Strongly minimal groups in o-minimal structures

We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a 2-dimensional group definable in M, and D = (G;+,...) a strongly minimal structure, all of whose atomic relations are definable in M. If D is not locally modular, then an algebraically closed field K is interpretable in D, and the group G, with all its induced D-structure, is definably isomorphic in D to an algebraic K-group with all its induced K-structure.

Zilber's Trichotomy Conjecture. The geometry of every strongly minimal structure D is either (i) trivial, (ii) non-trivial and locally modular, or (iii) isomorphic to the geometry of an algebraically closed field K definable in D. Moreover, in (iii) the structure induced on K from D is already definable in K (that is, the field K is "pure" in D).
The conjecture reduces by [6] to: If a strongly minimal structure D is not locally modular, then it interprets a field K, and the field K is pure in D.
In the early 1990s, Hrushovski refuted both parts of the conjecture. Using his amalgamation method he showed the existence of a strongly minimal structure which is not locally modular and yet does not interpret any group (so certainly not a field), see [8]. In addition he showed the existence of a proper strongly minimal expansion of a field, see [7], thus disproving also the purity of the field. Nevertheless, Zilber's Conjecture stayed alive since it turned out to be true in various restricted settings, and moreover its verification in those settings gave rise to important applications (such as Hrushovski's proof of the function field Mordell-Lang conjecture in all characteristics [9]).
A common feature to many cases where the conjecture is true is the presence of an underlying geometry which puts strong restrictions on the definable sets in the strongly minimal structure D. This is for example the case, when D is definable in an algebraically closed field ( [3], [13] and [28]), differentially closed field (see, for example, [14]), separably Date: October 4, 2018. The first author was supported by an Independent Research Grant from the German Research Foundation (DFG) and a Zukunftskolleg Research Fellowship. The second author was partially supported by an Israel Science Foundation grant number 1156/10. 1 closed field ( [9]), or algebraically closed valued field ( [12]). This is also the case when D is endowed with a Zariski geometry ( [10]).
Thus, it is interesting to examine the conjecture in various geometric settings. One such setting is that of o-minimal structures.
1.2. The connection to o-minimality. The complex field is an example of a strongly minimal definable in the o-minimal structure R; +, ·, < , and indeed, the underlying Euclidean geometry is an important component in understanding complex algebraic varieties. This leads to examining in greater generality those strongly minimal structures definable in o-minimal ones, and to the following restricted variant of Zilber's Conjecture, formulated by the third author in a model theory conference at East Anglia in 2005.
The o-minimal ZC. Let M be an o-minimal structure and D a strongly minimal structure whose underlying set and atomic relations are definable in M. If D is not locally modular then an algerbaically closed field K is interpretable in D, and moreover, K is a pure field in D.
(1) Because every algebraically closed field of characteristic zero (ACF 0 ) is definable in an o-minimal real closed field, Zilber's Conjecture for reducts of algebraically closed fields of characteristic zero is a special case of the o-minimal ZC. This variant of the conjecture is still open for reducts whose universe is not an algebraic curve. (2) The purity of the field in the o-minimal setting was already proven in [21], thus the o-minimal ZC reduces to proving the interpretability of a field in D.
(3) Since every definable algebraically closed field in an o-minimal structure has dimension 2 (see [23]), it is not hard to see that the above conjecture implies that the underlying universe of D must be 2-dimensional in M. Therefore, it is natural to consider the o-minimal ZC under the 2-dimensional assumption on D, which is the case of our Theorem 1.3 below. (4) By [5], if D is strongly minimal, interpretable in an o-minimal structure and in addition dim M D = 1, then D must be locally modular, thus trivially implying the o-minimal ZC in the case when dim M D = 1. (5) The theory of compact complex manifolds, denoted by CCM, (see [33]) is the multisorted theory of the structure whose sorts are all compact complex manifolds, endowed with all analytic subsets and analytic maps. It is known ([33, Theorems 3.4.3 and 3.2.8]) that each sort in this structure has finite Morley Rank, and also that the structure is interpretable in the o-minimal R an . Hence, every sufficiently saturated structure elementarily equivalent to a CCM is interpretable in an o-minimal structure. By [16], every set of Morley rank one in any model of CCM is definably isomorphic to an algebraic curve. Thus, Zilber's conjecture for reducts of CCM whose universe is analytically 1-dimensional reduces to the work in [3]. The higher dimensional cases may also reduce to the conjecture for ACF 0 but this is still open.
In [4] the following case of the o-minimal ZC was proven. Theorem 1.2. Let f : C → C be a definable function in an o-minimal expansion of the real field. If D = C; +, f is strongly minimal and non-locally modular (equivalently, f is not an affine map) then up to conjugation by an invertible 2 × 2 real matrix and finitely many corrections, f is a complex rational function. In addition, a function ⊙ : C 2 → C is definable in D, making C; +, ⊙ an algebraically closed field.
In our current result below we replace the additive group of C above by an arbitrary 2-dimensional group G definable in an o-minimal structure. Moreover, we let D be an arbitrary expansion of G and not only by a map f : G → G. Since strongly minimal groups are abelian ([27, Corollary 3.1]), we write the group below additively. Here is the main theorem of our article. Then there are in D an interpretable algebraically closed field K, a K-algebraic group H with dim K H = 1, and a definable isomorphism ϕ : G → H, such that the definable sets in D are precisely those of the form ϕ −1 (X) for X a K-constructible subset of H n .
In fact, the structure D and the field K are bi-interpretable.
Note that the theorem implies in particular that G is definably isomorphic in D to either K; + , K × ; · or to an elliptic curve over K.
1.3. The general strategy: from real geometry and strong minimality to complex algebraic geometry. Let M, G and D be as in Theorem 1.3. Since G is a group definable in an o-minimal expansion of a real closed field R, it admits a differentiable structure which makes it into a Lie group with respect to R (see [25]). We let F 0 be the collection of all differentiable (with respect to that Lie structure) partial functions f : G → G, with f (0 G ) = 0 G , such that for some D-definable strongly minimal S f ⊆ G 2 , we have graph(f ) ⊆ S f . We let J 0 f denote the Jacobian matrix of f at 0. The following is easy to verify, using the chain rule for differentiable functions: where on the left we use the group operation and functional composition, and on the right the usual matrix operations in M 2 (R). Let also The key observation, going back to Zilber, is that via the above equations we can recover a ring structure on R by performing addition and composition of curves in D. Most importantly, for the ring structure to be D-definable, one needs to recognize tangency of curves at a point D-definably. The geometric idea for that goes back to Rabinovich's work [28], and requires us to develop a sufficient amount of intersection theory for D-definable sets, so as to recognize "combinatorially" when two curves are tangent. This paper establishes in several distinct steps the necessary ingredients for the proof. In each of these steps we prove an additional property of D-definable sets which shows their resemblance to complex algebraic sets. We briefly describe these steps.
We call S ⊆ G 2 a plane curve if it is D-definable and RM(S) = 1 (we recall the definition of Morley Rank in Section 2.1). In Section 4 we investigate the frontier of plane curves, where the frontier of a set S is cl(S) \ S. We prove that every plane curve has finite frontier.
In Section 5 we consider the poles of plane curves, where a pole of S ⊆ G 2 is a point a ∈ G, such that for every neighborhood U ∋ a, the set (U × G) ∩ S is "unbounded". We prove that every plane curve has at most finitely many poles.
As a corollary of the above two results we establish in Section 6 another geometric property which is typically true for complex analytic curves. Namely, we show that every plane curve S whose projection on both coordinates is finite-to-one, is locally, outside finitely many points, the graph of a homeomorphism.
Next, we discuss the differential properties of plane curves, and consider in Section 8 the collection of all Jacobian matrices at 0 of local smooth maps from G to G whose graph is contained in a plane curve. Using our previous results we prove that this collection forms an algebraically closed subfield K of M 2 (R), and thus up to conjugation by a fixed invertible matrix, every such Jacobian matrix at 0 satisfies the Cauchy-Riemann equations.
In Section 9 we establish elements of complex intersection theory, showing that if two plane curves E and X are tangent at some point then by varying E within a sufficiently wellbehaved family, we gain additional intersection points with X. This allows us to identify tangency of curves in D by counting intersection points.
Finally, in Section 10 we use the above results in order to interpret an algebraically closed field in D and prove our main theorem.
Acknowledgements. The first author wishes to thank Rahim Moosa for many enlightening discussions on the subject, as well as the model theory groups at Waterloo and Mc-Master for running a joint working seminar on the relevant literature during the academic year 2011-2012. The authors wish to thank Sergei Starchenko for his important feedback. Finally, many thanks to the Oberwolfach Institute for bringing the authors together during the Workshop in Model Theory in 2016, and to the Institute Henri Poincare in Paris, for its hospitality during the trimester program "Model theory, Combinatorics and valued fields" in 2018.

