Blurred complex exponentiation

Jonathan Kirby

version 1.4, 18^{th} January 2007

Abstract

I describe a way of blurring an exponential field in a way which preserves the natural closure operator. The results of blurring the complex exponential field and of blurring Zilber’s exponential field are isomorphic.

1 Introduction

Zilber constructed an exponential field K_{exp} = ⟨K; +, ·, exp⟩ which is defined up to isomorphism by a list of axioms which is essentially as short and simple as possible [Zil05]. He called this field pseudoexponentiation. However, from an algebraic perspective it is a perfectly genuine exponential field so in this paper I will call it Zilber’s exponential field. The purpose of the construction was to make the following conjecture:

Conjecture 1.1 (Zilber). The complex exponential field ℂ_{exp} is isomorphic to K_{exp}.

This conjecture contains Schanuel’s Conjecture:

Conjecture 1.2 (Schanuel). If a_{1}, …, a_{n} are complex numbers then

td_{ℚ}(a_{1}, …, a_{n}, e^{a_{1}}, …, e^{a_{n}}) − ldim_{ℚ}(a_{1}, …, a_{n}) ⩾ 0

where ldim_{ℚ} is ℚ-linear dimension.

This conjecture answers essentially all number theoretic questions relating to exponentiation. As a simple example, taking a_{1} = 1 and a_{2} = 2iπ, it implies that e and π are algebraically independent. Even this very special case has been an open question for over a century. Faced with this, Zilber’s conjecture seems out of reach.

However, Schanuel’s conjecture is not the main idea in Zilber’s conjecture. If, for example, it were discovered that e and π were algebraically dependent then it should be possible to adapt Zilber’s conjecture accordingly. The main idea is rather that the “large scale” behaviour of K_{exp} is controlled by a pregeometry, that is, a closure operator satisfying the Steinitz exchange property. This pregeometry gives rise to a dimension theory in the same way that algebraic closure gives a dimension theory on a pure field.

There is a notion of exponential algebraic closure, ecl, on ℂ_{exp} which also defines a pregeometry. In this paper I show that ecl can be studied without answering any of the difficult number theoretic questions. I replace K_{exp} and ℂ_{exp} by the blurred exponential fields K_{B} = ⟨K; +, ·, B⟩ and ℂ_{B} = ⟨ℂ; +, ·, B⟩. In each case B is a “blurring” of the graph of exponentiation defined by (x, y) ∈ B iff exp(x)/y is exponentially algebraic. The main theorem of this paper can be viewed as a blurred version of Zilber’s conjecture.

Theorem 1.3 (Main theorem). The blurred exponential fields K_{B} and ℂ_{B} are isomorphic.

I also show that the blurring process preserves the exponential algebraic closure operator, ecl. That is, ecl is definable from the blurred structure and controls that structure in the same way the it controls Zilber’s exponential field. Hence it follows that the large scale behaviour of complex exponentiation is controlled by a pregeometry, and furthermore that this pregeometry is isomorphic to the pregeometry controlling Zilber’s field. This can be viewed as establishing the “geometric part” of Zilber’s conjecture. Unfortunately, although ecl controls ℂ_{B} very well (for example every permutation of a basis extends to an automorphism of the structure) one cannot immediately deduce that this also holds for ℂ_{exp} itself. Indeed, Mycielski’s question of whether ℂ_{exp} has more than the two obvious automorphisms remains open.

In section 2, I define exponential algebraic closure and give some basic properties. In section 3, I describe the Schanuel predimension and the Schanuel property, not just in their usual absolute form but also relative to a subfield. This relative Schanuel predimension gives rise to a pregeometry which I compare with the exponential algebraic closure. Section 4 describes Zilber’s exponential field and its theory. In section 5, I describe the process of blurring an exponential field and give an axiomatization of K_{B}. Section 6 shows that ℂ_{B} also satisfies these axioms.

The deduction that K_{exp} and ℂ_{exp} are isomorphic relies on the axiomatization being uncountably categorical. This is proved in a separate paper [Kir] using the same method of quasiminimal excellence that Zilber used to show that his description of K_{exp} is categorical.

This paper could be viewed as an exploration of the consequences for the complex exponential field itself of James Ax’s work [Ax71] on power series rings over ℂ.

Some of this work was done as part of my DPhil thesis in Oxford, under the supervision of Boris Zilber. I would like to thank him for many useful conversations. Theorem 6.5 was conceived and proved at the Isaac Newton Institute. The remainder of the work was carried out at the University of Illinois at Chicago.

2 Exponential algebraic closure

The definition of exponential algebraic closure is taken from the Angus Macintyre’s survey paper [Mac96].

