ≤ ℂ there are countably many 0dimensional analytic subsets S_{i} of ℂ^{n}, for all n, such that any b̄ ∈ ℂ^{n} satisfying δ(b̄/A) = 0 belongs to one of the S′_{i}s.

Proof. We may assume that A is closed under taking K-spans and under the operation ln(acl(exp(A))). We also assume that b̄ is ℚ-independent over A. Let V ⊆ ℂ^{n} be the minimal K-affine space over A containing b̄, and W ⊆ ℂ^{n} be the minimal algebraic variety definable over exp(A) containing exp(b̄). Since δ(b̄/A) = 0, we have

dim V + dim W = n.

If the dimension of the analytic set V ∩ ln W is 0, then we take S_{i} to be this set. Otherwise, by Corollary 2 of [Z2] (under SchC+CIT), there are finitely many tori (of the form exp(M_{i} + c_{i}) for M_{i} a ℚlinear subspace, c_{i} ∈ ℂ^{n}, i = 1, …, l) such that any infinite analytic component of exp(V)∩W belongs to one of the tori. Moreover, Lemma 3.1 of [Z2] proves that any such torus intersects W atypically. It follows immediately that exp(c_{i}) can be chosen in acl(exp(A)), thus c_{i} ∈ A^{n}. Then exp(b̄) ∉ exp(M_{i} + c_{i}), by our assumptions. It follows that b̄ belongs to

V ∩ ln W \ ⋃_{i=1}^{l} M_{i} + c_{i} + 2πiℤ^{n}

which is a countable analytic subset of ℂ^{n}. Claim proved.

It follows immediately from the claim that for countable A ≤ ℂ the ∂-closure of A is countable. Hence the ∂-basis of ℂ is uncountable. In particular, ID holds. □

REFERENCES

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[Z1] B.Zilber, Analytic and pseudo-analytic structures, To appear in Proceedings of European Logic Colloquium, Paris 2000

[Z2] B.Zilber, Exponential sums equations and the Schanuel conjecture, J. London Math. Soc. (2) 65 (2002)