Exponential and Weierstrass Equations

Jonathan Kirby Mathematical Institute, University of Oxford

Version 2.7 7th September 2005

Abstract

We axiomatize the theories of the exponential and the Weierstrass differential equations and show that they can be obtained from amalgamation constructions in the style of Hrushovski.

Contents

1 Introduction 2

2 Differentials 3 2.1 Derivations and Kähler differentials . . . . . . . . . . . . . . . 3 2.2 Extensions of differential fields . . . . . . . . . . . . . . . . . . 6 2.3 Kinds of differentials . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Linear dependence results . . . . . . . . . . . . . . . . . . . . 9

3 The universal theory 13 3.1 Group structures . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Schanuel conditions . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Uniformity and definability . . . . . . . . . . . . . . . . . . . . 15

4 Amalgamation and axiomatization 16 4.1 Axiomatization . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 The existentially closed condition 23

1 Introduction

In this paper we axiomatize the theories of the exponential and the Weierstrass differential equations and show that they can be obtained from amalgamation constructions in the style of Hrushovski. More precisely, let ⟨F; +, ·, D⟩ be a differentially closed field and define a binary predicate E(x, y) by Dy = yDx. By the theory of the exponential equation we mean the theory of the reduct ⟨F; +, ·, E⟩, and similarly for Weierstrass equations.

Hrushovski originally produced his constructions in [6] as counterexamples to Zilber’s trichotomy conjecture on strongly minimal sets. The impetus behind the conjecture was the idea that strongly minimal sets might all come from mainstream mathematical objects. The results presented here show that it is possible to view some of these constructions as natural mathematical objects and not just as pathologies. Only the amalgamation part of Hrushovski’s construction is involved here, not the collapse to finite Morley Rank.

The amalgamation constructions make use of a predimension notion, and the first indication that they might be related to other parts of mathematics was the observation that this predimension also appears in Schanuel’s conjecture of transcendental number theory. If for complex numbers a_{1}, …, a_{n} we define

δ(a_{1}, …, a_{n}) = td(a_{1}, …, a_{n}, e^{a_{1}}, …, e^{a_{n}}) − ldim_{ℚ}(a_{1}, …, a_{n})

then Schanuel’s conjecture states precisely that δ(a_{1}, …, a_{n}) ⩾ 0.

In [1], James Ax proved a differential field version of Schanuel’s conjecture (“Schanuel condition”). This theorem plays an important role in the current work. Brownawell and Kubota extended this work to Weierstrass ℘- functions in [2] and, using this result and a theorem of Seidenberg, I proved a version for Weierstrass equations in arbitrary differential fields in [8]. Here I give a different proof of this result, essentially adapting Brownawell and Kubota’s proof rather than using Seidenberg’s theorem. The use of some model theoretic ideas makes the resulting proof shorter than the original.

Section 2 of this paper gathers together background on differentials and differential forms in differential algebra. Amongst other sources for this are [1], [2], and David Pierce’s paper [12]. This section also includes some technical lemmas towards the proofs in the later sections.

The third section gives the universal theory of the reducts, including the Schanuel conditions mentioned above. Section 4 does the amalgamation construction to produce the complete first order theories. The ideas here were refined after discussions with Assaf Hasson, and can be seen as an example of the general procedure described in [5].

The final section proves that the reducts of differentially closed fields satisfy the existentially closed condition which completes the theory of the amalgams. Cecily Crampin proved essentially this for the exponential equation, and it is to appear in her thesis. The current proof is adapted from hers.

The presentation of this paper is not as clear as I would like. However, I will not now improve it substantially until I have written up my DPhil, and after some requests I have decided to make it available as it is.

Acknowledgements This is work carried out as part of my DPhil supported by the EPSRC and under the supervision of Boris Zilber. I thank him for many useful discussions. I would also like to thank Rachel for inspiration.

2 Differentials

2.1 Derivations and Kähler differentials

Let R be a ring and M an R-module. A derivation of R into M is a function R ⟶^{D} M such that for every a, b ∈ R, D(a + b) = Da + Db and D(ab) = aDb + bDa (the Leibniz rule).

An element a ∈ R is called a constant of D iff Da = 0, and the set of all constants is denoted C, or C_{D} if the derivation has to be specified. A well known fact (see for example [4, section 16], which is a good reference for this section) is that if R is a field of characteristic 0 then C is a subfield of R and is algebraically closed in R.

From now on we assume all rings are entire (have no zero divisors) and are of characteristic zero. We restrict attention to derivations which are constant on a subfield C which by the above we may assume to be relatively algebraically closed.

Given a ring R and a subfield C there is a universal derivation from R which is constant on C constructed as follows. Let Ω(R/C) be the R-module which is generated by the set of symbols {da | a ∈ R} subject to the relations that d : R → Ω(R/C) is a derivation (is additive and satisfies the Leibniz rule) and that da = 0 for each a ∈ C. So Ω(R/C) is given by the set of terms of the form ∑_{i=1}^{n} a_{i}db_{i} for n ∈ ℕ and a_{i}, b_{i} ∈ R, quotiented out by the equivalence relation generated by the relations described. The R-module structure and the map d are defined in the obvious way.

The universal property satisfied by d is the following. If R ⟶^{D} M is any derivation of R constant on C then there is a unique R-linear map Ω(R/C) ⟶^{D^{∗}} M such that the following diagram commutes.

[figure: R; d; Ω(R/C); M; D^{∗}; D]

The map D^{∗} is given by setting D^{∗}(da) = Da and extending R-linearly, and the universality follows immediately from the construction.

The elements of Ω(R/C) are called the Kähler differentials of R/C or just differentials.

Lemma 2.1. Suppose b_{1}, …, b_{n} ∈ R, f is a polynomial over C and f(b̄) = 0. Then ∑_{i=1}^{n} \frac{∂f}{∂x_{i}} db_{i} = 0.

Proof. By induction on f using the Leibniz and addition rules.

Suppose that C ⊆ R ⊆ F with F and C fields (of characteristic 0) and R a ring and consider the following diagram.

[figure: R; d_{R}; Ω(R/C); F; ι; ∩; d_{F}; Ω(F/C); θ; . . . . . . . . . . . . . . . .]

The maps d_{R} and d_{F} are the universal derivations of R/C and F/C respectively. The F -vector space Ω(F/C) is also an R-module and the map d_{F} ∘ ι is a derivation. The map θ is that arising from the universal property of d_{R}, and is given by θ(d_{R}a) = d_{F} a and R-linearity.

Proposition 2.2. The kernel of θ consists of the torsion elements of Ω(R/C).

Proof. Since F is a field, Ω(F/C) has no torsion, and so every torsion element of Ω(R/C) must be in the kernel.

Now suppose that ω = ∑_{i=1}^{n} a_{i}d_{R}b_{i} ∈ Ω(R/C) and that θ(ω) = ∑_{i=1}^{n} a_{i}d_{F} b_{i} = 0.

By reordering, we may assume that b_{1}, …, b_{m} form a transcendence base for C(b_{1}, …, b_{n})/C, for some m ⩽ n. For i = m + 1, …, n let f_{i} ∈ C[x_{1}, …, x_{m}, y] be a polynomial giving the algebraic dependence of b_{i} on b_{1}, …, b_{m}. That is, f_{i}(b_{1}, …, b_{m}, b_{i}) = 0 and \frac{∂f_{i}}{∂y} ≠ 0. These are polynomial equations in R, so from lemma 2.1 we have \frac{∂f_{i}}{∂x_{i}} d_{R}b_{i} = − ∑_{j=1}^{m} \frac{∂f_{i}}{∂x_{j}} d_{R}b_{j} for each i. Thus

( ∏_{i=m+1}^{n} \frac{∂f_{i}}{∂x_{i}} ) ω = ∑_{j=1}^{m} (a_{i} ( ∏_{i=m+1}^{n} \frac{∂f_{i}}{∂x_{i}} ) + ∑_{k=m+1}^{n} a_{k} \frac{∂f_{k}}{∂x_{j}} ∏_{i=m+1,i≠k}^{n} \frac{∂f_{i}}{∂x_{i}} ) d_{R}b_{j}

and I’ll abbreviate the right hand side as ∑_{j=1}^{m} e_{j}d_{R}b_{j}. Note that each e_{j} ∈ R. Applying θ, we have

( ∏_{i=m+1}^{n} \frac{∂f_{i}}{∂x_{i}} ) θ(ω) = ∑_{j=1}^{m} e_{i}d_{F} b_{j}

but θ(ω) = 0 and thus ∑_{j=1}^{m} e_{j}d_{F} b_{j} = 0. Since b_{1}, …, b_{m} are algebraically independent over C, it follows that d_{F} b_{1}, …, d_{F} b_{m} are linearly independent over F. Thus each e_{j} = 0 and so ∑_{j=1}^{m} ω = 0 in Ω(R/C). Thus ω is a torsion element of Ω(R/C).

