Analytic continuation and Zilber’s quasiminimality conjecture

A.J. Wilkie Mathematical Institute University of Oxford

Abstract

In this article, which is dedicated to my friend and colleague Boris Zilber on the occasion of his 75th birthday, I put forward a strategy for proving his quasiminimality conjecture for the complex exponential field. That is, for showing that every subset of ℂ definable in the expansion of the complex field by the complex exponential function is either countable or cocountable. In fact the strategy applies to any expansion of the complex field by a countable set of entire functions (in any number of variables) and is based on a certain property-an analytic continuation property-of the o-minimal structure obtained by expanding the ordered field of real numbers by the restrictions to compact boxes of the real and imaginary parts of the functions in the given set.

In a final section I discuss briefly the (rather limited) extent of our unconditional knowledge in the area.

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After some reflection we, Boris and I, agreed that it was in July 1993 that he first asked me whether I had thought about the model theory of the complex exponential field. The occasion was Logic Colloquium ’93, the European Summer Meeting of the Association for Symbolic Logic, which was held that year at Keele University in the UK. We were both invited to give plenary lectures. Boris spoke about his results on model-theoretic dimensions in complex (and ultrametric) analysis and I gave a talk on recent work with Angus (eventually published in the paper [MW]) concerning the decidability of the real exponential field, which was still my main concern. I certainly hadn’t even considered the complex exponential. In fact the idea seemed absurd; the set of integers was definable and so as far as I was concerned it was not a tame structure. But Boris, then as now, had considerably more imagination, and deeper insights than I did into potentialy good modeltheoretic behaviour of familiar mathematical structures. If the set of reals was also definable then, he agreed, the situation would indeed be hopeless. But what if it wasn’t? In fact, could it not be the case that every definable subset of ℂ was either countable or co-countable? That remark had a profound effect on my mathematical life and was the motivational force for much of my research since that moment almost thirty years ago. It still is.

I soon realised that I could make progress on the problem-the quasiminimality problem as Boris called it-only through o-minimal theory and hard analysis: my knowledge of abstract stability theory and its generalizations was simply not strong enough, and certainly not up to a level to be able to follow and make a meaningful contribution to Boris’s ingenious, beautiful and eventually highly successful construction of a “pseudo-exponential” field using a combination of Hrushovski’s predimensions and Shelah’s theory of excellent classes. So my own approach started with the observation that the complex exponential function is definable in an o-minimal structure (when considered as a function from ℝ^{2} to ℝ^{2}) provided that the imaginary part of its argument is restricted to a bounded interval. (One may take this structure to be, for example, the real ordered field expanded by both the real exponential function and by the sine function restricted to the interval [0, π].) After several years I came up with about forty pages of notes purporting to contain, amongst other things, a proof that the complex exponential field was quasiminimal and I intended to present a sketch of the argument at an Oberwolfach meeting in July 2004.

Most readers of this article will know that Boris’ conjecture is still unresolved and hence that there must have been a mistake, as indeed there was. But three things did come out of those notes. Firstly, I had established (correctly) a positive solution of Schanuel’s conjecture for “generic” finite sequences of real numbers, and this rescued my Oberwolfach talk. ( The result was eventually generalized and published in the paper [BKW] written jointly with two of the editors of this volume.) Secondly, in order to cope with the fact that the 2πiℤ-periodicity of the complex exponential function is at complete variance with the whole ethos of o-minimality, I was forced to investigate whether, and if so how, points with integer coordinates could lie in sets defined by algebro-exponential equations, and thereby obtain a measure of “non-tameness”. I certainly did not want to assume the full version of Schanuel’s Conjecture (which would have settled this particular issue), but if it could be shown that either there were few such points or, if not, then at least one such point behaved generically then the “generic Schanuel theorem” just mentioned could be used and this might be sufficient to control the periodicity. It didn’t work but, and I apologize for a small personal digression at this point, the idea of quantitatively investigating the occurrence of points with integer coordinates in sets defined as above did appeal to me. In fact, why not do the same in the framework of general o-minimal structures? So I temporarily abandoned work on the quasiminimalty problem and initiated the project of counting integer points lying in sets definable in a fixed, but arbitrary, o-minimal structure. I had limited success at the time, only managing to prove a result in the case of one dimensional sets. The paper ([W1]) was published in 2004 and came to the attention of Jonathan Pila who had been working on similar issues. He had obtained bounds of the same general nature as mine for rational points lying in low dimensional, globally subanalytic sets (i.e. sets definable in the o-minimal structure ℝ_{an}), but was finding it difficult to put his arguments into a general setting that would smoothly facilitate an induction on dimension. Jonathan and I met soon after and agreed that the setting had to be o-minimality. Our point counting theorem was published in 2006 and this is how, at least from my point of view, that result came about: my motivation for studying integer points in o-minimally definable sets was, apart from the fun of it, completely motivated by Boris’ quasiminimality problem. I had no inkling of how, over the following fifteen years, the result would be applied, with huge success, by Jonathan and many others to problems in diophantine geometry.

Serendipity? Not exactly. For while Boris, as far as I know, did not foresee such applications, his conjectures, often fearless but always with sound intellectual bases, have been an inspirational source of research throughout our community even when, and perhaps especially when, that research takes unexpected directions.

I mentioned that three things came out of my 2004 notes. The third occupies the remainder of this paper. It concerns my strategy, which I still believe to be plausible, for proving the quasiminimality conjecture for the complex exponential field and, possibly, for many other expansions of the complex field by entire functions. I am very grateful to the editors for giving me this opportunity to explain the strategy despite the fact that, at the time of writing, it has not resulted in any definite results. The main theorems do establish the quasiminimality of a certain class of structures expanding the complex field but is conditional on a property of locally definable holomorphic functions, namely that they have analytic continuations along “generic” paths (i.e. those avoiding obvious singularities). The first theorem is precisely this, while the second provides a criterion for the continuation in purely complex analytic terms that avoid notions of general definability.

So let ℝ̃ be a fixed o-minimal expansion of the real ordered field \overline{ℝ} := ⟨ℝ; <, +, ·⟩. Following [PS1, PS2] we say that a complex valued function F of the n complex variables z_{1}, …, z_{n} is definable if the real and imaginary parts of F are definable in ℝ̃ (without parameters) when considered as functions of the 2n real variables x_{1}, y_{1}, …, x_{n}, y_{n}, where z_{j} = x_{j} + iy_{j} for j = 1…. , n. If F is holomorphic, its domain will always be assumed to be, without further mention, a connected, open subset (a fortiori a definable subset if F is definable) of ℂ^{n}. We write dom(F) for the domain of F.

Peterzil and Starchenko develop complex analysis in this definable context, but in fact many of the subtleties of their work will not be needed here since we will only be working with the standard structure ℝ̃. However, one of their results is still worth mentioning, namely that if {F_{t} : t ∈ ℝ^{d}} is a definable family of n-variable, holomorphic functions, then there exists N such that for all t ∈ ℝ^{d} and all a ∈ dom(F_{t}), if \frac{∂^{α}F}{∂z^{α}} vanishes at a for each α ∈ ℕ^{n} with |α| ≤ N, then F_{t} is identically zero. (This is, of course, obvious if ℝ̃ is polynomially bounded, but not so clear otherwise. For example, and of particular relevance to Zilber’s problem, the F_{t}’s could range over polynomials in z_{1}, …, z_{n}, log z_{1}, …, log z_{n} of some fixed degree (and with suitable simply connected domains).)

Anyway, returning to definitions we consider the closure operator LD(·)

- “locally definable from” - where we specify that for all n ≥ 1 and for all a_{1}, …, a_{n+1} ∈ ℂ,

(1) a_{n+1} ∈ LD(a_{1}, …, a_{n}) :⟺ F (a_{1}, …, a_{n}) = a_{n+1} for some definable, holomorphic function F with ⟨a_{1}, …, a_{n}⟩ ∈ dom(F).

It is straightforward to show that LD is a pregeometry and, further, that for all b_{1}, …, b_{m} ∈ ℂ,

(2) b_{1}, …, b_{m} are LD-independent if and only if whenever G is a nonzero, definable, holomorphic function with ⟨b_{1}, …, b_{m}⟩ ∈ dom(G), then G(b_{1}, …, b_{m}) ≠ 0.