Definition 2.1. An exponential ring is a structure ⟨R; +, ·, exp⟩ which is a field together with a homomorphism from the additive group ⟨R; +⟩ to the multiplicative group ⟨R^{×}, ·⟩. If R is a field this is an exponential field.

Given a set X, the exponential polynomial ring over X is ℤ[X]^{E}, the free exponential ring generated by X. If R is an exponential ring we can also consider the exponential ring R[X]^{E} of exponential polynomials with coefficients from R. As with polynomial rings, when X = {x_{1}, …, x_{n}} these exponential polynomial rings act as rings of functions from R^{n} to R.

Definition 2.2. Let R be an exponential ring, let B be a subset of R and let α ∈ R. Then α is exponentially algebraic over B iff there are n, m ∈ ℕ, tuples a = (a_{1}, …, a_{n}) ∈ R^{n}, b = (b_{1}, …, b_{m}) ∈ B^{m}, and functions f_{1}, …, f_{n} ∈ ℤ[x_{1}, …, x_{n}, y_{1}, …, y_{m}]^{E} such that:

i) a_{1} = α,

ii) For each i = 1, …, n, f_{i}(a, b) = 0, and

iii) |\begin{matrix}\frac{∂f_{1}}{∂x_{1}} & ⋯ & \frac{∂f_{1}}{∂x_{n}} \\ ⋮ & ... & ⋮ \\ \frac{∂f_{n}}{∂x_{1}} & ⋯ & \frac{∂f_{n}}{∂x_{n}} \end{matrix} | (a, b) ≠ 0

Define ecl(B), the exponential algebraic closure of B in R, to be the set of elements of R which are exponentially algebraic over B.

Most of the following basic properties of exponential algebraic closure were given in [Mac96].

Proposition 2.3. Suppose that R has no zero divisors. Then ecl is a closure operator, and, for any subset B of R, ecl(B) is an exponential subring of R which is relatively algebraically closed and contains the kernel of exp. Furthermore, ecl has finite character, that is ecl B = ⋃ {ecl(B_{0}) | B_{0} ⊆_{fin} B }.

Proof. All simple calculations from the definition.

Macintyre also notes the following.

Proposition 2.4. If R is ℂ_{exp} and B is an exponential subfield then α ∈ ecl(B) iff there is an n-tuple a containing α which is an isolated zero of an exponential polynomial map f, whose n components lie in B[x_{1}, …, x_{n}]^{E}.

Proof. Take a and f witnessing that α ∈ ecl B. By the implicit function theorem for holomorphic maps, f is locally invertible at a, and so a is an isolated zero.

Alex Wilkie [Wil05] has shown that ecl in ℂ_{exp} is equal to the definable closure operator in the real field with restricted real exponentiation and restricted sine, when ℂ is interpreted as ℝ^{2}. In particular, this shows that ecl is a pregeometry on ℂ_{exp}. It appears to be an open question whether ecl is a pregeometry on any exponential field. We will not use Wilkie’s result as our methods will be those of complex analytic geometry. Indeed, we will later give an independent proof that ecl is a pregeometry on ℂ_{exp}. However, all the results proven here could presumably also be obtained using Wilkie’s methods and real analytic geometry.

3 The Schanuel property

From now on, we assume that R is an exponential ring with no zero divisors and, furthermore, that it is a ℚ-algebra.

Definition 3.1. Let R be an exponential ring as above, and let a = (a_{1}, …, a_{n}) be an n-tuple from R. Define the absolute Schanuel predimension

Sδ(a) = td_{ℚ}(a, exp(a)) − ldim_{ℚ}(a),

and, for a subset C ⊆ R, the relative Schanuel predimension

Sδ(a/C) = td_{C}(a, exp(a)) − ldim_{ℚ}(a/C)

where td_{C}(x) is the transcendence degree of the extension ℚ(C, x)/ℚ(C) and ldim_{ℚ}(x/B) is the ℚ-linear dimension of the quotient ℚ-vector space ⟨C, x⟩/⟨C⟩.

Then R has the absolute Schanuel property, SP, iff for each n ∈ ℕ and every a ∈ R^{n},

Sδ(a) ⩾ 0,

and it has the Schanuel property relative to C iff for each such n and a,

Sδ(a/C) ⩾ 0.