In general Ω(R/C) may have torsion, but if R is a field then it is a vector space so this can’t happen. Indeed lemma 1 of [2] shows that if R is a valuation ring of F containing C, or more generally is a local ring, then Ω(R/C) is torsion-free. Thus if R is a subfield of F or a valuation ring of F, the natural map Ω(R/C) ⟶ Ω(F/C) is an embedding of R-modules.

It is easy to see that the set of derivations from R to R over a subfield C forms an R-module, which we write Der(R/C). The universal property of d gives a natural isomorphism between Der(R/C) and the R-module of R-linear maps Ω(R/C) ⟶ R. When R is a field, F, this module is the dual vector space Ω(F/C)^{∗}.

Any vector space over a field embeds naturally in its double dual, so we may consider Ω(F/C) to be embedded in Der(F/C)^{∗}. From this point of view, the differential dx is the map which sends a derivation D to its value at x, and other differentials are F -linear combinations of such maps. When the transcendence degree of F/C is finite, the embedding is an isomorphism and every linear map in Der(F/C)^{∗} is of this form, but when the transcendence degree is infinite this is not true.

One advantage of thinking about differentials as elements of Der(F/C)^{∗} is that it gives a “coordinate free” representation of them. If we define something for Der(F/C)^{∗} then we don’t have to check that it is well-defined under a change of representation as ∑ a_{i}db_{i}.

The Schanuel conditions of this paper are relationships between transcendence degree (algebraic dimension) and linear dimension. Differentials can be used as a tool for proving these because of the following well-known fact, which can be found for example as Theorem 16.14 of [4] (although since the map d is not injective, the precise statement there is incorrect).

Theorem 2.3. If td(F/C) is finite and B is a transcendence base of F over C then {db | b ∈ B } is a basis for Ω(F/C) as an F -vector space. In particular dim Ω(F/C) = td(F/C).

2.2 Extensions of differential fields

In order to prove existentially closed properties of solutions to differential equations, we need to see what extensions a given differential field can have. For this we need a more general notion of differential. This is basically the idea described in [12] and independently of this was also used by Cecily Crampin and Alex Wilkie.

Let ⟨F_{0}, D_{0}⟩ be a differential field of characteristic zero, which we assume to be algebraically closed and with constant field C, and let F be a field extension of F_{0}. We want to consider the set of extensions of D_{0} to F, but this isn’t a vector space so instead we consider a slightly larger set.

Define Der(F/D_{0}) = {D ∈ Der(F/C) | ∃λ ∈ F (D ↾_{F_{0}}= λD_{0})}. This is an F -vector subspace of Der(F/C) and the set of actual extensions of D_{0} is a codimension 1 affine subspace (assuming that F_{0} ≠ C).

If we choose one extension D_{1} of D_{0} to F, this gives a bijection between the set of all extensions of D_{0} and Der(F/F_{0}) given by D ↦ D − D_{1}. From this we see that dim Der(F/D_{0}) = td(F/F_{0}) + 1.

Lemma 2.4. The space Der(F/D_{0}) is a Lie algebra over C (so with the F -vector space structure is a Lie ring).

We assume for convenience that td(F/F_{0}) is finite. Define Ω(F/D_{0}) to be the dual space of Der(F/D_{0}). We can consider this as a quotient of Ω(F/C) by the restriction map ω ↦ ω ↾_{Der(F/D_{0})}. From another point of view, this is the map arising from the universal property of Ω(F/C). There is a derivation

F ⟶^{d_{F/D_{0}}} Ω(F/D_{0}) and the diagram

[figure: F; d; Ω(F/C); Ω(F/D_{0}); d_{F/D_{0}}]

commutes. Where no ambiguity is likely to arise I will write the restriction of ω just as ω, and d_{F/D_{0}} as d.

Suppose that b_{1}, …, b_{n} is a transcendence base for F/F_{0}, and let b_{0} ∈ F_{0}∖C. Then db_{0}, …, db_{n} is a basis for Ω(F/D_{0}). (If F_{0} = C then Ω(F/D_{0}) = Ω(F/C) and db_{1}, …, db_{n} is a basis.)

Lemma 2.5. Ω(F/D_{0}) ≅ Ω(F/F_{0}) ⊕ ⟨dx⟩ as F -vector spaces, where x ∈ F_{0} ∖ C.

Proof. There is a natural restriction map from Ω(F/D_{0}) to Ω(F/F_{0}) which is surjective and has kernel ⟨dx⟩.

2.3 Kinds of differentials

The cohomological notions of exact and closed differential are standard, but we give the definitions for completeness.

Definition 2.6. A differential ω ∈ Ω(F/C) is exact iff there is x ∈ F such that ω = dx.

There is a natural map Ω(F/C) ⟶ Ω^{(2)}(F/C), also denoted d, where Ω^{(2)}(F/C) is the space of alternating bilinear forms on Der(F/C). A differential ω is closed iff dω = 0.

The ideas surrounding valuation rings and their importance in this context are sketched in detail, but with some proofs missing, in [7], which is motivated in large part by [9] where the remaining proofs can be found.

Definition 2.7. Let F/C be a field extension and R a valuation ring of F/C. A differential ω ∈ Ω(F/C) is said to be regular at R iff it is in Ω(R/C), considered as a sub R-module of Ω(F/C) via the canonical embedding. Otherwise ω is said to be singular at R.

Definition 2.8. A differential ω is of the first kind (a dfk) iff it is regular at every valuation ring of F/C. It is of the second kind (a dsk) iff for every valuation ring R of F/C there is x ∈ F such that ω − dx ∈ Ω(R/C).

An exact differential is of the second kind but is not of the first kind unless it is the zero differential.

Until now we have been looking at the algebraic objects differentials. We also need to consider the geometric notion of differential forms which, roughly speaking, are the assignment of a differential to each point on an algebraic variety. More precisely, the algebraic notion of derivation corresponds to the geometric notion of a tangent vector at a point of a variety. The dual notion of a differential corresponds to a cotangent vector. Thus a differential form on a variety V is simply an element of the contangent bundle T^{∗}(V). For a more detailed explanation, see [13, p195].

The simplest differential form is the exact form dx on the variety F. This has a pole at ∞ and is regular elsewhere. Another important differential form is the logarithmic form λ on F or F^{∗} which sends x to the logarithmic differential \frac{dx}{x}. This has poles at 0 and ∞, and indeed has residues there so it is not of the second kind unless it is zero.

Let E be an elliptic curve given by

E = {(y, z) ∈ F^{2} | z^{2} = f(y)}

for a given cubic f ∈ C[t]. The Weierstrass differential form associated with E is the form w(y, z) = \frac{dy}{z}. It is of the first kind, that is it has no poles on E, and it is closed. In fact w spans the C-space of differential forms of the first kind on E. See [10, p163] for details.

It is easy to see that the logarithmic differential form λ is a group homomorphism from the multiplicative group F^{∗} to the additive group Ω(F/C). Correspondingly, Proposition 2.1 of [8] shows that the Weierstrass differential form w is a group homomorphism from the algebraic group E to the additive group Ω(F/C). Furthermore, if E has complex multiplication by τ and is thus a ℤ[τ]-module then w is a homomorphism of ℤ[τ]-modules.

The following relationship between differential forms and differentials will be important.

Lemma 2.9. Suppose ω is a differential form of the first kind defined on a variety V, that y is a generic point of V and that ω(y) = 0. Then ω is zero on all of V.