(See [W3] for details.)

Strictly speaking LD(X) should be specified for all subsets X of ℂ and this is done in the usual way by taking LD(X) to be the union of all LD(X_{0})’s as X_{0} ranges over finite subsets of X. There is a small issue concerning the LD-closure of the empty set which is taken to be {s + it : s, t ∈ ℝ, both definable in ℝ̃}. One easily checks (using the Cauchy-Riemann equations) that if G is a nonzero, definable, holomorphic function with s + it ∈ dom(G) and G(s + it) = 0, then both s and t are indeed definable in ℝ̃, so this choice of LD(∅) is consistent with (2) above. However, the relationship between the LD-dimension of an arbitrary finite subset of ℂ and the dimension of the corresponding set of real and imaginary parts (for the usual pregeometry of the o-minimal structure ℝ̃) is more complicated and is resolved in section 4 of [W3].

Examples

(a) If we take ℝ̃ to be just \overline{ℝ}, then LD is just algebraic closure (over ℚ) in the field ℂ . More generally, if we expand \overline{ℝ} by a set A of constants, then LD is algebraic closure over the subfield ℚ(A) of ℂ . (Two remarks: firstly, I do not find this at all obvious (the proof can be found in [W2]), and secondly, recall that our holomorphic functions are allowed to be definable in ℝ̃ so, for example we can distinguish (uniformly in parameters) between different roots of a polynomial.)

(b) Let ℝ̃ = ℝ_{exp} := ⟨\overline{ℝ}, exp⟩ where exp : ℝ → ℝ : x ↦ e^{x}. Then, by a result of Bianconi (see [B]), LD is still just algebraic closure (but this time it’s over the minimal model K ≼ ℝ_{exp} of the theory of ℝ_{exp}). Of course, things are different, and of great relevance to Zilber’s problem, if we expand ℝ_{exp} by the restricted sine function and we will be discussing this situation later.

(c) For ℝ̃ = ℝ_{an} (where the definable sets are the globally subanalytic sets, see for example, [DD]), LD is trivial: LD(X) = LD(∅) = ℂ for all X ⊆ ℂ simply because all r ∈ ℝ are definable.

In view of example (c) above we now assume that the language of ℝ̃ is countable so, in particular, LD(∅) is countable. We shall require rather more:

Lemma (Existence of generic lines)

Let A be a countable subset of ℝ. Then there exists ω ∈ ℂ (in fact an uncountable, dense set of them) such that for all continuous functions ϕ : [0, 1] ↦ ℂ definable in the structure ⟨ℝ̃, s⟩_{s∈A} and satisfying ϕ(0) ≠ 0, we have that ϕ(t) ≠ tω for all t ∈ [0, 1]. Proof

For each such ϕ let

S_{ϕ} := {ω ∈ ℂ : for all t ∈ [0, 1], ϕ(t) ≠ tω}.

It clearly follows from the continuity of ϕ and the compactness of the closed unit interval that S_{ϕ} is an open subset of ℂ. So by the Baire Category Theorem we will be done if we show that S_{ϕ} is a dense subset of ℂ (because there are only countably many ϕ’s).

So let ω_{0} be an arbitrary complex number and let ϵ > 0. Let Δ be the open disc in ℂ centered at ω_{0} and of radius ϵ and suppose, for a contradiction, that Δ ⊆ ℂ ∖ S_{ϕ}, i.e. that for all ω ∈ Δ, there exists t ∈ [0, 1] such that ϕ(t) = tω. Then, by definable choice and the non-decreasing of dimension under definable, injective maps (working in the o-minimal structure ⟨ℝ̃, s⟩_{s∈A}), there exist ω_{1}, ω_{2} ∈ Δ with ω_{1} ≠ ω_{2} such that for some t_{0} ∈ [0, 1], ϕ(t_{0}) = t_{0}ω_{1} and ϕ(t_{0}) = t_{0}ω_{2}. This is absurd unless t_{0} = 0; but this is also ruled out since ϕ(0) ≠ 0. □

We will be considering analytic continuations of definable functions along generic paths in ℂ^{n}. In fact, we will only need to consider linear paths: for a, ω ∈ ℂ^{n} (n ≥ 1), define the map λ_{a,ω} : ℂ ↦ ℂ^{n} by λ_{a,ω}(z) := a + zω (for z ∈ ℂ). We say that λ_{a,ω} is generic on a set T ⊆ ℂ if λ_{a,ω}(t) is a generic ntuple for each t ∈ T, i.e. if a_{1}+tω_{1}, …, a_{n}+tω_{n} are LD-independent complex numbers for each t ∈ T (where a = ⟨a_{1}, …, a_{n}⟩ and ω = ⟨ω_{1}, …, ω_{n}⟩). The set T will, almost always, be the interval [0, 1] so that for any a, ω we have that λ_{a,ω}(0) = a.

We now come to our main definitions.

Definition 1

We say that the structure ℝ̃ has the Analytic Continuation Property (ACP) if for all LD-independent a_{1}, …, a_{n} ∈ ℂ, all definable, holomorphic functions F with a = ⟨a_{1}, …, a_{n}⟩ ∈ dom(F) and all ω ∈ ℂ^{n} with λ_{a,ω} generic on [0, 1], there exists a definable, holomorphic function G with λ_{a,ω}([0, 1]) ⊆ dom(G) ⊆ ℂ^{n} such that G(a) = F (a). (And hence, by (2), G(λ_{a,ω}(z)) = F (λ_{a,ω}(z)) for all z ∈ ℂ such that λ_{a,ω}(z) lies in the connected component of dom(F) ∩ dom(G) containing the point a. So the function G ∘ λ_{a,ω} analytically continues, in the usual sense, the function F ∘ λ_{a,ω} (restricted to a sufficiently small open neighbourhood of 0 ∈ ℂ) to an open set containing the interval [0, 1].)

Definition 2

Let 1 ≤ l ≤ n and let M ⊆ ℂ^{n}. Then we say that M is an l-dimensional, locally definable, complex submanifold of ℂ^{n} (or just an l-manifold for short) if

(a) M is a closed subset of ℂ^{n}, and

(b) for all a ∈ M, there exist a definable open set W with a ∈ W ⊆ ℂ^{n} and a holomorphic, definable map G = ⟨G_{1}, …, G_{n−l}⟩ : W ↦ ℂ^{n−l} such that a is a non-singular point of the zero set of G (i.e. G(a) = 0 and the vectors ⟨\frac{∂G_{j}}{∂z_{1}} (a), …, \frac{∂G_{j}}{∂z_{n}} (a)⟩ (for 1 ≤ j ≤ n − l) are linearly independent over ℂ ) and, further, M ∩W = Z_{reg}(G) where Z_{reg}(G) denotes the set of non-singular points of the zero set of G.

Example

If ℝ̃ = ⟨\overline{ℝ}, exp↾ [0, 1], sin↾[0, 2π]⟩, then the graph of the complex exponential function Γ_{exp} := {⟨z, e^{z}⟩ : z ∈ ℂ} is a 1-dimensional, locally definable, complex submanifold of ℂ^{2}.

Definition 3

The structure ℂ̃ is defined to be the expansion of the complex field by all l-dimensional, locally definable, complex submanifolds of ℂ^{n} (for all l, n with 1 ≤ l < n).

Remark

With ℝ̃ as in the example above we see that ℂ̃ is an expansion of ℂ_{exp} (:= the complex field expanded by the complex exponential function). I do not know if it is a proper expansion. For example, it is clear that every connected component of an l-manifold is also an l-manifold, but it seems to me (and, in fact, to Zilber) to be perfectly possible that some l-manifold, M say, is definable in ℂ_{exp} but that some connected component of M is not.