Schanuel’s conjecture is the assertion that ℂ_{exp} has the Schanuel property. The quantity Sδ gives rise to a pregeometry on any exponential ring. This

is the pregeometry used in the construction of Zilber’s exponential field. The method of construction is due to Hrushovski and is documented in many places, for example [Kir06]. We fix a subset C ⊆ R and first assume that Sδ(x/C) is bounded below on R, that is, there is N ∈ ℤ such that for any tuple a, Sδ(a/C) ⩾ N. In this case, for a finite tuple a and a subset B of R we define the Schanuel dimension (with respect to C) of a to be

Sd_{C}(a) = min {Sδ(ab/C) | b is a finite tuple from R} ,

the relative Schanuel dimension (with respect to C) of a over B to be

Sd_{C}(a/B) = min {Sd_{C}(ab) − Sd_{C}(b) | b is a finite tuple from B } ,

and we define the Schanuel closure (with respect to C) of B to be

Scl_{C}(B) = {x ∈ R | Sd_{C}(x/B) = 0} .

If Sδ(x/C) is not bounded below then we declare Scl_{C}(B) = R for every subset B of R.

We define the absolute Schanuel dimension and closure as above with C = ∅. Write Scl for the absolute Schanuel closure.

Proposition 3.2. Let C ⊆ R. Then Scl_{C} is a pregeometry on R.

Proof. If Sδ(x/C) is not bounded below this is trivial. Otherwise, it is immediate from the definition that Scl_{C} is increasing and monotone. It has finite character because td and ldim both do. It is standard to show that Sδ(x/C) is submodular: for any a, b,

Sδ(a ∩ b/C) + Sδ(ab/C) ⩽ Sδ(a/C) + Sδ(b/C),

and it follows quickly from this that Scl_{C} is idempotent. If a is a singleton then Sd_{C}(a/B) = 0 or 1, and the exchange property follows from this and submodularity.

However, even if Sδ(x/C) is bounded below, the pregeometry is only non-trivial if Sδ(x/C) is bounded below by 0, that is, if R has the Schanuel property relative to C.

Lemma 3.3. If R is an exponential ring which does not satisfy the Schanuel property relative to C then Scl_{C}(∅) = R.

Proof. Let a be such that Sδ(a/C) < 0, and let b ∈ R. Then Sd_{C}(b) ⩽ Sδ(ab/C) ⩽ Sδ(a/C) + 1 ⩽ 0. Hence b ∈ Scl_{C}(∅).

Suppose that C is a subring of R and C′ = Scl_{C}(C). Then the predimensions Scl_{C} and Scl_{C′} coincide. Furthermore, C′ contains all of the zero dimensional tuples over itself. That is, if a is a tuple from R then Sd_{C′}(a) ⩾ 1 or a lies in C′. In this case we say that R has the strong Schanuel property relative to C′. The proofs of the following two properties are standard predimension calculations such as those in [Kir06].

Lemma 3.4. If R has the Schanuel property relative to C and relative to C′, and C ⊆ C′ ⊆ R, then Scl_{C′}(A) = Scl_{C}(A ∪ C′).

If R has the strong Schanuel property relative to C and C ⊆ C′ ⊆ R then R has the strong Schanuel property relative to C′ iff C′ is Scl_{C}-closed.

Now suppose R has the Schanuel property relative to a given subring C, and C′ = Scl_{C}(C). Then for any intermediate ring C ⊆ S ⊆ R, the information in Scl_{S} is captured in Scl_{C}, and thus also in Scl_{C′}. So it is enough to look at Scl_{C} for subrings C relative to which R has the strong Schanuel property. However, there may be a proper subfield C of R such that R has the strong Schanuel property relative to C but nonetheless R does not have the absolute Schanuel property. The introduction of this relative Schanuel closure allows us to extend methods previously used only for exponential rings with SP to a more general setting.

We now have two closure operators on exponential rings: the exponential algebraic closure operator ecl and the Schanuel closure operator Scl. It is natural to ask how they are related. We can show inclusion in one direction. For the complex numbers, we will later give an analytic argument to show the reverse inclusion. As far as I know, the reverse inclusion is an open problem in general.

Proposition 3.5. If B ⊆ R then ecl(B) ⊆ Scl_{B}(B).

We first prove two lemmas. If a ∈ ecl(B) this is witnessed by a tuple of n exponential polynomials in n variables. If a ∈ Scl_{C}(B) this is witnessed by a tuple of polynomials in 2m variables, together with the fixed exponential dependencies h_{j} = Y_{m+j} = exp(Y_{j}) = 0 for j = 1, …, m. We must show that the exponential polynomials can be decomposed into this special form.