Preliminaries
We review briefly the basic model theoretic notions appearing in the text. We refer to any standard textbook in model theory (such as [15, §6, §7]) for more details. Standard facts on o-minimality can be found in [30] whose Sections 1.1 and 1.2 provide most of the basic background needed on structures and definability.
2.1. Strong minimality and related notions. Throughout the text, given a structure N , by N -definable we mean definable in N with parameters, unless stated otherwise. We drop the index 'N -' if it is clear from the context. In the next subsection, we will adopt a global convention about this index to be enforced in  Let N = N, . . . be an ω-saturated structure. A definable set S is strongly minimal if every definable subset of S is finite or co-finite. We call N strongly minimal if N is a strongly minimal set.
Let N = N, <, . . . be an expansion of a dense linear order without endpoints. We call N o-minimal if every definable subset of N is a finite union of points from N and open intervals whose endpoints lie in N ∪ {±∞}. The standard topology in N is the order topology on N and the product topology on N n . Now let N be a strongly minimal or an o-minimal structure. The algebraic closure operator acl in both cases is known to give rise to a pregeometry. We refer to [15, §6.2] and [25, §1] for all details, and recall here only some. Given A ⊆ N and a ∈ N n , we let dim(a/A) be the size of a maximal acl-independent subtuple of a over A. Given a definable set C ⊆ N n , we define dim(C) = max{dim(a/A) : a ∈ C}, and we call an element a ∈ C generic in C over A in N if dim(a/A) = dim(C).
If N is a strongly minimal structure, then dim(C) coincides with the Morley rank of C, and we denote dim(a/A) and dim(C) by RM(a/A) and RM(C), respectively. In the o-minimal case, dim C coincides with topological dimension of C, and we keep the notation dim(a/A) and dim(C).
Let N be any structure. Given a definable set X, a canonical parameter for X is an element in N eq which is inter-definable with the set X, namelyā is a canonical parameter for X if ϕ(x,ā) defines X and ϕ(x,ā ′ ) = X for allā ′ =ā. Any two canonical parameters are inter-definable over ∅, and so we use [X] to denote any such parameter. Note that if X = X t 0 for some definable family of sets over ∅, {X t : t ∈ T }, then [X] ∈ dcl(t 0 ), but t 0 need not be a canonical parameter for X.
A structure N = N, . . . is interpretable in M if there is an isomorphism of structures α : N → N ′ , where the universe of N ′ and all N ′ -atomic relations are interpretable M.
If N is interpretable in M via α and M is interpretable in N via β, and if in addition β • α is definable in N and α • β is definable in M, then we say that M and N are bi-interpretable.
Note that if M is an o-minimal expansion of an ordered group, then by Definable Choice, every interpretable structure in M is also definable in M.

2.2.
The setting. Throughout Sections 4 -10, we fix a sufficiently saturated o-minimal expansion of a real closed field M = R; +, ·, <, . . . , and a 2-dimensional definable group G. By [25], the group G admits a C 1 -manifold structure with respect to the field R, such that the group operation and inverse function are C 1 maps with respect to it. The topology and differentiable structure which we refer to below is always that of this smooth group structure on G. As noted in [1,Lemma 10.4], the group G is definably isomorphic, as a topological group, to a definable group whose domain is a closed subset of some M r , endowed with the M r -topology. Thus, we assume that G is a closed subset of M r and its topology is the subspace topology.
Throughout Sections 4 -10, we also fix a strongly minimal non-locally modular structure D = G; . . . . We treat M as the default structure and thus use "definable" to mean "definable in M", and use "D-definable" to mean "definable in D". Similarly, we use acl, dim and 'generic' to denote the corresponding notions in M, and let acl D , RM, 'D-generic' and 'D-canonical parameter' denote the corresponding notions in D.
Since the underlying universe of the strongly minimal D is the 2-dimensional set G, it follows that for every D-definable set X ⊆ G n , we have dim X = 2 RM(X).
Also, for a ∈ G n and A ⊆ G, we have dim(a/A) ≤ 2 RM(a/A), and in particular, if X ⊆ G n is definable in D and a ∈ X is generic in X over A then it is also D-generic in X over A. The converse fails: Indeed, let M be the real field and D the complex field, interpretable in the real field M. The element π ∈ C is D-generic in C over ∅ but it is not generic in C over ∅ because it is contained in the definable, 1-dimensional set R.

2.3.
Notation. If S is a set in a topological space, its closure, interior, boundary and frontier are denoted by cl(S), int(S), bd(S) := cl(S) \ int(S) and fr(S) := cl(S) \ S, respectively. Given a group G, + and sets A, B ⊆ G, we denote by A − B the Minkowski difference of the two sets, A − B = {x − y : x ∈ A, y ∈ B}. Given a set X and S ⊆ X 2 , we denote S op = {(y, x) ∈ X 2 : (x, y) ∈ S}. The graph of a function f is denoted by Γ f . If γ(t) is map with image in the domain of a function f , we often write f (γ) instead of f (Im(γ)).

Plane curves
In this section, we work in a strongly minimal structure D and prove some lemmas about the central objects of our study, plane curves. When D expands a group G and is nonlocally modular, we construct in Subsections 3.3 and 3.4 two special definable families of plane curves which will be used in the subsequent sections.
3.1. Some basic definitions and notations. Let D be a strongly minimal structure.
Note that this gives a D-definable equivalence relation on any D-definable family of plane curves. A D-definable family of plane curves It is almost faithful if all ∼-equivalence classes are finite.
Note that if F = {C t : t ∈ T } is a faithful family of plane curves, then t is a canonical parameter for C t . If F is almost faithful, then t is inter-algerbaic with a canonical parameter of C t .
Given a D-definable family of plane curves F, there exists assuming acl D (∅) is infinite (if not, name any element not in acl D (∅)) a D-definable almost faithful family of plane curves F ′ = {C ′ t : t ∈ T ′ }, such that every curve in F has an equivalent curve in F ′ and vice versa (see, for example, [7], p.137) 1 . It is not hard to see that RM(T ′ ) is independent of the choice of F ′ . Thus we can make the following definition. Definition 3.3. A D-definable family of plane curves F as above is said to be n-dimensional, written also as RM(F) = n, if in the corresponding almost faithful family F ′ as above, we have RM(T ′ ) = n. We call F stationary if Morley degree(T ) = 1.
We call D a non-locally modular structure if there exists a D-definable family of plane curves F with RM(F) ≥ 2.
In fact, [26,Proposition 5.3.2], if D is non-locally modular, then for every n there exists an n-dimensional D-definable family of plane curves. We will sketch a proof of a slightly stronger result in Proposition 3.20 below.
The following terminology is inspired by [10]: We say that F is (generically) very ample if for every p = q ∈ G 2 (each D-generic over the parameters defining F), In the rest of this section, D = G; +, . . . denotes a strongly minimal expansion of a group G.