Proof. Let z ∈ V. Then there is a specialization (field morphism) F ⟶^{π} F fixing C such that π(y) = z. Let R be the associated valuation ring. ω(y) ∈ Ω(C(y)/C), and is a differential of the first kind with respect to this field extension, since being a dfk is independent of the choice of field extension. Suppose ω is given by x ↦ ∑_{i=1}^{n} f_{i}(x)dx_{i}. Since ω(y) is a dfk it is regular at R′ = R ∩ C(y), that is it has a representation as ω(y) = ∑_{i=1}^{m} a_{i}db_{i} with each a_{i}, b_{i} ∈ R′. We can deduce that ∑_{i=1}^{n} f_{i}(y)dy_{i} = ∑_{i=1}^{m} a_{i}db_{i} from the algebraic relations on y over C, so it follows that ∑_{i=1}^{n} π(f_{i}(y))dπ(y_{i}) = ∑_{i=1}^{m} π(a_{i})dπ(b_{i}). Since the f_{i} are rational functions over C we have that π(f_{i}(y)) = f_{i}(π(y)) = f_{i}(z), and thus ω(z) = ∑_{i=1}^{m} π(a_{i})dπ(b_{i}). In particular, it is a well defined differential in Ω(F/C). Now (π(a), π(b)) satisfies all the equations over C that (a, b) does, so the same proof that ∑_{i=1}^{m} a_{i}db_{i} = 0 shows that ω(x) = ∑_{i=1}^{m} π(a_{i})dπ(b_{i}) = 0.

2.4 Lie Derivatives

If Δ ⊆ Der(F/C) is a sub Lie ring (in particular, when Δ = Der(F/D_{0})) then there is a Lie derivative

Δ × Δ^{∗} ⟶^{L} Δ^{∗}

given for D, D_{1} ∈ Δ and ω ∈ Δ^{∗} by

L_{D}ω(D_{1}) = D(ωD_{1}) − ω[D, D_{1}]

This can be thought of as a map Δ^{∗} ⟶^{L_{D}} Δ^{∗} for each D ∈ Δ. Recall that when Δ = Der(F/D_{0}), Δ^{∗} = Ω(F/D_{0}).

When Δ is finite dimensional, the image of the universal derivation F ⟶^{d} Δ^{∗} spans Δ^{∗}, and every differential can be written in the form ∑_{i=1}^{n} a_{i}db_{i} for some a_{i}, b_{i} ∈ F. The form so represented will be the image under the quotient map Ω(F/C) ⟶ Ω(F/C)/ Ann(Δ) of the Kähler differential with the same representation. In these terms the Lie derivative is given by

L_{D} ∑_{i=1}^{n} a_{i}db_{i} = ∑_{i=1}^{n} Da_{i}db_{i} + ∑_{i=1}^{n} a_{i}d(Db_{i})

Lemma 2.10. Let Δ be a sub Lie ring of Der(F/C), D ∈ Δ, ω ∈ Δ^{∗} and a ∈ F. The Lie derivative has the following properties.

• L_{D} is an additive function

• L_{D}(aω) = aL_{D}ω + (Da)ω

• L_{D} is C_{D}-linear

• If ω is closed then L_{D}ω = d(ωD)

• L_{D}(da) = d(Da)

Proof. These are straightforward calculations from the definitions, and are done for example in [14].

2.5 Linear dependence results

If Δ ⊆ Der(F/C), write C_{Δ} = {x ∈ F | (∀D ∈ Δ)Dx = 0}, and write Ann(Δ) = {ω ∈ Ω(F/C) | (∀D ∈ Δ)ωD = 0}.

Lemma 2.11. If Δ ⊆ Der(F/C) and x ∈ F then x ∈ C_{Δ} iff dx ∈ Ann(Δ).

Proof. x ∈ C_{Δ} iff for all D ∈ Δ, Dx = 0 iff for all D ∈ Δ, (dx)D = 0 iff dx ∈ Ann(Δ).

If we take Δ = Der(F/D_{0}) for a field extension F of a differential field ⟨F_{0}, D_{0}⟩ with constants C then C_{Δ} is the algebraic closure of C in F. Also Ω(F/D_{0}) ≅ Ω(F/C)/ Ann(Der(F/D_{0})).

We use the Lie derivative for the following result.

Lemma 2.12. Suppose Δ_{0} is a sub Lie ring of Der(F/C), that Δ ⊆ Δ_{0}, and that η_{1}, …, η_{m} ∈ Ω(F/C) are closed and lie in Ann(Δ). Then the η_{i} are linearly independent over F iff they are linearly independent over C_{Δ}.

Proof. One direction is immediate. For the other, suppose there are α_{i} ∈ F such that ∑_{i=1}^{n} α_{i}ω_{i} = 0 and some α_{i} ≠ 0. We may choose the α_{i} such that the least possible number s of them is non-zero, and that some α_{i} = 1. For each D ∈ Δ we get

0 = L_{D} ∑_{i=1}^{n} α_{i}ω_{i} = ∑_{i=1}^{n}(Dα_{i})ω_{i} + ∑_{i=1}^{n} α_{i}L_{D}ω_{i}

= ∑_{i=1}^{n}(Dα_{i})ω_{i} + ∑_{i=1}^{n} α_{i}d(ω_{i}D)

= ∑_{i=1}^{n}(Dα_{i})ω_{i}

since ω_{i}D = 0 for each i, and using the prevous lemma on properties of the Lie derivative. Some Dα_{i} = 0 but then, by our minimal choice of α_{i}, we have that Dα_{i} = 0 for every i, that is that every α_{i} ∈ C_{D}. This holds for every D ∈ Δ, so every α_{i} ∈ ⋂_{D∈Δ} C_{D} = C_{Δ}. So the ω_{i} are C_{Δ}-linearly dependent.

In particular, if dim_{F} (η_{1}, …, η_{n}) = n − 1, and for some c_{i} ∈ F we have ∑ c_{i}η_{i} = 0 with one of the c_{i} equal to 1 and all nonzero, then all of the c_{i} lie in C_{Δ}.

Lemma 2.13. Let ⟨F, D⟩ be a differential field with constants C, and let E be a subfield of F properly containing C. (E is not necessarily a differential subfield.) Then Ann(D) ∩ Ω(E/C) is an E-subspace of Ω(E/C) of codimension 1.

Proof. It is easy to see that it is a subspace. For the codimension, we have the diagram

[figure: F; d; Ω(F/C); F; D^{∗}; D]

from the universality of d, and Ann(D) is the kernel of the linear map D^{∗}. The diagram restricts to

[figure: E; d; Ω(E/C) ⊗_{E} F; F; D^{∗}; D]

where again D^{∗} is F -linear, with kernel

Ann(D) ∩ (Ω(F/C) ⊗_{E} F) = (Ann(D) ∩ Ω(E/C)) ⊗_{E} F

Since E ≠ C there is x ∈ E ∖ C. Then D^{∗}(dx) = Dx ≠ 0, so D^{∗} is not the zero map and thus has rank 1, and so its kernel has codimension 1. But

dim_{E}(Ann(D) ∩ Ω(E/C)) = dim_{F} (Ann(D) ∩ Ω(E/C)) ⊗_{E} F

and so Ann(D) ∩ Ω(E/C) has codimension 1 in Ω(E/C), as claimed.

It is possible to consider fields with a family of r independent derivations. In this case the result (and proof) generalize immediately to get that the common annihilator of the derivations has codimension r.

Write K(E) for the field of fractions of the endomorphism ring of the elliptic curve E. So K(E) is ℚ if E has no complex multiplication or ℚ(τ) for some imaginary quadratic τ if E has complex multiplication by τ.

Lemma 2.14. The algebraic subgroups of E^{n} are given by equations of the form ⨁_{i=1}^{n} m_{i}X_{i} = O for m_{i} ∈ K(E).

Proof. Algebraic subgroups are closed in the Zariski topology, so their complex realizations must be closed in the complex topology. Thinking of E as the quotient of the complex plane by a lattice, the result follows.

We make use of the following general theorem which was the inspiration for Zilber’s better known model theoretic version.

