STRONGLY MINIMAL EXPANSIONS OF (ℂ, +) DEFINABLE IN O-MINIMAL FIELDS

ASSAF HASSON^{∗} AND PIOTR KOWALSKI^{♣,†}

Abstract. We characterize those functions f : ℂ → ℂ definable in o-minimal expansions of the reals for which the structure (ℂ, +, f) is strongly minimal – such functions must be complex constructible, possibly after conjugating by a real matrix. In particular we prove a special case of the Zilber Dichotomy – an algebraically closed field is definable in certain strongly minimal structures which are definable in an o-minimal field.

1. Introduction

This paper is concerned with the question of recognizing algebraicity in model theoretic terms. In its broadest context, the problem amounts to recognizing when a geometry given, say, in real analytic terms is in fact an algebraic geometry over the complex numbers. Such is, for example, Riemann’s theorem, identifying a compact Riemann surface with an algebraic curve. Within model theory, this line of research is motivated by a conjecture of Zilber in [29] suggesting that certain combinatorial geometries arising naturally in model theory must be either of linear type (locally modular) or coincide with the Zariski geometry coming from an (irreducible) algebraic curve over an algebraically closed field. Although in general false ([7]) Zilber’s conjecture has been proved under additional assumptions in [10] and played a crucial role in several applications outside model theory, the first and arguably the best known of which is [8].

Aside from [10] there are several other instances under which Zilber’s conjecture has been proved (usually, but not always, by reducing the problem to one close enough to the context treated in [10]). In [9] Hrushovski and Sokolovic prove the conjecture for differentially closed fields of characteristic 0; in [8] the same is done for separably closed fields, and in [4] for difference fields. In [25] Zilber’s conjecture has been proved for those combinatorial geometries interpretable in algebraically closed fields (under the additional assumption that the interpretation is rank preserving).

These results suggest that Zilber’s conjecture tends to be true in contexts of geometrical flavor. It seems therefore natural to look for a proof of Zilber’s conjecture for o-minimal structures. This question has been formulated by Y. Peterzil as:

Problem 1.1. Let D be a strongly minimal non-locally modular structure interpretable in an o-minimal structure. Does D interpret an algebraically closed field?

The combinatorial geometries referred to in Zilber’s conjecture are precisely those associated with strongly minimal structures. In [20] Peterzil and Starchenko prove a complementary result. They show that if R is an o-minimal expansion of a real closed field and K = R(\sqrt{−1}) then K has no proper expansions which are strongly minimal (or stable). In the present paper we make a first step toward proving a generalization of Peterzil and Starchenko’s result and give a positive answer to problem 1.1 in a special case. Our main result is:

Mathematics Subject Classification: 03C64, 03C45, 14P25.

∗Supported by the EPSRC grant no. EP C52800X 1.

♣Supported by a MODNET (European Commission Research Training Network) grant.

†Supported by the Polish grants: KBN no. 2P03A 018 24, MEN no. N201 032 32/2231.

Theorem 1.2. Let f : ℂ → ℂ be definable in an o-minimal expansion of the Reals and such that ℂ_{f} := (ℂ; +, f) is strongly minimal and non-locally modular. Then there is M ∈ GL_{2}(ℝ) such that ℂ_{MfM^{−1}} is interdefinable with the complex field. In particular, ℂ_{f} is biinterpretable with the complex field.

Note that the assumption of the theorem above are a special case of the assumptions in 1.1, but our conclusion is sharper than the one in 1.1. Not only do we find a ℂ_{f}-definable field, but we also show that the entire structure of ℂ_{f} comes from that field. One can not expect such a strengthening of the conclusion of 1.1 in general, see the Remark on page 9 of [30].

Proving Zilber’s conjecture for o-minimal expansions of the additive group of the complex field reduces by standard model theoretic arguments to proving it for (ℂ; +, X) for arbitrary strongly minimal X ⊆ ℂ^{2} definable in an o-minimal expansion of the Reals such that (ℂ; +, X) is strongly minimal. Hence, our theorem deals with the special case where X is the graph of a function. Some additional comments regarding the assumptions of the theorem are in place. First, note that if f is a conjugate of a complex rational function by a real matrix then ℂ_{f} is a strongly minimal structure and by [14] is interdefinable with the complex field. However, examples due to Hrushovski (essentially along the lines of [7]) show that, in general, if f : ℂ → ℂ is a set theoretic function such that ℂ_{f} is strongly minimal no field need be interpretable in the structure even if it is non-locally modular. In view of the above discussion the o-minimality assumption is quite natural. It is also clear that o-minimality is a natural assumption in the (less natural) case of an arbitrary real closed field, since the natural topology in such a field is usually not good enough for any arguments.

Generally speaking, our proof of Theorem 1.2 uses mild consequences of ominimality, which suggests that considerable parts of the proof may be generalized to weaker contexts. One possible such context is weak o-minimality. Another one (perhaps more natural and challenging) would be to assume only that f is continuous (or C^{1}). For example, if we assume f to be holomorphic, we get an immediate proof, since strong minimality forbids any essential singularities (infinity included), hence f is in fact rational.

The idea for our proof is based on a theorem of Marker and Pillay in [13] where they prove - as a special case - the same result as that of our theorem under the assumption that f is definable in the complex field. A key feature in their proof is that f is meromorphic, hence

(∗)  f′(c) = 0 ⟺ f is not a local homeomorphism at c.

The main part of the paper is aimed at proving (∗) in our setting. This turned out to be a non-trivial task. While the assumption of o-minimality assures that f is quite close to being C^{1} (as a function from ℝ^{2} to itself), in general, f could be far from analytic at any point, let alone at special points such as the ones in (∗).

The paper is structured as follows. In Section 2 we give the (rather soft) model theoretic background we need and gather a few easy, though useful, facts concerning our function f. Section 4 is dedicated to the study of the topological properties of our function. We show that (up to finitely many corrections) f is a continuous, open ramified covering of the Riemann sphere. In order to prove these properties of f we use ideas of Peterzil and Starchenko (§3 in [20]) to show that strongly minimal subsets of ℂ^{2} are closed (up to possibly finitely many corrections), which is the main result of Section 3. This readily gives us the continuity of f, its properness and its openness. In the second part of Section 4 we use ideas from topological analysis to conclude that f is a ramified covering. We then show that f admits a topological degree and the remainder of the section is dedicated to the investigation of the local behavior of this degree in definable families of definable functions. In Section 5 we extend yet further our use of topological analysis to obtain a weak version of the Cauchy-Riemann equation for f, namely the right-to-left implication in (∗): if f is not a local homeomorphism at c and c is a smooth point of f then f′(c) = 0. Finding such a point c turned out, unexpectedly, to be a major difficulty.

In Section 6 we prove a local version of the left-to-right implication in (∗). The proof makes extensive use of the results of Section 5, and goes through local Lie group theory and the classification of maximal solvable Lie subgroups of GL_{2}(ℝ). This allows us to return to the line of proof of [13] in order to produce in Section 7 a certain combinatorial configuration, which by a well known variant of the Hrushovski-Weil group configuration theorem assures the existence of an interpretable field in ℂ_{f}. Sections 8 and 9 are dedicated to generalizing the proof to o-minimal expansions of other real closed fields.

Acknowledgment: We would like to thank Y. Peterzil for suggesting the problem to us, and for his helpful remarks and ideas; G. Cherlin for directing us to some useful references; C. Mora-Corral for saving us from trying to re-invent degree theory and directing us to the right books; K. Dymara and J. Dymara for supplying us with examples which kept us off wrong tracks; and the Oxford Logic Group for many tea time discussions.

2. Model theoretic background and preliminaries

In this section we give the formal definitions required for an understanding of Problem 1.1 and the statement of Theorem 1.2. We also gather some basic facts concerning strongly minimal and o-minimal structures which we use throughout the paper, and prove some immediate implications to our function f.

The key to all the definitions that follow are the notions of structure and definability. Roughly, a structure is a non empty set M with a collection M of distinguished functions and relations (possibly on powers of M) and a set S ⊆ M^{k} is definable if it can be obtained from M by applying logical operations. More formally:

Definition 2.1. Let M be an non-empty set. A structure S on M is a sequence {S_{n} : n ⩾ 0} of subsets of P(M^{n}) satisfying:

(1) Each S_{n} is a boolean algebra of sets (i.e. closed under finite unions intersections and under taking complements).

(2) S is closed under cartesian powers (i.e. If A ∈ S_{n} and B ∈ S_{m} then A × B ∈ S_{n+m}).

(3) For all n the diagonal Δ_{i,j}^{n} ⊆ M^{n} given by x_{i} = x_{j} is a set in S_{n}.

(4) S is closed under projections (i.e. if A ∈ S_{n+1} and π : S_{n+1} → S_{n} is the projection map on the first n coordinates then π(A) ∈ S_{n}).

A set A ⊆ M^{n} will be called ∅−definable if A ∈ S_{n}. A is definable if it is a fiber of some ∅−definable set. A function f : M^{n} → M^{k} is definable if its graph is.

For basic properties of structures we refer the reader to [26]. By the Tarski-Seidenberg theorem the field of real numbers equipped with the semi-algebraic sets is a structure which we will refer to as the Reals. Similarly, by a theorem of Chevalley’s the projection of an algebraically constructible set is constructible, whence any algebraically closed field K equipped with the K−constructible sets is a structure. If K = ℂ we will call the resulting structure the complex field. To simplify the notation we will usually refer to “a structure M” without denoting explicitly the pair (M; S). With this notation, for a structure M the underlying set will be denoted M. We will have two exceptions, ℝ and ℂ will denote the (structures of) the real and complex fields, as well as their underlying sets. To get a better feeling of (non-)definability, it is worth remarking that the S_{n} appearing in the definition, are usually not σ-algebras. For example, the integers are not definable in neither the Reals nor the complex field.

Given a non-empty set M and some S ⊆ ⋃_{n⩾0} P(M^{n}) we denote (M; S) the smallest structure in which every S ∈ S is definable. Note that if S is countable then the class of definable subsets in (M; S) is countable. Identifying functions with their graphs, it is clear by what we have just said that, e.g., ℝ = (ℝ; +, ·).

To simplify the exposition in this section, it will be very useful for us to assume that we consider only such structures (M; S) for which S is countable and M uncountable. Note this assumption is not harmful at all for our problem – the structure ℂ_{f} clearly satisfies it and we can always replace the ambient structure on the Reals by (ℝ, +, ·, f).

Whereas the notion of definability is essential to the understanding of the proofs in this papers, the remaining model theoretic concepts are - on the technical level - of less importance. Readers who feel uncomfortable with this sort of concepts can skip the next set of definitions and be content with Facts 2.8 and 2.9 which sum up all the consequences of the model theoretic assumptions which will be used in this paper (at least up to Section 7).

The following definition will not be of great importance in this paper, but since it appears in the statement of Problem 1.1 we give it nonetheless:

Definition 2.2. The structure (N, R) is definable in the structure (M, S) if there exist an injection f : R → S which preserves projections and boolean operations. (M, S) is a an expansion of (N, R) if N = M and S ⊇ R.

We will mostly be interested in structures (M, S) for which a structure is interpretable in (M, S) if and only if it is definable there. To avoid technicalities we omit the definition of the former. Section 7 is the only place in the text where a short excursion into the interpretable, rather than the definable, realm seems unavoidable. Roughly, a structure (N, R) is interpretable in the structure (M, S) if it is definable in a (canonical) expansion of (M, S) in which every class of every ∅-definable equivalence relation is given a name. We refer the reader to 1.1 of [22] or 4.3 of [6] for more details.

Note that if (ℝ, S) is any structure expanding the Reals then the complex numbers are naturally definable in (ℝ, S). In fact, as in this example, in the present paper a structure (N, R) will be definable in (M, S) if and only if N ∈ S and R ⊆ S.

We are now able to introduce the two main model theoretic notions which will be used in this paper:

Definition 2.3. Let M be a structure:

(1) M is minimal if every definable subset of M is either finite or co-finite.

(2) M is strongly minimal if it is minimal and every definable family of finite subsets of M is uniformly bounded, i.e. for all definable X ⊂ M^{n+1} such that for all ā ∈ M^{n}

Xā := {b ∈ M : (b, ā) ∈ X}.

is finite, there exists N ∈ ℕ such that |Xā| < N for all ā ∈ M^{n}.

(3) If < is a dense linear definable ordering on M, we say that (M, <) is ominimal if the definable subsets of M are precisely those definable using the ordering alone (i.e. S_{1} is precisely the boolean algebra generated by intervals and points in M).

Let M be any structure and A ⊆ M any set. Define the algebraic closure of A to be the union of all finite sets definable with parameters from A. Formally,

acl(A) := {a ∈ M : (∃b̄ ⊆ A, S ∈ S_{1+|b̄|})((a, b̄) ∈ S ∧ |S(x, b̄)| < ∞}

Definition 2.4. Let M be a structure and A ⊆ M. a ∈ M is algebraic over A if a ∈ acl(A).

Note that if, e.g. M is the Reals or the complex field then a is algebraic over A in the above sense if and only if it is algebraic over A in the usual sense. In general if M expands the Reals, say, then the model theoretic algebraic closure contains the standard algebraic closure. In this paper we will only be interested in the former notion, so no confusion can arise.

Since, when M is either a strongly minimal or an o-minimal structure, acl is a closure operator (in the sense of Steinitz) we can define for any A ⊆ M:

rk A := min{|a| : a ⊆ A ∧ acl(a) ∩ A = A}

which gives a notion of independence in M in the following way:

Definition 2.5. Let (A, Cl) be a combinatorial pre-geometry and rk the associated rank.

(1) If B, C ⊆ A are finite we write rk(C/B) = rk(BC) − rk(B). For arbitrary B we define rk(C/B) = min{rk(C/B′) | B′ ⊆ B finite}.

(2) B, C ⊆ A are independent over D ⊆ A if rk(C/BD) = rk(C/D).

(3) (A, Cl) is locally modular, if there is a finite set F ⊂ A such that for each pair of finite sets B, C ⊂ A, Cl(B ∪ F) is independent from Cl(C ∪ F) over Cl(B ∪ F) ∩ Cl(C ∪ F).

Definition 2.6. We assume now that a combinatorial geometry comes from a strongly minimal structure or an o-minimal structure.

(1) A strongly minimal structure is locally modular if its pre-geometry is.

(2) If X is a D-definable set we define

rk(X) = max{rk(d/D) | d ∈ X}.

(3) For a D-definable set X and D ⊆ D′ an element d̄ ∈ X is D′-generic if rk(d̄/D) = rk(X).

(4) For definable sets X, Y we denote X ∼ Y if rk(X △ Y) < rk(X ∪ Y).

(5) If P is any property (not necessarily definable) a definable set X almost satisfies property P if there exists a definable Y such that X ∼ Y and Y satisfies property P.

(6) If M is strongly minimal rk is known as Morley rank and is denoted RM.

(7) If M is o-minimal rk is known as o-minimal dimension and is denoted dim.

(8) A strongly minimal set in a strongly minimal structure M is a definable set X with RM(X) = 1 such that for every definable Y ⊆ X either Y is finite or X ∼ Y.

Remark 2.7. We collect some observations regarding the above definitions.