Local modularity.
Here we recall some basic facts about local modularity. Definition 3.5. A D-definable set is G-affine if it is a finite boolean combination of cosets of D-definable subgroups of G.
Lemma 3.8. For C a strongly minimal D-plane curve, and p, q ∈ G 2 , (1) C + p ∼ C + q if and only if p − q ∈ Stab * (C).
(2) Stab * (C) is trivial if and only if {C + p : p ∈ G 2 } is a faithful family.
We only use here the following characterization of non-local modularity in expansions of groups, which follows from [11]. Note that if F = {C + p : p ∈ T } is a D-definable family of plane curves, with C strongly minimal and T = G 2 , then for every p ∈ G 2 , In particular T (p) is strongly minimal so that, in fact, if RM(T (p) ∩ T (q)) = RM(T (p)) then T (p) ∼ T (q). We thus have the following lemma.
Lemma 3.10. If C is a strongly minimal plane curve and F = {C + p : p ∈ G 2 }, then the following are equivalent: (1) F is very ample Finally, given a D-definable family of plane curves F = {C t : t ∈ T }, we call C t a D-generic curve in F over A, if t is D-generic in T over A. We say that F is generically strongly minimal if every D-generic curve in F is strongly minimal.

3.3.
Dividing by a finite subgroup of G. The main goal of this subsection is to prove Lemma 3.11 below, which will be used in the proof of Theorem 1.3 in Section 10. It will also allow us to assume, without loss of generality, the existence of a D-definable faithful, very ample family of strongly minimal plane curves of Morley rank two (Proposition 3.13 below).
Given a strongly minimal plane curve C which is not G-affine, we plan to work with the family F = {C + p : p ∈ G 2 }. We know that Stab * (C) cannot be infinite but it can be a finite, non-trivial, group in which case F is neither faithful nor very ample. We next want to prove that dividing the structure D by a finite group is harmless.
We let D F be the structure whose universe is G/F and whose atomic relations are all sets of the form π F (S) for S ⊆ G n a ∅-definable set in D. The structure D F is again an expansion of a group.
The following result implies that for the purpose of our main theorem we may work with D F instead of D. Lemma 3.11. Assume that the group G has unbounded exponent. Then the structures D and D F are bi-interpretable, without parameters. In particular, D is bi-interpretable with an algebraically closed field if and only if D F is.
Proof. Because F and π F are ∅-definable in D, the structure D F is interpretable, with no additional parameters, in D, via the identity interpretation α(g + F ) = g + F .
Next, let us see how we interpret D in D F . Let n = |F |, and let π * F : G/F → G be the map defined as follows: Given y ∈ G/F , and x ∈ G for which π F (x) = y, let Since G is strongly minimal and has unbounded exponent, the group G[n] is finite and hence ker(π * F ) is finite, so dim Im(π * F ) = dim G/F = dim G. Because G is definably connected, π * F is surjective. Thus the homomorphism π * F induces an isomorphism of where g/n is any element h ∈ G such that nh = g. By our assumptions, π F (G[n]) is ∅-definable in D F , and therefore the quotient To see that this is indeed bi-interpretation, we first note that the isomorphism between D and its interpretation in D F is α • β, which equals β. It is clearly definable in D.
Let us examine the map induced on G/F by β • α and prove that it is definable in D F . We denote by F/n the preimage of F in G under the map g → ng. It is not hard to see that the image of F inside β(G) is the group π F (F/n) + π F (G[n]) = π F (F/n), and hence the isomorphism which β • α induces on G/F is This map is definable in the group G/F by sending g + F to the unique coset h + F/n such that nh + F = g + F .
This completes the proof that D and D F are bi-interpretable over ∅.
Note that in our case, when the group G is abelian and definable in an o-minimal structure then by [29], the group G has unbounded exponent, so the above result holds.
For the rest of this subsection, assume that D is non-locally modular, and fix (after possibly absorbing into the language a finite set of parameters) a strongly minimal plane curve C ⊆ G 2 which is D-definable over ∅ and not G-affine. Assume that F ′ = Stab * (C) is a finite group and let F ⊆ G be a D-∅-definable subgroup such that F ′ ⊆ F × F . Consider the structure D F expanding G/F, + as above.
Claim 3.12. π F (C) is strongly minimal in D F and Stab * (π F (C)) in (G/F ) 2 is trivial.
Proof. The strong minimality of π F (C) is immediate from the strong minimality of C in D.
We thus showed that Stab * (π F (C)) is trivial.
Combining Lemmas 3.10, 3.11 and Claim 3.12, we can conclude: Proposition 3.13. Assume D is non-locally modular, expanding a group G of unbounded exponent. Then there exists a finite group F ⊆ G, possibly trivial, and in the structure D F defined above there exists a definable family L 0 = {l t : t ∈ Q 0 }, of strongly minimal plane curves, which is faithful, very ample, and has RM(Q 0 ) = 2.
The structures D and D F are bi-interpretable, over the parameters defining F .
Assumption: For the rest of the article, we replace the structure D with the structure D F , and thus assume that a family L 0 as above is definable in D.