Theorem 2.15 (Indecomposability Theorem). Let G be an algebraic group and V an irreducible subvariety of G, containing the identity. Then the subgroup generated by V is algebraic.

Proof. See for example [11, p261].

James Ax proved the following by an argument using Puiseux series.

Proposition 2.16. Let y_{1}, …, y_{n} ∈ F^{∗} and suppose that λ(y_{1}), …, λ(y_{n}) are linearly dependent over C. Then they are linearly dependent over ℚ.

In analogy, we prove the following.

Proposition 2.17. Let Y_{1}, …, Y_{n} ∈ E and suppose that w(Y_{1}), …, w(Y_{n}) are linearly dependent over C. Then they are linearly dependent over K(E).

Proof. Suppose Y_{1}, …, Y_{n} ∈ E are such that ∑_{i=1}^{n} c_{i}w(Y_{i}) = 0 for some c_{i} ∈ C, not all zero. Consider the differential form

η : E^{n} → Ω(F/C)

given by η(X_{1}, …, X_{n}) = ∑_{i=1}^{n} c_{i}w(X_{i}). Then the assumption is that Ȳ lies in the kernel of η, which is a proper subgroup of E^{n} since η is a non-zero group homomorphism.

Let V be the irreducible subvariety of E^{n} over C which has Ȳ as a generic point. Then V contains some element P̄ ∈ E_{C}^{n}, which is contained in the kernel of η. Let Ȳ′ = Ȳ ⊖ P̄ and V′ be the irreducible subvariety of E^{n} over C which has Ȳ′ as a generic point. Then Ȳ′ ∈ ker η and the identity O of E^{n} lies in V′. Also w(Y′_{i}) = w(Y_{i}) for each i.

The form η is a C-linear combination of Weierstrass differentials. These are of the first kind and the set of all differentials of the first kind is a Clinear subspace of Ω(F/C), so η is a differential form of the first kind. Thus its restriction to V′ is also of the first kind. By assumption, η is zero at a generic point of V′, so by lemma 2.9, V′ lies in the kernel of η.

Now V′ is an algebraic subvariety of E^{n} containing the identity so, by the Indecomposability Theorem, the subgroup H generated by V′ is an algebraic subgroup. Since V′ is contained in the proper subgroup ker η of E^{n} it follows that H is also contained in ker η, and so H is a proper algebraic subgroup of E^{n}. From the description of the algebraic subgroups of E^{n} given in lemma 2.14 we see that there are m_{1}, …, m_{n} ∈ K such that ⨁_{i=1}^{n} m_{i}Y_{i} = O. Since η is a homomorphism this gives a K-linear dependence between the w(Y_{i}).

3 The universal theory

3.1 Group structures

In this section as before, ⟨F, D⟩ is a differential field of characteristic 0, and C is the constant subfield.

Consider the algebraic groups ⟨F; +⟩, ⟨F^{∗}; ·⟩ and elliptic curves ⟨E; ⊕⟩ where the affine part of E is given by E = {(y, z) ∈ F^{2} | z^{2} = f(y)} for a cubic f with coefficients in C and distinct roots (in a splitting field, although usually C and F will be algebraically closed).

We consider the exponential differential equation

\frac{Dy}{y} = Dx

for x ∈ F and y ∈ F^{∗}, and the Weierstrass differential equation

\frac{Dy}{z} = Dx

for x ∈ F and (y, z) ∈ E, for different elliptic curves E. This equation can also be written just in terms of x and y as

\frac{(Dy)^{2}}{f(y)} = (Dx)^{2}.

Let E ⊆ F × F^{∗} be the solution set to the exponential equation and W ⊆ F × E be the solution set to the Weierstrass equation for a given elliptic curve E. The main aim of this project is to study the reducts of differentially closed fields with the field structure and these relations but without the derivation.

Theorem 3.1. E is a subgroup of F × F^{∗} and W is a subgroup of F × E. Where E has complex multiplication by τ, W is a sub ℤ[τ]-module. The fibres of the projections of E and W are cosets of the constant subgroups C of F, C^{∗} of F^{∗} and E_{C} of E.

Proof. By the comments in section 2.3, the differential forms

F × F^{∗} ⟶^{ω} Ω(F/C)
(x, y) ⟼ dx − \frac{dy}{y}

and

F × E ⟶^{ω} Ω(F/C)
(x, (y, z)) ⟼ dx − \frac{dy}{z}

are group homomorphisms. In the case of an elliptic curve with complex multiplication the differential form is also a ℤ[τ]-module homomorphism. The sets E and W are the kernels of these hence are subgroups. The kernels of the exact, logarithmic and Weierstrass differential forms are the constant subgroups, hence the fibres of points are the cosets of these.

3.2 Schanuel conditions

The main result of James Ax’s paper [1] is the following.

Theorem 3.2 (Ax). Suppose n ⩾ 1 and x_{i}, y_{i} ∈ F such that y_{i} ≠ 0 and (x_{i}, y_{i}) ∈ E for i = 1, …, n. Then

td_{C}(x_{1}, …, x_{n}, y_{1}, …, y_{n}) − dim_{ℚ}(Dx_{1}, …, Dx_{n}) ⩾ 1

We prove the corresponding Schanuel condition for Weierstrass equations.

Theorem 3.3. Let E be an elliptic curve with associated cubic f, and let n ⩾ 1. Suppose that x_{1}, y_{1}, …, x_{n}, y_{n} ∈ F such that f(y_{i}) ≠ 0 and (x_{i}, y_{i}) ∈ W for each i = 1, …, n. Then

td_{C}(x_{1}, …, x_{n}, y_{1}, …, y_{n}) − dim_{K(E)}(Dx_{1}, …, Dx_{n}) ⩾ 1

Proof. Let L be the subfield (not necessarily differential subfield) C(x_{1}, y_{1}, …, x_{n}, y_{n}) of F. It is enough to assume that td(L/C) < n + 1 and show that the Dx_{i} are linearly dependent over K(E).

For each i, there are two Y_{i} ∈ E such that Y_{i} = (y_{i}, z_{i}) for some z_{i}. Choose the one such that (w(Y_{i}) − dx_{i})D = 0, and let η_{i} = w(Y_{i}) − dx_{i}. Then for each i we have η_{i} ∈ Ann(D) ∩ Ω(L/C). Suppose td(L/C) < n + 1. Then by lemma 2.13, dim_{L}(Ann(D) ∩ Ω(L/C)) < n. So η_{1}, …, η_{n} must be L-linearly dependent and thus F -linearly dependent. They are closed differentials and so by lemma 2.12 they are C-linearly dependent. Say

∑_{i=1}^{n} c_{i}η_{i} = 0

with the c_{i} ∈ C, not all zero. Then

∑_{i=1}^{n} c_{i}w(Y_{i}) = ∑_{i=1}^{n} c_{i}dx_{i} = d ∑_{i=1}^{n} c_{i}x_{i}

Each w(Y_{i}) is of the first kind and differentials of the first kind form a Csubspace, so the left hand side here is a differential of the first kind. The right hand side is exact, and an exact differential can only be of the first kind if it is zero, so both sides are zero. So ∑_{i=1}^{n} c_{i}w_{i} = 0, that is the Weierstrass differentials w(Y_{i}) are linearly dependent over C. By proposition 2.17 they are linearly dependent over K(E), so we may assume the c_{i} ∈ K(E). Then ∑_{i=1}^{n} c_{i}Dx_{i} = (∑_{i=1}^{n} c_{i}dx_{i})D = 0, so the Dx_{i} are K(E)-linearly dependent as required.

Essentially the same proof also gives the following generalization.

Theorem 3.4. Let f_{1}, …, f_{m} ∈ C[t] be cubics with distinct roots such that the corresponding elliptic curves are non-isogenous. For i = 1, …, m, let K_{i} be ℚ(τ_{i}) if the elliptic curve corresponding to f_{i} has complex multiplication by τ_{i} or ℚ otherwise, and let K_{0} = ℚ. For each i = 0, …, m, let n_{i} ∈ ℕ, not all zero, and x_{i1}, y_{i1}, …, x_{in_{i}}, y_{in_{i}} ∈ F such that Dx_{i1}, …, Dx_{in_{i}} are K_{i}-linearly independent. Suppose for each i > 0 and each k = 1, …, n_{i} that f_{i}(y_{ik}) ≠ 0 and

\frac{(Dy_{ik})^{2}}{f_{i}(y_{ik})} = (Dx_{ik})^{2}

and that for each k = 1, …, n_{0} that y_{0k} ≠ 0 and that

\frac{Dy_{0k}}{y_{0k}} = Dx_{0k}.