(1) Throughout this paper our usage of the term “almost” will be reserved exclusively to the one given in the above definition. Thus, an almost continuous (definable) function is one all of whose discontinuities are removable (and not, e.g., one which is continuous outside a set of measure 0).

(2) For a finite set X, a possible confusion may arise – when we write e.g. dim(X) it could mean either its dimension as a definable set (which is always 0) or its rank as a tuple (which is usually not 0). To prevent this confusion, from now on we distinguish between finite tuples, denoted by lowercase letters a, b, c, … and definable (possibly finite) sets denoted by uppercase letters A, B, …, X, Y, . . ..

(3) Usually, the Morley rank and o-minimal dimension for definable sets are defined otherwise, using properties of definable sets rather then the combinatorial geometry on the universe (see, e.g. [3] and [12], [26] respectively). Then it is shown (see same references) that in particular under our assumptions (uncountable structure, countably many ∅-definable sets) our definition coincides with the standard one. Our countability assumption is also used for 2.8(5), 2.8(6) and 2.9(2) below.

(4) It may be worth mentioning that locally modular strongly minimal structures (in any context) are well characterized as, essentially, covers of linear spaces. Hence the assumption of non-local modularity is necessary in Problem 1.1.

We will now collect those facts concerning o-minimal structures which we will use. For a more detailed discussion (and for the proofs) we refer to [12] and [26].

Fact 2.8. Let M be an o-minimal expansion of a real closed field, f : M → M a (partial) definable function then:

(1) M can be partitioned into finitely many open intervals I_{0}, …, I_{k} and finitely many points a_{1}, …, a_{k} such that for each 0 ≤ i ≤ k one of the following holds:

(a) f is not defined on I_{i}.

(b) f is constant on I_{i}.

(c) f is monotone on I_{i}.

(2) The limit lim_{t→1^{−}}f(t) exists in M ∪ {±∞}.

(3) For every n, m, k ∈ ℕ, any definable g : M^{m} → M^{k}, there is X ⊆ M^{m} such that g is C^{n} on X and dim(M^{m} \ X) ⩽ m − 1.

(4) If X ⊆ Y × Z are all definable and Y_{X} is the projection of X onto Y, then there exists a definable function f : Y_{X} → Z (called Skolem function) such that for all y ∈ Y_{X}, we have (y, f(y)) ∈ X.

(5) If X is a definable set of dimension n, then for any definable Y ⊆ X, dim(Y) = n if and only if Y contains a relatively open subset of X if and only if Y contains a generic point of X. In particular if M is the field of real numbers dim and the topological dimension coincide.

(6) If dim(a/A) = n, then there is an A-definable, n-dimensional set X such that a ∈ X. If a ∈ Y, where Y is A-definable, then we can find such X with X ⊆ Y.

The properties of strongly minimal sets we will be using are considerably simpler and will consist mainly in:

Fact 2.9. Let X be definable in a strongly minimal structure M and assume that X (with induced structure) is strongly minimal.

(1) Let Y be a strongly minimal set in M such that X ∩ Y is infinite, then X ∼ Y.

(2) If a_{1}, a_{2} ∈ X are such that RM(a_{1}/A) = RM(a_{2}/A) = 1 for some parameter set A then a_{1} ∈ Y ⟺ a_{2} ∈ Y for every Y ⊆ X definable over A.

(3) For any definable X ⊆ M^{n+m}, a ∈ M^{n}, let X_{a} := {b ∈ 𝕄^{m} | (a, b) ∈ X}. Then the set

{a ∈ M^{n} | X_{a} is finite (resp. infinite)}

is definable.

From now till Section 8 we fix R, an o-minimal expansion of the Reals. We identify ℂ with ℝ^{2} and fix a function f : ℂ → ℂ definable in R such that the reduct ℂ_{f} := (ℂ; +, f) is strongly minimal and non-locally modular. For simplicity of notation we will assume that f is ∅−definable in R. Note that dim(c) ≤ 2 RM(c) for every c ∈ ℂ^{n} and that if X is a ℂ_{f}−definable set then dim(X) = 2 RM(X).

We will fix now some (pretty standard) notation.

Notation 2.10. Assume X is a topological space, Y ⊆ X and g : ℝ^{n} → ℝ^{n}. (1) We will write g′(c) for the Jacobian matrix of g at c (if it exists).

(2) cl(Y) is the topological closure of Y in X, Int(Y) is its interior.

(3) ∂(Y) := cl(Y) \ Int(Y) is the border of Y, fr(Y) := cl(Y) \ Y its frontier.

(4) For x ∈ ℝ^{n}, ∥x∥ denotes the norm of x and for r > 0, B_{r}(x) denotes the open ball centered at x and of radius r.

(5) For a ∈ ℝ^{n} and R-definable X, Y ⊂ ℝ^{n} we say that X is transversal to Y at a, denoted X ⋔_{a} Y, if a ∈ X ∩ Y and there is an open U ∋ a such that X ∩ U, Y ∩ U are C^{1}-submanifolds of ℝ^{n} which are transversal at a.

(6) For a field K, 𝔾_{m}(K) is the multiplicative group of K and 𝔾_{a}(K) is the additive group of K

We finish this section with some easy consequences of strong minimality.

Fact 2.11. f is not almost ℝ-affine.

Proof. If f is ℝ-affine then the structure ℂ_{f} is definable in the structure ℝ_{vect}, whose universe is ℝ and the definable subsets of ℝ^{n} are ℝ-linear subspaces. Then ℝ_{vect} is locally modular (see 2.7(4)). But then ℂ_{f} is also locally modular, as is any strongly minimal structure definable in a locally modular one, see 6.3 on p. 182 of [22]. □

By strong minimality we know that f has finite fibers (otherwise it would be almost constant) and the size of fibers is uniformly bounded. Indeed, if a ∈ ℂ is generic then d := |f^{−1}(a)| = |f^{−1}(a′)| for all but finitely many a′ ∈ ℂ. We call d the degree of f. We will show in Section 4 that f admits a topological degree, and that the two notions agree up to sign (a posteriori, having proved the main theorem, we know that they agree unconditionally). The next fact follows easily from Sard’s Lemma and the assumption that f is finiteto-one. We still give another proof, because it is also valid in the more general context considered in Section 8. We are not aware whether it is even possible to state Sard’s Lemma in that context, since there is no good notion of a measure zero set in the case of an arbitrary real closed field. If we replace “measure zero” with “smaller dimension”, then Sard’s Lemma (provided the function is finite-to-one) would look like the statement below.

Fact 2.12. Assume U ⊆ ℂ is open and f is C^{1} on U. Let X denotes the set of critical points of f (i.e. those c ∈ ℂ such that det(f′(c)) = 0) on U. Then dim(X) ⩽ 1.

Proof. If not, we get by 2.8(5) an open V ⊆ U such that for each c ∈ V, f is critical at c. But then, dim(f(V)) ⩽ 1, hence f has an infinite fiber. Therefore, f is almost constant and ℂ_{f} is locally modular, a contradiction. □

One of the most useful characterizations of non-locally modular strongly minimal structures is the existence of a large definable family of essentially distinct plane curves. In the next lemma we show that in ℂ_{f} we have a natural choice of such a family:

Lemma 2.13. For each a ∈ ℂ \ {0} and each open U ⊆ ℂ, f(x + a) − f(x) is not constant on U.

Proof. Assume there existed b ∈ ℂ such that f(x + a) − f(x) = b on U. Strong minimality would imply f(x + a) − f(x) = b for almost all x ∈ ℂ. So for almost all x we have

f(x + 2a) − f(x) = f(x + a + a) − f(x + a) + f(x + a) − f(x) = 2b.

Inductively, for each n ∈ ℕ, f(x + na) − f(x) = nb for almost all x ∈ ℂ. So by 2.9(3), there is a cofinite set X ⊆ ℂ such that for all c ∈ X there exists l(c) such that f(x + c) − f(x) = l(c) for almost all x ∈ ℂ. Claim l is an almost ℝ-linear ℂ_{f}-definable function.

Proof of Claim. By the uniformity part in the definition of strong minimality, l is a ℂ_{f}-definable function from X to ℂ. Take any c ∈ X and d ∈ ℂ such that RM(d/c) = 1. Then RM(d + c/c) = RM(d/c) = 1, so d + c, d ∈ X and we have for almost all x ∈ ℂ:

l(d+c) = f(x+d+c)−f(x) = f(x+d+c)−f(x+d)+f(x+d)−f(x) = l(c)+l(d).

Therefore, l is “generically-additive”, but we still need to find an actual additive function coinciding almost everywhere with l. Let L denote the graph of l and consider the following set:

S := {(v, w) ∈ ℂ^{2} | ((v, w) + L) ∼ L}.

By 2.9(3), S is ℂ_{f}-definable and it is clearly a subgroup of (ℂ, +) × (ℂ, +). Let us take (v, w) ∈ S such that v ∈ X and a ∈ ℂ generic over {v, w}. Then (v, w) + (a, l(a)) ∈ L and v + a ∈ X. Therefore,

l(v) + l(a) = l(v + a) = l(a) + w,

so (v, w) ∈ L. Therefore, S is the graph of an additive function l̄ which coincides with l almost everywhere. By 2.8(3), there is an R-definable, open U ⊆ ℂ such that dim(ℂ \ U) ⩽ 1 and l̄ is continuous on U. For any a ∈ ℂ, if we take b ∈ U such that dim(b/a) = 2, then dim(b + a/a) = 2, so b + a ∈ U by 2.8(5). Hence U + U = ℂ. Now, it is an easy exercise (left to the reader) to show that an additive function on ℂ which is continuous on an open set U ⊆ ℂ satisfying U + U = ℂ is ℝ-linear. □

By the Claim we may assume that l is ℝ-linear. For a generic a ∈ ℂ and for almost all x ∈ ℂ, we have:

f(a + x) − l(x) = f(a).

Therefore, f is almost ℝ-affine, contradicting 2.11. □

Let l_{a,c} denote the graph of f(x + a) + c. We get 5 corollaries (the first one is immediate) which will be of use:

Corollary 2.14. F := {l_{a,c} : a, c ∈ ℂ} is a normalized 2-dimensional family of plane curves, i.e. (a, c) ≠ (a′, c′) implies l_{a,c} ≁ l_{a′,c′}.

Corollary 2.15. Let a ∈ ℂ\{0} and f_{a}(x) = f(a+x)−f(x). Then the conclusion of 2.12 holds for f_{a}.

Proof. We can proceed as in the proof of 2.12 using 2.13. □

Corollary 2.16. Assume S ⊂ ℂ^{2} is strongly minimal, definable over ∅ and a, c ∈ ℂ such that RM(a, c) > 0. Then S ≁ l_{a,c}.

Proof. Assume not and take

X := {(x, y) ∈ ℂ^{2} | S ∼ l_{x,y}}.

By 2.9(3), X is ℂ_{f}-definable over ∅. Since (a, c) ∈ X and RM(a, c) > 0, X is infinite. In particular, there are (a, c) ≠ (a′, c′) with l_{a,c} ∼ S, l_{a′,c′} ∼ S. Hence l_{a,c} ∼ l_{a′,c′} contradicting 2.14. □

Corollary 2.17. For any open U ⊆ ℂ, dim(f′(U)) ∈ {1, 2}.

Proof. Since f′ is R-definable and dim(U) = 2, dim(f′(U)) ⩽ 2. If f′(U) is finite, then without loss f′ is constant on U. For simplicity, we may assume that 0 ∈ U, so choose any c ∈ U and consider the function f_{c}(x) := f(x) − f(x + c). We get an open V ⊆ U such that f_{c}|_{V} is constant, which contradicts 2.13. □

Corollary 2.18. There is no open U ⊆ ℂ, (α, β) ∈ ℝ^{2} \ {0} and i ∈ {1, 2} such that α\frac{∂f_{i}}{∂x} + β \frac{∂f_{i}}{∂y} is defined and constantly 0 on U.

Proof. Assume not. Let f = (f_{1}, f_{2}) : ℝ^{2} → ℝ^{2}. Without loss there is α ∈ ℝ such that

\frac{∂f_{1}}{∂x} (a) = α\frac{∂f_{1}}{∂y} (a), ∀a ∈ U.

Take an open U′ ⊂ U and ε > 0 such that

U′ + B_{2ε}(0) ⊆ U.

For any (v, w) ∈ U′ consider the function

h_{v,w} : (−2ε, 2ε) → ℝ, h_{v,w}(γ) = f_{1}(v + γ, w − αγ).

By the Chain Rule, h′_{v,w} is constantly 0, so h_{v,w} is constant. Consider now the following ℂ_{f}-definable function

f_{α,ε}(x, y) := f(x + ε, y − αε) − f(x, y).

and let g : ℂ → ℝ be its first coordinate function. Take (v, w) ∈ U′. Since h_{v,w} is constant, we have

g(v, w) = h_{v,w}(ϵ) − h_{v,w}(0) = 0.

Therefore, g is constant on U′, hence

dim(f_{α,ε}(U′)) ⩽ 1.

Therefore, f_{α,ε} has an infinite fiber, so it is almost constant contradicting 2.13. □

3. Transversality

In this section we prove that strongly minimal subsets of ℂ^{2} are almost closed. The idea of the proof is taken from [20] where for a strongly minimal X ⊂ ℂ^{2} the finiteness of its frontier is proved through an investigation of the possible ways X can meet complex lines. In the present context we do not have enough ℂ_{f}-definable complex lines, so we replace them by the family of plane curves, F, introduced in the previous section. The main difference with [20] is the fact that our curves need not be smooth. The basic difficulty in the situation here is showing that any fixed strongly minimal X ⊂ ℂ^{2} can moved in a controlled way outside the set of “bad points” (i.e. points where the proof from [20] does not work).

Let us fix b = (b_{1}, b_{2}) ∈ ℂ^{2} and consider the family {f_{a}^{b} : ℂ → ℂ}_{a∈ℂ}:

f_{a}^{b}(x) := f(a + x) − f(a + b_{1}) + b_{2}.

By 2.14, we know that for each a ≠ a′ we have f_{a}^{b} ≁ f_{a′}^{b} and f_{a}^{b}(b_{1}) = b_{2}. We call any graph(f_{a}^{b}) the f-line through b parameterized by a and denoted it l_{a}^{b}. We will usually denote the graph of a function g by Γ_{g}.

Consider a b-definable (in ℂ_{f}) relation between ℂ^{2} and ℂ:

Φ_{b}(s; a) ⇔ s ∈ l_{a}^{b}.

We say that a point s ∈ ℂ^{2} is b-good if the following holds:

(1) Φ_{b}(s; x)^{ℂ} ≠ ∅,

(2) For each a ∈ ℂ such that Φ_{b}(s; a), there is a neighborhood U ∋ (s, a) such that Φ_{b}^{ℂ^{3}} ∩ U is the graph of a C^{1}-function which is a submersion at s.

In other words, if s is a b-good point, we have neighborhoods V ∋ s, W ∋ a and a C^{1} function ϕ_{b} : V → ℂ being a submersion at s such that for any s′ ∈ V, a′ ∈ W, we have ϕ_{b}(s′) = a′ if and only if s′ ∈ l_{a′}^{b}. We say that a point is b-bad if it is not b-good. Let B^{b} denote the set of all b-bad points.