3.4.
Very ample families of high dimension. The goal of this subsection is to construct a larger family L ′ of plane curves which still has the geometric properties of the family L 0 from Proposition 3.13. The main method is to use composition of binary relations and families of plane curves. Recall the notion of a composition of binary relations, extending composition of functions: Given S 1 , S 2 ⊆ G 2 , we let Clearly, if S 1 , S 2 are D-definable, then so is S 1 • S 2 . We will be mostly interested in the composition of plane curves, and even more so, in the composition of families of plane curves: if L 1 , L 2 are D-definable families of plane curves, we let L 1 • L 2 := {C 1 • C 2 : C 1 ∈ L 1 , C 2 ∈ L 2 }. As a rule, geometric properties are not preserved under compositions of (families of) curves. The composition of two strongly minimal curves has, indeed, Morley rank 1, but it need not be strongly minimal. More generally, a D-generic curve of L 1 • L 2 need not be strongly minimal, and even if it were, L 1 • L 2 need not be faithful. In fact, although the dimension of L 1 • L 2 cannot decrease, it need not be greater than that of L 1 or L 2 . For example, if both families are the family of affine lines in A 2 then L 1 • L 2 = L 1 .
We will need a series of lemmas to address these issues. We start with the following easy observation: Lemma 3.14. Assume that L 1 = {C t : t ∈ T } and L 2 = {D r : r ∈ R} are two D-definable faithful, generically very ample families of plane curves. Then L 1 • L 2 is also generically very ample.
Proof. Let L = L 1 • L 2 and (a, b), (c, d) ∈ G 2 be distinct D-generic points. Let C = C 1 • C 2 for C i ∈ L i . Then (a, b) ∈ C if and only if there is e ∈ G such that (a, e) ∈ C 2 and (e, b) ∈ C 1 . For a D-generic e independent over all the data, This is true since D-genericity of (a, b) implies that a, b are D-generic in G and by choice of e the points (a, e) and (e, b) are, therefore, D-generic in G 2 . So RM(T (a, e)) = RM(T ) − 1, and RM(R(e, b)) = RM(R) − 1, with the desired conclusion.
Note that if e is non-D-generic, and since (a, e) and (e, b) are not D-algebraic over ∅, we get -from rank considerations -that RM(T (a, e)) < RM(T ) − 1 for all e ∈ G, and similarly RM(R(e, b)) < RM(R). So by the above calculations, varying e ∈ G we get by faithfulness of the families, Similarly, (c, d) ∈ C if and only if there exists e ′ such that (c, e ′ ) ∈ C 2 and (e ′ , d) ∈ C 1 . By assumption, fixing e, e ′ ∈ G, the Morley rank of R(a, e) ∩ T (c, e ′ ) is smaller than that of R(a, e), T (c, e ′ ) and R(e ′ , b) ∩ T (d, e ′ ). So Definition 3.15. Given two D-definable family of plane curves, L and L ′ , we say that L extends L ′ if for every C ′ ∈ L ′ there exists C ∈ L such that C ′ ⊆ C.
In the next couple of lemmas we show that although the composition of two families of curves need not preserve the properties of the original families (as already discussed), it extends a family of curves which does. Proof. Fix some C ∈ L which is D-generic over [E] and C E ⊆ E • C strongly minimal. Note that (E −1 • C E ) ∩ C is infinite, and since C is strongly minimal E −1 • C E is a set of Morley rank 1, containing the set C, up to a finite set. It follows that Absorbing [E] into the language, we can findc ∈ acl D ([C]) and a formula ϕ(x,c) defining C E . By compactness, there is a formula θ ∈ tp(c) such that wheneverc ′ |= θ there is some C ′ ∈ L such that ϕ(x,c ′ ) ⊆ E • C ′ , and for all D-genericc ′ |= θ the formula ϕ(x,c ′ ) is strongly minimal. We may further require -by compactness, again -that if ϕ(x,c ′ )∧ϕ(x,c ′′ ) is infinite then the symmetric difference ϕ(x,c ′ )△ϕ(x,c ′′ ) is finite for allc ′ ,c ′′ |= θ. By rank considerations the family ϕ(x, y)∧θ(y) is almost faithful, and therefore satisfies the required properties.
As an immediate application we get the following statement.
Corollary 3.17. Let L 1 , L 2 be faithful k-dimensional D-definable families of plane curves. Then L 1 • L 2 extends an almost faithful, stationary, generically strongly minimal family of plane curves of dimension at least k.
We can now formulate a definition.
Definition 3.18. Let L 1 , L 2 be faithful k-dimensional D-definable families of plane curves. We say that D-definable family L is a composition subfamily of L 1 • L 2 if (i) L is an almost faithful, stationary, generically strongly minimal family, extended by L 1 • L 2 , and (ii) L is such family of maximal dimension.
The last technical lemma we need is the following. (1) If some composition subfamily of L 1 • L 2 is k-dimensional then D interprets an infinite field. Proof.
(1) Assume that there exists a k-dimensional composition subfamily L of L 1 • L 2 . We will construct a group configuration (see [2]). Let S i be the parameter set of L i and S the parameter set of L. By strong minimality of D and the fact that L 1 and L 2 are both k-dimensional there is a finite-to-finite correspondence α between S 1 and S 2 . We may assume that α is ∅-definable in D and since field configurations are invariant under inter-algebraicity, we may also assume that S 1 = S 2 . By the same token there is also a finite-to-finite correspondence between S and S 1 , so we may also assume that S = S 1 . By assumption there is a D-definable correspondence β : S × S → S mapping (s, t) to a finite subset of S corresponding to C s • C t . Now let t, s ∈ S be independent D-generics, C s ∈ L 1 , C t ∈ L 2 . Let u = β(s, t), x ∈ G D-generic over all the data, y ∈ C t (x) and z ∈ C op s (x). Note that y ∈ C t • C s (z) so y is inter-algerbraic with z over u. This implies that {s, t, u, x, y, z} form a group configuration. The D-canonical base of tp D (x, y/t) is inter-algebraic with t, by almost faithfulness of L 1 , and similarly for the D-canonical bases of tp D (z, x/s) and tp D (z, x/u). This implies, see the concluding paragraph of [2] that k = 2, 3 that an infinite field is interpretable in D. This ends the proof of (1).
We can finally conclude the last result of this section. Proposition 3.20. There exists a D-definable almost faithful family of generically strongly minimal plane curves, F = {C t : t ∈ T }, which is generically very ample, and RM(T ) ≥ 3.
Proof. Let L 0 be as in Proposition 3.13. By Corollary 3.17, there is a composition subfamily F of L 0 • L 0 . If RM(F) = 2, then by Lemma 3.19(1) a field K is interpretable in D, and then the conclusion is obvious (take the family of polynomials of degree d > 1 over K). If RM(F) ≥ 3, then by Lemma 3.19 (2), F is generically very ample.
From now on, until the end of the paper, we fix a sufficiently saturated ominimal expansion of a real closed field M = R; +, ·, <, . . . , and a 2-dimensional group G = G; ⊕ definable in M. We also fix a strongly minimal non-locally modular structure D = G; ⊕, . . . definable in M. As discussed in Section 2.2, we include the index D when referring to definability, genericity and such in the structure D, and omit the index when referring to M. We also assume the existence of a D-definable, very ample family of plane curves L 0 , as noted after Proposition 3.13. In Sections 4 -7, we denote ⊕ by +, for simplification.

Frontiers of plane curves
4.1. Strategy. Our goal is to show (Theorem 4.9) that if S ⊆ G 2 is a D-definable set with RM(S) = 1, then its frontier fr(S) is finite and in fact contained in acl D ([S]). The geometric idea originates in [21] and it is implemented in Lemma 4.7 below, as follows. We consider the family L 0 from the assumption following Proposition 3.13. We also fix b ∈ fr(S) and consider a curve l q ∈ L 0 going through b with q generic over [S]. If l q meets S transversely at every point of intersection and b is sufficiently generic in G 2 , then by moving l q to an appropriate l q ′ close to l q , the curve l q ′ will intersect S near all points of l q ∩ S, and in addition at a new point near b. Since b itself was not in S it follows that a generic l q through b intersects S at fewer points than a generic curve in L 0 . Thus b is D-algebraic over [S] and in particular fr(S) is finite.
While this strategy works well when the curves in L 0 are complex lines in C 2 , the problem becomes more difficult when they are arbitrary plane curves and b is not necessarily generic in G 2 . To get around this problem, the idea in [4] was to replace S by its image under composition with a "generic enough" curve from a new "large" family L ′ (Proposition 3.20). We carry out this replacement in Lemma 4.8 below. An additional complication of this strategy in the current setting is that instead of the functional language in [4] we need to work with arbitrary curves, and control their composition 4.2. Two technical lemmas about 2-dimensional sets in G 2 . The following lemmas will be used in the sequel.
Assume that for all e ∈ E there are at most finitely many e ′ ∈ E, such that |Y e ∩ Y e ′ | = ∞. Then dim( t∈E Y t ) = 4.

Proof. The set
{(e, s) : e ∈ E , s ∈ Y e } has dimension k + 2. Therefore, if the union of the Y e had dimension smaller than 4, then for a generic s in this union, the dimension of E(s) = {e ∈ E : s ∈ Y e } is at least k − 1 ≥ 1, and in particular, is infinite. Hence, there are e 1 , e 2 ∈ E(s), independent and generic over s. Therefore, dim(e 1 , e 2 /s) = 2k − 2 and hence dim(e 1 , e 2 , s) = 2k − 2 + 3 = 2k + 1. But this is impossible since dim(e 1 , e 2 /∅) ≤ 2k and, by our assumption on the family, the set Y e 1 ∩ Y e 2 is finite, so s ∈ acl(e 1 , e 2 ).
Definition 4.2. We say that two 2-dimensional sets C 1 and C 2 intersect transversely at p ∈ C 1 ∩ C 2 if C 1 and C 2 are both smooth at p, and their tangent spaces at p generate the full tangent space of G 2 at p, namely Lemma 4.3. Let L = {l q : q ∈ Q} be a definable family of 2-dimensional subsets of G 2 , and S ⊆ G 2 a definable 2-dimensional set, all ∅-definable. Let q be generic in Q over ∅ and assume that l q and S intersects transversely at s. Then for every neighborhood U ⊆ G 2 of s, there exists a neighborhood V ⊆ Q of q, such that for every q ′ ∈ V , we have l q ′ ∩ S ∩ U = ∅.
Proof. Without loss of generality, U is definable over ∅ and l q ∩ U is smooth (we can shrink it so that q is generic in Q over the parameters defining it). Reducing U further, if needed, we may -by cell decomposition, and the assumption that l q is smooth at s -write l q ∩ U as the zero set of a definable C 1 -map F q : U → R 2 , and similarly write S as the zero set of a C 1 -map G : U → R 2 . The transversal intersection of l q and S implies that the joint is in its open image. We may choose U 0 so q is still generic over the parameters defining U 0 . It follows that there is a neighborhood

Bad points.
Recall that L 0 = {l q : q ∈ Q} is a faithful and very ample D-definable family of strongly minimal plane curves, with RM(Q) = 2. Notice that for b ∈ G 2 generic, the set Q(b) = {q ∈ Q : b ∈ l q } has Morley rank 1. We denote L 0 (b) = {l q : q ∈ Q(b)}.
Definition 4.4. Let U ⊆ G 2 be an open set and b ∈ G 2 . We say that L 0 (b) fibers U if for every s ∈ U there exists a unique q ∈ Q(b) such that s ∈ l q , the set Q(b) is smooth at q and furthermore the function s → q : U → Q(b) is a submersion at s (that is, the differential map between the tangent spaces is surjective).
Clearly, the set of b-bad points in G 2 in definable over b.
Lemma 4.6. For every b ∈ G 2 , the set of b-bad points has dimension at most 3.
By cell decomposition, for a fixed generic q ∈ Q, the set of points s ∈ l q failing (2) is at most 1-dimensional. So the set of all points s failing (2) is at most 3-dimensional.
We now fix a generic s ∈ G 2 over b, and show that it satisfies (1). The set of singular points q on Q(b) has dimension one, and for every such q, l q has dimension 2. Thus, the union of all such l q has dimension at most 3, and does not contain s. So if s ∈ l q for some q ∈ Q(b) then q is a smooth point on Q(b).
Since s is generic in G 2 , there are at most finitely many curves in L 0 (b) containing s.
We may choose W to be definable over generic parameters. Hence the first-order property over b: , and by the genericity of s in dom(g), the function g is a submersion at s, thus s is a b-good point.