Lemma 3.6. Let B be an exponential subring of R containing ker exp. Suppose ā = (a_{1}, …, a_{n}) ∈ R^{n} and f_{1}, …, f_{n} ∈ B[X̄]^{E} are such that f̄(ā) = 0. Then there is a tuple ᾱ = (α_{1}, …, α_{2m}) extending ā, say with ᾱ = āb̄, such that

• each b_{i} lies in the exponential subring of R generated by B(ā),

• there are g_{j} ∈ B[Z_{1}, …, Z_{m}, Y_{1}, …, Y_{m}] for j = 1, …, m such that ḡ(ᾱ) = 0, h̄(ᾱ) = 0 and

|\frac{∂f̄}{∂X̄} | (ā) = |\begin{matrix}\frac{∂ḡ}{∂Z̄} & \frac{∂ḡ}{∂Ȳ} \\ \frac{∂h̄}{∂Z̄} & \frac{∂h̄}{∂Ȳ} \end{matrix} | (ᾱ).

Proof. Each exponential polynomial f_{i} is constructed from variables and elements of B by finitely many additions, multiplications and applications of exponentiation. By induction, it suffices to reduce the number of applications of exponentiation by one.

If all the f_{i} are polynomials, set g_{i} = f_{i}(Z̄) (replacing the variable X_{i} by Z_{i}). Otherwise, without loss of generality, f_{1} has the form p(X̄, exp(q(X̄))) where p is an exponential polynomial and q is a polynomial. Replace f_{1} by the exponential polynomial g_{1} = p(Z̄, Y_{n+1}), and set g_{i} = f_{i}(Z̄) for i = 2, …, n. Then, for i = 1, …, n + 1, let h_{i} = Y_{i} − exp(Z_{i}). Let

α_{i} = \begin{cases}a_{i} & for i = 1, …, n \\ q(ā) & for i = n + 1 \\ exp(α_{i}) & for i = 1, …, n \end{cases} .

Then ḡ(α) = h̄(α) = 0 and a standard Jacobian calculation with the chain rule shows that

|\frac{∂f̄}{∂X̄} | (ā) = |\begin{matrix}\frac{∂ḡ}{∂Z̄} & \frac{∂ḡ}{∂Ȳ} \\ \frac{∂h̄}{∂Z̄} & \frac{∂h̄}{∂Ȳ} \end{matrix} | (ᾱ)

as required.

Lemma 3.7. Let B be as above, and suppose a ∈ ecl B. Then there is m ∈ ℕ, an m-tuple s̄ from ecl(B) and g_{1}, …, g_{m} ∈ B[X̄, Ȳ] (with X̄, Ȳ both m-tuples of variables) such that

• a lies in the exponential subring of R generated by B and s̄,

• the s_{i} are ℚ-linearly independent over B,

• taking t_{i} = exp(s_{i}), we have ḡ(s̄, t̄) = 0 and

|\begin{matrix}\frac{∂ḡ}{∂X̄} & \frac{∂ḡ}{∂Ȳ} \\ \frac{∂h̄}{∂X̄} & \frac{∂h̄}{∂Ȳ} \end{matrix} | (s̄, t̄) ≠ 0

Proof. Take a tuple ā containing a and exponential polynomials f_{i} ∈ B[X̄]^{E} witnessing that a ∈ ecl B. Then apply lemma 3.6 to produce a 2m′ tuple ᾱ. Let σ̄ be the first m′ elements of ᾱ and let τ̄ be the latter m′ elements. Let s̄ be a maximal subset of σ̄ which is ℚ-linearly independent over B, and let t̄ be the corresponding tuple from τ̄. Replace each g_{i} by a polynomial in the variables corresponding to s̄ and t̄ by substituting in for the other variables according to the ℚ-linear dependencies amongst the σ_{i} and the multiplicative dependencies amonst the τ_{i}. The result will be m′ + m equations in 2m variables for some m ⩽ m′. The rank of the Jacobian matrix as above will be 2m by lemma 3.6 and the fact that |\frac{∂f̄}{∂X̄} | (ā) ≠ 0,since the original f_{i} were witnesses to a ∈ ecl B. Thus we may throw away some of the equations g_{i} to leave 2m equations in 2m variables, with nonzero Jacobian as desired.

Proof of proposition 3.5. Suppose a ∈ ecl(B). We may replace B by the exponential subring of R generated by B and ker exp, since ecl(B) and Scl_{C}(B) both contain this subring. Apply lemma 3.7 to produce s̄ and ḡ, with

|\begin{matrix}\frac{∂ḡ}{∂X̄} & \frac{∂ḡ}{∂Ȳ} \\ \frac{∂h̄}{∂X̄} & \frac{∂h̄}{∂Ȳ} \end{matrix} | (s̄, exp(s̄)) ≠ 0.

Then the matrix (\frac{∂ḡ}{∂X̄} \frac{∂ḡ}{∂Ȳ} ) (s̄, exp(s̄)) has rank m. Hence td(s̄, exp(s̄/B) ⩽ m. The s_{i} are ℚ-linearly independent over B, and so Sδ(s̄/B) ⩽ 0. Hence a ∈ Scl_{B}(B).