Lemma 3.1. For each b, b′ ∈ ℂ^{2}, dim(B^{b}) ⩽ 3 and B^{b+b′} = B^{b} + b′.

Proof. Fix s_{1} ∈ ℂ \ {b_{1}} and consider the function:

h_{s_{1}} : ℂ → ℂ, h_{s_{1}}(x) = f(s_{1} + x) − f(b_{1} + x) + b_{2}.

By 2.15, there is a closed set B_{s_{1}} ⊂ ℂ of dimension at most 1 such that outside B_{s_{1}} the function h_{s_{1}} is C^{1} and regular. Let

A_{s_{1}} := ℂ \ h_{s_{1}}^{−1}(h_{s_{1}}(B_{s_{1}})).

By 2.14, h_{s_{1}} is finite-to-one, hence codim(A_{s_{1}}) ⩽ 1. It easy to see that for any s_{2}, a ∈ ℂ, we have:

(s_{1}, s_{2}) ∈ l_{a}^{b} ⇔ s_{2} = h_{s_{1}}(a).

For any a ∈ ℂ \ A_{s_{1}}, a′ ∈ ℂ such that h_{s_{1}}(a) = h_{s_{1}}(a′), we have a′ ∉ B_{s_{1}}, so h_{s_{1}} is C^{1} at a′ and det(h′_{s_{1}}(a′)) ≠ 0. Let H(s_{1}, s_{2}, a) = h_{s_{1}}(a) − s_{2}. Therefore for each a ∈ A_{s_{1}}, a′ ∈ ℂ such that h_{s_{1}}(a) = h_{s_{1}}(a′), we have:

\frac{∂}{∂a}H(s_{1}, s_{2}, a) ≠ 0, \frac{∂}{∂s_{2}} H(s_{1}, s_{2}, a)) ≠ 0.

By the Implicit Function Theorem (s_{1}, h_{s_{1}}(a)) is b-good. Let B_{s_{1}}^{b} be the section of B^{b} over s_{1}. We have just proved that for s_{1} ≠ b_{1}, we have B_{s_{1}}^{b} ⊆ ℂ \ h_{s_{1}}(A_{s_{1}}). Hence dim(B_{s_{1}}^{b}) ⩽ 1 and dim(B^{b}) ⩽ 3. The moreover part is clear from the definition of b-bad points. □

Now, given a strongly minimal S ⊂ ℂ^{2}, and b ∈ ℂ^{2} we cannot hope to be able to say much on the possible intersections of S with f-lines through b, if S meets B^{b} in a large set. Therefore, we first move S almost outside B^{b} for a right choice of b.

Lemma 3.2. Assume S ⊂ ℂ^{2} is strongly minimal and with infinite frontier. Then, there is a strongly minimal S′ ⊂ ℂ^{2} definable over a certain set A and b ∈ fr(S′) such that dim(b/A) = 1 and dim(S′ ∩ B^{b}) ⩽ 1.

Proof. For any a ∈ ℂ let us define

F_{a} : ℂ^{2} → ℂ^{2}, F_{a}(x, y) = (x, f(y + a)).

Without loss, S is ∅-definable. Take c = (c_{1}, c_{2}) ∈ fr(S) with dim(c) = 1 and let

X := {a ∈ ℂ | F_{a}(c) ∈ F_{a}(S)}.

Claim: X is finite.

Proof of Claim. Assume not. Let S_{2} denote the section of S over c_{1}. If S_{2} is infinite, then S ∼ {c_{1}} × ℂ, so S has finite frontier. Hence S_{2} is finite. We get that for each a ∈ X there is d_{2} ∈ S_{2} such that

f(a + c_{2}) = f(a + d_{2}).

Thus there is a fixed d_{2} ∈ S_{2} such that for an infinite X′ ⊆ X the functions f(x + c_{2}) and f(x + d_{2}) coincide on X′, which is impossible by 2.13, since c ∉ S, so c_{2} ≠ d_{2}. □

Let

Y := {a ∈ ℂ \ X | F_{a} is C^{1} at c and det(F′_{a}(c)) ≠ 0}.

Y is of codimension 1 at most by 2.8(3) and 2.12. By 3.1, for any a ∈ ℂ we have:

dim(F_{a}(S)∩B^{F_{a}(c)}) = dim[F_{a}(S)∩(B^{c}+F_{a}(c)−c)] = dim[(F_{a}(S)+c−F_{a}(c))∩B^{c}].

For any a ∈ ℂ consider the function

F̄_{a} = F_{a} + c − F_{a}(c).

In a similar way as in the proof of the claim, we can prove that for all a′ ≠ a′′

F̄_{a′}(S) ≁ F̄_{a′′}(S).

Therefore we have:

dim{a ∈ Y | dim(F̄_{a}(S) ∩ B^{c}) = 2} ⩽ 1,

since otherwise, we would have a 2-dimensional subfamily of (F̄_{a}(S) ∩ B^{c})_{a∈Y} consisting of 2-dimensional subsets of B^{c}, hence some a′ ≠ a′′ with F̄_{a′}(S) ∼ F̄_{a′′}(S). Take a ∈ Y such that dim(a, c) = 3 and

dim(F_{a}(S) ∩ B^{F_{a}(c)}) = dim(F̄_{a}(S) ∩ B^{c}) ⩽ 1.

By the definition of Y and the Inverse Function Theorem, there is an open set U ∋ c such that F_{a}|_{U} is a diffeomorphism. For a topological space Z, the following is trivial:

(∗)  For any C ⊂ D ⊂ Z, z ∈ D \ C, z ∈ fr(C) if and only if z ∈ fr_{D}(C).

Therefore, since c ∈ fr(S) and c ∈ U, we get c ∈ fr_{U}(S ∩ U). Since F_{a}|_{U} is a homeomorphism, we get:

(∗∗)  F_{a}(c) ∈ fr_{F_{a}(U)}(F_{a}(S ∩ U)).

Let us set

b := F_{a}(c), S′ := F_{a}(S), A := {a}.

Since F_{a} has finite fibers and dim(c/a) = 1, we get dim(b/a) = 1. By the claim and dim(a/c) > 0, we have a ∉ X. Therefore b ∉ S′ and (∗), (∗∗) imply b ∈ fr(S′). □

We can now proceed in a similar fashion as in [20], but we have to be always careful about the singular loci of the sets we consider.

Theorem 3.3. Let S ⊂ ℂ^{2} be strongly minimal. Then fr(S) is finite.

Proof. After replacing S with S′ from 3.2, we may assume that S is defined over ∅ and there is b ∈ fr(S) such that dim(b) = 1 and dim(S ∩ B^{b}) ⩽ 1. Take a ∈ ℂ such that dim(a, b) = 3 and consider the f-line l_{a}^{b} through b. Recall that l_{a}^{b} is the graph of the function f_{a}^{b}(x) := f(a + x) + c_{a}^{b} (c_{a}^{b} = −f(a + b_{1}) + b_{2}, but it will not be important for us). Claim 1

RM(c_{a}^{b}, a) = 2.

Proof of Claim 1. Note that l_{a}^{b} ∩ fr(S) is finite, since otherwise, we would have a 2-dimensional subfamily of (l_{a}^{b} ∩ fr(S))_{a∈ℂ} consisting of infinite subsets of a 1dimensional set fr(S), yielding the infinite intersection of a certain couple of different f-lines, which contradicts 2.14. Since fr(S) is ∅-definable, we get that dim(b/c_{a}^{b}, a) = 0. If RM(c_{a}^{b}, a) ⩽ 1, then dim(c_{a}^{b}, a) ⩽ 2, so dim(c_{a}^{b}, a, b) ⩽ 2, which contradicts the choice of a. □

By 2.16 and Claim 1, S ∩l_{a}^{b} is finite, so S ∩l_{a}^{b} ⊂ acl(a, b). Take any s ∈ S ∩l_{a}^{b}. Since there are finitely many f-lines through b and s, we also have a ∈ acl(s, b). Hence

dim(s/b) = dim(a/b) = 2.

Claim 2

S ⋔_{s} l_{a}^{b}

(in the sense of 2.10).

Proof of Claim 2. Since dim(B^{b} ∩S) ⩽ 1 and dim(s/b) = 2, s is b-good. Therefore, there are U ∋ s, W ∋ a and a C^{1} parameter choice function ϕ_{b} : U → ℂ such that:

• For each s′ ∈ U, a′ ∈ W, we have s′ ∈ l_{a′}^{b} if and only if ϕ_{b}(s′) = a′.

• ϕ_{b}(s) = a.

• ϕ_{b} is a submersion at s.

In particular, ϕ_{b}^{−1}(a) = l_{a}^{b} ∩ U. Since ϕ_{b} is a submersion at s, there is an open s ∈ U′ ⊆ U such that ϕ_{b}^{−1}(a) ∩ U′ = l_{a}^{b} ∩ U′ is a 2-dimensional C^{1}-submanifold of ℂ^{2} and we have:

ker(ϕ′_{b}(s)) = T_{s}(ϕ_{b}^{−1}(s) ∩ U′) = T_{s}(l_{a}^{b} ∩ U′).

Let us denote T_{s}(l_{a}^{b} ∩ U′) just by T_{s}(l_{a}^{b}) and similarly with T_{s}(S) (s is a generic point of S, so S is, locally around s, a 2-dimensional C^{1}-submanifold of ℂ^{2}). Let V := T_{s}(l_{a}^{b}) ∩ T_{s}(S). Assume now that S is not transversal to l_{a}^{b} at s, i.e. dim V ⩾ 1. Let ψ denote ϕ_{b}|_{S}. Since T_{s}(l_{a}^{b}) = ker(ϕ′_{b}(s)), we have

dim(im(ψ′(s))) = dim(ϕ′_{b}(s)(T_{s}(S))) ⩽ 1.

Since s is a b-generic point of S, we get an open s ∈ U_{s} ⊆ U such that for each s′ ∈ U_{s} we have dim(im(ψ′(s′))) ⩽ 1. We consider now two cases: Case 1 im(ψ) is infinite. There is a′ ∈ im(ψ) such that ψ^{−1}(a′) is infinite and dim(a′) = 1. This means that l_{a′}^{b} ∩ S is infinite, which contradicts 2.16. Case 2 im(ψ) is finite. To obtain contradiction as in Case 1, we can shrink S around s to make it connected and such that ψ is continuous on S. Hence im(ψ) is connected, so it coincides with {a}. Then l_{a}^{b} ∩ S is infinite, which contradicts 2.16 again (by Claim 1). □

We can now conclude as in [20]. Consider a strongly minimal subfamily of f-lines from 2.14, which is ℂ_{f}-definable over a:

F_{a} := (l_{a,c})_{c∈ℂ} , l_{a,c} := Γ_{f(a+x)+c},

which are merely translations of l_{a}^{b} = l_{a,c_{a}^{b}} (so there is no fixed point now through which they all pass). By Claim 1, l_{a}^{b} is a generic f-line in F_{a}. Therefore, there are cofinitely many c ∈ ℂ such that

|l_{a}^{b} ∩ S| = |l_{a,c} ∩ S| =: n < ∞.

Consider now l_{a}^{b} ∩ S = {s_{1}, …, s_{n}}. By Claim 2, there is ε > 0 such that for any c′ ∈ ℂ, if |c′ − c_{a}^{b}| < ε, then l_{a,c′} still intersects S near each s_{i}. But there are infinitely many c′ as above such that l_{a,c′} intersects S near b as well (since b ∈ fr(S)), contradicting the choice of n. □

4. Topology

4.1. The topological properties of f. Because even rational functions are only continuous on the Riemann sphere, this will be the natural domain for us to work in. We will use the finiteness of the frontier to show that allowing the value ∞, f is an almost continuous, proper open map.

The key to the proof is Theorem 3.3 which will be used repeatedly to obtain various finiteness results. First we apply the theorem to show that f is locally injective on a cofinite set.

Definition 4.1. We say that c is a non-injective point of f if for any open U ∋ c, f|_{U} is not injective. Let N (f) denote the set of non-injective points of f.

Fact 4.2. N (f) is finite.

Proof. Let

S := {(x, y) ∈ ℂ^{2} : f(x) = f(y) ∧ x ≠ y}.

Clearly, RM(S) = 1 and fr(S) is contained in the diagonal. For c ∈ ℂ, we have (c, c) ∈ fr(S) if and only if c ∈ N (f). But fr(S) is finite by 3.3, so we are done. □

Let S^{2} denote the 2-sphere. We view it as a 1-point compactification of ℂ, so S^{2} = ℂ ∪ {∞}. Our first task is to show that ∞ ∈ S^{2} can not be obtained as a limit infinitely often. Let us define

L := {a ∈ ℂ | ∃a_{n} → a, lim(f(a_{n})) = ∞}.

We need a lemma giving a partial description of the topological properties of f.

Lemma 4.3. There is a finite F ⊂ ℂ such that f is continuous and open on ℂ \ (L ∪ F).

Proof. Let F be the union of N (f) and the projection to the first coordinate of fr(Γ_{f}). By 3.3 and 4.2, F is finite. It is clear from our choices that f is continuous and locally injective on ℂ \ (L ∪ F). By Brouwer’s Invariance of Domain Theorem [2], f is open on ℂ \ (L ∪ F). □

Fact 4.4. L is finite.

Proof. Assume not. We represent L as the following union of R-definable sets:

L = L_{1}^{+} ∪ L_{2}^{+} ∪ L_{1}^{−} ∪ L_{2}^{−},

where e.g.

L_{1}^{+} := {x ∈ L| ∃x_{n} → x, lim(f_{1}(x_{n})) = +∞}.

Without loss, we may assume that L_{1}^{+} is infinite. By 2.8(3,4), for each c ∈ L_{1}^{+}, there is an R-definable, C^{1} curve γ : (0, 1) → ℂ such that

lim_{x→1} γ(x) = c, lim_{x→1} f_{1}(γ(x)) = +∞.

We will sometimes identify γ with its image. Take ℓ ⊆ L, which is a 1-dimensional C^{1}-submanifold of ℂ being homeomorphic to a closed interval such that there is an open W ⊇ ℓ such that f|_{W} is 1-to-1 and f|_{W \L} is a homeomorphism (this is possible by 4.2 and 4.3). Take R-generic c_{1}, c_{2} ∈ Int_{L} ℓ and γ_{1}, γ_{2}, R-definable continuous curves witnessing that c_{1}, c_{2} ∈ L_{1}^{+}. We may assume that γ_{1}, γ_{2} are “on the same side” of ℓ. Take T := T_{ε}(ℓ), an open tubular neighborhood of ℓ of width ε such that:

(1) ∂T ∩ ℓ = ∅.

(2) f is a homeomorphism on Cl(T) \ ℓ.

(3) γ_{1} ∩ γ_{2} ∩ T = ∅.

(4) |γ_{i} ∩ ∂T | = 1.

(5) Γ_{i} := f(γ_{i} ∩ T) is the graph of the continuous function g_{i} : (r_{i}, ∞) → ℝ. We replace γ_{i} with γ_{i} ∩ T. Let U be the open set bounded by γ_{1}, ℓ, γ_{2} and ∂T. U is clearly connected and simply connected. Now, we want to show that:

(∗)  f(U) is the entire area bounded by Γ_{1} ∪ Γ_{2} ∪ f(Cl(U) ∩ ∂T).