4.4.
Finiteness of the frontier. The heart of the geometric argument is contained in the following lemma which shows that in a generic enough setting the frontier of S is indeed contained in acl D ([S]).
Proof. We may assume first that S t 0 is strongly minimal. Indeed, S t 0 is a finite union of strongly minimal sets, each definable over acl D (t 0 ). Clearly, b is in the frontier of one of those so we may replace S t 0 by this strongly minimal set, and modify the family F accordingly.
Denote S = S t 0 and B = Bad(b).
Proof of Claim 2. If not then by strong minimality of S, we would have S ∼ l q ′ for some q ′ ∈ Q, implying -since S = S t 0 and F is almost faithful -that t 0 ∈ acl(q ′ ). However, we assumed that dim(t 0 /∅) ≥ 6, while dim(q ′ /∅) ≤ 4, a contradiction.
We fix an element q ∈ Q(b) generic over t 0 and b. Since dim(b/∅) = 4, q is generic in Q over ∅, hence we have dim(q/∅) = 4.
Since L 0 is very ample, no two points in G 2 belong to infinitely many curves in L 0 , and hence each s ∈ S ∩ l q is inter-algebraic with q over t 0 and b. Thus such an s is generic in S over t 0 and b. So in particular S is smooth at s. It is not hard to see now (using the fact that F is almost faithful) that dim(s/b) = 4.
For the rest of this proof, we fix an element s ∈ S ∩ l q . Claim 3. The curve l q is smooth at s, and the intersection of S and l q is transversal at s.
Proof of Claim 3. Because dim(s/b) = 4, it follows from Claim 1 that s is b-good, so in particular l q is smooth at s and there exist neighborhoods U ⊆ G 2 of s and W ⊆ Q(b) of q, and a D-definable a parameter choice function is a 1-dimensional manifold (or finite), and it follows that for some q ′ in this image, l q ′ ∩ S is infinite. This contradicts Claim 2.
Proof of Claim 4. By assumptions, b ∈ l q is generic in G 2 over ∅. Thus, by shrinking V if needed, we may assume b is still generic in G 2 over the parameters defining V . Since Q(b) ∩ V is infinite, the first order statement: holds for b and therefore there is a neighborhood U ∋ b for which it holds.
Let N be the number of intersection points with S of a generic curve from L 0 over t 0 (recall that L 0 has Morley degree 1, so it has a unique generic type).
Claim 5. The curve l q intersects S in less than N points.
Proof of Claim 5. We write l q ∩ S = {s 1 , . . . , s n } (note that b is not among them). We first fix some open disjoint neighbourhoods U 1 , . . . , U n ⊆ G 2 , of s 1 , . . . , s n , respectively. By Claim 3 and Lemma 4.3, applied to each of the s i , there is a neighbourhood V ⊆ Q of q such that for every q ′ ∈ V , the curve l q ′ intersects S at least n times -at least once in each of the U i , i = 1, . . . , n. Next, we apply Claim 4 to V and find U 0 ∋ b, which we may assume is disjoint from all the U i , as in Claim 4.
Because b is in cl(S) \ S, we can find in S ∩ U 0 a D-generic element s ′ of S over t 0 , and by Claim 4, we can find in V a generic q ′ ∈ Q(s ′ ) over s ′ and t 0 . But now l q ′ intersects S at least n + 1 times: at s ′ and in each of U 1 , . . . , U n . Since S ∩ l q ′ is finite, the curve l q ′ is generic in L 0 over t 0 . So we have N ≥ n + 1 > n = |l q ∩ S|.
Finally, let us see that b ∈ acl D (t 0 ). By Claim 5 no generic curve in L 0 (b) intersects S t 0 in a generic number of points. So b is contained in the set Y of all those b ′ ∈ G 2 such that for all but finitely many q 1 ∈ L 0 (b ′ ), we have |l q 1 ∩ S| < N . The set Y is D-definable over ∅ and has Morley rank at most 1. Since t 0 is generic in T over ∅ and RM(T ) ≥ 3 we get that In our next step we show that the assumptions of Lemma 4.7 can be met for a D-definable set S of RM(S) = 1, after replacing S by its composition with a generic enough curve in a family L ′ as in Proposition 3.20.
If we write c = (c 1 , c 2 ) then by assumption, c 2 is generic in G over ∅. Fix an element b 2 ∈ G, which is generic over c 2 ∪ acl D ([S]) (abusing notation, in the present proof we will write [S] for acl D ([S])), and let t 0 be generic in and RM(C t 0 ) = 1, b 2 and c 2 are inter-algebraic in D over t 0 and [S], and hence so are (c 1 , b 2 ) and (c 1 , c 2 ).
Proof of Claim. Since c 2 is generic in G over ∅, (c 2 , b 2 ) is generic in G 2 over ∅ and therefore, by our choice of t 0 , the point (c 2 , b 2 ) is also generic in C t 0 over t 0 . Hence, the curve C t 0 is a homeomorphism at (c 2 , b 2 ). Denote this local map by f 0 . It follows that the map (x, y) → (x, f 0 (y)) is a local homeomorphism on a neighborhood W of (c 1 , c 2 ), sending (c 1 , c 2 ) to (c 1 , b 2 ). It is easy to verify that it sends every point in S ∩ W to a point in C t 0 • S, and therefore sends every point in cl(S) ∩ W to a point in cl and is contained, therefore, in an almost faithful family S ′ of the same rank. This gives condition (1) of the lemma, (2) is by the choice of t 0 , (3) is the line before the above claim, and (4) is what we just showed. So the lemma is proved.
We can now conclude the main result of this section.
Proof. Fix c ∈ fr(S). Replacing S by S + p for p generic in G 2 over c and [S], we may assume that dim(c/∅) = 4. We can now apply Lemma 4.8 and obtain t 0 , S ′ t 0 and b ∈ fr(S ′ t 0 ) as in the lemma. Working first in a richer language where [S] is ∅-definable, we may apply Lemma 4.7, and then conclude that b ∈ acl D (t 0 , [S]).
For p ∈ S, let U ∋ p be any neighborhood such that U ∩ fr(S) = ∅, and then S ∩ U is closed in U .

4.5.
A structural corollary on plane curves.
is the graph of an injective function in the x variable (we do not specify its domain).
We call x ∈ π(S) a non-injective point of S otherwise.
Note that at this point, if x is injective then it might be the case that for some y, the point (x, y) is an isolated point of S, or that S is locally at (x, y) 1-dimensional.
We now prove: Corollary 4.11. Let S ⊆ G 2 be strongly minimal and assume S is not ∼-equivalent to any fiber G × {a} or {a} × G. Then the set of non-injective points of S is finite and contained in acl D ([S]).
Proof. By Theorem 4.9, we may assume that S is closed. Let It is definable in D over the same parameters as S and RM(S 1 ) ≤ 1. Note that (a, a) / ∈ fr(S 1 ) if and only if there exists an open U ∋ a such that for all x ∈ G there exists at most one y ∈ U such that (x, y) ∈ S. This is equivalent to saying that S ∩ (G × U ) is the graph of a function.
Similarly, let Then . By Theorem 4.9, these frontiers are finite and contained acl D ([S]).
Finally, by our assumptions on S, for every a, b ∈ G the sets S a and S b are finite and therefore the set of (a, b) which are non-injective is finite and contained in acl D ([S]).