4 Zilber’s exponential field

Zilber considered exponential fields with SP and three further properties: standard kernel, the countable closure property (CCP), and strong exponential closedness (SEC).

Standard kernel An exponential ring R is said to have standard kernel iff the kernel of exponentiation is ηℤ for a transcendental element η.

Note that ℂ_{exp} has standard kernel, since 2iπ is transcendental.

CCP If B ⊆ R is finite then Scl(B) is countable.

Note that CCP makes sense for any closure operator.

The Schanuel property gives a restriction on what systems of equations can have solutions, and the SEC property effectively says that this is the only restriction. It uses the notion of rotundity of a subvariety, which we must define. G is an abelian group, hence a ℤ-module. Thus G^{n} is naturally a module over the ring Mat_{n×n}(ℤ) of n × n matrices with integer coefficients. The action of Mat_{n×n}(ℤ) is by linear combinations in the additive part of G and by multiplicative combinations in the multiplicative part. Given a subset A of G^{n}, and a matrix M ∈ Mat_{n×n}(ℤ), write M · A for the image of A under the action. If V is an algebraic subvariety of G^{n} then M · V is also an algebraic subvariety.

Definition 4.1. A subvariety V of G^{n} is said to be rotund iff for every M ∈ Mat_{n×n}(ℤ), dim M · V ⩾ rk M.

Each M · V is a projection of V in some direction that makes sense in the group, and the idea is that a rotund subvariety is “large in every direction”. (Zilber’s terminology is normal, but this is not so descriptive.)

SEC If V is a rotund subvariety of G^{n} then there is a generic point of V in the intersection V ∩ G^{n}, where G is the graph of the exponential map.

Theorem 4.2 (Zilber). For each uncountable cardinal κ, there is a unique exponential field of cardinality κ with standard kernel satisfying SP + SEC + CCP.

In some sense, these are the “simplest” nontrivial exponential fields, or at least those admitting the simplest possible description. Zilber’s conjecture is that the unique model K_{exp} of cardinality 2^{ℵ_{0}} is isomorphic to ℂ_{exp}.

5 Blurred exponentiation

Let F be an exponential field and let G be the group F × F^{×}. Let θ be the group homomorphism G ⟶^{θ} F^{×} given by θ(x, y) = y/ exp(x). The graph of exp is of course the kernel of this map: G_{exp} = θ^{−1}(1). Let ℋ be a subgroup of F^{×}. We define the blurred graph of exponentiation, with respect to ℋ, to be

B_{ℋ} = θ^{−1}(ℋ)

or, in other words, B_{ℋ} = {(x, y) ∈ G | ∃h ∈ ℋ[y = h · exp(x)]}. We call the structure ⟨F; +, ·, B_{ℋ} ⟩ a blurred exponential field.

Now consider K_{0} = Scl(∅) in Zilber’s exponential field K. Let K_{B} = ⟨K; +, ·, B⟩ be the result of blurring Zilber’s exponential field with respect to K_{0}^{×}. Observe that K_{0} is recoverable from B as {x | (x, 1) ∈ B }.

Theorem 5.1. This blurred exponential field K_{B} satisfies the following axioms.

Field ⟨K; +, ·⟩ is an algebraically closed field of characteristic zero, and K_{0} is a proper algebraically closed subfield.

Group B is a subgroup of the algebraic group G = K × K^{×}.

Fibres The fibres of the projection of B to K are cosets of K_{0}^{×}. The fibres of the projection to K^{×} are cosets of K_{0}.

Blurred Schanuel Property (BSP) Let n ∈ ℕ and let V be an algebraic subvariety of G^{n}, defined over K_{0}. Then if (ā, b̄) ∈ V ∩B^{n} and dim V <

n + 1 then there are m_{1}, …, m_{n} ∈ ℤ such that ∑_{i=1}^{n} m_{i}a_{i} ∈ K_{0}.

Existential Closedness (BEC) For each n ∈ ℕ, and each rotund algebraic subvariety V of G^{n}, the intersection V ∩ B^{n} is nonempty.

Proof. The Field, Group, and Fibres axioms are immediate from the definition.

Let (a, b) ∈ V ∩ B^{n} with V a subvariety of G^{n} defined over K_{0} with dim V < N + 1. Then there is c ∈ (K_{0}^{×})^{n} such that b = c · exp(a), so td_{C}(a, exp(a)) < n+1. Thus either Sδ(a/C) ⩽ 0 or a is ℚ-linearly dependent over K_{0}. But K_{0} is Scl-closed, so in the first case a ∈ K_{0}^{n}. Thus, either way, there are m_{i} ∈ ℤ, not all zero, such that ∑_{i=1}^{n} m_{i}a_{i} ∈ K_{0}. So BSP holds.