To prove (∗), it is enough to notice that f(l) ∩ ∂f(U) = ∅. But f(U) is simply connected and Γ_{1}, Γ_{2} ⊆ fr(f(U)), hence f(l)∩∂f(U) is either unbounded or empty. By 2.8(3), f|_{ℓ} is almost continuous, so f(ℓ) is bounded, therefore f(l) ∩ ∂f(U) = ∅. Without loss, for each x_{1} ⩾ r_{1}, r_{2}, we have g_{2}(x_{1}) ⩾ g_{1}(x_{1}). Let

l := lim_{x_{1}→∞}(g_{2}(x_{1}) − g_{1}(x_{1})) ⩾ 0.

Case 1 l ≠ 0 Let us divide f(U) into V_{1}, V_{2} – “half-tubular neighborhoods” of Γ_{1}, Γ_{2} of width l/2. Without loss, the set

ℓ_{l} := Cl(f^{−1}(V_{1})) ∩ ℓ

is infinite. For b ∈ ℝ let

F_{b} := {(x, y) ∈ ℂ^{2} | f(x) + (0, b) = f(y)}.

By (∗), for all v ∈ f^{−1}(V_{1}) and b < l/2, there is t_{v} ∈ U such that

f(v) + (0, b) = f(t_{v}).

If x → v ∈ ℓ_{l}, then t_{x} → v′ ∈ ℓ. Therefore, for each b < l/2, the set (ℓ_{l} × ℓ) ∩ ∂F_{b} is infinite and projects onto ℓ_{l}. By 3.3, there is an infinite ∅-definable (in R) I ⊆ ℓ_{l} such that (I × ℓ) ∩ F_{b} is infinite and projects onto I. Therefore, for all b < l/2, we get f(I) + (0, b) ⊆ f(ℓ). Since dim(f(ℓ)) = 1, f(I) is an interval parallel to the y-axis. Let c ∈ ℂ be R-generic and f^{c}(x) := f(x) + f(x + c). Then (c + I) ∩ L is finite, so there is I′ ⊆ I such that (c + I′) ∩ L = ∅ and (f^{c}, I′) satisfies assumptions as those imposed on (f, ℓ). Now we repeat the same process with (f^{c}, I′) in place of (f, ℓ). It is easy to see that the corresponding l^{c} is still different from 0. Hence we get an infinite ∅-definable I′′ ⊆ I′ such that f^{c}(I′′) is an interval parallel to the y-axis. Without loss, I′′ is a C^{1}-submanifold of ℂ. Since |f_{1}^{c}(I′′)| = |f_{1}(I′′)| = 1, we have |f_{1}(c + I′′)| = 1. Since dim(c) = 2, there is an open U ⊆ ℂ such that for each c′ ∈ U, we have |f_{1}(c′ + I′′)| = 1. Fix a ∈ I′′ and let α ∈ S^{1} such that T_{a}(I′′) = αℝ. Since for each a′′ ∈ I′′ and each c′ ∈ U we have f_{1}(a + c) = f_{1}(a′′ + c), we get \frac{∂}{∂α}f_{1}(a + c) = 0. Therefore, \frac{∂}{∂α}f_{1} is constantly 0 on U + a contradicting 2.18. Case 2 l = 0. Take an R-generic c ∈ ℂ and infinite ∅-definable ℓ′ ⊆ ℓ such that

(ℓ′ + c) ∩ L = (ℓ′ − c) ∩ L = ∅.

Let us define now:

f^{c}(x) := f(x) + f(x + c), f_{c}(x) := f(x) − f(x + c).

Now we can proceed as in Case 1 again, simultaneously with f^{c}, ℓ′ and f_{c}, ℓ′ in place of f, ℓ. It is again easy to see that l^{c}, l_{c} ≠ 0. Hence we get α ∈ S^{1}, a ∈ ℂ and an open U ⊆ ℂ such that \frac{∂}{∂α}f_{1}^{c} and \frac{∂}{∂α}(f_{c})_{1} are constantly 0 on U + a. But then \frac{∂}{∂α}f_{1} is constantly 0 on U + a contradicting 2.18. □

It seems that from this point on the assumption that f is definable in an o-minimal field R is used very mildly, except for the fact that f is C^{1} on a large set. We will try to point out all the places where o-minimality is used. It is not clear to us whether o-minimality is crucial for the conclusion of the proof.

We need a simple topological lemma:

Fact 4.5. Let D = B\{a} be a punctured disc, Y a topological space and f : D → Y a continuous function. Then the set

E := {e ∈ Y : (a, e) ∈ fr(Γ_{f})}

is connected.

Proof. Suppose not and let U, V be open and disjoint subsets of Y such that E = (U ∩ E) ∪ (V ∩ E). Consider D̃ := f^{−1}(U) ∪ f^{−1}(V). By continuity, D̃ is open, and by the definition of E, B̃ := D̃ ∪ {a} is open in B, since for any sequence B ∋ x_{n} → a we know that lim f(x_{n}) ∈ E so x_{n} ∈ D̃ for n big enough. B has a basis of open discs, so there must be a disc C centered at a which is contained in B̃. Therefore C \ {a} ⊆ D̃ and C \ {a} ⊈ f^{−1}(U), f^{−1}(V), which contradicts the connectedness of C \ {a}. □

Gathering all this information we easily get:

Proposition 4.6. Let A be the set of points at which f is not continuous. Then A is finite and there is a continuous function f̂: ℂ → S^{2} which coincides with f on ℂ \ A.

Proof. By 3.3, 4.4 and 4.5, for each a ∈ ℂ,

f̂(a) := lim_{x→a}f(x)

exists in S^{2}. For a ∈ ℂ \ A, f̂(a) = f(a). If a ∈ A, then either a is contained in the projection of the frontier of the graph of f, which is finite by 3.3, or f̂(a) = ∞, so a ∈ L which is finite by 4.4. □

Since there is no harm in changing f on a finite set, from now on we replace f with f̂. Therefore f is a continuous function into S^{2}. We call each point a ∈ ℂ such that f(a) = ∞, a pole of f.

Proposition 4.7. f is an open map.

Proof. By Brouwer’s Invariance of Domain Theorem [2], a locally injective function f : V → ℝ^{2} is open for any open V ⊆ ℝ^{2}. Hence for any a ∉ N (f) there exists a neighborhood V_{a} such that f(V_{a}) is open. So we focus on a ∈ N (f). Without loss, a = 0 = f(0). Since N (f) is finite we can find an open disc V around 0 such that cl(V) ∩ N (f) = {0}. We may also assume that V is so small that f^{−1}(0) ∩ cl(V) = {0}. Denote D := V \ {0}.

By the Invariance of Domain f|_{D} is open so ∂f(D) ⊆ f(∂D). By our choice of V we know that 0 ∉ f(∂D \ {0}) and as the latter is closed, 0 is an isolated point of ∂f(D). Hence for a small enough ball B ∋ 0 we get that B ∩ ∂f(D) = {0}.

Therefore (B \ {0}) ∩ f(D) is non-empty and clopen in B \ {0}, so (since B \ {0} is connected) B \ {0} ⊆ f(D) and B ⊆ f(V). So 0 ∈ Int(f(V)), which is what we needed. □

We can now show that f is proper, i.e.:

Proposition 4.8. f continuously extends to S^{2}.

Proof. We need to show that lim_{x→∞}f(x) exists. By 4.5, it is enough to show that

E := {e ∈ ℂ : (∞, e) ∈ fr_{S^{2}×S^{2}} Γ_{f}}

is finite. Since {z : |f^{−1}(z)| ≠ d} is finite (d is the degree of f), it will suffice to show that there does not exists e ∈ E such that |f^{−1}(e)| = d. Assume the contrary, and let e_{1}, …, e_{d} be distinct such that f(e_{i}) = e for i ⩽ d. Let U_{i} ∋ e_{i} be pairwise disjoint open sets. Since f : ℂ → ℂ is open, V := f(U_{1}) ∩ … ∩ f(U_{d}) is an open set which is non-empty, as it contains e. But e ∈ E implies that f^{−1}(V) is unbounded. Therefore, the set

A := f^{−1}(V) \ U_{1} ∪ … ∪ U_{d}

is infinite. But for any a ∈ A and i ⩽ k, f(a) has a preimage by f in U_{i}, therefore each element of the infinite set f(A) has at least d + 1 preimages by f, a contradiction. □

Question 4.9. In the presence of a ℂ_{f}-definable algebraically closed field Peterzil and Starchenko [20] use similar arguments to conclude that g is C^{1} in all but finitely many points. Can such a result be obtained already at this stage?

Before we proceed we note that what we have proved for f is also valid for any (non-almost constant) ℂ_{f}-definable function g : ℂ → ℂ, i.e. any such function is almost continuous, open and with finitely many non-injective points as a function from S^{2} to S^{2}. Therefore, even though complex addition does not extend to S^{2}, we can still add continuous ℂ_{f}-definable functions from S^{2} to S^{2}, understanding e.g. f(x + 1) − f(x) as the continuous correction of a ℂ_{f}-definable function. Whenever we say that a ℂ_{f}-definable function is not injective, we mean that its continuous correction is not. Since a ℂ_{f}-definable function can have finitely many “accidental” values, non-injectivity does not say much before this correction. Note however, that if a ℂ_{f}-definable function has at least 2 poles, then it is necessarily non-injective in the right sense described above.

Fact 4.10. If f is injective, then g(x) := f(x + 1) − f(x) is not.

Proof. By 2.13, g is not almost constant. Therefore the image of g is cofinite, hence g is onto as a continuous function on the compact space S^{2}. Since f is injective it has only one pole at a ∈ S^{2}, say. If a ∈ ℂ, then g has poles at a and a − 1, so it is not injective. Assume a = ∞. Then g has no poles on ℂ, so g(ℂ) = ℂ. In particular, there is c ∈ ℂ such that g(c) = 0. But then f(c + 1) = f(c), so f is not injective, a contradiction. □

In the case f was injective (e.g. if f(x) = 1/x), we replace f with the g above, so we may assume f is not injective. The reader may wonder if we do not lose any information by replacing ℂ_{f} with this reduct ℂ_{g}. However, the course of the proof is to find a ℂ_{f}-definable field (so replacing f with g is fine) and then show that all the extra structure must be already definable in that field. Hence, there is no danger in replacing f with g, as long as ℂ_{g} is not locally modular, which is guaranteed e.g. if g is continuous and not injective. We will often make such replacements.

In a similar way, we may also assume that f(∞) = ∞. If not, there exists some a′ ∈ ℂ such that f(a′) = ∞. Replacing f with g := f(f(x) − f(∞) + a′) we get a function with the desired property. Note that if f was not injective clearly so is g.

From now on whenever we consider a ℂ_{f}-definable function g : S^{2} → S^{2} which is not almost affine we will implicitly assume that g has the same topological properties as f, i.e. we will always assume that g is continuous, open, non-injective with finite fibers and with finitely many non-injective points. The following result, which combines §V III.4.5 and §V III.6.1 of [27] will be crucial in our analysis of such functions:

Theorem 4.11. Assume that g : S^{2} → S^{2} has all the above properties, then:

(i) g is a ramified covering, i.e. for each c ∈ S^{2} there exists a neighborhood V ∋ c and homeomorphisms ϕ : V → B_{1}(0), ψ : g(V) → B_{1}(0) such that ϕ(c) = ψ(g(c)) = 0 and there is k ⩾ 1 such that the following diagram commutes:

\xymatrix{V \ar[r]^{g} \ar[d]_{ϕ} & g(V) \ar[d]^{ψ} \\ B_{1}(0) \ar[r]^{g_{k}} & B_{1}(0),}

for g_{k}(z) = z^{k}. The number k is called the multiplicity of g at c and denoted mlt_{c} g. (ii) For y ∈ S^{2}, the number

deg(g) := ∑_{c∈g^{−1}(y)} mlt_{c} g

does not depend on y and is called the degree of g.

Remark 4.12. Note that the above theorem implies in particular that the size of a generic fiber of f is maximal (which can be also inferred easily from the openness of f). In particular, since we know that f is not injective, we get that deg(f) > 1.

Question 4.13. By Stoïlow’s theorem (see, e.g. [27]) if g is a ramified covering of S^{2}, say, there exists a unique complex structure on S, denoted 𝕊, such that

g : (S^{2}, 𝕊) → (S^{2}, Riemann sphere structure)

is a holomorphic function. Moreover, (S^{2}, 𝕊) is biholomorphic with the Riemann sphere. Can those two facts be used to shorten the proof of our main result?

Definition 4.14. We call a point y ∈ S^{2} a ramification point of g if |g^{−1}(y)| < deg(g).

By strong minimality we obtain immediately that the set of ramification points is finite. We have a relation between non-injective points and ramification points.

Fact 4.15. y ∈ S^{2} is a ramification point of g if and only if there is c ∈ g^{−1}(y), which is a non-injective point of g.

Proof. By 4.11(ii), y is a ramification point of g if and only if there is c ∈ g^{−1}(y) such that mlt_{c} g > 1. Such c is clearly a non-injective point of g. □

We prove that non-injective points of non-injective functions exist.

Lemma 4.16. If deg g > 1, and g(∞) = ∞ then g has at least 2 ramification points. Therefore, there is c ∈ ℂ which is a non-injective point of g and not a pole.

Proof. The proof is actually an application of the Hurwitz formula which is valid in our context. However, we need a very weak form of it, so we show the easy calculation. Let d := deg(g), R the set of ramification points of g, r := |R| and X := S^{2}\g^{−1}(R). Since f|_{X} is d-to-1, we get E(X) = dE(S^{2} \R), where E is the Euler characteristic. Therefore

d(2 − r) = dE(S^{2} \ R) = E(X) ⩽ 2 − r.

Since d > 1, d(2 − r) ⩽ 2 − r implies r > 1. By 4.15, we have 2 non-injective points of g on which g attains different values, so one of these values is from ℂ and since g(∞) = ∞ we get what we wanted. □

By the last lemma, we can add to all our assumptions about f that it has a noninjective point at 0 and that f(0) = 0 (replacing f with f(x) − f(0), since we know 0 is not a pole).

4.2. Degree theory. We want to investigate the way multiplicities vary in continuous families, which we will do using the notion of topological degree. We start with some notation. For a simple closed curve C, let C^{0} denote the interior of the region bounded by C and C̄ := C ∪ C^{0}. For a continuous function h : B → ℂ from a bounded open subset of ℂ and y ∈ ℂ \ ∂B, let d(h, B, y) denote the topological degree as in [5]. We need a lemma about the relation between the topological degree and the degree we have defined earlier.

Lemma 4.17. Let B, y be as above and h : B → ℂ be an R-definable ramified covering. Then

∑_{x∈h^{−1}(y)} mlt_{x}(h) = ±d(h, B, y).

Proof. By Theorem 2.9(1) of [5], we have

∑_{x∈h^{−1}(y)} i(h, x, y) = d(h, B, y),

where i(h, x, p) is the index of h with respect to x, p defined in 2.8 of [5].