Poles of plane curves
Recall that we assume that G is a definable closed subset of some R n , equipped with the subspace topology, making it a topological group.
The goal of this section is to prove that just like affine algebraic curves in C 2 , every plane curve has at most finitely many poles. We may assume that 0 G = 0 ∈ R n , and for x ∈ G, we write B(x; ǫ) for all g ∈ G whose Euclidean distance from x is smaller than ǫ. We write B ǫ for B(0; ǫ). For A ⊆ G, and ǫ > 0, we let B(A; ǫ) = {y ∈ G : ∃x ∈ A , y ∈ B(x; ǫ)}.
Definition 5.1. Let S ⊆ G 2 be a definable set. We call a ∈ G a pole of S if for every open U ⊆ R n containing a, the set (U × G) ∩ S is an unbounded subset of R n . We denote the set of poles of S by S pol .
Note that then a ∈ S pol if and only if for every open U ⊆ R n containing a, S(U ) is unbounded. Another remark is that if S is G-affine then S pol = ∅. Indeed, if S is a subgroup of G 2 or its coset then its projection onto the first coordinate is a finite-to-one topological covering map, and hence S has no poles.
The main result of this section is the following.
Theorem 5.2. If S ⊆ G 2 is a D-definable set and RM(S) = 1, then S pol is finite.
Notice that if G is a definably compact group (for example, a complex elliptic curve) then G is a closed and bounded subset of R n , and hence S pol = ∅. So the theorem is about those G which are not definably compact.
Let us first introduce the key notion of "approximated points" and then discuss the strategy of our proof. Recall that for S ⊆ G 2 and x ∈ G, we let S x = {y ∈ G : (x, y) ∈ S}.
The set of such points b is denoted by A(S, I). We omit S from the above notation whenever it is clear from the context.
The following claim is immediate from the definitions. Here is a simple example.
Example 5.5. Let G = C, + and consider the complex algebraic curve The following are easy to verify: S pol = {0}, every b ∈ C is attained near 0, and thus A(S, {0}) = C.
The strategy of the proof of Theorem 5.2 is as follows. Assume towards a contradiction that the theorem fails. It is easy to see that we may assume that S is closed, strongly minimal and not G-affine. Now, for any such D-definable set S and infinite definable I ⊆ G, we first find an infinite definable set I 0 ⊆ I and an open bounded ball B ⊆ R n , such that the set A(S, I 0 ) \ B is at most 1-dimensional (Proposition 5.6(1)). Then, using further that S pol is infinite, we construct (Proposition 5.10) another D-definable setŜ, again closed, strongly minimal and not G-affine, and an infinite definableÎ ⊆ G, such that for every infinite definable set T ⊆Î and open bounded ball B, the set A(Ŝ, T ) \ B is 2-dimensional. A contradiction.

5.1.
Upper bound on dimension of the set of approximated points. The goal of this subsection is to prove the following proposition.
Proposition 5.6. Assume that S ⊆ G 2 is a D-definable strongly minimal closed set which is not G-affine, and let I ⊆ G be an infinite definable set. Then there is a definable 1- The rest of this subsection is devoted to the proof of Proposition 5.6. We fix throughout S and I as in its assumptions. By o-minimality dim S pol ≤ 1. Without loss of generality, S is defined over ∅.
We begin with an observation regarding the notions of Definition 5.3.
(2) Fix b generic in G over ∅, and assume that it is approximated near I. It follows from the definition that for every ǫ > 0, the element b is in the closure of Notice that the collection of Y ǫ forms a definable chain of definable sets decreasing with ǫ. We may now take ǫ sufficiently small, so that b is still generic in G over ǫ, and therefore b is generic in cl(Y ǫ ) over ǫ. Hence, b / ∈ fr(Y ǫ ), a set of dimension at most 1. It follows that b ∈ Y ǫ for all sufficiently small ǫ, and so b is attained near I.
The following technical claim about definable and D-definable sets will be used in the subsequent lemma.
Proof. Notice that if either I or J are finite then the result follows from our assumption on R. So we may assume that dim I = dim J = 1. By assumption on R, the projection of R on one of the coordinates of G 2 has infinite image. Let us assume it is the projection on the first coordinate. Hence for every D-generic a ∈ G, the set {(w, z) ∈ G × G : (a, w, z) ∈ R} is finite. Since I ⊆ G is infinite every generic of I is also D-generic in G. But then, for such an a ∈ I the set {(w, z) ∈ J × G : (a, y, z) ∈ R} is finite. Because of our assumption on R, it follows that dim(R ∩ (I × J × G)) ≤ 1.
We proceed with the proof of Proposition 5.6.
, and a definable set X ⊆ G, with dim X ≤ 1, such that for every b ∈ G \ X and for every ( Proof. Consider the D-definable set Since every generic fiber S x is finite, RM(T ) = 2. Also, it is easy to see that the projection of T on the last coordinate is infinite and hence for every D-generic b ∈ G the set . We also note, although we will not use this, that ( Claim 1. For b ∈ G and x 1 , x 2 ∈ G, the following are equivalent: , and by our assumption, (x 1 , x 2 ) / ∈ T b , so belongs to fr(T b ). Notice that y 1 (t) is bounded if and only if y 2 (t) is bounded, in which case, since S is closed, their limit points y 1 , y 2 satisfy (x 1 , y 1 ), (x 2 , y 2 ) ∈ S and y 2 − y 1 = b, so b is attained at (x 1 , x 2 ). Because we assumed that this is not the case, y 1 (t) and y 2 (t) are unbounded, hence x 1 , x 2 are both in S pol .
The other implications are easy, thus ending the proof of Claim 1.
By Theorem 4.9, for each b ∈ G, every element of fr(T b ) is contained in acl D (b). By compactness we may find then a set R ⊆ G 2 × G, D-definable over [S], such that for every b ∈ G the set R b is finite and contains fr(T b ). It follows that RM(R) = 1. Note however that we do not claim that for every b ∈ G, we have R b = fr(T b ). Thus, for example, we allow at this stage the possibility that the set of Assume first that the image of R under the projection onto the G 2 -coordinates, call it F 1 , is finite, and let F ⊆ G be a finite set, D-definable over acl D ([S]), such that F 1 ⊆ F 2 . We may take X = ∅ and complete the proof of the lemma in this case. Assume then that F 1 is infinite.
Let F 0 ⊆ G 2 be the set of all p ∈ G 2 such that the fiber R p ⊆ G is infinite. This is a finite set, D-definable over acl D ([S]), and because we assumed that F 1 is infinite, the set R * := (G 2 \ F 2 0 ) × G still has Morley rank 1, and the projection map from R * onto the This contradicts Claim 5.8.
By Claim 1, for every b ∈ G and for every ( Thus, we may take X = Y and F = F 0 and complete the proof of Lemma 5.9. Proof of Proposition 5.6 (1) . Fix a finite F ⊆ G as in Lemma 5.9, and a definable 1- we may shrink I 0 further and assume that the set S ∩ (I 0 × G) is closed and bounded. Thus, the set is a closed and bounded subset of G. By Lemma 5.9 and the choice of I 0 , there is a definable . Assume towards contradiction that the set A(S, I 0 ) \ B has dimension 2. By Lemma 5.7 (2), the set L of all b ∈ G \ B which are attained near I 0 has dimension 2, and therefore there is some b ∈ L which is not in X. By Lemma 5.7 (1), b is attained near some (x 1 , x 2 ) ∈ cl(I 0 ) = I 0 , and since b / ∈ X it is attained at (x 1 , x 2 ).
The rest of this subsection is devoted to the proof of Proposition 5.6 (2). Fix an open V ⊆ G containing 0. We may assume that V is bounded and symmetric, namely −V = V . Given r > 0, let P r = cl(B r ) ∩ G and S r = fr(B r ) ∩ G, the intersection of the r-sphere with G. Let B be as in Proposition 5.6(1).
Fix such r 0 , r 1 . For ǫ > 0 again let Proof of Claim 2. The family of Y ǫ decreases with ǫ, and we have already seen above that We restrict our attention to the definably compact set P r 1 \ int(B) and let ǫ is definably compact, and hence A r 1 (S, I 0 ) is also definably compact.
By the choice of B, Proposition 5.6(1) implies that dim(A(S, I 0 ) \ B) ≤ 1 and hence, since the boundary of B is at most 1-dimensional, also dim(A(S, I 0 )\int(B)) ≤ 1. It follows that A r 1 (S, I 0 ) is a definably compact set which is at most 1-dimensional. Using that, it is not hard to see that for sufficiently small open W ∋ 0 the set A r 1 (S, I 0 ) + W does not contain any translate of our open set V . Fix such a set W .
Because A r 1 (S, I 0 ) = ǫȲ r 1 ǫ it is not hard to see that there exists ǫ 0 > 0,Ȳ r 1 ǫ 0 ⊆ A r 1 (S, I 0 ) + W . It follows that the setȲ r 1 ǫ 0 does not contain any translate of V , thus proving Claim 2.
It is left to show that setting ǫ := ǫ 0 for ǫ 0 as provided by Claim 2, the requirements of Proposition 5.6 (2) are satisfied.
Proof of Claim 3. Assume towards a contradiction that no such r exists. Then we can find an unbounded, definably connected curve Γ ⊆ G such that Γ + V ⊆ S(B(I 0 , ǫ 0 )). It follows from the definition that Fix any γ 0 ∈ Γ and let Γ 0 = Γ − γ 0 . The curve Γ 0 is unbounded, definably connected, with 0 ∈ Γ 0 and in addition Γ 0 Indeed, although S r 0 = fr(B r 0 ) ∩ G need not be definably connected, by considering the curve Γ 0 in R n which is definably connected and unbounded, containing 0 we observe that Γ 0 must intersect the sphere fr(B r 0 ) ⊆ R n . This intersection point necessarily lies in S r 0 . Fix x 0 ∈ Γ 0 ∩ S r 0 .
By our choice of Γ 0 , x 0 + V ⊆ Γ 0 + V ⊆ Y ǫ 0 and by our choice of r 0 in Claim 1, Choose r as in Claim 3. Setting B ′ = P r and ǫ = ǫ 0 finishes the proof of Proposition 5.6 (2).