Theorem 1

Suppose that ℝ̃ has the ACP. Then the structure ℂ̃ is quasi-minimal. That is, for any subset S of ℂ which is definable in the language L(ℂ̃) of ℂ̃, and we do allow parameters here, we have that either S is countable or its complement ℂ \ S is countable. In fact, the same is true for sets S defined by a formula of the infinitary language L(ℂ̃)_{∞,ω} provided, of course, that the formula contains only countably many parameters. Proof

We first show that if u and v are elements of ℂ \ LD(∅) then there exists a back-and-forth system (for the structure ℂ̃) containing the pair ⟨u, v⟩.

Then by a classical result of Karp ([K]) the quasiminimality condition for parameter-free formulas follows from this since it implies that if S ⊆ ℂ is a parameter-free L(ℂ̃)_{∞,ω}-definable set, then either S ⊆ LD(∅) or else ℂ \ LD(∅) ⊆ S, and LD(∅) is countable.

So suppose that u, v ∈ ℂ \ LD(∅). We may assume that λ_{u,v−u} is generic on [0, 1]. For otherwise, by the countability of LD(∅), there exists some w ∈ ℂ \ LD(∅) such that both λ_{u,w−u} and λ_{w,v−w} are generic on [0, 1] and we prove the result for the pair u, w and for the pair w, v.

We now set up a back-and-forth argument.

For n ≥ 1 and a = ⟨a_{1}, …, a_{n}⟩, b = ⟨b_{1}, …, b_{n}⟩ ∈ ℂ^{n} we write a ∼_{n} b if

(i)_{n} a_{1} = u and b_{1} = v ;

(ii)_{n} for some m with 1 ≤ m ≤ n and some i_{1}, …, i_{m} with 1 ≤ i_{1} < ⋯ < i_{m} ≤ n we have that λ_{a′,b′−a′} is generic on [0, 1], where a′ := ⟨a_{i_{1}}, …, a_{i_{m}}⟩ and b′ := ⟨b_{i_{1}}, …, b_{i_{m}}⟩;

(iii)_{n} there exists a definable connected, open set V ⊆ ℂ^{m} with λ_{a′,b′−a′}([0, 1]) ⊆ V and, for each j = 1, …, n, a definable, holomorphic function F_{j} : V ↦ ℂ such that F_{j}(λ_{a′,b′−a′}(0)) = F_{j}(a_{i_{1}}, …, a_{i_{m}}) = a_{j} and F_{j}(λ_{a′,b′−a′}(1)) = F_{j}(b_{i_{1}}, …, b_{i_{m}}) = b_{j}.

We clearly have ⟨u⟩ ∼_{1} ⟨v⟩ (where, in this case, n = m = 1, V = ℂ and F_{1} is the identity function on ℂ ) .

In order to establish the back-and-forth property, suppose that n ≥ 1 and that a ∼_{n} b as above, with m, a′, b′ as in (ii)_{n}. We write λ for λ_{a′,b′−a′}.

So let a_{n+1} ∈ ℂ. We must find b_{n+1} ∈ ℂ such that ⟨a, a_{n+1}⟩ ∼_{n+1} ⟨b, b_{n+1}⟩. There are two cases.

Case 1. a_{n+1} ∉ LD(a_{i_{1}}, …, a_{i_{m}}).

Let A be a finite subset of ℝ containing the real and imaginary parts of a_{i_{1}}, …, a_{i_{m}}, a_{n+1}, b_{i_{1}}, …, b_{i_{m}}. Apply the generic lines lemma to obtain some ω ∈ ℂ such that for all continuous ϕ : [0, 1] → ℂ definable in the structure ⟨ℝ̃, c⟩_{c∈A} with ϕ(0) ≠ 0, we have ϕ(t) ≠ tω for all t ∈ [0, 1] and, further, such that ω does not lie in the (countable) set LD(b_{i_{1}}, …, b_{i_{m}}, a_{n+1}). Let b_{n+1} := a_{n+1} + ω. We show that ⟨a, a_{n+1}⟩ ∼ ⟨b, b_{n+1}⟩. Now (i)_{n+1} is obvious and for (ii)_{n+1}, (iii)_{n+1} we replace m by m + 1 (≤ n + 1) and let i_{m+1} := n + 1. We also replace a′ by ⟨a′, a_{n+1}⟩ and b′ by ⟨b′, b_{n+1}⟩ so that λ (= λ_{a′,b′−a′}) becomes λ^{∗} : ℂ ↦ ℂ^{m+1} given by λ^{∗}(z) := ⟨λ(z), a_{n+1} + zω⟩. We must show that λ^{∗} is generic on [0, 1]. So suppose, for a contradiction, that for some t_{0} ∈ [0, 1] the (m+1)-tuple λ^{∗}(t_{0}) has LD-dimension < m+1. Now since λ(t_{0}) has LD-dimension m (because λ is generic on [0, 1]) we must have a_{n+1} + t_{0}ω ∈ LD(λ(t_{0})). So there exists a definable, holomorphic function F with λ(t_{0}) ∈ dom(F) and such that

(*) F (λ(t_{0})) = a_{n+1} + t_{0}ω.

Now we cannot have t_{0} = 0 for then we would have a′ = λ(0) ∈ dom(F) and F (a′) = a_{n+1} which contradicts the hypothesis of Case 1. Also, t_{0} ≠ 1 for otherwise we would have that ω = F (λ(1)) − a_{n+1} = F (b′) − a_{n+1} ∈ LD(b_{i_{1}}, …, b_{i_{m}}, a_{n+1}). So 0 < t_{0} < 1 and since dom(F) is open and λ is continuous, there exist rationals q_{1}, q_{2} with 0 < q_{1} < t_{0} < q_{2} < 1 such that λ([q_{1}, q_{2}]) ⊆ dom(F).

Define ϕ : [0, 1] → ℂ to be the continuous function taking the value −a_{n+1} + F (λ(t)) for q_{1} ≤ t ≤ q_{2} and which is linear on each of the intervals [0, q_{1}], [q_{2}, 1] with, say, ϕ(0) = 1 and ϕ(1) = 0.

Then ϕ is definable in the structure ⟨ℝ̃, c⟩_{c∈A} because λ is, a_{n+1} is, and F is definable in ℝ̃ without parameters. Further, ϕ(0) ≠ 0 so by the construction of ω we have that ϕ(t) ≠ tω for all t ∈ [0, 1]. In particular, ϕ(t_{0}) ≠ t_{0}, i.e. −a_{n+1} + F (λ(t_{0})) ≠ t_{0}ω, which contradicts (*) and establishes (ii)_{n+1}.

As for (iii)_{n+1}, we take our new V to be V ×ℂ and our new F_{j}’s, call them F_{j}^{∗} (for j = 1, …, n+1), to be specified (for ⟨z_{1}, …, z_{m+1}⟩ ∈ V ×ℂ) by setting F_{j}^{∗}(z_{1}, …, z_{m+1}) := F_{j}(z_{1}, …, z_{m}) if 1 ≤ j ≤ n and F_{n+1}^{∗}(z_{1}, …, z_{m+1}) := z_{m+1}.

Then the required conditions for (iii)_{n+1} carry over from (iii)_{n} for j = 1, …, n, and for j = n + 1 we have, for each t ∈ [0, 1],

F_{n+1}^{∗}(λ_{⟨a′,a_{n+1}⟩,⟨b′,b_{n+1}⟩−⟨a′,a_{n+1}⟩}(t)) = a_{n+1} + t(b_{n+1} − a_{n+1})

which takes the value a_{n+1} for t = 0 and b_{n+1} for t = 1. So we have that ⟨a, a_{n+1}⟩ ∼_{n+1} ⟨b, b_{n+1}⟩ as required.

Case 2. a_{n+1} ∈ LD(a_{i_{1}}, …, a_{i_{m}}).

In this case there exists a definable, holomorphic function F such that a′ ∈ dom(F) and F (a′) = a_{n+1}. Apply the ACP to obtain a definable, holomorphic function G with λ([0, 1]) ⊆ dom(G) ⊆ ℂ^{m} satisfying G(a′) = F (a′), i.e. G(a′) = a_{n+1}. Now with V ⊆ ℂ^{m} as given by (iii)_{n}, note that λ([0, 1]) ⊆ V ∩ dom(G). Let U be a definable, connected open subset of V ∩ domG such that λ([0, 1]) ⊆ U. (Clearly such a U exists and may be taken, for example, to be a certain finite union of polydiscs with Gaussian rational centres and rational radii.)