Since d(h, B, y) is continuous with y and the smooth points of h are dense, it will be enough to show the equality for y ∈ ℂ such that h is C^{1} on h^{−1}(y) and h′(x_{i}) ≠ 0 for all x_{i} ∈ h^{−1}(y). Since h is locally injective on a co-finite set, by 3.2 of [21] h′ has constant sign (whenever it is defined). Therefore i(h, x, y) has constant sign. Because we restrict to those points where f′ ≠ 0 we get that i(h, x, y) = sgn(h′(y)) and mlt_{x}(h) = ±i(h, x, y), with the sign independent of y. Hence the desired equality. □

In [21] o-minimality is used to prove that sgn(h′) is constant. It seems, however, that o-minimality is not really needed for the present context, besides the fact that C^{1}-points are dense (see also the proof of Theorem IX(2.1) of [27]).

Remark 4.18. We know a posteriori that any ℂ_{f}-definable function is orientation preserving, so the topological degree and our degree do, in fact, agree (not only up to sign). But we are unable to show it at this stage.

Let G : ℂ^{n} × S^{2} → S^{2} be a ℂ_{f}-definable continuous family of continuous functions. In the next two lemmas we gather some facts on the trajectories of non-injective points in the family

(g_{a}(x) := G(a, x))_{a∈ℂ^{n}} .

These results will be crucial in the next section. We note that both lemmas follow immediately from the Argument Principle (applied to g′_{a}) if g is holomorphic. It is possible that they are still folklore even in this more general topological context, but we could not find a reference, so we give the proofs instead.

The function g_{0} is either orientation preserving everywhere or orientation reversing everywhere. If it is orientation preserving (resp. reversing) then for each a close enough to 0, g_{a} is orientation preserving (resp. orientation reversing). Since, in the proofs below we are interested only in a close enough to 0, we may assume that all our functions are orientations preserving and that we actually have the equality in the statement of 4.17 (otherwise, we just use the negative topological degree, which has all the properties we want as well).

In the next lemma we show that if 0 is a non-injective point of g_{0} then for every a close enough 0, g_{a} has a non-injective point near 0. In other words, we cannot lose non-injective points by changing our functions within a continuous family of continuous functions.

Lemma 4.19. Assume 0 ∈ N (g_{0}) and 0 is not a pole. Then for each open U ∋ 0, there is ε > 0 such that for each a ∈ B_{ε}(0), g_{a} has a non-injective point in U.

Proof. Set g := g_{0}. We may assume g(0) = 0. By 4.11(i), we can find a simple closed curve 0 ∈ C^{0} ⊂ U such that g(C) = D is a simple closed curve, g(C^{0}) = D^{0} and g|_{C^{0}} is homeomorphic to z^{n}. Take a closed 2-ball (i.e. a set homeomorphic to a closed disc) 0 ∈ B ⊂ D^{0} such that g|_{g^{−1}(B)∩C̄} is still homeomorphic to z^{n}. Take ε > 0 such that for each |a| < ε we have

g_{a}(D) ∩ B = ∅.

Since the topological degree is invariant under homotopy, (see D4 from the Introduction to [5]), we have (by 4.17) for each |a| < ε and for each p ∈ B

d(g_{a}, C^{0}, p) = n.

Because g|_{g^{−1}(B)∩C} is homeomorphic to z^{n} we know that ∂g^{−1}(B) ∩ C̄ is a simple closed curve. In particular g|_{g^{−1}(B)∩C} is connected. Therefore, there is 0 < ε′ < ε such that g_{a}^{−1}(B) ∩ C̄ is connected for all |a| < ε′. To simplify the notation we will from now on identify g_{a} with its restriction to C̄. Assume g_{a} has no non-injective points in C^{0} for some |a| < ε′. In particular, by 4.11(i), g_{a} is a local homeomorphism on C^{0}. Therefore g_{a}^{−1}(B) = cl(Int(g_{a}^{−1}(B))), and ∂g_{a}^{−1}(B) is a 1-dimensional set which by 4.11 is everywhere locally homeomorphic to an interval (since, by assumption, there are no non-injective points of g_{a} in C^{0}). Case 1 ∂g_{a}^{−1}(B) is connected Then, g_{a}^{−1}(B) is homeomorphic to a closed disc and g_{a}|_{g_{a}^{−1}(B)} can not be n-to-1, therefore, by the definition of the topological degree, there are non-injective points.

Case 2 ∂g_{a}^{−1}(B) is not connected. Each connected components of ∂g_{a}^{−1}(B) is not self intersecting, and therefore homeomorphic to S^{1}. Therefore g_{a}^{−1}(B) is homeomorphic to a closed disc with finitely many closed discs removed. Hence, the Euler characteristic of g_{a}^{−1}(B) is not positive, so g_{a}|_{g_{a}^{−1}(B)} is not n-to-1, and again N (g_{a}) ∩ C^{0} ≠ ∅ (since d(g_{a}, C^{0}, p) = n for every p ∈ g^{−1}(B)). In both cases we reach a contradiction. □

In the next lemma we investigate the behavior of the local degree, as we trace it along the trajectories of non-injective points within our family G. Roughly, our statement is that if this trajectory splits at some point into several trajectories then the local degree must go down.

Lemma 4.20. Assume c ∈ N (g_{0}), g_{0}(c) ≠ ∞ and for all open balls B ∋ 0, C ∋ c there exists a ∈ B such that |N (g_{a}) ∩ C| > 1, then there exist a ∈ ℂ and d ∈ N (g_{a}) arbitrarily close to 0 and c respectively, such that mlt_{d}(g_{a}) < mlt_{0}(g_{0}).

Proof. Without loss c = 0. Take a simple closed curve C such that g_{0}|_{C^{0}} is topologically equivalent to z ↦ z^{n}. Take a smaller closed 2-ball D ⊂ C^{0} as above and ε > 0 such that for each a ∈ B_{ε}(0) and each x ∈ D

d(g_{a}, C^{0}, g_{a}(x)) = n.

We may assume that ε was also chosen in such a way that for each a ∈ B_{ε}(0) there are s_{1}, s_{2} ∈ D^{0} ∩ N (g_{a}) and g_{a}(s_{1}) and g_{a}(s_{2}) belong to the same component of ℂ \ g_{a}(C). By 4.17 and a remark before 4.19, we have:

n = d(g_{a}, C, g_{a}(s_{i})) = ∑_{x∈g_{a}^{−1}(g_{a}(s_{i}))∩D} mlt_{x} g_{a}

for i = 1, 2. So, we are done if there is i ∈ {1, 2} and s_{0} ≠ s_{i} such that g_{a}(s_{0}) = g_{a}(s_{i}). If not, we are in the situation of [27, VIII(5.21)] for E = D, p_{i} = s_{i}, q_{i} = g_{a}(s_{i}). Take any closed region B ⊂ g_{a}(D) \ g_{a}(C) bounded by a simple closed curve and containing g_{a}(s_{1}) and g_{a}(s_{2}) (it can be done since g_{a}(s_{1}) and g_{a}(s_{2}) belong to the same component of ℂ \ g_{a}(C)). By [27, VIII(5.21)], there exist components B_{1} ∋ s_{1}, B_{2} ∋ s_{2} of g_{a}^{−1}(B) such that g_{a}|_{B_{i}} is topologically equivalent to z ↦ z^{n_{i}} and g_{a}(B_{i}) = B. Clearly B_{1} ≠ B_{2}, therefore n ⩾ n_{1} + n_{2}, so n_{1} < n. □

5. Smoothness

The main result of this section is a weak version of the Cauchy-Riemann theorem. We show that if g is a ℂ_{f}-definable function with the usual properties and g is not locally injective at some C^{1} point c then g′(c) = 0. Rather surprisingly, proving that such functions existed (namely, one with a smooth non-injective point) turned out to be one of the hardest parts of the proof.

Lemma 5.1. Let g : S^{2} → S^{2} be a ramified covering and g(0) = 0. Then 0 ∈ N (g) if and only if there are an open set U ∋ 0 and a natural number k > 1 such that for some ε > 0,

g^{−1}([−ε, ε] × {0}) ∩ U = ⋃_{i=1}^{k} C_{i},

where the C_{i} are distinct arcs (i.e. each C_{i} is homeomorphic to a closed interval) such that for i ≠ j, we have C_{i} ∩ C_{j} = {0} and 0 ∈ Int(C_{i}) for all i.

Proof. For the right-to-left direction is enough to see that the above condition implies that g is not a local homeomorphism at 0. For the left-to-right direction, note that the complex mapping h_{k}(z) := z^{k} on B_{1}(0) has the corresponding property for any arc J with 0 ∈ Int(J). By 4.11(i), there exists U ∋ 0 and homeomorphisms h_{1} : U → B_{1}(0), h_{2} : B_{1}(0) → g(U) such that g = h_{1} ∘ h_{k} ∘ h_{2} on B_{1}(0). Necessarily h_{1}(0) = h_{2}(0) = 0. The result follows, if we take J := h_{2}^{−1}([−ε, ε] × {0}). □

From the above lemma we get.

Proposition 5.2. Let g : S^{2} → S^{2} be a ramified covering, g(0) = 0 and 0 ∈ N (g). Assume that g is C^{1} at 0 then g′(0) = 0.

The proof follows immediately from the more general statement:

Lemma 5.3. Let g : S^{2} → S^{2} be a ramified covering, g(0) = 0 and 0 ∈ N (g). For all but finitely many a ∈ S^{1} if the partial derivative \frac{∂g}{∂α} exists and is continuous at 0 then \frac{∂g}{∂α}(0) = 0.

Proof. Let g_{2} be the second coordinate function of g (locally around 0). It is enough to focus on g_{2}. It will be enough to prove that for each ε > 0 for cofinitely many directions α there is c ∈ B_{ε}(0) where \frac{∂g_{2}}{∂α} (c) = 0. Let U, C_{1}, …, C_{k} be as provided by 5.1 so g_{2}|_{C_{i}} = 0. Fix a direction α such that (possibly after shrinking U) \frac{∂g_{2}}{∂α} exists and is continuous on U. Let L_{α} ⊂ ℝ^{2} be the line through 0 given by α. By the mean value theorem it will be enough to find d ∈ ℂ such that

|(L_{α} + d) ∩ (⋃_{i=1}^{k} C_{i}) ∩ B_{ε}(0)| > 1.

Take any parameterizations τ_{i} : [−1, 1] → C_{i} such that τ(0) = 0. By Fact 2.8(5) for any i the limits

c_{i}^{+} := lim_{t→0^{+}} τ′_{i}(t), c_{i}^{−} := lim_{t→0^{−}} τ′_{i}(t)

exist. We are done if L_{α} meets C_{1} and C_{2} “both-sided transversally” at 0, i.e. if α ∉ {c_{1}^{+}, c_{1}^{−}, c_{2}^{+}, c_{2}^{−}}, which is a finite set. □

In the last proof we used o-minimality in order to assure that the (one-sided) tangents to the curves C_{i} existed at 0. We are not sure whether o-minimality can be avoided:

Question 5.4. Is 5.2 true without the o-minimality assumption?

To apply the above proposition, we need to have a ℂ_{f}-definable continuous function with a non-injective point at which the function is C^{1}. Toward this end we will use the degree theory developed in the previous section. We introduce some notation. For a function h : ℂ → ℂ, we denote by S(h) the set of all points z ∈ ℂ such that h is not C^{1} at z. By definition, S(h) is closed. If h is R-definable, then S(h) is also R-definable and (by 2.8(3)) dim S(h) ⩽ 1. We fix the continuous ℂ_{f}-definable family of continuous functions

f_{a}(x) := f(a + x) + f(x).

If N (f_{a}) ⊈ S(f_{a}) for some a ∈ ℂ we are done. So we will now investigate the bad case that N (f_{a}) ⊆ S(f_{a}) for all a. Note that S(f_{a}) ⊆ S(f) ∪ (S(f) − a). Note that S(f) is relatively small (i.e. at most 1-dimensional). In the next lemma, we show that we can replace S(f) with a finite set.

Lemma 5.5. If N (f_{a}) ⊆ S(f) ∪ (S(f) − a) for all a ∈ ℂ, then there is a finite S ⊂ ℂ such that for each a, we have N (f_{a}) ⊆ S ∪ (S − a).

Proof. Let d_{n} : ℂ → S(f) be the (partial) R-definable function such that d_{n}(a) is the n^{th}-smallest (in the lexicographical order) element of N (f_{a})∩(S(f)∪(S(f)−a)), if it exists. Claim: For all n such that dim(dom(d_{n})) = 2 there is a finite S_{n} ⊆ S such that dim(d_{n}^{−1}(S \ S_{n})) ⩽ 1.

Proof of Claim. Assume not. Since the set of ramification points of f_{a} is uniformly ℂ_{f}-definable, by strong minimality (2.9(3)) we obtain

N := max{|f_{a}(N (f_{a}))| | a ∈ ℂ} < ∞.

Since dim S(f) < 2 and dim(dom(d_{n})) = 2, there are infinitely many x ∈ S(f) such that d_{n}^{−1}(x) is infinite. For any such x, the set

R_{x} := {a ∈ ℂ | f_{a}(x) is a ramification point of f_{a}}

is infinite and ℂ_{f}-definable, hence cofinite. Let

D := max{deg(f_{a}) | a ∈ ℂ}

and take DN + 1 distinct x_{i} such that d_{n}^{−1}(x_{i}) is infinite. Then for any

a ∈ R_{x_{1}} ∩ … ∩ R_{x_{DN+1}},

we get that f_{a} has at least N + 1 ramification points, which contradicts the choice of N. □

Note that |N (f_{a})| is uniformly bounded by N in the proof of the Claim and the uniform bound for deg(f_{a}). Therefore, there is n ∈ ℕ, the least such that dim(dom(d_{n})) < 2. Let

X := ℂ \ ⋃_{i<n}(d_{i}^{−1}(S \ S_{i})), S := ⋃_{i<n} S_{i}.

So dim(X) ⩽ 1 and S is finite. For each a ∈ ℂ \ X, we have N (f_{a}) ⊆ S ∪ (S − a). It remains to show that it still holds for a ∈ X. Assume there is c ∈ N (f_{a}) such that c ∉ S ∪ (S − a). Therefore, there is an open U ∋ c and ε > 0 such that for each a′ ∈ B_{ε}(a), we have

U ∩ (S ∪ (S − a′)) = ∅.

By 4.19, there is ε′ < ε such that for each a′ ∈ B_{ε′}(a), there is a non-injective point of f_{a′} in U. But then N (f_{a′}) ⊈ S ∩ (S − a′), which is a contradiction, since B_{ε′}(a) is 2-dimensional. □

Theorem 5.6. There is a ℂ_{f}-definable function g : ℂ → ℂ such that 0 is a noninjective point of g and g is C^{1} at 0.

Proof. We may assume that for all a ∈ ℂ we have N (f_{a}) ⊆ S(f_{a}). Since S(f_{a}) ⊆ S(f) ∪ (S(f) − a), we get by 5.5 a finite S such that N(f_{a}) ⊆ S ∪ (S − a) for each a ∈ ℂ. By 4.16, N (f) ∩ ℂ ≠ ∅ so without loss (after replacing f with f(x − a)) we have 0 ∈ N (f) ⊆ S. Take a disc D around 0 with boundary C such that D ∩ S = {0}. Since f_{0} = 2f, so 0 ∈ N (f_{0}), we get by 4.19 that there is ε > 0 such that for all a ∈ B_{ε}(0) we have:

N (f_{a}) ∩ D ≠ ∅ and D ∩ (S − a) = {−a}.