Let L be the field generated over C by all the x_{ik} and y_{ik}.

Then td_{C} L ⩾ ∑_{i=0}^{m} n_{i} + 1.

3.3 Uniformity and definability

It has been known and indeed used by Poizat, Wilkie and Zilber that the statement of Ax’s theorem is first order in the language of differential fields, that is it can be expressed as a collection of first order sentences in that language. I show below that essentially the same argument gives this for the language ⟨+, ·, E⟩ of the reduct. This is due to a certain uniformity in parameters property that strengthens the statement of the Schanuel condition. I give the proof for the Weierstrass case, and the same proof works for the exponential case.

The usual language of fields is equivalent to a relational language with one symbol for each irreducible variety V defined over the prime subfield, ℚ. A variety defined over the constant field C is just a fibre of a variety over ℚ. Note that the constant field is definable in ⟨F; +, ·, W ⟩, for example by x ∈ C ⟺ W (x, O). The field K(E) has an embedding into ℂ (unique up to complex conjugation) which gives rise to a norm on its cartesian powers.

Theorem 3.5 (Uniform Schanuel condition). Let V (x, y, z) ⊆ F^{n} × E^{n} × F^{k} be an irreducible variety over ℚ, and p the projection onto the z coordinates.

Then there is a finite set H_{V} ⊆ R^{n} ∖ {¯0} where R is the endomorphism ring of E such that for all (a, b, c) ∈ V with (a, b) ∈ W^{n}, c ∈ C^{k}, and dim p^{−1}(c) ⩽ n there is m ∈ H_{V} such that ∑_{i=1}^{n} m_{i}a_{i} ∈ C and ⨁_{i=1}^{n} m_{i} · b_{i} ∈ E_{C}.

Proof. We use a simple compactness argument. Let

A = {(a, b, c) ∈ V | (a, b) ∈ W^{n}, c ∈ C^{k}, and dim p^{−1}(c) ⩽ n} .

Then A is definable, the dimension part by the theorem on dimension of fibres (see [13, p76]). For N ∈ ℕ, let

φ_{N}(x, y, z) ≡ [(x, y, z) ∈ A ∧ ¬ ⋁_{\begin{matrix}m∈R^{n} \\ 0<∥m∥⩽N \end{matrix}} ∑_{i−1}^{n} m_{i}x_{i} ∈ C.

This is a first order formula since the disjunction is finite. Suppose there is no H_{V} with the properties required for the theorem. Then the type {φ_{N} | N ∈ ℕ} is finitely satisfiable and hence, by compactness, is satisfied in some differential field. But this contradicts the Schanuel condition.

For definiteness we take H_{V} to be the set of those vectors from R^{n} which actually arise. We don’t extract any information about the size or nature of H_{V} from the proof, although some effectivity is probably possible.

Now for each V the sentence

θ_{V} ≡ ∀xyz[(x, y, z) ∈ A → ⋁_{m∈H_{V}} ∑_{i=1}^{n} m_{i}a_{i} ∈ C

is first order in the language ⟨+, ·, W ⟩ and the set of θ_{V} over varieties V in the relational language expresses the uniform Schanuel condition.

4 Amalgamation and axiomatization

We produce a structure by amalgamation intended to have the same theory as the reduct of a differentially closed field, and give a complete axiomatization of its theory.

Take a language L = ⟨+, ·, (−)^{−1}, 0, 1, g_{2}, g_{3}, W, C⟩ where (−)^{−1} is a function giving field inverse (with the convention that 0^{−1} = 0), C is a unary relation symbol and W is a ternary relation symbol, a subset of F × E where E is the elliptic curve related to g_{2} and g_{3}. Having the inverse means that structures will be fields, not rings. This is convenient but not important.

Let T_{0} be the L-theory given for a structure F as follows.

(Field) F is a field of characteristic zero.

(C) C is a relatively algebraically closed subfield.

(E) We fix constants g_{2} and g_{3} in C.

(W) – W is a subgroup (or sub ℤ[τ]-module) of F × E.

– If W (x_{0}, Y_{0}) then W (x, Y_{0}) ⟺ x − x_{0} ∈ C.

– If W (x_{0}, Y_{0}) then W (x_{0}, Y) ⟺ Y ⊖ Y_{0} ∈ E_{C}.

(SC) For each variety V over ℚ, the sentence θ_{V} expressing the uniform Schanuel condition for V in the language L.

If E has complex multiplication by τ then we add τ to the language as a constant, as it is convenient to have the complex multiplication field K = K(E) in each model of T_{0}.

For a model F of T_{0}, let F̂ be the projection of W onto the first coordinate (the additive group). In other words, F̂ = {x ∈ F | ∃Y ∈ E_{F} [W (x, Y)]}. F̂ is always a K-vector space containing C. Let the group rank of W in the model F, grk_{C} F be defined as the K-linear dimension of F̂ modulo C, that is the size of a minimal subset B of F̂ such that every element of F̂ can be written as a finite sum of the form ∑ m_{i}b_{i} + c with m_{i} ∈ K, b_{i} ∈ B and c ∈ C.

If F is a model of T_{0} such that td(F/C) is finite then we can define a predimension function

δ(F) = td_{C} F − grk_{C} F.

This is finite and non-negative by the Schanuel condition, and is zero precisely when F = C.

We define an L-embedding F_{1} ^{⊂} ^{}F_{2} to be strong iff for every finitely generated substructure X of F_{2} we have δ(X ∩ F_{1}) ⩽ δ(X). Write F_{1} ◁ F_{2} for a strong embedding. Let K_{0} be the category of finitely generated models of T_{0} and strong embeddings.

The category K_{0} has the hereditary property (if F_{2} ∈ K_{0} and F_{1} is a strong substructure of F_{2} then the embedding F_{1} ◁ F_{2} is in K_{0}). We want to show that K_{0} also has the joint embedding property (JEP) and the amalgamation property (AP). The field K(g_{2}, g_{3}) is a model of T_{0} and strongly embeds into every model of T_{0}, so the JEP is a special case of the AP.

We will show below that the category K_{0} has the amalgamation property. Assuming this for now, by the Fraissé amalgamation theorem (see for example [3]) there is a unique countable L-structure, M, such that the strong substructures are exactly the countable models of T_{0} and whenever F_{1}, F_{2} ◁ M are isomorphic and finitely generated then the isomorphism extends to an automorphism of M. This says that M is ℵ_{0}-homogeneous for strong substructures, or that it is K_{0}-saturated or K_{0}-existentially closed. (“All possible finite strong extensions happen inside M.”)

The theory T_{0} is universal so every substructure of a model of T_{0} is also a model of T_{0}, in particular every substructure (not necessarily strong) of M is a model of T_{0}.

4.1 Axiomatization

Our aim now is to find axioms which, together with T_{0}, give the complete theory of M. We would like an axiom scheme expressing the K_{0}-existential closedness.

• If F_{1} ◁_{f.g.} M and F_{2} is a finitely generated model of T_{0}, with F_{1} ◁ F_{2} in K_{0} then there is a copy of F_{2} in M such that F_{1} ◁ F_{2} inside M.

One way of looking at this is that the Schanuel condition (hereditary positivity of the relative predimension function) gives a sufficient condition (as well as a necessary one) for a system of equations to have solutions.

There are three problems with this. Firstly, it is not possible to tell with a finite number of formulas whether or not a substructure F ⊆ M is strong in M. Because of this, our axiom scheme must say something stronger – the assumption that F_{1} ◁ M must be relaxed to F_{1} ⊆ M. Fortunately this stronger condition does hold in M as we will show.