We now take the same m as in (ii)_{n} so that (i)_{n+1} and (ii)_{n+1} are obviously satisfied. For (iii)_{n+1} we take the F_{j}’s as given by (iii)_{n} for j = 1, …, n and restrict them to the set U. For F_{n+1} we take G restricted to U so that F_{n+1}(λ(0)) = F_{n+1}(a′) = G(a′) = a_{n+1}. Finally, taking b_{n+1} := F_{n+1}(λ(1)) completes the construction in Case 2.

Of course we also need to consider the “back” case-where we take some b_{n+1} ∈ ℂ and have to find some a_{n+1} ∈ ℂ satisfying ⟨a, a_{n+1}⟩ ∼ ⟨b, b_{n+1}⟩. But this follows in exactly the same way upon noting that for any c, d ∈ ℂ^{l} we have λ_{c,d−c} is generic on [0, 1] if and only if λ_{d,c−d} is (because the ranges on [0, 1] are the same).

So, our system {⟨a, b⟩ : n ≥ 1, a, b ∈ ℂ^{n} and a ∼_{n} b} has the back-and-forth property.

We must now show that atomic formulas of L(ℂ̃) are preserved from a to b whenever a ∼_{n} b.

So let 1 ≤ l ≤ n and suppose that M ⊆ ℂ^{n} is an l-manifold. Suppose a ∈ M and b ∈ ℂ^{n} are such that a ∼_{n} b. We must show that b ∈ M (and similarly for a and b interchanged, for which the proof is the same). Write a = ⟨a_{1}, …, a_{n}⟩ and b = ⟨b_{1}, …, b_{n}⟩. Let m, 1 ≤ i_{1} < ⋯ < i_{m} ≤ n, V and the F_{j}’s be as in (ii)_{n} and (iii)_{n}, and write F for the map ⟨F_{1}, …, F_{n}⟩ : V → ℂ^{n}.

Define T := {t ∈ [0, 1] : F (λ(t)) ∈ M} where, as before, λ = λ_{a′,b′−a′} and a′ = ⟨a_{i_{1}}, …, a_{i_{m}}⟩, b′ = ⟨b_{i_{1}}, …, b_{i_{m}}⟩,

Then T is not empty because 0 ∈ T. Also, T is a closed subset of [0, 1] because M is closed (2(a) of Definition 2). So we shall be done if we can show that T is open.

So let t_{0} ∈ [0, 1] be such that F (λ(t_{0})) ∈ M. Say F (λ(t_{0})) = c = ⟨c_{1}, …, c_{n}⟩. Choose G = ⟨G_{1}, …, G_{n−l}⟩ and W as in (2b) of Definition 2 for this particular c ∈ M. Then c ∈ Z_{reg}(G), and by reducing W (definably) if necessary we may suppose that w is a non-singular point of the zero set of G for all w ∈ W satisfying G(w) = 0. By continuity, there exists ϵ > 0 such that F (λ(t)) ∈ W for all t ∈ [t_{0}−ϵ, t_{0}+ϵ]. Thus {z ∈ V : F (z) ∈ W } is a definable, open subset of V containing λ([t_{0}−ϵ, t_{0}+ϵ]) and G∘F : {z ∈ V : F (z) ∈ W }* → ℂ is a definable holomorphic function such that G ∘ F (λ(t_{0})) = 0, where the * denotes taking the connected component of the set {z ∈ V : F (z) ∈ W } that contains the point λ(t_{0}) (and hence the set λ([t_{0} − ϵ, t_{0} + ϵ])). However, λ is generic on [0, 1] and so it follows from (2) that G ∘ F is identically zero. In particular G(F (λ(t))) = 0 for all t ∈ [t_{0} − ϵ, t_{0} + ϵ]. But Z_{reg}(G) ⊆ M ∩ W ((2b) of Definition 2) and so F (λ(t)) ∈ M for all t ∈ [t_{0} − ϵ, t_{0} + ϵ], and this shows that T is open, as required.

The proof of our present aim is now complete apart from one small detail. The reader may have noticed that, strictly speaking, atomic formulas of L(ℂ̃) of ℂ̃ have the form Φ(v_{j_{1}}, …, v_{j_{p}}) for some 1 ≤ l < p where Φ is the symbol of the language L(ℂ) interpreting an l-submanifold of ℂ^{p} (for some l, p with 1 ≤ l < p). But we have tacitly assumed in our proof above that j_{k} = k and that p is (an arbitrarily large) n. But this assumption can easily be arranged (at least for j_{1}, …, j_{p} distinct) by “adding vacuous variables” and observing that the set

{⟨a_{1}, …, a_{n}⟩ ∈ ℂ^{n} : ℂ̃ ⊧ Φ[a_{j_{1}}, …, a_{j_{p}}]}

is an (n − p + l)-submanifold of ℂ^{n} .

Notice also that the graph of equality is a 1-submanifold of ℂ^{2} (so we may indeed assume in the above discussion that j_{1}, …, j_{p} are distinct) and that the graphs of addition and multiplication are 2-submanifolds of ℂ^{3}, so that equality of polynomial terms is also preserved by the ∼_{n} relation.

We now need to deal with case that the formula defining the set S contains a countable set, X say, of parameters. But for this we simply apply the result above with ℝ̃ replaced by the structure, ℝ̆ say, obtained by expanding ℝ̃ by a constant for each element of X′ where X′ is the set of real and imaginary parts of elements of X. The required result follows since it is easy to check (upon denoting by ℂ̆ the corresponding complex structure as given by Definition 3) that for any formula of L(ℂ̃)_{∞,ω} with parameters in X there exists a parameter-free formula of L(ℂ̆)_{∞,ω} that defines the same set. □

As a test for quasi-minimality Theorem 1 has limited use because in order to prove that a given structure has the ACP one still needs some reasonable mathematical description of the definable sets and functions. We now turn to this problem in the case of expansions of the complex field by entire functions and we look for a complex analytic criterion for such a structure to have the ACP. To this end, suppose that we are given, for each n ≥ 0, a countable ring H_{n} of entire functions of the n complex variable z_{1}, …, z_{n}. We assume that H_{n} ⊆ H_{n+1} (in the obvious sense) and that each H_{n} contains the projection functions and is closed under partial differentiation and Schwarz reflection (i.e. if f ∈ H_{n} , then \frac{∂f}{∂z_{j}} ∈ H_{n} for j = 1, …, n and f^{SR} ∈ H_{n}, where

f^{SR}(z) := \overline{f(\overline{z})} for z ∈ ℂ^{n} (and the bar denotes co-ordinatewise complex conjugation)). We then call the sequence H := ⟨H_{n} : n ≥ 0⟩ of rings a suitable sequence and associate to such an H a certain expansion ℝ̃(H) of \overline{ℝ} as follows:-

For each n ≥ 0, f ∈ H_{n} and discs D_{1}, …, D_{n} in ℂ with Gaussian rational centres and rational radii we denote by f̃ the restriction of f to the polydisc D_{1} × ⋯ × D_{n} (and define f̃(z) to be 0 for z ∉ D_{1} × ⋯ × D_{n}). (For n = 0, f̃ is taken to be the element f of H_{0} (⊆ ℂ).)

Now, with the usual convention concerning the identification of ℂ with ℝ^{2}, we define the structure ℝ̃(H) to be the expansion of \overline{ℝ} by all such f̃.

Then ℝ̃(H), being a reduct of ℝ_{an}, is a polynomially bounded, o-minimal expansion of \overline{ℝ} and its language is countable (since each H_{n} is and there are only countably many polydiscs to which we restrict the functions therein).

In [W3] I gave a characterization of the definable, holomorphic functions of ℝ̃(H) around generic points of ℂ^{n} . This characterization has been shown to be insufficient around non-generic points (see [JKLS]), but at least it does give an alternative description of the LD-pregeometry in terms that avoid notions of general definability. The characterization is as follows.