We need the following Claim: There is a ∈ B_{ε}(0) such that |N(f_{a}) ∩ D| > 1.

Proof. N (f_{a}) ∩ D ⊆ {0, −a} for each a ∈ B_{ε}(0). Let

X := {a ∈ B_{ε}(0) \ {0} | 0 ∈ N (f_{a})}.

By 4.19, X is an open subset of B_{ε}(0) \ {0}. The same applies to

Y := {a ∈ B_{ε}(0) \ {0} | − a ∈ N (f_{a})}.

By 4.19, B_{ε}(0) \ {0} = X ∪ Y. Since f_{a} = f_{−a} ∘ L_{a}, where L_{a}(x) = a + x, we get

N (f_{a}) = L_{a}^{−1}(N (f_{−a})) = N (f_{−a}) + a.

Therefore X = −Y, so they are both non-empty. Since B_{ε}(0) \ {0} is connected, X ∩ Y ≠ ∅. □

By 4.20, there is a ∈ ℂ such that

min{mlt_{f_{a}}(x) | x ∈ N (f_{a})} < min{mlt_{f}(x) | x ∈ N (f)}.

Since this minimum can not go down for ever, at some stage we get a function with the required property. □

Replacing f, if needed, we will from now on assume that f (aside from all the usual properties) is C^{1} at 0 and 0 ∈ N (f). We will also assume that f(0) = 0. For c ∈ ℂ let f_{c}(x) := f(x + c) and for a := (a_{1}, …, a_{n}), n ≥ 2 denote

f_{a} := f_{a_{n−1}} ∘ … ∘ f_{a_{1}} ∘ f − f_{a_{n}}.

For example, if n = 2 we get f_{a}(x) = f(f(x) + a_{1}) − f(x + a_{2}). With this notation, an immediate but important application of the last theorem is:

Corollary 5.7. There is a ball B ⊆ ℂ^{n} with 0 ∈ B such that f_{a} has a C^{1} noninjective point for all a ∈ B. Moreover, for each a if c ∈ N (f_{a}) then

(∗)  f′_{a_{n}}(c) = f′_{a_{n−1}}(f_{a_{n−2}} ∘ … ∘ f_{a_{1}}(f(c))) · . . . · f′_{a_{1}}(f(c)) · f′(c).

Proof. The existence of such B is immediate from 4.19 and the last theorem. The equality (∗) follows immediately from 5.2 using the Chain Rule. □

We denote

R(a; c) ⟺ f_{a}(c) is a ramification point of f_{a},
R′(a; c) ⟺ R(a; c) ∧ f′_{a}(c) = 0.

Note that R is ℂ_{f}-definable and R′ is R-definable. We claim that:

Lemma 5.8. Let

E := {(y, f(y) + a_{1}, …, f_{a_{n−2}} ∘ … ∘ f_{a_{1}}(f(y)) + a_{n−1}) | (∃a_{n} ∈ ℂ)R′(a_{⩽n}; y)}.

Then dim(E) = 2n.

Proof. Let B be as provided by Corollary 5.7 and a_{1}, …, a_{n−1} ∈ ℂ be such that for

a_{<n} := (a_{1}, …, a_{n−1})

we have dim(a_{<n}) = 2(n − 1) and a_{<n} is in the projection of B. Let

R′_{a_{<n}} := {y ∈ ℂ | (∃x ∈ ℂ)R′(a_{<n}, x; y)}.

Claim: dim(R′_{a_{<n}}) = 2.

Proof of Claim. Let U := {x ∈ ℂ : (a_{<n}, x) ∈ B}. Then U is open and there is an o-minimally definable function c : U → ℂ such that for each x ∈ U, R′(a_{<n}, x; c(x)). It is enough to show that dim(c(U)) = 2. Assume not. Case 1: dim(c(U)) = 0 Then, there is y_{0} ∈ ℂ such that dim(c^{−1}(y_{0})) = 2, hence there is an open U ⊆ ℂ such that U ⊆ c^{−1}(y_{0}). Therefore, for each a_{n} ∈ U, (∗) from 5.7 holds for a_{<n}, a_{n}, y_{0} (with a_{<n}, y_{0} fixed). But the right hand side of (∗) does not depend on a_{n}, implying that f′ is constant on an open set, which contradicts 2.17. Case 2: dim(c(U)) = 1 Then, there is an infinite set Y ⊆ c(B) such that for each y ∈ Y, c^{−1}(y) is infinite. Since R′ ⊆ R, we get that for each y ∈ Y the set

R^{y} := {x ∈ ℂ | R(a_{<n}, x, y)}

is cofinite. Let

R_{x} := {y ∈ ℂ | R(a_{<n}, x, y)}, n := max{|R_{x}| | x ∈ ℂ}.

If we take y_{1}, … y_{n+1} ∈ Y, then for any x ∈ R^{y_{1}} ∩ … ∩ R^{y_{n+1}}, we have |R^{x}| > n, a contradiction. □

By the claim, there are a_{n}, c ∈ ℂ such that dim(c/a_{<n}) = 2 and R(a_{<n}, a_{n}; c), so

(c, f(c) + a_{1}, …, f_{a_{n−2}} ∘ … ∘ f_{a_{1}}(f(c)) + a_{n−1}) ∈ E.

Since clearly,

dim(f(c) + a_{1}, …, f_{a_{n−2}} ∘ … ∘ f_{a_{1}}(f(c)) + a_{n−1}/c) = dim(a_{<n}/c) = 2n − 2,

we get that

dim(c, f(c) + a_{1}, …, f_{a_{n−2}} ∘ … ∘ f_{a_{1}}(f(c)) + a_{n−1}) = 2n,

which is exactly what we want. □

It follows that for some ℂ_{f}-definable functions g : ℂ → ℂ the set g′(ℂ) ⊂ M_{2}(ℝ) has at least some of the flavor of a group:

Corollary 5.9. Suppose g is a ℂ_{f}-definable function and V ⊆ ℂ is open such that g is C^{1} on V and there is v ∈ V such that g is non-injective at v. Then for any n > 1, there are open U_{1}, …, U_{n} ⊆ V such that for i ⩽ n, g′(U_{i}) ⊆ GL_{2}(ℝ) and

dim(g′(U_{1}) · . . . · g′(U_{n})) ⩽ 2.

Proof. Let E be as a in the previous lemma, V the set of C^{1} points of g, which is open. Since dim(E) = 2n (by 5.8) and dim(ℂ \ V) = 1, we may assume that E ⊆ V^{n}. By 2.8(5), there are open U_{i} ⊆ ℂ such that U_{1} × ⋯ × U_{n} ⊆ E. The result follows from the definition of E, the moreover part of 5.7 and 2.12. □

6. Analyticity

The aim of this section is to prove the converse of Proposition 5.2, namely that if f′(c) = 0 then f is not a local homeomorphism at c. We will do it by proving that f is analytic on some open set. The proof will go as follows. As a first step we show that from 5.9 we obtain a local Lie group X ⊂ GL_{2}(ℝ) and an open set U ⊆ ℂ such that f′(U) ⊆ XM for some M ∈ GL_{2}(ℝ). We then use local Lie group theory to prove that there exists a Lie subgroup H < GL_{2}(ℝ) such that H ∩ X is a relatively open subset of X. Then, we use the classification of maximal solvable Lie subgroups of GL_{2}(ℝ) to show that H = GL_{1}(ℂ) (naturally embedded into GL_{2}(ℝ)). Then we use 5.9 again to show that we can take M = I and therefore f is holomorphic on U.

We will try to keep the following notational conventions: U, V, W will denote (usually open) subsets of ℂ, a, b, c elements of ℂ, A, B, C, D subsets of GL_{2}(ℝ), H, G Lie subgroups of GL_{2}(ℝ), X, Y local Lie subgroups of GL_{2}(ℝ) and M, N elements of GL_{2}(ℝ) (with the exception of the more general 6.1 below). In this section we omit the group multiplication symbol.

Lemma 6.1. Let B, C, D be k−dimensional, R-definable subsets of an R-definable group G with dim(G) > k. Assume that dim(BCD) = k. Then there is a definable relatively open A ⊆ B such that

dim(AA^{−1}AA^{−1}) = k.

Proof. Choose b_{0} ∈ B, c_{0} ∈ C, d_{0} ∈ D independent generics, and set z_{0} = b_{0}c_{0}d_{0}. Since z_{0} is interdefinable with d_{0} over {b_{0}, c_{0}} we get that dim(z_{0}) = k, i.e. it is generic in BCD. By the definability of dimension and the genericity of b_{0}, c_{0} there are B_{0} ∋ b_{0}, C_{0} ∋ c_{0} relatively open such that for all b ∈ B_{0} and c ∈ C_{0} there is some d ∈ D such that z_{0} = bcd. It follows that

C_{0}^{−1}B_{0}^{−1}z_{0} ⊆ D
B_{0}C_{0}C_{0}^{−1}B_{0}^{−1}z_{0} ⊆ BCC_{0}^{−1}B_{0}^{−1}z_{0} ⊆ BCD
dim(B_{0}C_{0}C_{0}^{−1}D_{0}^{−1}) = k.

Since dim(B_{0}C_{0}) = k, we get in exactly the same way that for a relatively open B_{00} ⊆ B_{0} and some z_{00} ∈ B_{0}C_{0}, we have B_{00}^{−1}z_{00} ⊆ C_{0}, and therefore z_{00}^{−1}B_{00} ⊆ C_{0}^{−1}. It follows that

B_{00}B_{00}^{−1}z_{00}z_{00}^{−1}B_{00}B_{00}^{−1} ⊆ B_{00}C_{0}C_{0}^{−1}B_{00}^{−1}.

Therefore, dim(B_{00}B_{00}^{−1}B_{00}B_{00}^{−1}) = k and A := B_{00} works. □

It may be worth mentioning that in the stable case (and in particular in the strongly minimal case) the analogue with just 2 subsets of G would have already produced a definable group. An o-minimal analogue of this result was recently proved by Otero and Peterzil [16, 2.7]. It is unclear to us whether the o-minimality assumption on top of the fact that G is a Lie group (which is the case in practice) is needed at all.

Question 6.2. Is 6.1 true if we replace “definable” by “submanifold” (so, no ominimal theory around)?

Proposition 6.3. Let X ⊂ GL_{2}(ℝ) be local Lie subgroup, i.e. a submanifold such that X = X^{−1}, XX is a submanifold and dim(XX) = dim(X). Assume dim(X) ⩽ 2. Then, there is a Lie subgroup H < GL_{2}(ℝ) such that H ∩ X is open in X. Moreover, H is a conjugate of either the upper-triangular group or GL_{1}(ℂ).

Proof. It is easy to see that 𝔥 := T_{1}(X) is a Lie subalgebra of 𝔤𝔩_{2}(ℝ). Since dim(𝔥) = 2, 𝔥 is solvable. Hence 𝔥 is contained in 𝔤, a maximal solvable subalgebra of 𝔤𝔩_{2}(ℝ). We want to show that 𝔤 is conjugated to GL_{1}(ℂ) or the algebra of upper-triangular matrices. This should be well-known but we could not find a direct reference, so we present a quick argument. All elementary facts on Lie algebras we use here can be found in [15]. Since 𝔤𝔩_{2}(ℝ) splits into the product of its center and 𝔰𝔩_{2}(ℝ), we may assume that 𝔤 is a Lie subalgebra of 𝔰𝔩_{2}(ℝ). If dim(𝔤) = 1, then 𝔤 is commutative, so it must be a Cartan subalgebra of 𝔰𝔩_{2}(ℝ). Therefore, 𝔥 is conjugated to 𝔰𝔬_{2}(ℝ) (which is contained in 𝔤𝔩_{1}(ℂ)) or to the subalgebra of diagonal matrices. But the latter is not maximal among solvable subalgebras of 𝔰𝔩_{2}(ℝ). If dim(𝔤) = 2, then (by e.g. the Lie Theorem) 𝔤 contains a semi-simple element M. If M is not diagonalizable over ℝ, then the ℝ-span of M is conjugated to 𝔰𝔬_{2}(ℝ). But 𝔰𝔬_{2}(ℝ) is maximal in 𝔰𝔩_{2}(ℝ) even as a Lie subalgebra (by a direct computation or e.g. exercise 1 on p.265 of [15]), a contradiction. Hence, M is diagonalizable over ℝ, so we may assume it is diagonal. It is easy to see that if N ∈ 𝔰𝔩_{2}(ℝ) is not triangular, then the Lie algebra generated by M and N is the entire 𝔰𝔩_{2}(ℝ), so 𝔤 consists of triangular matrices. Taking one more conjugation (if needed), we make sure that these matrices are upper-triangular. We have obtained that 𝔤 is a Lie algebra of a Lie subgroup H < GL_{2}(ℝ) and H is conjugated to either the upper-triangular group or to GL_{1}(ℂ). To finish, it is enough to notice that for local Lie subgroups X, Y, we have T_{1}(X) ⊆ T_{1}(Y) if and only if X ∩ Y is open in X (see Theorem 74 on page 258 of [24]). □

Remark 6.4. This is the place in the proof of the main theorem where we find the matrix A (as in the statement of 1.2) – it will be the conjugating matrix appearing in the moreover part of the above Proposition. This will be made clearer in the last three statements in this section and Remark 7.5.

Proposition 6.5. There are M, M′ ∈ GL_{2}(ℝ) and an open U ⊆ ℂ such that M ∘ f ∘ M′ is holomorphic on U. Moreover, we may assume that M′ = I.

Proof. Take n = 9 in 5.9 to obtain open sets U_{1}, … U_{9} such that f is C^{1} and regular on U_{1} ∪ ⋯ ∪ U_{9} and

dim(f′(U_{1}) … f′(U_{9})) ⩽ 2.

We want to be under the assumptions of 6.1. By 2.17, dim(f′(U_{i})) ⩾ 1 for each i ⩽ 9. If there is i ⩽ 7 such that

dim f′(U_{i}) = dim f′(U_{i+1}) = dim f′(U_{i+2}) = dim f′(U_{i})f′(U_{i+1})f′(U_{i+2})

we take B = f′(U_{i}), C = f′(U_{i+1}), D = f′(U_{i+2}). If there is no such i, we get in particular

dim f′(U_{1})f′(U_{2})f′(U_{3}) = dim f′(U_{4})f′(U_{5})f′(U_{6}) = dim f′(U_{7})f′(U_{8})f′(U_{9}) = 2.

Then we take

B = f′(U_{1})f′(U_{2})f′(U_{3}), C = f′(U_{4})f′(U_{5})f′(U_{6}), D = f′(U_{7})f′(U_{8})f′(U_{9}).

By 6.1, there is a relatively open A ⊆ B such that X := AA^{−1} satisfies the assumptions of 6.3. Take H < GL_{2}(ℝ) as given by 6.3. By shrinking A we may assume that X ⊆ H. For any a ∈ A, Aa^{−1} ⊆ H, therefore A ⊆ Ha. By our choice of A, there is i ⩽ 7 and b ∈ GL_{2}(ℝ) such that A ∩ f′(U_{i})b is open in f′(U_{i})b, so A′ := Ab^{−1} is open in f′(U_{i}). Since f is C^{1} on U_{i}, f′|_{U_{i}} : U_{i} → GL_{2}(ℝ) is continuous, therefore U := (f′|_{U_{i}})^{−1}(A′) is open in ℂ. By our choices we obtain

f′(U) ⊆ Ab^{−1} ⊆ Hab^{−1}.