5.2.
Lower bound on dimension of the set of approximated points. In this subsection, we modify the set S from Lemma 5.11, using an idea from [4,Section 4]. The proof of Theorem 5.2 in the next subsection is by contradiction, and towards that we prove here the following lemma.
Lemma 5.10. Let S ⊆ G 2 be a D-definable strongly minimal, closed set which is not Gaffine, and assume that S pol is infinite. Then there is a strongly minimal closed setŜ ⊆ G 2 which is not G-affine, definable in D (over additional parameters), and there exists an infinite definableÎ ⊆ G, such that for every infinite definable set T ⊆Î and any bounded ball B, the set A(Ŝ, T ) \ B is 2-dimensional.
The rest of this subsection is devoted to the proof of Lemma 5.10. We apply Proposition 5.6 to the fixed set S and the infinite set S pol . We thus obtain and fix a definable one dimensional I 0 ⊆ S pol satisfying Clause (1) and (2) of the proposition. f (γ x (t)) = ∞, and for every x 1 , Proof. Using o-minimality and the fact that the projection of S onto G is finite-to-one, we may partition S and I 0 into finitely many cells and reach the following situation. There is a definable, definably connected bounded open U ⊆ G and a definable 1-dimensional smooth I 1 ⊆ I 0 , with I 1 on the boundary of U and U ∪ I 1 a manifold with a boundary. We may assume that cl(U ) ∩ S pol = cl(I 1 ). Furthermore, there is a definable, injective, continuous function f : U → G whose graph is contained in S, such that for every x 0 ∈ I 1 and every curve γ : (0, 1) → U tending to x 0 at 0, the image of γ under f is unbounded. After applying a definable local diffeomorphism, we may assume that I 1 = (a, b) × {0} ⊆ R 2 and U = (a, b) × (0, 1) ⊆ R 2 . By shrinking I 1 if needed we may assume that f is defined on the box [a, b] × (0, 1]. For ǫ ≤ 1, let × (0, ǫ)). When ǫ = 1, we denote C 1 , Γ 1,1 and Γ 1,2 by C, Γ 1 and Γ 2 , respectively. For every ǫ ≤ 1, the set C ǫ is a bounded set and Γ ǫ,i , i = 1, 2, are unbounded curves. Recall that ∂f (U ǫ ) denotes the boundary of f (U ǫ ) (which is contained in G). Because f : U → G is continuous and injective it is in fact a homeomorphism, by [?], hence ∂f (U ǫ ) = Γ ǫ,1 ∪ Γ ǫ,2 ∪ C ǫ (we use here the fact that the limit of |f (x)| as x tends to any point in I 1 is ∞).
The next claim roughly says that for an infinitesimal ǫ, the set f (U ǫ ) is contained in two infinitesimal tubes around Γ 1 and Γ 2 .
Proof of Claim 1. We fix ǫ 1 > 0. Using Proposition 5.6(2), we can find ǫ > 0 and a bounded neighborhood of 0, B ′ ⊆ G, such that for every y ∈ G \ B ′ , y + B ǫ 1 ⊆ f (U ǫ ). Next, choose 0 < ǫ 2 < min{ǫ, ǫ 1 }, such that f (U ǫ 2 ) does not intersect the bounded sets B ′ and C ǫ + B ǫ 1 . This can be done since the limit of |f (x)| is ∞ as x tends in U to any point in I 1 . We claim that this ǫ 2 satisfies our requirements.
By Claim 1, for each x ∈ I 1 , the curve f (γ x (t)) approaches one of the Γ i as t tends to 0, and therefore, after possibly re-parameterizing γ x , we can find γ i , i = 1, 2, such that lim t→0 f (γ x (t)) − γ i (t) = 0. The re-parametrization can be done uniformly in x. We can now find an infinite subinterval I 2 ⊆ I 1 and i ∈ {1, 2} such that if x ∈ I 2 then lim t→0 f (γ x (t)) − γ i (t) = 0.
Replacing I 1 by I 2 finishes the proof of Lemma 5.11.
The rest of this subsection is devoted to the proof of Lemma 5.10. We fix I 1 ⊆ I 0 , U, f, {γ x : x ∈ I 0 } as in Lemma 5.11. Without loss of generality, S is defined over ∅.
Because I 0 is smooth on the boundary of U , we can find an infinite sub-cellÎ ⊆ I 0 and c ∈ G generic over ∅ such that cl(Î + c) is contained in U . We fix suchÎ and c. A key initial observation is the following. Proof. Since f is continuous and cl(Î + c) ⊆ U , it follows that V c is bounded. Assume now towards contradiction that dim V c = 1. By [17,Lemma 2.7], the one dimensional set A c = f (T + c) is a translate of local subgroup H c of G (by that we mean that for all x, y, z ∈ A c sufficiently close to each other, x − y + z ∈ A c ).
By shrinking T if needed we may assume that c is still generic in G over the parameters defining T . It follows that for all We say that A c 1 and A c 2 have the same germ at 0 if there exists a neighborhood W 0 ∋ 0 such that W 0 ∩ A c 1 = W 0 ∩ A c 2 . Note that A c 1 and A c 2 have the same germ if and only if their intersection is infinite. This is an equivalence relation on (definable families of) definable sets. The collection of germs of the various A c ′ at 0 is the collection of equivalence classes of this equivalence relation. Because G is a two dimensional group, the collection of germs at 0 of the local subgroups A c ′ can be at most 1-dimensional. Indeed, we may choose V a sufficiently small neighborhood of 0 such that every element of V has infinite order in G. Thus, if A 1 and A 2 are definable and inequivalent local subgroups of G then Thus, the family of germs of the A c ′ is at most one dimensional.
Because c ′ varies in a two dimensional set, there are infinitely many c ′ for which the germ of A c ′ is the same. If we now fix generic and independent x, y, z ∈ T sufficiently close to each other, then there is w ∈ T and there are infinitely many c ′ such that It easily follows (see Lemma 3.10) that S is G-affine, contradicting our assumptions.
Consider now the D-definable set S ′ = {(x, y 1 − y 2 ) : (x + c, y 1 ), (x, y 2 ) ∈ S}. and the continuous functionf : Clearly, RM(S ′ ) = 1 and Γ(f ) ⊆ S ′ . We now want to replace S ′ with a D-definable strongly minimal subset containing the graph off . This is not hard to do and the details are given in Lemma 7.2. Thus we may conclude that Γ(f ) is contained in a strongly minimal set S ⊆ G 2 . Clearly, Γ(f ) pol ⊆Ŝ pol . Since fr(Ŝ) is finite, we may assume thatŜ is closed.
We can now proceed with the proof of Lemma 5.10. Let T be any infinite definable subset ofÎ, and B any open bounded ball. We want to prove that A(Ŝ, T ) \ B has dimension 2. Claim 1. There is a definable unbounded 1-dimensional group H ⊆ G, such that for every x ∈ T and h ∈ H, there is a definable π : (0, 1) → (0, 1), with π(0 + ) = 0 + and Proof of Claim 1. We first recall a theorem from [23]: Given a definable curve σ : (0, 1) → G with lim t→0 |σ(t)| = ∞, the set of all limit points of σ(t) − σ(s), as s and t tend to 0, forms an 1-dimensional torsion-free unbounded subgroup H σ ⊆ G. In particular, for each h ∈ H σ there is a definable function π h : (0, 1) → (0, 1) with π h (0 + ) = 0 + such that lim t→0 σ(π h (t)) − σ(t) = h. It follows from the definition of H σ that for every other definable We now apply this result to the unbounded curves f (γ x (t)), x ∈ T , and obtain the desired H.
We have: As t tends to 0, for i = 1, 2, the curve γ x i (π(t)) + c) still tends to x i + c so its image under f tends to f (x i + c). By Lemma 5.11 (3), lim t→0 f (γ x 2 (t)) − f (γ x 1 (t)) = 0. Thus, the above expression tends to f (x 1 + c) − f (x 2 + c) + h = b + h, proving that b + h can be approximated near T .
We can now conclude the proof of Lemma 5.10, as follows. Because V 1 and B are bounded, we can find r 0 > 0 such that for every h ∈ G \ B r 0 , the set V 1 + h ⊆ G \ B. In particular, for every b ∈ V 1 , b+ (H \B r 0 ) ⊆ G\B. Moreover, since H is unbounded, H \B r 0 has dimension 1. Hence, by Claim 2, the 2-dimensional set V 1 + (H \ B r 0 ) is contained in A(Ŝ, T ) \ B, as needed.