Consider an LD-generic point a = ⟨a_{1}, …, a_{n}⟩ ∈ ℂ^{n} and let F be a definable, holomorphic function (definable, that is, in the structure ℝ̃(H) without parameters). Then, as is proved in [W3], there exist disks D_{1}, …, D_{n} in ℂ (with centres and radii as specified above) and, for some N ≥ 1, functions f_{1}, …, f_{N} ∈ H_{n+N}, and definable, holomorphic functions ϕ_{1}, …, ϕ_{N} : D_{1} × ⋯ × D_{n} → ℂ such that

(3) a ∈ D_{1} × ⋯ × D_{n} ⊆ dom(F);

(4) for all z ∈ D_{1}×⋯×D_{n} we have that ⟨ϕ_{1}(z), …, ϕ_{N}(z)⟩ is a non-singular zero of the map f_{z} : ℂ^{N} → ℂ^{N} : w ↦ ⟨f_{1}(z, w), …, f_{N}(z, w)⟩;

(5) F (z) = ϕ_{1}(z) for all z ∈ D_{1} × ⋯ × D_{n}.

In other words, F arises, at least close to the generic point a, as a coordinate function of a map given by an application of the Implicit Function Theorem applied to functions from H. (Actually, the characterization from [W3] (see 1.5 and 1.6 of [W3]) makes use of just the one (dependent) variable version of the Implicit Function Theorem together with composition, but it is easy to see that this formulation is equivalent. Note also that the operations 1.2 and 1.3 from [W3] follow from the corresponding closure conditions that we have placed on H. (One can consult [S] for more on this,))

For later use we remark now that (4) is equivalent to

(4*) for all z ∈ D_{1} × ⋯ × D_{n} and j = 1, …, N we have that f_{j}(z, ϕ_{1}(z), …, ϕ_{N}(z)) = 0 and J(z, ϕ_{1}(z), …, ϕ_{N}(z)) ≠ 0 where J is the determinant of the Jacobian matrix (\frac{∂f_{j}}{∂w_{i}})_{1≤i,j≤N}. (Note that J ∈ H_{n+N}.)

In [W3] I define a pregeometry D̃ on ℂ associated with implicit functions as discussed above. Namely, a d-tuple ⟨b_{1}, …, b_{d}⟩ ∈ ℂ^{d} is declared to be D̃- generic if there do not exist k ≥ 0 and b_{d+1}, …, b_{d+k} ∈ ℂ and g_{1}, …, g_{k+1} ∈ H_{d+k} such that ⟨b_{1}, …, b_{d}, b_{d+1}, …, b_{d+k}⟩ is a non-singular zero of the map ⟨g_{1}, …, g_{k+1}⟩ : ℂ^{d+k} → ℂ^{k+1}.

It is shown in [W3] (Theorem 1.10) that LD and D̃ are identical pregeometries and as regards to our present aim this leads to our next theorem. It states that one only needs to check that implicitly defined functions, such as the ϕ_{i}’s mentioned above, have analytic continuations along generic paths. To be more precise we make the following

Definition 4

We say that the suitable sequence H has the Weak Analytic Continuation Property (WACP) if for all n, N ≥ 1, all a, ω ∈ ℂ^{n} such that λ_{a,ω} is D̃-generic on [0, 1], all r ∈ [0, 1] and all f ∈ (H_{n+N})^{N}, if γ : [0, r) → ℂ^{N} is a continuous map such that ⟨λ_{a,ω}(t), γ(t)⟩ ∈ Z_{reg}(f) for all t ∈ [0, r) then ||γ(t)|| ↛ ∞ as t → ∞. (The non-singularity here is with respect to the last N variables (as in (4*)), and || · || is some standard norm on ℂ^{N}.)

Notice that there is no explicit mention of definability here. Nevertheless, we have the following

Theorem 2

If H has the WACP then ℝ̃(H) has the ACP. Hence, by Theorem 1, the corresponding expansion ℂ̃(H) of the complex field is quasiminimal (even for the language L(ℂ̃(H)_{∞,ω}).

Reminder and Remark

The structure ℂ̃(H) is defined to be the expansion of the complex field by all l-dimensional, locally definable, complex submanifolds of ℂ (for all l, n with 1 ≤ l < n) with respect to the expansion ℝ̃(H) of the real field. It is trivial to show that all functions in H are definable in ℂ̃(H).

Proof of Theorem 2

Let a ∈ ℂ^{n} be an LD-generic point and F a definable holomorphic function with a ∈ dom(F). Let ω ∈ ℂ^{n} be such that λ_{a,ω} is generic on [0, 1]. We write λ for λ_{a,ω}. We must find G satisfying the conclusion of Definition 1.

Choose ⟨D_{1}, …, D_{n}⟩, N, f = ⟨f_{1}, …, f_{N}⟩ and ϕ = ⟨ϕ_{1}, …, ϕ_{N}⟩ as in (3),

(4) and (5).

Now we may suppose that every zero, ⟨z^{(0)}, w^{(0)}⟩ say, of f satisfies J(⟨z^{(0)}, w^{(0)}⟩) ≠ 0 (see (4*)). Indeed, let f_{N+1} ∈ H_{n+N+1} be defined by

f_{N+1}(z, w, w_{N+1}) := w_{N+1} · J(z, w) − 1.

Then, letting f^{∗} := ⟨f_{1}, …, f_{N}, f_{N+1}⟩, one easily calculates that the Jacobian of f^{∗} (with respect to w_{1}, …, w_{N+1}) has determinant J(z, w)^{2}, which is nonzero whenever f_{N+1}(z, w, w_{N+1}) is zero. Further, any non-singular (with respect to w_{1}, …, w_{N+1}) zero of f gives rise to a (unique) zero of the map f^{∗}. So we may replace f by f^{∗} and (3), (4) and (5) remain true (by setting ϕ_{N+1}(z) = J(z, ϕ_{1}(z), …, ϕ_{N}(z))^{−1}). And now, all zeros of f^{∗} are nonsingular (with respect to w_{1}, …, w_{N+1}).

So we continue our proof with this non-singularity assumption.

Let T be the set of all those t ≥ 0 having the following properties:-

(6)_{t} there exists a definable, open, connected set U_{t} ⊆ ℂ with [0, t] ⊆ U_{t};

(7)_{t} there exists a definable, holomorphic map ψ^{(t)} with range contained in ℂ^{N} and with λ(U_{t}) ⊆ dom(ψ^{(t)}) which satisfies

(8)_{t} for all u ∈ U_{t} we have that ⟨λ(u), ψ^{(t)}(λ(u)⟩ is a non-singular zero of f (with respect to w_{1}, …, w_{N});

(9)_{t} ψ^{(t)}(λ(0)) = f(λ(0)). (So, in particular, the first coordinate of ψ^{(t)}(λ(0)) is F (a).)

We shall be done if we can show that 1 ∈ T, for then we take G to be the first coordinate of ψ^{(1)} to satisfy the conclusion of Definition 1.

Notice that we certainly have 0 ∈ T by taking U_{0} to be a (definable) disk around 0 ∈ ℂ which is small enough to satisfy λ(U_{0}) ⊆ D_{1} × ⋯ × D_{n}, and then taking ψ^{(0)} := ϕ.

Notice also that if t_{1}, t_{2} ∈ T and t_{1} ≤ t_{2} then ψ^{(t_{1})} ∘ λ and ψ^{(t_{2})} ∘ λ must agree on (U_{t_{1}} ∩ U_{t_{2}})^{∗}, the connected component of U_{t_{1}} ∩ U_{t_{2}} containing the interval [0, t_{1}]. This is because they agree at 0 (by (9)_{t_{1}} and (9)_{t_{2}}) and λ(0) (= a) is LD-generic, so the definable holomorphic map (ψ^{(t_{2})} − ψ^{(t_{1})})↾(U_{t_{1}} ∩ U_{t_{2}})^{∗} must be identically zero (the zero, that is, of ℂ^{N}).