Any strong extension F_{1} ◁ F_{2} can be split into two parts, the first just extending the subfield C and the second keeping C fixed. Existential closedness for extensions of C corresponds to saying that C is algebraically closed and of infinite transcendence degree. The latter part we ignore because it is not first order and because it doesn’t hold in all the reducts of a differentially closed field in which we are interested. From the general existential closedness condition it follows that the whole field M will be algebraically closed, and T_{0} says that C is relatively algebraically closed, so to capture the rest of existential closedness under extensions of C we take the usual axiom scheme expressing the following.

(EC_{1}) M is an algebraically closed field

For extensions which keep C fixed we start by considering the strong existential closedness axiom scheme.

(SEC) If F_{1} ⊆_{f.g.} M, F_{2} is a finitely generated model of T_{0} with F_{1} ◁ F_{2} in K_{0} and containing no new constants then there is a copy of F_{2} in M such that F_{1} ◁ F_{2} inside M.

The word “strong” here refers both to the notion of strong embeddings and also to the fact that this axiom is stronger than the existential closedness axiom scheme (EC) which we will give soon.

The second problem we must address is that to express this axiom in a first order way we have to replace finitely generated structures with generating sets for them. We give a normal form for a generating set.

Take F_{1} ⊆_{f.g.} M generated by a finite tuple a. Some of the coordinates of a are in C, some are in a projection of W ∩ F_{1}^{3} and some are in neither, so we may write a as (a_{1}, a_{2}, a_{3}) where a_{1} ∈ C^{t}, a_{2} ∈ W^{n} and a_{3} ∈ F_{1}^{s}, and W is generated by a_{2}, under the rules specified by the theory T_{0}. Recall that R is the endomorphism ring of E, with field of fractions K. The group W is an R-module, and the quotient W/C is a K-vector space, the dimension of which is grk_{C} F. By removing some coordinates of a_{2} we may assume that the remainder are K-linearly independent in the group W over W_{C}. The part of T_{0} which specifies the fibres of W ensures that F_{1} is still well-defined by this reduced set of generators. A generating set of this form (a_{1}, a_{2}, a_{3}) is said to be in normal form.

Suppose F_{1} ◁ F_{2} with no extra constants, that (a_{1}, a_{2}, a_{3}) is a generating set for F_{1} in normal form, and b is a generating set for F_{2} over F_{1}. There are no extra constants, so there is no need for a b_{1} part of the normal form of b. Since W projects surjectively onto the field M in all three coordinates, we may extend b so that every coordinate is part of some triple in W. Thus we may also do away with the b_{3} part of b. This may have the effect of extending F_{2}, but this is no problem. Thus we say that the normal form of a generating set for an extension F_{2} over F_{1} is a tuple b ∈ W^{m} which is Klinearly independent over C and over a_{2}. This says simply that (a_{1}, a_{2}b, a_{3}) is a generating set in normal form.

For a tuple a in normal form, let V = loc(a), the algebraic locus of a over ℚ, and let Ṽ be given as follows.

Ṽ = {(x_{1}, x_{2}, x_{3}) ∈ V | x_{1} ∈ C^{t}, x_{2} ∈ W^{n}, dim V_{x_{1}x_{3}} = dim V_{a_{1}a_{3}}, and grk_{C}(x_{2}) = n}

Note that this is first-order definable, the dimension of the variety by the fibre condition of algebraic geometry and the linear independence of x_{2} by the argument giving the uniformity of the Schanuel condition.

Let U = loc(ab), the algebraic locus of ab over ℚ, and define

Ũ = {(x, y) ∈ U | (x) ∈ Ṽ , y ∈ W^{m}, dim U_{x} = dim U_{a}, and grk_{C}(x_{2}y) = n + m}

which again is first order definable. If (x, y) ∈ Ũ, the predimension δ(x, y) = td_{C}(x_{2}, x_{3}, y) − (n + m). Thus genericity in the sense of the predimension coincides with algebraic genericity. By assumption, (a, b) is generic in Ũ, and b is generic in Ũ_{a} over the field C together with the parameters a.

Thus (SEC) can be restated as the following.

• For each strong extension F_{1} ◁ F_{2} in K_{0}, with generators in normal form and with Ṽ and Ũ as defined above,

∀x∃y [x ∈ Ṽ → (y is generic in Ũ_{x} over C(x))] .

The third and final problem with giving an axiomatization is that this is still not a first order statement, because we cannot insist that y is generic in a first order way. Instead we must give all first order approximations to this. If the predimension of F_{2} is equal to that of F_{1} this does not matter, because the Schanuel condition ensures that y must be generic. However, the first order theory of M does not imply that the dimension of a model (in the sense of the pregeometry arising from the predimension) is infinite, or indeed that it is greater than 1.

To approximate this in a first order way, we say that for each proper subvariety U′ of U_{x}, there is a y satisfying the conditions but not living in U′. The existential closedness axiom scheme is as follows.

(EC_{2}) For each strong extension F_{1} ◁ F_{2} in K_{0}, with generators in normal form and with Ṽ and Ũ as defined above, for each proper subvariety U′ of pr U definable without parameters,

∀x∃y [x ∈ Ṽ → (y ∈ Ũ_{x} ∖ U′)] .

Take (EC) = (EC_{1}) + (EC_{2}), and T_{W} = T_{0} + (EC). We must show that M ⊧ T_{W} and that T_{W} is complete.

It is easy to see that (EC_{1}) holds in the amalgam M. We prove that (SEC) holds in M. Suppose that a is a tuple such that M ⊧ Ṽ (a), and let A = ⟨a⟩. (SEC) says that we can find a tuple b in M which is generic in Ũ_{a} over C(a). This is equivalent to finding a strong extension B of A inside M, corresponding to a generating set b, and it does not matter than a may not be a generic point of Ṽ since B will still be a strong extension of A. If A ◁ M, the existence of the strong extension B follows from the fact that M is K_{0}-homogeneous and K_{0}-universal. We have to show that the same holds even if A is not strong in B. For this, let Ā be a hull of A, that is A ⊆ Ā ⊆_{f.g.} M with δ(Ā) = d(A). (We can also ask for Ā to be minimal such, but it is not necessary to prove that such a minimal hull exists so we don’t do that here.) In other words, Ā is a finitely generated extension of A in M which is strong in M. We now have the situation where A embeds in Ā and B, strongly in B but not in Ā, and we want to find an amalgamation of this.

Proposition 4.1. The class K_{0} of finitely generated models of T_{0} has the asymmetric amalgamation property (AAP). That is, any diagram of the form

\begin{matrix}Y_{1} \\ X & ^{^{∪} } & ◁ & Y_{2} \end{matrix}

can be completed to a commuting diagram

\begin{matrix}Y_{1} & ◁ & Z \\ X & ^{^{∪} } & ◁ & Y_{2} & ^{^{∪} } \end{matrix}

in the class K_{0}. In particular, the category K_{0} has the AP.

Assuming this proposition, take Z to be an amalgam of Ā and B. Then by the K_{0}-saturatedness of M, there is a copy of Z extending Ā in M. This Z contains a copy of B which is a strong extension of A in M, and so M satisfies (SEC) as required.

Proof of Proposition 4.1. As before, we may assume that the constant field C remains the same in all extensions. Take X′ to be the algebraic closure of X as a field, with the group W on X′ generated by its restriction to X according to the rules specified in the theory T_{0}. Similarly define Y′_{1} and Y′_{2}. The embeddings X ↪ Y_{1} and X ◁ Y_{2} extend to X′ ↪ Y′_{1} and X′ ◁ Y′_{2}, uniquely up to isomorphism. As a field, define Z′ to be the algebraically closed “free amalgam” of the underlying fields Y′_{1} and Y′_{2} over X′. That is, if Y_{i} has transcendence base B_{i} over X then the transcendence base of Z′ over X′ is the disjoint union B_{1} ⊔ B_{2}, and we choose embeddings of Y′_{1} and Y′_{2} into Z′ extending the embeddings X′ ↪ Z′ and B_{i} ↪ Z′.