We want to show now that H is a conjugate of GL_{1}(ℂ). By 6.3 and since real conjugation is an isomorphism of ℂ_{f}, it is enough to show that f′(B) is not contained in a single coset of the upper-triangular group. However, this follows immediately from Corollary 2.18 since if B were contained in a coset of the upper triangular matrices, it would imply that some partial derivative of f is constant on B. The moreover part follows, since again we can replace f with its conjugate. □

Strangely enough, despite our good understanding of f we were not, until this stage, able to show that f′ does not have infinite fibers, which is the reason we had to use n = 9 (rather that n = 3) in the proof of the last proposition.

Our next step is to show that in fact f is holomorphic (up to conjugation) on an open set, i.e. that M = I (for M as in the statement of the last proposition). First, we note that as an immediate application of the last result we do get the desired control on the dimension of f′(B).

Lemma 6.6. There is an open W ⊆ U such that f′(W) is an open subset of M GL_{1}(ℂ) (U, M are from 6.5).

Proof. By 6.5, M^{−1} ∘ f is holomorphic on U. In particular dim(M^{−1} · f′(U)) = 2, since (M^{−1} ∘ f)′ is again holomorphic, hence open. Therefore,

B := Int_{M GL_{1}(ℂ)} (f′(B)) ≠ ∅.

Take W := (f′|_{U})^{−1}(B) (it is open, since f is C^{1} on U). □

Proposition 6.7. f is holomorphic on an open V ⊆ ℂ.

Proof. Take W from 6.6. We may assume that 0 ∈ W. By 6.6, f′(W) − f′(0) is an open subset of MM_{1}(ℂ) for some M ∈ GL_{2}(ℝ) (we work in M_{2}(ℝ) now). Take 0 ≠ A ∈ f′(W) − f′(0) such that −A ∈ f′(W) − f′(0). Take a, b ∈ W such that

f′(a) − f′(0) = A, f′(b) − f′(0) = −A.

Then we have

f′(a) + f′(b) − 2f′(0) = 0.

Consider the ℂ_{f}-definable function

g(x) := f(a + x) + f(b + x) − 2f(x).

Then g′(0) = 0 and there is an open V ⊆ W such that 0 ∈ V, g is C^{1} on V and g′(V) ⊆ MM_{1}(ℂ). Therefore, M^{−1}g is holomorphic on V and (M^{−1}g)′(0) = 0. Thus 0 is a non-injective point of M^{−1}g, hence 0 is also a non-injective point of g. Therefore g satisfies the assumptions of Corollary 5.9 whence there are U_{1}, U_{2} ⊆ V open such that

dim(g′(U_{1}) · g′(U_{2})) ≤ 2.

By the argument as in 6.6, we have

dim(g′(U_{1})) = dim(g′(U_{2})) = 2.

Since the set of critical points of g|_{V} can not be 2-dimensional, we may assume that

g′(U_{1}), g′(U_{2}) ⊆ M GL_{1}(ℂ).

Therefore, it is enough to show: Claim: For any open B_{1}, B_{2} ⊆ GL_{1}(ℂ) if dim(MB_{1}MB_{2}) = 2 then M ∈ GL_{1}(ℂ).

Proof of Claim. For h ∈ M_{2}, let X_{h} := MB_{1}Mh. Since

MB_{1}MB_{2} = ⋃_{h∈B_{2}} X_{h},

there is h ∈ B_{1} such that

dim({h′ ∈ B_{2} | X_{h} ∩ X_{h′} ≠ ∅}) = 2.

Then for any such h′ we have

MB_{1}Mh ∩ MB_{1}Mh′ ≠ ∅
B_{1}Mh ∩ B_{1}Mh′ ≠ ∅
GL_{1}(ℂ)Mh = GL_{1}(ℂ)Mh′
Mh′h^{−1}M^{−1} ∈ GL_{1}(ℂ).

Hence the group

H := {h′′ ∈ GL_{1}(ℂ) | Mh′′M^{−1} ∈ GL_{1}(ℂ)}

contains an open set, so (since GL_{1}(ℂ) is connected) H = GL_{1}(ℂ) and M GL_{1}(ℂ) = GL_{1}(ℂ)M. Since GL_{1}(ℂ) is self-normalizing in GL_{2}(ℝ), we get M ∈ GL_{1}(ℂ). □

7. Rationality

In this section we finish the proof of our main result. For the first time in this paper we will have to use some non-trivial model theoretic results. To obtain an infinite field interpretable in ℂ_{f} we will use a variant of the Hrushovski-Weil group configuration theorem. We will then have to use known results of Hrushovski’s to show that in fact the field is definable in ℂ_{f} in order to conclude that it is ominimally definably isomorphic to ℂ. To conclude that f is rational with respect to this field structure we will have to use the purity of the field, as proved in [20]. The following will be the key to our argument:

Definition 7.1. Let M be a strongly minimal structure. A field configuration is a collection of tuples such that RM(g_{1}, g_{2}, g_{3}, b_{1}, b_{2}, b_{3}) = 5 and:

• RM(g_{i}) = 2 and RM(b_{i}) = 1 for all 1 ≤ i ≤ 3.

• The g_{i} are pairwise independent but g_{3} ∈ acl(g_{1}, g_{2}).
• RM(g_{1}, b_{1}, b_{2}) = RM(g_{2}, b_{2}, b_{3}) = RM(g_{3}, b_{1}, b_{3}) = 3.

• There are no other dependencies.

For a proof of the theorem below see 4.5 on page 206 of [22] and 3.27(2) of [23].

Theorem 7.2 (Hrushovski). If M is strongly minimal and admits a field configuration then there is a strongly minimal field interpretable in M.

Note that if K is an algebraically closed field then a field configuration can be easily constructed using the action of 𝔾_{m}(K) ⋉ 𝔾_{a}(K) on 𝔾_{a}(K). More precisely, take g_{j} := (α_{j}^{i}, α_{j}^{i}) for independent generic elements of K for i, j ∈ {1, 2}. Thinking of g_{j} as the affine transformation x ↦ α_{j}^{1}x + α_{j}^{2} we let g_{3} = g_{2} ∘ g_{1}. Take b_{1} ∈ K any generic element independent over g_{1}, g_{2} and let b_{2} = g_{1}(b_{1}) and b_{3} = g_{3}(b_{1}). It is easy to check that this gives us a field configuration in the structure K.

We will now use the analyticity of f on an open set together with the field configuration theorem to show:

Theorem 7.3. There is a ℂ_{f}-interpretable strongly minimal field.

Proof. Our aim is to construct a field configuration. Take V from 6.7. Without loss of generality, 0 ∈ V and let M := f′(0). We will find a field configuration of 𝔾_{m}(ℂ) ⋉ 𝔾_{a}(ℂ) acting on 𝔾_{a}(ℂ) within f′(V) and use it to construct a similar field configuration in ℂ_{f}. Take a_{1}, a_{2} ∈ ℂ o-minimally independent generics very close to 1 ∈ ℂ and b_{1}, b_{2}, b o-minimally independent generics over a_{1}, a_{2} in a small ball around 0 ∈ ℂ. Let

g := (a_{1}, b_{1}), h := (a_{2}, b_{2}) ∈ 𝔾_{m}(ℂ) ⋉ 𝔾_{a}(ℂ)

(note that g, h are very close to the identity of 𝔾_{m}(ℂ) ⋉ 𝔾_{a}(ℂ)). We get the afore mentioned group configuration

X := (g, h, hg, b, g · b, hg · b)

where · denotes the action of 𝔾_{m}(ℂ) ⋉ 𝔾_{a}(ℂ) on 𝔾_{a}(ℂ). Denoting, for (a, b) ∈ 𝔾_{m}(ℂ) ⋉ 𝔾_{a}(ℂ),

M(a, b) = (M · a, M + b),

(where on the right-hand side a, b ∈ GL_{1}(ℂ) ↪ GL_{2}(ℝ)) we get that

M(a, b) ∈ f′(V)

for all a close enough to 1 and b close enough to 0. Therefore, we get a group configuration MX ⊂ f′(V). We aim to show that for any six-tuple Y ⊂ V such that f′(Y) = MX, Y is a group configuration in ℂ_{f}. Since R-independence is stronger than ℂ_{f}-independence, we need to check only the algebraicity conditions. They come from either a group operation in 𝔾_{m}(ℂ)⋉𝔾_{a}(ℂ) or a group action of 𝔾_{m}(ℂ) ⋉ 𝔾_{a}(ℂ) on 𝔾_{a}(ℂ). We consider only the latter case, so we take g, b, g · b ∈ X. We have

g = (a_{1}, b_{1}), g · b = a_{1}b + b_{1}
Mg = (Ma_{1}, M + b_{1}), Mb = M + b, M(g · b) = M + a_{1}b + b_{1}.

Take α, β, γ, δ ∈ V such that

(f′(α), f′(β)) = Mg, f′(γ) = Mb, f′(δ) = M(g · b).

We want to show δ ∈ acl(α, β, γ).

f′(δ) = M + a_{1}b + b_{1} = M + M^{−1}f′(α)(f′(γ) − M) + f′(β) − M

= M^{−1}f′(α)f′(γ) − M^{−1}f′(α)M + f′(β).

Since we have set M = f′(0) and M ∈ GL_{1}(ℂ), M commutes with f′(α) (this is one of the reasons we needed f to be analytic on V and not only a translate of one) and we get

f′(0)f′(δ) = f′(α)f′(γ) − f′(0)f′(α) + f′(0)f′(β).

For each μ, ν ∈ ℂ there is a ℂ_{f}-definable function f_{μ,ν} which is holomorphic in a neighborhood of 0 such that

f_{μ,ν}(0) = 0, f′_{μ,ν}(0) = f′(μ)f′(ν).

Consider the function

g_{y} := f_{α,γ} − f_{α,0} + f_{β,0} − f_{y,0}.

We will show that g_{δ} is not a generic function in the strongly minimal family {g_{y} : y ∈ ℂ}. Since this family is defined over (α, β, γ) this will prove that indeed δ ∈ acl(α, β, γ). It will be enough to find infinitely many y such that |g_{y}^{−1}(0)| > |g_{δ}^{−1}(0)|. Since g_{δ} is holomorphic (which is another reason we needed f to be analytic on V) in a neighborhood of 0, and g′_{δ}(0) = 0, we get that g_{δ} has a multiple 0 at 0. Let {a_{1}, …, a_{k}} = g_{δ}^{−1}(0) and without loss of generality assume that a_{1} = 0. Let V_{i} ∋ a_{i} be pairwise disjoint open sets. Choose δ′ close enough to δ such that g′_{δ′}(0) ≠ 0. Therefore g_{δ′} has a single zero at 0. We use the Argument Principle to find c ∈ V_{1} \ {0} such that g_{δ′}(c) = 0. It remains to check that for δ′ close enough to δ, |g_{δ′}^{−1}(0) ∩ V_{i}| > 0 for all i > 1. By 4.17, d(0, g_{δ}, V_{i}) ≠ 0. By Axiom (D4) of the degree theory from the introduction to [5], d(0, g_{δ′}, V_{i}) ≠ 0 for δ′ close enough to δ. By 4.17 again, |g_{δ′}^{−1}(0) ∩ V_{i}| > 0. By Theorem 7.2, a strongly minimal field is interpretable in ℂ_{f}. □

It remains to prove:

Theorem 7.4. There is A ∈ GL_{2}(ℝ) such that ℂ_{A∘f∘A^{−1}} is interdefinable with the field of complex numbers.

Proof. We will need a little more model theory for this proof. Let K be the strongly minimal field interpretable in ℂ_{f} by the previous theorem. Note that in particular dim(K) = 2. By [23, 2.27], (ℂ, +) is K-internal, i.e. there exists a partial ℂ_{f}-definable surjective map g : K^{n} → (ℂ, +). Therefore the structure (ℂ, +) is interpretable in the structure K_{ℂ_{f}} whose universe is K and the definable sets are those which are ℂ_{f}-definable.

By Theorem 1.1. of [17] (and dim(K) = 2), there is an R-definable isomorphism of fields Φ : K → ℂ. Then Φ(K_{ℂ_{f}} ) is an R-interpretable, strongly minimal structure and the complex field is definable in Φ(K_{ℂ_{f}} ). By Theorem 1.3 of [20], the structure Φ(K_{ℂ_{f}} ) coincides with the complex field. Therefore, K_{ℂ_{f}} = K (as structures). Hence (ℂ, +) is interpretable in K. By [23, 3.1], [23, 4.13] and the remark on p. 69 of [23], (ℂ, +) is ℂ_{f}-definably isomorphic to a K-algebraic group. Since dim ℂ = dim K = 2 we get that also dim_{K}(ℂ, +) = 1. Since (ℂ, +) has no torsion, it is ℂ_{f}-definably isomorphic to 𝔾_{a}(K). Using the above isomorphism we can find a ℂ_{f}-definable ⋆ : ℂ^{2} → ℂ such that (ℂ, +, ⋆) is a field. Claim: There is A ∈ GL_{2}(ℝ) such that

A : (ℂ, +, ·) ≅ (ℂ, +, ⋆).

Proof. Let 1_{⋆} be the neutral element for ⋆ and for any a ∈ ℂ let us denote a_{⋆} := a·1_{⋆}. For any a ∈ ℂ, we have

a_{⋆} ⋆ 2_{⋆} = a_{⋆} ⋆ (2 · 1_{⋆}) = a_{⋆} ⋆ (1_{⋆} + 1_{⋆}) = a_{⋆} + a_{⋆} = 2 · a_{⋆} = 2 · a · 1_{⋆} = (a · 2)_{⋆}.

We can replace 2 above with any rational number to obtain

∀a ∈ ℂ, b ∈ ℚ a_{⋆} ⋆ b_{⋆} = (a · b)_{⋆} .

Fix a ∈ ℂ and consider 2 additive, ℂ_{f}-definable functions

ϕ_{a}(x) := a_{⋆} ⋆ x_{⋆}, ψ_{a}(x) := (a · x)_{⋆}.