We now set r := sup{t : [0, t] ⊆ T } and we need to show that r ≥ 1. So suppose, for a contradiction, that r < 1. By the extension property just proved, it follows that χ := ⋃_{t<r}(ψ^{(t)} ↾ λ([0, t])) is a continuous map with domain λ([0, r)) such that for all t ∈ [0, r) we have that ⟨λ(t), χ(λ(t))⟩ is a non-singular zero of f (with respect to w_{1}, …, w_{N}) - see (8)_{t}. So by applying the WACP (with γ = χ ∘ λ) we see that there exists some positive R and an increasing sequence ⟨t_{p} : p ≥ 0⟩ in [0, r) converging to r such that ||ψ(λ(t_{p}))|| ≤ R for all p ≥ 0.

Let w^{(0)} ∈ ℂ^{N} be a limit point of the sequence ⟨χ(λ(t_{p})) : p ≥ 0⟩. Then ⟨λ(r), w^{(0)}⟩ is a zero of f (since f is certainly a continuous map throughout ℂ^{n+N}) and by our non-singularity assumption, it is a non-singular zero (with respect to w_{1}, …, w_{N}). So by the Implicit Function Theorem there exist an open polydisc Δ ⊆ ℂ^{n} (which we may take to be definable) with λ(r) ∈ Δ, and a holomorphic map θ : Δ → ℂ^{N} satisfying θ(λ(r)) = w^{(0)} and such that for all z ∈ Δ, the (n + N)-tuple ⟨z, θ(z)⟩ is a non-singular zero of f (with respect to w_{1}, …, w_{N}). Further, we may assume that Δ has been chosen small enough for there to exist a (definable) open polydisk E ⊆ ℂ^{N} with w^{(0)} ∈ E such that for all z ∈ Δ, w = θ(z) is the one and only solution in E of the equation f(z, w) = 0. (This follows from the uniqueness condition in the conclusion of the Implicit Function Theorem.) It follows from this that θ is definable.

Now choose p large enough so that t_{p} is close enough to r to satisfy

(10) λ(t_{p}) ∈ Δ and

(11) χ(λ(t_{p})) ∈ E (and hence ψ^{(t_{p})}(λ(t_{p})) ∈ E).

Now choose ϵ > 0 so small that the rectangle

ρ := {x + iy ∈ ℂ : t_{p} − ϵ < x < t_{p} + ϵ, −ϵ < y < ϵ}

is contained in U_{t_{p}}. (See (6)_{t_{p}}.)

Since λ is linear, λ(ρ) is a convex, open subset of ℂ^{n}(=ℝ^{2n}) and since Δ is too, it follows that λ(ρ) ∩ Δ is convex and, in particular, connected.

Now λ(t_{p}) ∈ λ(ρ) ∩ Δ (see (10)) and both θ(λ(t_{p})) and ϕ^{(t_{p})}(λ(t_{p})) lie in E (see (11)). So by the uniqueness condition we have that ϕ^{(t_{p})}(λ(t_{p})) = θ(λ(t_{p})) (see (8)_{t_{p}}).

But λ is generic (for either pregeometry) on [0, 1] so, by (2), ϕ^{(t_{p})} and θ must agree on a sufficiently small, definable open polydisk containing the point λ(t_{p}) and contained within λ(ρ) ∩ Δ. But then, by the principle of analytic continuation, they must agree throughout the connected set λ(ρ)∩Δ. (Note that λ(ρ)∩Δ ⊆ dom(ϕ^{(t_{p})})∩dom(θ)-see (7)_{t_{p}}.) So we may consistently define a holomorphic map, Γ say, with domain λ(ρ) ∪ Δ and taking values in C^{N} by specifying Γ(z) to be ϕ^{(t_{p})}(z) for z ∈ λ(ρ) and θ(z) for z ∈ Δ.

Finally, since λ(r) ∈ Δ we may choose r′ with r < r′ ≤ 1 such that λ(r′) ∈ Δ. We now obtain a contradicion by showing that [0, r′] ⊆ T.

Indeed, suppose that s ∈ [0, r′]. Then λ([0, s]) ⊆ λ(ρ) ∪ Δ (because λ([0, t_{p}]) ⊆ λ(ρ) and λ([t_{p}, s]) ⊆ Δ (since Δ is convex)). So if we take U_{s} to be a definable, open, connected subset of ℂ containing the interval [0, s] and contained within λ^{−1}(λ(ρ) ∪ Δ), and set ψ^{(s)} := Γ ↾ λ(U_{s}), we see that (6)_{s}-(9)_{s} are all satisfied. So s ∈ T as required. □

Exponentiation

We now consider the setting appropriate, with respect to the preceeding discussion of analytic continuation, for Zilber’s quasiminimality conjecture for the complex exponential field.

We let K be a fixed countable, real closed subfield of ℝ and let E_{0} be the algebraically closed subfield K[i] of ℂ . Then, for each n ≥ 1 we let E_{n} be the ring E_{0}[z_{1}, …, z_{n}, e^{z_{1}}, …, e^{z_{n}}] of exponential polynomials over E_{0} in the complex variables z_{1}, …, z_{n}. It is clear that E := ⟨E_{n} : n ≥ 0⟩ is a suitable sequence and one easily checks that ℝ̃(E) is essentially (i.e. has the same definable sets as) as the structure ℝ_{K}^{RE} considered by Binyamini and Novikov in [BN]. The superscript “RE” stands for “restricted elementary”: ℝ_{K}^{RE} is the expansion of \overline{ℝ} by the restricted functions exp↾[0, 1] and sin↾[0, π] and by a constant for each element of K.

Thus, in the present context, a subset M of ℂ^{n} is an l-dimensional, locally definable, complex submanifold of ℂ^{n} if it satisfies (2a) and (2b) of Definition 2, where definability is now with respect to the structure ℝ_{K}^{RE} (and, as before, is without parameters).

Let us refer to such M simply as elementary complex manifolds (ECM’s). We write ℂ_{K}^{ECM} for the structure denoted ℂ̃ in Definition 3; that is, ℂ_{K}^{ECM} is the expansion of the complex field by all ECM’s.

ℂ_{K}^{ECM} is certainly an expansion of ℂ_{exp}. (See the remark following Definition 3.) Notice also that the notion of an ECM is, at least apparently, more general than if we had required the functions G_{1}, …, G_{n−l} of (2b) (of Definition 2) to lie in E_{n}. Perhaps the methods of [JKLS] could be used to produce an ECM which is not of this latter kind.

Theorem 2 implies

Theorem 3

If E has the WACP then the structure ℂ_{K}^{ECM} (and so, in particular, the structure ℂ_{exp}) is quasiminimal (even for the language L(ℂ_{K}^{ECM})_{∞,ω}).

□

I conclude this article with a proof that E has the WACP in the case N = 1 in the hope that others might find a way to generalize the method, which uses the (algebraic) valuation inequality.

So, suppose that n ≥ 1 and a, ω ∈ ℂ^{n} are such that λ := λ_{a,ω} is D̃-generic on [0, 1]. Suppose further that f ∈ E_{n+1}, r ∈ [0, 1] and that γ : [0, r) → ℂ is a continuous function such that

(12) f(λ(t), γ(t)) = 0 ≠ \frac{∂f}{∂z_{n+1}} (λ(t), γ(t))

for all t ∈ [0, r).

We must show that |γ(t)| ↛ ∞ as t → r.

In order to set up a use of the valuation inequality we require the following general

Lemma

Suppose that r > 0 and that ϕ : [0, r) → ℂ is a continuous function such that |ϕ(t)| → ∞ as t → r. Then either (A) for all integers k, l, not both 0 with k ≥ 0, we have either |ϕ(t)^{k} exp(lϕ(t))| → 0 as t → r or |ϕ(t)^{k} exp(lϕ(t))| → ∞ as t → r, or (B) for some integers k, l, not both 0 with k ≥ 0 we have that for all countable sets S ⊆ ℂ, there exists α ∈ ℂ \ S and an increasing sequence 0 ≤ t_{0} < t_{1} < … < t_{p} < … converging to r such that lim_{j→∞} ϕ(t_{j})^{k} exp(lϕ(t_{j})) = α. Proof

Set J := {⟨k, l⟩ ∈ ℤ^{2} : k ≥ 0 and k, l not both 0}.