Since we are assuming the constant field C is contained in X′, this is well defined in Z′. We take W in Z′ to be generated by the images of W in Y′_{1} and Y′_{2}. Finally take Z to be the substructure of Z′ which is (finitely) generated by the images of Y_{1} and Y_{2}. This satisfies all the conditions of being an amalgam except possibly that Y_{1} ◁ Z. First note that

td_{C}(Z) = td(Y_{1}/X)+td(Y_{2}/X)+td(X/C) = td(Y_{1}/C)+td(Y_{2}/C)−td(X/C)

and similarly grk_{C} Z = grk_{C} Y_{1} + grk_{C} Y_{2} − grk_{C} X, so we have

δ(Z) = δ(Y_{1}) + δ(Y_{2}) − δ(X)

and thus δ(Z) − δ(Z ∩ Y_{1}) = δ(Z) − δ(Y_{1}) = δ(Y_{2}) − δ(X) ⩾ 0. The same equality holds taking the intersection of the diagram with any substructure of Z, so the fact that X ◁ Y_{2} forces Y_{1} ◁ Z as required.

To show that T_{W} is complete, we prove that it is near model complete, that is, it has quantifier elimination to the level of existential formulas and their negations (which are universal formulas). This is also a useful fact in its own right.

Proposition 4.2. T_{W} is near model complete and complete.

Proof. Write etp(a) for the set of existential formulas or their negations satisfied by a. Suppose M, M′ are ω-saturated models of T_{W}, and a, a′ are tuples from M, M′ respectively such that etp(a) = etp(a′). Let b ∈ M. We want to find b′ ∈ M′ such that etp(ab) = etp(a′b′). By the standard back and forth argument, this will show that tp(a) = tp(a′), and so T_{W} is near model complete. We allow the tuples a, a′ to be empty, so this also shows that tp(∅) is the same in M and M′, that is that Th(M) = Th(M′), and so T_{W} is complete.

We prove this in three steps.

Step 1 If a, a′ have the same etp then they have isomorphic hulls.

Take A = ⟨a⟩. We may assume that a is in the normal form of a generating set. Suppose that Ā is a hull of A in M, and V is the locus of a generating set in normal form of Ā over A. Then etp(a) contains the formulas ∃yṼ (y) and ¬∃y(Ṽ (y) ∧ V′(y)) for each proper subvariety V′ of V, defined over the parameters a. Any realisation of Ṽ in M′ will give an isomorphic copy of Ā containing a′ in M′.

Take isomorphic hulls Ā of A = ⟨a⟩ and Ā′ of A′ = ⟨a′⟩. Note that Ā◁M and Ā′ ◁ M′.

Step 2 If A ◁_{f.g.} M, A′ ◁_{f.g.} M′ and A ≅ A′ then the isomorphism can be extended to any b ∈ M.

Take B to be a hull of Ab in M, and b̄ a generating set for B over A, in normal form. Since A ◁ M it follows that A ◁ B. The structure M′ is ω-saturated and satisfies (EC) so it satisfies (SEC), and hence there is b̄′ in M′ generating B′ isomorphic over A′ to B over A. This isomorphism restricts to Ab ≅ A′b′ for some b′ (which may not be one of the generators in b̄′).

Step 3 If A ◁ M, A′ ◁ M′ and A ≅ A′ then etp(A) = etp(A′).

A ≅ A′ is the same as equality of the quantifier-free types, qftp(A) = qftp(A′). The etp of A also contains information about what finitely generated extensions of A exist in M, and this determines etp(A). Since A ◁ M the only extensions of A which exist in M are strong extensions. By (SEC), all finitely generated strong extensions do exist. The same holds for A′, and so etp(A) = etp(A′).

Continuing from step 2, we deduce that etp(ab) = etp(a′b′), and so T_{W} is near model complete and complete as required.

We may define T_{E} and T_{EW} for the exponential equation and for both equations in a precisely analogous way, and the results and proofs go through essentially unchanged.

5 The existentially closed condition

We show that the reducts of differentially closed fields in question satisfy the existential closedness condition (EC). The theory T_{0} was chosen specifically as the universal theory of these reducts, so this will show that the theory of the reducts is T_{W}.

To show (EC), we use the differential nullstellensatz which says that if a finite system of differential equations and inequations over a differentially closed field F has a solution in some extension of F then it has a solution already inside F. We also use the characterization of differential field extensions in terms of differential forms.

Any differentially closed field F is algebraically closed, so (EC_{1}) holds. It remains to show (EC_{2}). That is, if F_{1} ◁ F_{2} is a strong extension in K_{0} with generators in normal form with locuses V and U, U′ is a proper subvariety of pr U, and a is a tuple in F satisfying Ṽ then there is a tuple b in F such that (a, b) ∈ Ũ and b ∉ U′. By the differential nullstellensatz it is enough to find b in some extension of F. F_{1} can be subsumed into F as a set of parameters. Take F_{2} to have generators b over F, where b ∈ W^{n} and is free in this group over F. Let U_{a} = loc(b)/F, the algebraic locus of b over F, which has parameters a from the subfield F_{1} of F. Take

Ũ_{a} = {y ∈ U_{a} | y ∈ W^{n} and is free over F } .

This is definable in the language ⟨+, ·, W, 0, 1⟩ as explained before. We prove that in some differential field extension of F there is b ∈ Ũ_{a}, generic over F.

Note that although this appears to prove (SEC) not just (EC), when we use the differential nullstellensatz to pull this solution back into F we only retain the first order part of the existential closedness condition.

The assumptions on F_{2} are that it satisfies T_{0}, that it is a strong extension of F and that the chosen generating set is independent in W over F. In terms of the variety Ũ_{a}, this means that a point y of Ũ_{a}, generic over F, does not satisfy any equation of the form ⨁_{i=1}^{n} m_{i}y_{i} ∈ F with m_{i} ∈ R and that for any matrix M ∈ Mat_{r×n}(R),

td_{C}(My) − grk_{C}(My) ⩾ 1.

In Boris Zilber’s terminology, these two conditions are called respectively freeness over F and normality of the variety Ũ_{a}.

We change notation slightly to match the earlier section on Schanuel conditions. The subvariety U_{a} ⊆ (F × E)^{n} becomes V, and a generic point of V is written (a, b) where a ∈ F^{n} and b ∈ E^{n}. The differentially closed field is now F_{0}, and the extension F ◁ F_{2} becomes F_{0} ◁ F. The result we now need, stated in terms of differentials, is as follows.

Proposition 5.1. If V is free and normal over F_{0}, (a, b) is generic in V over F_{0} and F = F_{0}(a, b) then there is a derivation D on F which extends the original derivation D_{0} on F_{0} with no new constants and such that w(b_{i})D = da_{i}D for each i = 1, …, n.

In order to prove this we use the following two lemmas which are essentially the same as lemma 3.1 of [15]. For x, y ∈ F^{n}, write x · y for the dot product ∑_{i=1}^{n} x_{i}y_{i}. For any y ∈ F^{n}, write H_{y} for the affine hyperplane {x ∈ F^{n} | x · y = 1}. It is easy to see that if every coordinate of y is nonzero and F^{n} ⟶^{p} F^{m} is any coordinate projection with m < n then the image of H_{y} under p is all of F^{m}. The following lemma generalises this observation.

Lemma 5.2. Let F be an algebraically closed field of infinite transcendence degree over a set b of parameters, possibly infinite. Let V ⊆ F^{n} be any irreducible variety defined over b and let d = dim V. Let c ∈ F^{n} be algebraically independent over b, and let V′ be an irreducible component of V ∩ H_{c}. Let a be generic in V′ over bc. Suppose that e ∈ F^{m} for some m with e ∈ acl(ba) but a ∉ acl(be).

Then td(e/bc) = td(e/b), that is e is free from c over b.

Proof. By the independence of c over b, V is not contained in H_{c}. Thus dim V′ < dim V, and so dim V′ = d − 1. Thus td(a/bc) = d − 1, and td(a/b) = d since a is necessarily generic in V over b.

Let U = loc(c/be) be the irreducible variety over be with generic point c. Suppose for a contradiction that td(e/bc) < td(e/b). Then e is not free from c over b, and so dim U < |c| = n. But dim U = td(c/be) ⩾ td(c/ba) = n − 1, so dim U = n − 1.