By 4.6, ϕ_{a} and ψ_{a} are continuous on a cofinite set. But any additive function g continuous on a cofinite set is continuous everywhere – for c ∈ ℂ and c_{n} → c take N such that g is continuous at Nc, then Nc_{n} → Nc and f(Nc_{n}) → f(Nc), so f(c_{n}) → f(c). Therefore ϕ_{a}, ψ_{a} are continuous and they coincide on ℚ, so they coincide on ℝ. Take i^{⋆} ∈ ℂ such that i^{⋆} ⋆ i^{⋆} = −1_{⋆}. By an argument as above, we get that for all r ∈ ℝ, we have r · i^{⋆} = r_{⋆} ⋆ i^{⋆}. Let the first column of A consists of the real and imaginary part of 1_{⋆} and the second column of A consists of real and imaginary part of i^{⋆}. Take any 2 complex numbers a = x + iy, b = w + iv. We have

A(a · b) = A(xw − vy + i(xv + yw)) = (xw − vy)_{⋆} + (xv + yw) · i^{⋆}

= (xw − vy)_{⋆} + (xv + yw)_{⋆} ⋆ i^{⋆} = x_{⋆} ⋆ w_{⋆} − v_{⋆} ⋆ y_{⋆} + (x_{⋆} ⋆ v_{⋆} − y_{⋆} ⋆ w_{⋆})_{⋆} ⋆ i^{⋆}

= (x_{⋆} + i^{⋆} ⋆ y_{⋆}) ⋆ (w_{⋆} + i^{⋆} ⋆ v_{⋆}) = (x_{⋆} + y · i^{⋆}) ⋆ (w_{⋆} + v · i^{⋆}) = Aa ⋆ Ab. □

By the claim the complex field is definable in ℂ_{AfA^{−1}}. By [20, 1.3], AfA^{−1} is also definable in the complex field. Therefore, ℂ_{AfA^{−1}} is interdefinable with the complex field. □

Remark 7.5. (1) Note that in the end of the proof of 7.4, we get that AfA^{−1} is ℂ-rational, hence there is an open set U ⊆ ℂ on which both f and AfA^{−1} are holomorphic. As in the proof of 6.7 this implies that A is multiplication by a non-zero a ∈ ℂ. Since af(a^{−1}z) is rational, f is also rational and we can take A = I in the statement of 7.4. Therefore 6.3 is the only place of the proof where conjugation by a real matrix takes place. Hence A from the statement of 1.2 is indeed as was mentioned in 6.4.

(2) In particular we have shown the following statement:

Let f : ℂ → ℂ be R-definable such that ℂ_{f} is strongly minimal and not locally modular. Assume f is holomorphic on an open set. Then f is an almost ℂ-rational function and ℂ_{f} is interdefinable with the complex field.

(3) Note, however, that we still needed the claim in the proof of 7.4, since [17, 1.1] gives only a certain R-definable isomorphism between (ℂ, +, ·) and (ℂ, +, ⋆) and we want this isomorphism to be ℝ-linear. It is also easy to find ⋆ ≠ · such that (ℂ, +, ⋆) is isomorphic to the complex field by a ℂ-linear map (it must be of the form x ⋆ y = axy for a fixed a ∈ ℂ \ {0}).

Let us forget now all the changes we have done to f and just take it as it was in the statement of 1.2. We easily obtain:

Corollary 7.6. f is a conjugate of a complex constructible function.

Note that this is the best possible description of f, since any such a function yields a strongly minimal structure ℂ_{f}.

8. O-minimality

This section is dedicated to generalizing our main result to any o-minimal (real closed) field. More precisely, assume R is an ω-saturated o-minimal expansion of a real closed field R. Let K = R[i] be the algebraic closure of R and f : K → K an R-definable function such that K_{f} := (K, +, f) is strongly minimal and not locally modular. Theorem 1.2 generalizes naturally to:

Theorem 8.1. K_{f} is biinterpretable with the field K.

Before we give the details of the proof, we would like to mention that many classical geometrical and topological theorems translate naturally into our present context. Theorems such as the Mean Value Theorem, the Implicit Function Theorem (together with its relatives, the Open Mapping Theorem and the Inverse Function Theorem) as well as deeper theorems from differential geometry can be proved in the o-minimal context by investigating classical proofs and replacing the geometrical notions involved with their definable counterparts.

We do the exact same thing here. There is one significant difference (referred to in the introduction) with the classical case. Working over ℝ, the o-minimality assumption simplifies the proofs, but it is unclear to what extent it is actually needed. In the more general context considered here it gives the natural working grounds, since the pure topology on R may be totally disconnected and the usual topological notions will not be meaningful anymore. For example, in the totally disconnected case the topological dimension of every set in R is 0, contrary to the classical case where it coincides with the o-minimal one. Hence, the o-minimal dimension is the meaningful one in this context, and because it only applies to definable sets, the natural framework is that of R-definable sets and functions.

Since the methods of translating proofs are pretty standard by now, and our original proofs were written with the abstract real closed field context in mind, we just sketch how the translation can be done section by section, focusing on two places where it may not straightforward.

Section 2 was written in the general context and since we assume R to be ωsaturated needs no modifications. The same applies to Section 3, having in mind the theory of smooth o-minimal manifolds as developed, e.g., by Berarducci and Otero in [1].

We focus a bit more on Section 4. The fact that f is open (Lemma 4.3) goes through by the o-minimal version of the Invariance of Domain Theorem (see [11]) and gives the finiteness of the number of poles of f as in Lemma 4.4. The (definable) connectedness of the set of possible limits of f at a point (Lemma 4.5) also goes through – replace sequences and their limits with definable curves and their limits. The proof of the continuity and properness of f need no modification.

We postpone the discussion of the transfer of crucial facts from topological analysis from [27] (notably Theorem 4.11) to the Appendix. Other parts of Section 4.1 go through. For the degree theory we use Peterzil and Starchenko’s development of the theory in the o-minimal setting ([20] and [21]). This allows for a smooth adaptation of the topological Lemmas in subsection 4.2 to the present setting. Section 5 is done purely in the definable realm so needs no adaptation.

We focus now on Section 6. The corresponding Lie algebra theory for definable groups was developed by Peterzil, Pillay and Starchenko in [18] and [19]. The algebraic facts on Lie algebras we needed (classification of maximal solvable subalgebras of 𝔤𝔩_{2}(ℝ)) hold for any real closed field and the proofs from [18] work in the local case to give an analogue of Theorem 74 of [24]. The rest of Section 6 and the entire Section 7 go through, using the theory of K-holomorphic functions as developed in [20].

We conclude this section by showing that some care is needed when translating proofs from the classical case to the o-minimal one. The key result in Section 6 is the proof in Proposition 6.3 of a correspondence between local Lie subgroups of GL_{2}(ℝ) and solvable Lie subgroups of GL_{2}(ℝ). Our original proof of 6.3 went along the following path – a local subgroup X produces a Lie subalgebra 𝔥. From 𝔥 we can only get a virtual Lie subgroup of GL_{2}(ℝ), namely a smooth injective map ϕ : H → GL_{2}(ℝ), which allows for a simple conclusion of the argument.

However, this map need not be definable in any o-minimal expansion of ℝ as the following example shows.

Example 8.2. Let α ∈ ℝ \ ℚ, T = S^{1} × S^{1} and 𝔥 := (1, α)ℝ be a Lie subalgebra of Lie(T). Then 𝔥 corresponds to a virtual Lie subgroup ϕ : ℝ → T and ϕ(ℝ) is dense in T. In particular, ϕ(ℝ) is not definable in any o-minimal expansion of ℝ.

Taking a piece of ϕ(ℝ) which is a local Lie subgroup of T definable in an ominimal theory, we get a definable local subgroup not corresponding to any global definable subgroup (in the definable category, the notion of definable subgroup and virtual definable subgroup clearly coincide).

So, in o-minimal theories there is no chance for a full Lie correspondence

Lie subalgebras of Lie(G) ↔ Virtual Lie subgroups of G.

As was pointed out by Kobi Peterzil, there is no Lie correspondence even on the level of local subgroups in the real field, since one can not define there even locally subgroups of 𝔾_{m}(ℝ)^{2} which are graphs of non-rational power function. However, there is a natural o-minimal expansion of ℝ (by analytic functions) for which we have the full local correspondence. Hence, we find it natural to ask the following question:

Question 8.3. Assume R is an o-minimal expansion of a real closed field. Is there an o-minimal expansion R′ ⊇ R such that we have the correspondence

Lie subalgebras of 𝔤𝔩_{n}(R) ↔ R′-definable local subgroups of GL_{n}(R) ?

9. Appendix

We give a short overview of the results in topological analysis used in the paper, setting them in the o-minimal context. Since the classical proofs (taken from [27]) are given in a purely topological setting their adaptation to the o-minimal case is rather straightforward. For the sake of completeness, and in view of its importance to our analysis we give a proof of Theorem 4.11. Other results taken from [27] and used in the text transfer to the o-minimal context as easily. We will use the o-minimal version of the Jordan plane curve theorem (see [28]) without reference. All references to [27] are to Chapter VIII unless stated otherwise. We keep the notation and assumptions from the previous section.

The proof of the following can be found in Whyburn (4.2) and needs no adaptation to the general context:

Lemma 9.1. Any definable open mapping of one simple definable closed curve onto another is definably homeomorphic to the mapping z ↦ z^{k} for |z| = 1 and some k ≥ 1.

Definition 9.2. A definable mapping satisfying the conclusion of the previous lemma will be said to be of degree k.

The following should be standard (though possibly out of style):

Definition 9.3. A set T homeomorphic to a straight line interval is called a simple arc. If a_{1}, …, a_{k} are any (distinct) points in R^{n} we sometimes denote by a_{1}a_{2} … a_{k} an arbitrary simple arc T through a_{1}, …, a_{k} such that if ϕ : [c, d] → T is a homeomorphism witnessing it then ϕ^{−1}(a_{1}) = c, ϕ^{−1}(a_{k}) = d and ϕ^{−1}(a_{i}) < ϕ^{−1}(a_{i+1}) for 1 ≤ i < k.

The second lemma we will need is:

Lemma 9.4. Let A, B be definably homeomorphic to a definable closed disc, g(A) = B a definable finite-to-one open mapping. Let C = ∂A and J = ∂B, and assume that g(A^{o}) = B^{o}, g(C) = J and g ↾ C is of degree k. Then |g^{−1}(p)| ≤ k^{2} for all p ∈ B.

Proof. It will be enough to show that if a, b ∈ J, p ∈ B^{o} and apb is a definable simple arc spanning J (i.e. having no common points with J but a, b) then |g^{−1}(p)| ≤ m(ap) · m(pb) where for any set S we define m(S) to be the number of definable components of g^{−1}(S).

If this were not true there would exist definable components K of g^{−1}(ap) and H of g^{−1}(pb) such that |H ∩ K ∩ g^{−1}(p)| > 1. Clearly H and K are infinite and 1-dimensional. By 1-dimensionality for any generic (a_{1}, a_{2}) ∈ H we may assume that a_{2} ∈ acl(a_{1}) so for any a_{1} ∈ π(H), the projection of H on the first coordinate, the set {a_{2} | (a_{1}, a_{2}) ∈ H} is finite. Similarly for K. By the existence of Skolem functions (Fact 2.8(4)) there are finitely many definable continuous (by 2.8(1)) functions h_{1}(x) … h_{i}(x) and k_{1}(x), …, k_{r}(x) whose domains are intervals in R and such that

H = ⋃ Γ_{h_{i}}, K = ⋃ Γ_{k_{i}}.

Let a ∈ H, and assume without loss that a ∈ Γ_{h_{1}}. Since h_{1}(x) is continuous, for all ε > 0 small enough, Γ_{h_{1}} ∩ B_{ε}(a) is connected. By the existence of Skolem functions again, if for all ε > 0 there exists b ∈ H ∩ B_{ε}(a) \ Γ_{h_{1}} then there exists a definable curve γ ⊆ H such that for every 0 < ε < ε_{0} (some ε_{0} small enough),

γ(ε) ∈ H ∩ B_{ε}(a) \ Γ_{h_{1}}.

It follows (by the definition of o-minimality) that for some ε > 0 small enough {γ(x) : x < ε} ⊆ Γ_{h_{i}} for some i ≠ 1. So by continuity of h_{i}, lim_{x→0}γ(x) ∈ Γ_{h_{i}},

implying that (Γ_{h_{1}} ∪ Γ_{h_{i}}) ∩ B_{ε}(a) is definably connected (because the Γ_{h_{j}} are definably connected and those two have a as a common point). By induction it is now easy to check that H (and similarly K) are definably locally connected, so H ∪ K is definably path connected. Since H ∩ K ⊆ g^{−1}(p) it is in particular finite. From the assumption that |H ∩ K| > 1 it follows that H ∪ K contains a definable simple closed curve. So there exists a definable component T of A \ (H ∪ K) such that T ⊆ A \ C. But fr(T) ⊆ H ∪ K so T ∩ g^{−1}(apb) = ∅ and so g(T) would have to be a definable component of B \ (apb) which is impossible, since g(T) ∩ J = ∅. □

The next corollary is very easy:

Corollary 9.5. Let g(A) = B be as in the previous lemma, k = deg(g ↾ C).

(1) If k = 1 then g is a homeomorphism.

(2) For any p_{1}, p_{2} ∈ B, |g^{−1}(p_{1})| ≤ k |g^{−1}(p_{2})|. In particular, if for some p ∈ B we have |g^{−1}(p)| = 1 then |g^{−1}(q)| ≤ k for all q ∈ B.

We can now prove the main result of Whyburn we are using:

Proposition 9.6. Let A, B be definably homeomorphic to a definable closed disc, g(A) = B a definable finite-to-one open mapping. Let C = ∂A and J = ∂B, and assume that g(A^{o}) = B^{o}, g(C) = J. If there exists a point q ∈ B such that g^{−1}(q) = {p} for some p ∈ A then g is definably homeomorphic to z ↦ z^{k} on |z| ≤ 1 for some k ≥ 1.

Proof. By Lemma 9.1, g|_{C} is definably homeomorphic to z ↦ z^{k} on |z| = 1 for some k ≥ 1. Let aqb be any definable simple arc spanning J and

{a_{1}, …, a_{k}} = g^{−1}(a), {b_{1}, …, b_{k}} = g^{−1}(b)

be cyclically ordered on C. Because g is finite-to-one dim(g^{−1}(apb)) = 1. By ominimality (as in the proof of the Lemma 9.4 above) we get that g^{−1}(apb) is locally definably connected and since p must be in all its definable components, it must be definably connected. Thus, g^{−1}(aq) contains arcs of the form a_{i}q and g^{−1}(qb) contains arcs qb_{i} (i = 1, …, k). Since a_{i}p ∪ a_{j}p separates b_{i} and b_{k} for i ≠ j, and

(a_{i}p ∪ a_{j}p) ∩ (pb_{i} ∪ pb_{n}) = {p}

it separates pb_{i} from pb_{j}. Whence pb_{i} ∩ pb_{j} = {p}. By (2) of the last corollary,

g^{−1}(apb) = ⋃_{i=1}^{k}(a_{i}p ∪ pb_{i}).

Denote

C_{i} := a_{i}p ∪ pb_{i} ∪ b_{i}a_{i},

where a_{i}b_{i} is the arc in C satisfying a_{i}b_{i} ∩ g^{−1}(a) = {a_{i}}. It is now straightforward to check that g|_{C} satisfies the assumptions of the previous corollary with k = 1, and so is a homeomorphism. The conclusion now follows readily. □

To obtain (i) of theorem 4.11 we still have to show that if g is a definable finiteto-one continuous open map, for every q ∈ K there exists a small enough closed disc D ∋ q such that every definable component of g^{−1}(D) is definably homeomorphic to a closed disc. This is the analogue of (3.1) of Whyburn, which is easily translated into the present context as well. Alternatively, this can easily be obtained using the properness of g. Part (ii) of 4.11 is an immediate corollary of part (i), as it shows that the degree function is continuous.

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∗Mathematical Institute, Oxford University, 24-29 St Giles’, Oxford OX1 3LB, UK

E-mail address: hasson@maths.ox.ac.uk
URL: http://www.maths.ox.ac.uk/~hasson/

†Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

E-mail address: pkowa@math.uni.wroc.pl
URL: http://www.math.uni.wroc.pl/~pkowa/