For ⟨k, l⟩ ∈ J write h_{k,l}(t) for ϕ(t)^{k} exp(lϕ(t)) (for t ∈ [0, r)), and define c_{k,l}^{+} := lim sup_{t→r} |h_{k,l}(t)| and c_{k,l}^{−} := lim inf_{t→r} |h_{k,l}(t)|.

Then 0 ≤ c_{k,l}^{−} ≤ c_{k,l}^{+} ≤ ∞.

If for each ⟨k, l⟩ ∈ J we have either 0 = c_{k,l}^{−} = c_{k,l}^{+} or c_{k,l}^{−} = c_{k,l}^{+} = ∞ then clearly (A) holds.

Otherwise, choose ⟨k, l⟩ ∈ J with c_{k,l}^{+} > 0 and c_{k,l}^{−} < ∞. Let S be a countable subset of ℂ. Write h for h_{k,l}.

Now either c_{k,l}^{−} < c_{k,l}^{+} or 0 < c_{k,l}^{−} = c_{k,l}^{+} < ∞.

In the first case choose c ∈ ℝ with c_{k,l}^{−} < c < c_{k,l}^{+} and c ∉ {|s| : s ∈ S}. By the continuity of |h| there clearly exists a sequence 0 ≤ t′_{0} < t′_{1} < …

converging to r such that |h(t′_{j})| = c for all j ∈ ℕ. But now ⟨h(t′_{j}) : j ∈ ℕ⟩ is a bounded sequence of complex numbers and hence has a convergent subsequence ⟨h(t_{j}) : j ∈ ℕ⟩ whose limit, α say, cannot lie in S because |α| = c.

In the second case we have lim_{t→r} |h(t)| = c, say, with 0 < c < ∞. There is no harm in assuming that both h and ϕ are nonzero throughout [0, r) and so there exist continuous functions θ, ψ : [0, r) → ℝ such that

h(t) = |h(t)| exp(iθ(t)), and
ϕ(t) = |ϕ(t)| exp(iψ(t)),

for all t ∈ [0, r).

So by definition of h we have

(13) |h(t)| exp(iθ(t)) = |ϕ(t)|^{k} exp(ikψ(t))·exp(l|ϕ(t)|(cos ψ(t)+i sin ψ(t))).

Hence |h(t)| = |ϕ(t)|^{k} exp(l|ϕ(t)| cos(ψ(t)) → c as t → r.

We cannot have l = 0, for then k > 0 and ϕ would be bounded. So l ≠ 0 from which it follows that cos ψ(t) → 0 as t → r (since c ≠ 0, ∞). Thus ψ is bounded and sin ψ(t) → ±1 as t → r. Equating arguments in (13) we obtain, for some fixed N_{0} ∈ ℤ and for all t ∈ [0, r):

θ(t) = kψ(t) + l|ϕ(t)| sin ψ(t) + 2πN_{0}.

It follows from this that θ(t) → ±∞ (the sign here depending on the eventual sign of l sin ψ(t)) as t → r.

Thus we may choose some θ_{0} ∈ ℝ such that c exp(iθ_{0}) ∉ S and for which there exists a sequence 0 ≤ t_{0} < t_{1} < … converging to r such that θ(t_{j}) = θ_{0} (mod 2πℤ) for all j ∈ ℕ. It follows that h(t_{j}) → c exp(iθ_{0}) as j → ∞, and we are done. □.

Now returning to the discussion before the statement of the lemma, suppose, for a contradiction, that (12) holds and that |γ(t)| → ∞ as t → r. By definition of E_{n+1} we see that f has the form

(14) f(z_{1}, …, z_{n}, z_{n+1}) = ∑_{⟨i,j⟩∈L} P_{i,j}(z_{1}, …, z_{n})z_{n+1}^{i} exp(jz_{n+1})

for some non-empty finite set L ⊆ ℕ^{2}, where P_{i,j} ∈ E_{n}\{0} for each ⟨i, j⟩ ∈ L. We must have L ≠ {⟨0, 0⟩} by (12).

By the genericity of λ on [0, r] it routinely follows that for all P ∈ E_{n} \{0} there exists some R_{P} ≥ 1 such that

(15) R_{P} ≥ |P (λ(t))| ≥ R_{P}^{−1} for all t ∈ [0, r].

Let us pass to a non-principal ultrapower ^{⋆}ℂ of ℂ (with corresponding ⋆ℝ, ^{⋆}ℤ, ^{⋆}ℕ). Then the functions λ_{1}, …, λ_{n} (where λ = ⟨λ_{1}, …, λ_{n}⟩) and all the P_{i,j}’s have natural extensions to the ultrapower and (keeping the same notation for the extended functions) (15) remains true for all t ∈ ^{⋆}ℝ with 0 ≤ t ≤ r.

For each such t consider the subfield

F_{t} := E_{0}(λ_{1}(t), …, λ_{n}(t), exp(λ_{1}(t)), …, exp(λ_{n}(t)))

of ^{⋆}ℂ.

Then by (the extension to the ultrapower of) (15) it follows that F_{t} is actually a subfield of the valuation subring Fin(^{⋆}ℂ) (:= {z ∈^{⋆} ℂ : |z| ≤ R for some R ∈ ℝ}) of ^{⋆}ℂ.

By the continuity of each P ∘ λ (for P ∈ E_{n}) it follows that for all t_{1}, t_{2} ∈^{⋆} ℝ with 0 ≤ t_{1}, t_{2} ≤ r and satisfying t_{1} ≈ t_{2} (i.e. t_{1} infinitesimally close to t_{2}) we have that P (λ(t_{1})) ≈ P (λ(t_{2})) and so the correspondence t_{1} ↦ t_{2} induces an isomorphism I_{t_{1},t_{2}} : F_{t_{1}} → F_{t_{2}} with I_{t_{1},t_{2}}(z) ≈ z for all z ∈ F_{t_{1}} (and so, in particular, I_{t_{1},t_{2}}(z) = z for all z ∈ E_{0}).

Further, by the continuity of roots of polynomials (see [HM] or, perhaps more appropriately in the present context, [R]) the map I_{t_{1},t_{2}} extends to an isomorphism Ĩ_{t_{1},t_{2}} : F̃_{t_{1}} → F̃_{t_{2}} of the algebraic closures and we still have (for t_{1} ≈ t_{2}):

(16) (a) F̃_{t_{1}}, F̃_{t_{2}} ⊆ Fin(^{∗}ℂ) and (b) Ĩ_{t_{1},t_{2}}(z) ≈ z for all z ∈ F̃_{t_{1}}.

Now choose any t^{∗} ∈ ^{∗}ℝ with 0 < t^{∗} < r and t^{∗} ≈ r. Extend the function γ : [0, r) → ℂ to the ultrapower and set

(17) H_{t^{∗}} := F̃_{t^{∗}}(γ(t^{∗}), exp(γ(t^{∗}))).

Then by (12) and (14), H_{t^{∗}} is a subfield of ^{∗}ℂ of transcendence degree at most 1 over F̃_{t^{∗}} and, in fact, exactly 1 because γ(t^{∗}) ∉ Fin(^{∗}ℂ) (and F̃_{t^{∗}} ⊆ Fin(^{∗}ℂ)). It also follows from this (and the valuation inequality for the valuation on ^{∗}ℂ (restricted to H_{t^{∗}}) associated to the valuation subring Fin(^{∗}ℂ) of^{∗}ℂ) that γ(t^{∗}) and exp(γ(t^{∗})) have ℚ-dependent valuations which, by an easy saturation argument, implies that (A) of the Lemma (with ϕ = γ) cannot hold (back in the standard situation). So (B) holds and we choose integers k, l not both 0 with k ≥ 0 and take S to be the subset F̃_{r} of ℂ . Let α ∈ ℂ \ S and 0 ≤ t_{0} < t_{1} < … be a sequence converging to r such that lim_{j→∞} ϕ(t_{j})^{k} exp(lϕ(t_{j})) = α.

Let p ∈ ^{∗}ℕ \ ℕ and set t^{∗} := t_{p}. Then t^{∗} ≈ r, t^{∗} < r and α^{∗} := γ(t^{∗})^{k} exp(lγ(t^{∗})) ≈ α.