If V is rotund then SEC says V ∩ G_{exp}^{n} ≠ ∅. But G_{exp} ⊆ B, so V ∩ B^{n} ≠ ∅. Thus BEC holds.

In my thesis [Kir06] I proved:

Theorem 5.2. The above axioms are expressible as first order axiom schemes in the language L = ⟨+, ·, B⟩ and the L-theory T which they define is complete.

We now construct a pregeometry on K_{B} along the same lines as the Schanuel pregeometry. The construction works on any model of the axioms Field, Group, Fibres and BSP. The last axiom, BEC, is not used in the construction.

For a tuple (a, b) ∈ B^{n}, let Bδ′(a, b) = td_{C}(a, b) − ldim_{ℚ}(a/C). Then for any finite subset A ⊆ K, define Bδ(A) = max {Bδ′(a, b) | each a_{i}, b_{i} ∈ A}.

For a finite subset A of R and a subset B of R we define the blurred Schanuel dimension of A to be

Bd(A) = min {Bδ(AX) | X is a finite subset of R} ,

the blurred Schanuel dimension of A over B to be

Bd(A/B) = inf {Bd(AX) − Bd(X) | X is a finite subset of B } ,

and we define the blurred Schanuel closure of B to be

Bcl(B) = {x ∈ R | Bd(x/B) = 0} .

The same proof as for proposition 3.2 shows:

Proposition 5.3. Bcl is a pregeometry on K.

Theorem 5.4. The pregeometries Scl and Bcl on Zilber’s field K are equal.

Proof. Suppose α ∈ Scl(B). Since K_{0} = Scl(∅), we may assume K_{0} ⊆ B. Then there is a finite tuple a from K containing α such that td(a, exp(a)/B)− ldim_{ℚ}(a/B) ⩽ 0. Since (a, exp(a)) ∈ B^{n}, this shows that α ∈ Bcl(B).

Conversely, suppose α ∈ Bcl(B). By BSP, K_{0} = Bcl(∅), so we may assume K_{0} ⊆ B. Then there are n-tuples a and b of K, one of them containing α, such that (a, b) ∈ B^{n} and td(a, b/B) − ldim_{ℚ}(a/B) ⩽ 0. Let c = exp(a)/b. Then c ∈ K_{0}^{n} and, since K_{0} ⊆ B, we get that td(a, exp(a)/B)−ldim_{ℚ}(a/B) ⩽ 0. Hence a ∈ Scl(B). But Scl(B) is an exponential subfield of K containing K_{0}, so b ∈ Scl(B) as well. Hence α ∈ Scl(B).

As a corollary, we see that Bcl on K_{B} has the countable closure property. The same arguments can be applied more generally, to prove the following.

Proposition 5.5. Let F be any exponential field and let C be an exponential subfield. Then the result F_{B_{C}} = ⟨F; +, ·, B_{C}⟩ of blurring F by C^{×} is a model of the axioms Field, Group and Fibres, except that F may not be algebraically closed and some fibres of B_{C} may be empty. Furthermore, ⟨F; +, ·, B_{C}⟩ satisfies BSP iff F has the strong Schanuel property relative to C. In this case, Bcl is a well-defined pregeometry on F and is equal to Scl_{C}.

We will also make use of the following theorem, proven in [Kir].

Theorem 5.6. For each uncountable cardinal κ, there is exactly one model of T of cardinality κ such that Bcl has the countable closure property and td(C/ℚ) = ℵ_{0}. In other words, the abstract elementary class defined by the L_{ω_{1}ω}(Q)-theory T + CCP + td(C/ℚ) = ℵ_{0} is uncountably categorical.

Thus to prove the main theorem it remains to show that we can blur complex exponentiation to produce a model of this theory.

6 Blurred complex exponentiation

We now turn to the complex exponential field.

Theorem 6.1. If C is a proper ecl-closed subfield of ℂ_{exp} then ℂ_{B_{C}} is a model of the theory T. If in addition C is countable then ℂ_{B_{C}} satisfies CCP.

Let C_{0} = ecl ∅, and define ℂ_{B} to be ℂ_{B_{C_{0}}}. Note that td(C_{0}/ℚ) = ℵ_{0}, and so the main theorem of this paper, Theorem 1.3, follows from theorems 6.1 and 5.6.

The remainder of the paper is devoted to proving theorem 6.1. An eclclosed subfield is an exponential subfield, so axioms Field, Group, and Fibres hold. We prove BSP, BEC and CCP below.