Now c ∈ H_{a} ∩ U, and dim H_{a} = n − 1, so by the genericity of c in U we see that U = H_{a}. Thus H_{a} is defined over be, and so (∀y ∈ H_{a})[y · x = 1] is expressible as a formula with parameters be. This formula defines a uniquely, and so a ∈ acl(be), which contradicts the assumption.

We apply this to reduce the dimension of a free and normal variety without losing freeness and normality.

Lemma 5.3. If (ā, b̄) is free and normal over A_{1} and td(ā, b̄/A_{1}) > n then there is A_{2} extending A_{1} such that td(ā, b̄/A_{2}) = n and (ā, b̄) is free and normal over A_{2}.

Proof. This is an immediate application of the previous lemma using the definitions of freeness and normality.

Proof of Proposition 5.1. We assume that td_{F_{0}}(a, b) = n, using lemma 5.3. For i = 1, …, n let ω_{i} = w(b_{i}) − da_{i}. Let H be the subspace ⟨ω_{1}, …, ω_{n}⟩ of Ω(F/C), and Δ = ker H ⊆ Der(F/C).

Step 1 Suppose x ∈ C_{Δ}. Then dx ∈ H so there are c_{1}, …, c_{n} ∈ F such that dx = ∑_{i=1}^{n} c_{i}ω_{i}.

Choose a K-basis γ_{1}, …, γ_{r} for c_{1}, …, c_{n}, such that there are m_{ij} ∈ R with c_{i} = ∑_{j=1}^{r} m_{ij}γ_{j}. Here R is the ring of endomorphisms of E and K is its field of fractions. Let

α_{j} = ∑_{i=1}^{n} m_{ij}a_{j},  β_{j} = ⨁_{i=1}^{n} m_{ij}b_{i},  η_{j} = w(β_{j}) − dα_{j}

for each j. Then η_{j} = ∑_{i=1}^{n} m_{ij}ω_{j} and dx = ∑_{j=1}^{r} γ_{j}η_{j}.

Let Δ′ = ker(η_{1}, …, η_{r}), a superspace of Δ in Der(F/C). If D ∈ Δ′ then dx(D) = ∑_{j=1}^{r} γ_{j}η_{j}D = 0, so dx ∈ Ann(Δ′) and x ∈ C_{Δ′}. By freeness of V over F_{0}, and nondegeneracy of the matrix (m_{ij}), the η_{j} are K-linearly independent. Thus by proposition 2.17 they are linearly independent over C_{Δ′} and by 2.12 they are linearly independent over F. Now applying 2.12 to dx, η_{1}, …, η_{r} we see that each γ_{i} ∈ C_{Δ′}.

Expanding out the η_{j}, we get

dx + ∑_{j=1}^{r} γ_{j}dα_{j} = ∑_{j=1}^{r} γ_{j}w(β_{j})

Applying the canonical map Ω(F/C) → Ω(F/C_{Δ′}) we get that

d_{F/C_{Δ′}}x + ∑_{j=1}^{r} γ_{j}d_{F/C_{Δ′}}α_{j} = ∑_{j=1}^{r} γ_{j}w_{F/C_{Δ′}}(β_{j})

and, since the γ_{j} ∈ C_{Δ′}, the left hand side is exact and the right hand side is a C_{Δ′}-linear combination of Weierstrass differentials, so is of the first kind. Thus both sides are zero. In particular, ∑_{j=1}^{r} γ_{j}w_{F/C_{Δ′}}(β_{j}) = 0.

By renumbering, we may suppose that the j for which w_{F/C_{Δ′}}(β_{j}) ≠ 0 are 1, …, s.

Then w_{F/C_{Δ′}}(β_{1}), …, w_{F/C_{Δ′}}(β_{s}) are Weierstrass differentials which are linearly dependent over C_{Δ′}, so by proposition 2.17 we may assume these γ_{j} ∈ K. But the γ_{j} were taken to be linearly independent over K, so we must have s = 0, that is each β_{j} ∈ C_{Δ′}. Then also each α_{j} ∈ C_{Δ′} by the definition of Δ′.

Now td(C_{Δ′}/C) = dim Ann(Δ′) ⩽ r, so td_{C}(ᾱ, β̄) ⩽ r and thus by the normality of (ā, b̄) we deduce that r = 0. Then each c_{i} = 0, so dx = 0 and x ∈ C. This shows that C_{Δ} = C as required. Since each c_{i} = 0 this also shows that the ω_{i} are linearly independent over F.

Step 2 We now show that they are even F -linearly independent modulo Ω(F_{0}/C) ⊗_{F_{0}} F. Suppose that for some γ_{i} ∈ F and η ∈ Ω(F_{0}/C) we have ∑_{i=1}^{n} γ_{i}ω_{i} = γ_{0}η. Apply the canonical map Ω(F/C) ⟶ Ω(F/F_{0}). The kernel of this map contains Ω(F_{0}/C) ⊗_{F_{0}} F, so we get ∑ γ_{i}ω̄_{i} = 0. Now ω̄_{i} = w_{F/F_{0}}b_{i} − d_{F/F_{0}}a_{i}, and so the argument of step 1 shows that the γ_{i} ∈ K.

Let a_{0} = ∑ γ_{i}a_{i} and b_{0} = ⨁ γ_{i} · b_{i}. Then γ_{0}η = ∑ γ_{i}ω_{i} = wb_{0} − da_{0}. By freeness, either γ_{1}, …, γ_{n} are all zero or at least one of a_{0}, b_{0} does not lie in F_{0}. Then applying the canonical map into Ω(F/F_{0}) we have η̄ = 0 and thus da_{0} = wb_{0} which is impossible when neither is in F_{0}. Thus each γ_{i} is zero and γ_{0}η = 0 as required.

Step 3 Consider the canonical map Ω(F/C) ⟶ Ω(F/D_{0}). Step 2 says that the images ω̄_{i} are F -linearly independent in Ω(F/D_{0}). Now dim Ω(F/D_{0}) = td(F/F_{0}) + 1 = n + 1. Also dim Δ = dim Ω(F/D_{0}) − dim⟨ω̄_{1}, …, ω̄_{n}⟩ = 1. Take D to be the unique D ∈ Δ such that D ↾_{F_{0}}= D_{0}. Then C_{D} = C_{Δ} since D spans Δ, and we have already shown that C_{Δ} = C. Thus there are no new constants, and this completes the proof.

References

[1] James Ax, On Schanuel’s conjectures, Annals of Mathematics 93 (1971), 252–268.

[2] W. Dale Brownawell and K. K. Kubota, The algebraic independence of Weierstrass functions and some related numbers, Acta Arithmetica XXXIII (1977), 111–149.

[3] Manfred Droste and Rüdiger Göbel, A categorical theorem on universal objects and its application in abelian group theory and computer science, Contemporary Mathematics 131 (Part 3) (1992), 49–74.

[4] David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995.

[5] Assaf Hasson and Ehud Hrushovski, DMP in strongly minimal structures, preprint, 2005.

[6] Ehud Hrushovski, A new strongly minimal set, Annals of Pure and Applied Logic 63 (1993), no. 2, 147–166.

[7] Jonathan Kirby, Valuations and field morphisms, expository article,

available on www.maths.ox.ac.uk/~kirby.

[8] , A Schanuel condition for Weierstrass equations, J. Symbolic Logic 70 (2005), no. 2, 631–638.

[9] Serge Lang, Introduction to algebraic geometry, Interscience, 1958.

[10] Solomon Lefschetz, Algebraic geometry, Princeton University Press, Princeton, N. J., 1953.

[11] David Marker, Model theory: An introduction, GTM, no. 217, Springer, 2002.

[12] David Pierce, Differential forms in the model theory of differential fields, Journal of Symbolic Logic 68 (2003), no. 3, 923–945.

[13] Igor R. Shafarevich, Basic algebraic geometry. 1, Springer-Verlag, Berlin, 1994.

[14] Alex Wilkie, Lecture notes on Marker’s lecture notes on Ax’s theorem, Logic Advanced Class, Oxford, 2003.

[15] B. Zilber, Bi-coloured fields on the complex numbers, J. Symbolic Logic 69 (2004), no. 4, 1171–1186.