Now for no β ∈ F̃_{t^{∗}} do we have α^{∗} ≈ β; for this would imply Ĩ_{t^{∗},r}(β) ≈ β ≈ α^{∗} ≈ α ∈ ℂ \ F̃_{r}, which is absurd since Ĩ_{t^{∗},r}(F̃_{t^{∗}}) = F̃_{r}.

Since F̃_{t^{∗}} is an algebraically closed subfield of ^{∗}ℂ contained in Fin(^{∗}ℂ) this now implies that F̃_{t^{∗}}(α^{∗}) is also a subfield of ^{∗}ℂ contained in Fin(^{∗}ℂ) and is of transcendence degree 1 over F̃_{t^{∗}}. However, certainly F̃_{t^{∗}}(α^{∗}) ⊆ H_{t^{∗}}. Further, γ(t^{∗}) ∈ H_{t^{∗}} and γ(t^{∗}) , being infinite, is transcendental over F̃_{t^{∗}}(α^{∗}) which forces H_{t^{∗}} to have transcendence degree at least 2 over F̃_{t^{∗}}. This contradiction shows that it cannot be the case that |γ(t)| → ∞ as t → r and completes the proof of the WACP in this rather simple situation.

With a little more care one can show that the function γ above has an extension to a definable (in the structure ℝ_{K}^{RE}), holomorphic function with [0, 1] ⊆ dom(γ) which (therefore) satisfies (12) for all t ∈ dom(γ). It is then an easy matter to set up an inductive argument (using the full version of the valuation inequality for polynomially bounded o-minimal structures, see for example [Sp]) to establish the WACP for diagonal systems of exponential polynomial equations:

Proposition

E does have the diagonal WACP, i.e. where the map f = ⟨f_{1}, …, f_{N}⟩ in Definition 4 satisfies the extra condition that f_{j} ∈ E_{n+j} for j = 1, …, N. Further, the map γ has an extension to a definable (in ℝ_{K}^{RE}), holomorphic function with [0, 1] ⊆ dom(γ). □

But, unfortunately, this is a very special case.

Further remarks on quasiminimality

In his paper [B], Boxall shows that every formula (parameters allowed) of the language L(ℂ_{exp}) having the form ∃z̄(P (w̄, z̄) = 0), where P (w̄, z̄) is a term of this language (the w̄, z̄ being sequences of variables, not necessarily of the same length), is equivalent (in ℂ_{exp}) to a countable boolean combination of formulas of the form (∃z̄ ∈ ℚ^{m})ϕ(w̄, z̄) where ϕ is a quantifier-free formula of L(ℂ_{exp}) (containing no parameters other than those used in P). This immediately implies that sets of the form

π_{1}(Z(P)) := {w ∈ ℂ : ∃z̄ ∈ ℂ^{m}P (w, z̄) = 0}

are either countable or cocountable.

It is worth mentioning here that, even for the case m = 1, this is not a property of entire functions in general. For in the paper [A], Alexander complements earlier work of Tsuji ([T]) by giving a complete characterization of sets of the form π_{1}(Z(F)) for F : ℂ^{2} ↦ ℂ an entire function and this characterization implies that there do exist such F with both π_{1}(Z(F)) and ℂ \ π_{1}(Z(F)) uncountable. In particular, there is an expansion ⟨\overline{ℂ}, F ⟩ of the complex field \overline{ℂ} by an entire function F of two variables which is not quasiminimal. However, as pointed out by P. Koiran, it is still not known whether there exists a non-quasiminimal such expansion by an entire function of one variable, or even by finitely many entire functions of one variable. On the other hand, at least we do know (using a combination of ideas from [Ko],

[W2] and [Z]) that there exists a transcendental entire function f : ℂ ↦ ℂ such that the expansion ⟨\overline{ℂ}, f⟩ of \overline{ℂ} is quasiminimal.

References

[A] H. Alexander, ’On a problem of Julia’, Duke Mathematical Journal, Vol. 42, No. 2, (1975), 327-332.

[B] Ricardo Bianconi, ’Undefinability results in o-minimal expansions of the real numbers’, Annals of Pure and Applied Logic, Vol. 134, Issue 1, June 2005, 43-51.

[Bo] Gareth Boxall, ’A special case of quasiminimality for the complex exponential field’, The Quarterly Journal of Mathematics, 71(3), (2020), 1065-1068.

[BKW] Martin Bays, Jonathan Kirby, A.J. Wilkie, ’A Schanuel property for exponentially transcendental powers’, Bull. Lond. Math. Soc. 42 (2010), no.5, 917-922.

[BN] Gal Binyamini and Dmitry Novikov, ’Wilkie’s conjecture for restricted elementary functions’, Annals of Mathematics Vol. 186 (2017), Issue 1, 237-275.

[DD] J. Denef and L. van den Dries, p-adic and real subanalytic sets, Annals of Mathematics 128 (1988), 79-138.

[HM] Gary Harris and Clyde Martin, ’The roots of a polynomial vary continuously as a function of the coefficients’, Proc. Amer. Math. Soc. Vol 100, Number 2, (1987), 390-392.

[JKLS] Gareth Jones, Jonathan Kirby, Olivier Le Gal and Tamara Servi, ’On local definability of holomorphic functions’, Quarterly Journal of Mathematics, 70(4), (2019), 1305-1326.

[K] Carol R. Karp, ’Languages with expressions of infinite length’, North-Holland Publishing Company, Amsterdam, (1964).

[Ko] Pascal Koiran, ’The theory of Liouville functions’, The Journal of Symbolic Logic, 68(2), (2003), 353-365.

[MW] Angus Macintyre and A. J. Wilkie, ’On the decidability of the real exponential field’, in ’Kreiseliana: about and around Georg Kreisel’, edited by Piergiorgio Odifreddie, A. K. Peters (1996),441-467.

[PS1] Ya’acov Peterzil and Sergei Starchenko, ’Expansions of algebraically closed fields in o-minimal structures’, Selecta Mathematica, New Series 7 (2001), 409-445.

[PS2] Ya’acov Peterzil and Sergei Starchenko, ’Expansions of algebraically closed fields II: functions of several variables’, J. Math. Logic, Volume 3, Number 1 (2003), 1-35.

[PW] J. Pila and A. J. Wilkie, ’The rational points of a definable set’, Duke Math. J. 133 (2006), no. 3, 591-616.

[R] David A. Ross, ’Yet another proof that the roots of a polynomial vary continuously as a function of the coefficients’, arXiv:2207.00123v2. (2022).

[S] H. Sfouli, ’On the elementary theory of restricted real and imaginary parts of holomorphic functions’, Notre Dame J. of Formal Logic, 53(1), (2012), 67-77.

[Sp] P. Speissegger,’Lectures on o-minimality’, in ’Lectures on Algebraic Model Theory’, edited by Bradd Hart and Matthew Valeriote, Fields Institute Monographs, AMS (2002), 47-65.

[T] M. Tsuji, ’Theory of meromorphic functions in a neighborhood of a closed set of capacity zero’, Japanese J. Math. 19, (1944), 139-154.

[W1] A. J. Wilkie, ’Diophantine properties of sets definable in o-minimal structures’, J. Symb. Logic, Vol 69, No. 3, (2004), 851-861.

[W2] A. J. Wilkie, ’Liouville functions’, in ’Logic Colloquium 2000: proceedings of the Annual European Summer Meeting of the Association of Symbolic Logic’, edited by René Cori, Alexander Razborov, Stevo Todorc̆ević and Carol Wood, Lecture Notes in Logic 19, (2005), 383-391.

[W3] A. J. Wilkie, ’Some local definability theory for holomorphic functions’, in ’Model Theory with Applications to Algebra and Analysis (Volume 1)’, Volume 349 of LMS Lecture Note Series, CUP, (2008), 197-214.

[Z] Boris Zilber, ’Analytic and pseudo-analytic structures’, in ’Logic Colloquium 2000: proceedings of the Annual European Summer Meeting of the Association of Symbolic Logic’, edited by René Cori, Alexander Razborov, Stevo Todorc̆ević and Carol Wood, Lecture Notes in Logic 19, (2005), 392-408.