Before continuing with the proof of theorem 6.1, we give an alternative characterization of ecl-closed subfields of ℂ in the style of a nullstellensatz.

Definition 6.2. An (affine complex) exponential-algebraic variety is the zero set of a finite set of exponential polynomials in ℂ^{n}. It is defined over an exponential field C iff the exponential polynomials may be taken to have coefficients in C.

Lemma 6.3. A subfield C of ℂ is ecl-closed iff for every exponential-algebraic subvariety A of ℂ^{n}, defined over C, every connected component X of A has a C-point.

Proof. First note that if X is a component of dimension zero, that is, a singleton {x}, then x is an isolated zero of the system of exponential polynomial equations. We may choose just n of the equations whose loci intersect transversally at x. Then, by the implicit function theorem, we see that this is the same thing as saying that x ∈ ecl(C). Thus C has this property for components of dimension 0 iff C is ecl-closed.

We now reduce the question to components of dimension 0. Suppose X is a connected component of dimension strictly greater than 0, and let x ∈ X be a regular point. X is an analytic subset of ℂ^{n}, so for small enough ε it contains a good approximation to a disc about x of radius ε. Let r, s be Gaussian rational points in ℂ^{n} which are very close to the disc, one on each side. If either of r, s ∈ X then we are done, since ℚ(i) ⊆ C. Suppose not. Let L be the (real) line joining r and s and let L′ be its Zariski closure. Then L′ ∩ A is an exponential-algebraic subvariety of dimension strictly less than the dimension of A, with nonempty intersection with X. By induction on dimension, each component has a C-point. In particular, L′ ∩ X has a C-point.

We give a consequence of Ax’s theorem for analytic sets.

Theorem 6.4. Let V be an algebraic subvariety of G^{n}(ℂ).

If W is a connected component of the analytic variety V ∩ G_{exp}^{n} with analytic dimension dim W satisfying dim W = (dim V − n) + t for some t > 0, then there are t linearly independent n-tuples of integers (m_{ij}) and c_{j} ∈ ℂ such that W is contained in the algebraic variety defined by

⋀_{j=1}^{t} ∑_{i=1}^{n} m_{ij}x_{i} = c_{j}  and  ⋀_{j=1}^{t} ∏_{i=1}^{n} y_{i}^{m_{ij}} = exp(c_{j})

Proof. Let w be a regular point of W and let F be the differential field of germs at w of meromorphic functions on G^{n}(ℂ), the differential operators being the usual 2n partial differentiation operators (with respect to a chosen basis z_{1}, …, z_{2n}). The field of constants is ℂ.

Let (x, y) be coordinate functions on W at w, with x = (x_{1}, …, x_{n}) being a basis of coordinate functions on 𝔾_{a}^{n} and y = (y_{1}, …, y_{n}) being a basis of coordinate functions on 𝔾_{m}^{n}. Then y_{i} = exp(x_{i}), and so \frac{∂y_{i}}{∂z_{j}} = y_{i} \frac{∂x_{i}}{∂z_{j}}, for each i and j. Also (x, y) ∈ V and the rank of the Jacobian matrix

(\frac{∂x_{i}}{∂z_{j}}  \frac{∂y_{i}}{∂z_{j}} )i=1,...,n;j=1,...,2n

is equal to dim W, by the definition of the analytic dimension of W. By Ax’s theorem [Ax71, Theorem 3], there are m_{ij} and c_{j} as described. These algebraic relations hold on the coordinate functions, and so they hold on the whole component W as required.

Proposition 6.5. ℂ_{B_{C}} satisfies BSP.

Proof. Let V be an irreducible subvariety of G^{n}, defined over C. Suppose that (a, b) ∈ V ∩ B^{n} and dim V < n + 1.

Let c = exp(a)/b and define V′ = {(x, y) ∈ G^{n} | (x, y/c) ∈ V }. Then V′ and V are isomorphic and dim V′ = dim V < n + 1. Let W be the connected component of V′ ∩ G_{exp}^{n} containing (a, exp(a)).

If dim W = 0 then W is the singleton {(a, exp(a))}. Now V′ ∩ G_{exp}^{n} is an exponential-algebraic variety definable over C, and C is ecl-closed, so (a, exp(a)) ∈ G^{n}(C). Also c ∈ C^{n} by definition of B, and so (a, b) ∈ G^{n}(C).

Otherwise, dim W > 0 ⩾ dim V′ − n. Hence, by theorem 6.4, there are m_{i} ∈ ℤ, not all zero, and γ ∈ ℂ such that ∑_{i=1}^{n} m_{i}x_{i} = γ. Since V′ is irreducible, and (x, y) is a generic point of V′ over ℂ, this equation holds for all points in V′. In particular, ∑_{i=1}^{n} m_{i}a_{i} = γ. Now W is a connected component of an exponential-algebraic variety defined over C, so contains a C-point, say (α, β). Then ∑_{i=1}^{n} m_{i}α_{i} = γ, but C is a field and so γ ∈ C.

Proposition 6.6. ℂ_{B_{C}} satisfies BEC.

Proof. Let V be an irreducible, rotund subvariety of G^{n}(ℂ). By intersecting with generic hyperplanes (which preserves rotundity) we may assume that dim V = n. We may also assume that V is free, that is there are no ℚ-linear dependences on its x-coordinates and no multiplicative dependences on its y-coordinates, over a field of definition.

Let v be a regular point of V (ℂ), and let U be a small neighbourhood of v in V. We may take U to be analytically diffeomorphic to an open ball in ℂ^{n}. We have the map G(ℂ) ⟶^{θ} ℂ^{×} given by θ(x, y) = y/ exp(x). Let ψ be its restriction to V (ℂ).

Let t = ψ(v) ∈ S(ℂ) and let A = ψ^{−1}(t) ∩ U. Then A is an analytic subset of U, so, by taking U sufficiently small, we may assume that A is connected.

Now let V′ be the translation of V given by

V′ = {(x, y/t) | (x, y) ∈ V } ,

let V ⟶^{τ} V′ be the translation map and let A′ = τA. By construction, A′ ⊆ G_{exp} ∩ V′. Now V is rotund and free, so V′ is also rotund and free. Thus by theorem 6.4 the intersection is typical, that is,

dim A′ ⩽ 2n − dim V′ − dim G_{exp} = 0.

Thus dim A = dim A′ = 0, and A is connected, so A is the singleton {v}.

By the inverse function theorem, ψ is locally invertible at v. Thus, choosing the neighbourhood U even smaller if necessary, ψ ↾_{U} is a topological embedding. So ψ(U) is open in ℂ^{×}^{n}. Now C is dense in ℂ and so there is c ∈ ψ(U) ∩ C^{×}^{n}. Then θ^{−1}(c) ∈ U ∩ B^{n} ⊆ V ∩ B^{n}, which is nonempty as required.

This completes the proof that ℂ_{B_{C}} is a model of T. Recall that C_{0} = ecl(∅), and ℂ_{B} = ℂ_{B_{C_{0}}}. Let Bcl be the blurred Schanuel pregeometry on ℂ_{B}.

Proposition 6.7. The blurred Schanuel pregeometry Bcl is equal to the exponential algebraic closure operator ecl on ℂ_{exp}. In particular, ecl is a pregeometry on ℂ.

Proof. By proposition 5.5, Bcl = Scl_{C_{0}}. Thus by proposition 3.5, for every A ⊆ ℂ, ecl(A) ⊆ Bcl(A). Thus every Bcl-closed subfield is ecl-closed. By lemma 3.4 and proposition 6.5, every ecl-closed subfield is Bcl-closed. Hence ecl = Bcl.

This provides the promised converse to proposition 3.5, in the complex case.

Proposition 6.8. Bcl satisfies CCP. More generally, if C is any countable ecl-closed subfield of ℂ then Bcl_{C} satisfies CCP.

Proof. It is enough to show that ecl has the CCP on ℂ, since Bcl_{C}(A) = Bcl(A ∪ C).

If A is a finite subset of ℂ then x ∈ ecl(A) iff x is a coordinate of an isolated zero of an exponential polynomial map defined over ℚ(A). There are only countably many such maps and each can have only countably many isolated zeros because ℂ^{n} is a countable union of compact subspaces. Hence ecl(A) is countable.

This completes the proof of theorem 6.1, and thus also of theorem 1.3.

References

[Ax71] James Ax. On Schanuel’s conjectures. Ann. of Math. (2), 93:pp. 252–268, 1971.

[Kir] Jonathan Kirby. On exponential differential equations. Forthcoming.

[Kir06] Jonathan Kirby. The theory of exponential differential equations. DPhil thesis, University of Oxford, 2006.

       http://eprints.maths.ox.ac.uk/275/.

[Mac96] Angus J. Macintyre. Exponential algebra. In Logic and algebra (Pontignano, 1994), volume 180 of Lecture Notes in Pure and Appl. Math., pp. 191–210. Dekker, New York, 1996.

[Wil05] A. J. Wilkie. Some local definability theory for holomorphic functions, October 2005. Available on the MODNET preprint server.

[Zil05] Boris Zilber. Pseudo-exponentiation on algebraically closed fields of characteristic zero. Ann. Pure Appl. Logic, 132(1):pp. 67–95, 2005.