Proof of Theorem 5.2.
Assume towards a contradiction that S pol is infinite. Since for any S 1 , S 2 ⊆ G 2 , (S 1 ∪ S 2 ) pol = S 1pol ∪ S 2pol , and S pol = cl(S) pol , we may assume that S is strongly minimal and closed. Since S pol = ∅, we have that S is not G-affine. By Lemma 5.10, there is a D-definable setŜ which is closed, strongly minimal and not G-affine, and an infinite definableÎ ⊆ G, such that for every infinite set T ⊆Î and open bounded ball B, A(Ŝ, T ) \ B is 2-dimensional. This contradicts Proposition 5.6(1) forŜ andÎ.
Example 5.14. One of the difficulties in the above proof was the need to replace the initial set S with a setŜ, in order to reach a situation where dim(A(Ŝ, T ) \ B) = 2, for every infinite T ⊆Î ⊆Ŝ pol . The following example shows that the initial S can indeed have infinitely many poles and yet dim A(S, I 0 ) = 1 for some (in fact, any bounded) infinite I 0 ⊆ S pol . Consider the graph of function f : with G = C, + . The function f is a bijection of C which is its own inverse. Its set of poles is the x-axis. For every x ∈ R, as (x, y) → (x, 0), f (x, y) approaches the y-axis, with |f (x, y)| → ∞. Thus, for any bounded I 0 ⊆ R × {0}, A(S, I 0 ) = y-axis. After moving tô S as in the proof of Lemma 5.10, we can see that dim(A(Ŝ, T ) \ B) = 2, for any infinite T ⊆Ŝ pol and bounded ball B.

Topological corollaries
We establish here several topological properties of plane curves, typically true for complex algebraic plane curves. These properties are used later on in our proof of the main theorem. Definition 6.1. For S ⊆ G 2 and a = (a 1 , a 2 ) ∈ S, we say that S is an open relation at a if for every open box B = B 1 × B 2 ∋ a, a 1 ∈ int(π 1 (B ∩ S)) and a 2 ∈ int(π 2 (B ∩ S)) (here π 1 and π 2 are the projections onto the first and second coordinates).
We say that S is open over a 1 ∈ π 1 (S) if for every a 2 such that (a 1 , a 2 ) ∈ S and every open box B = B 1 × B 2 ∋ (a 1 , a 2 ), a 1 ∈ int(π 1 (B ∩ S)).
Note that if there exists an open box B ∋ a = (a 1 , a 2 ) such that a 1 / ∈ int(π 1 (B ∩ S)) then the same remains true for all smaller open boxes. Lemma 6.2. Assume that S ⊆ G 2 is a plane curve. Then there are at most finitely many a 1 ∈ π 1 (S) such that S is not open over a 1 . In particular, S does not contain any 1-dimensional components.
If S is strongly minimal and its projection on both coordinates is finite-to-one then there are at most finitely points a ∈ S such that S is not open at a.
Proof. First note that if S = S 1 ∪ S 2 and S is not open over a 1 ∈ G then either S 1 is not open over a 1 or S 2 not open over a 1 . Thus we may assume that S is strongly minimal. Without loss of generality, S is defined in D over ∅.
Assume towards contradiction that the set N of all a in π 1 (S) over which S is not open is infinite. Pick a 1 generic in N over ∅. Because RM(π 1 (S)) = 1, the point a 1 is D-generic in π 1 (S) over ∅.
Fix a = (a 1 , a 2 ) ∈ S and B = B 1 ×B 2 ∋ a such that a 1 / ∈ int(π 1 (S ∩B)). Let B S = S ∩B and writeB S := cl(B S ). Note that a is D-generic in S over ∅.
By Theorem 5.2, S has finitely many poles and since dim(a 1 /∅) ≥ 1, the point a 1 is not a pole of S. By Corollary 4.11, there are at most finitely many points in π 1 (S) at which S is is non-injective and each one of those is in acl D ([S]). Thus a 1 is an injective point of S. By Theorem 4.9, fr(S) ⊆ acl D ([S]) and hence we have ({a 1 } × G) ∩ fr(S) = ∅.
Since a 1 / ∈ int(π 1 (B S )) there exists a definable curve γ : (0, 1) → B 1 \ π 1 (B S ) such that lim t→0 γ(t) = a 1 . Notice that for t small enough γ(t) must be D-generic in G, and therefore, because π 1 (S) is co-finite in G, γ(t) is D-generic in π(S) over ∅. So, we may assume that the fiber S γ(t) has constant size n ≥ 1. For each t, let y 1 (t), . . . , y n (t) ∈ G be distinct such that (γ(t), y i (t)) ∈ S. Because γ(t) / ∈ π 1 (B S ), none of the y i (t) is in B 2 . Since a 1 / ∈ S pol , each of the curves γ i (t) is bounded, and hence has a limit y i ∈ G \ B 2 . Since ({a 1 }×G)∩fr(S) = ∅ each of the limit points (a 1 , y i ) is in S and in addition (a 1 , a 2 ) ∈ S, with a 2 = y i for all i. However, since a 1 is D-generic we must have |S a 1 | = n. This implies that for some i = j, we have y i = y j , so S is non-injective at (a 1 , y i ), contradiction.
Assume now that S is strongly minimal and its projection on both coordinates is infinite. We apply the above to both S and S op , and then by removing from π 1 (S) and π 1 (S op ) finitely many points, we remain, by our assumption on S, with a co-finite subset of S at which it is an open relation.
We are ready to deduce two useful topological corollaries: Proof. Assume towards contradiction that S is not an open relation at a, say not open over a 1 (since our assumptions on a are symmetric, the argument for a 2 is identical). In order to reach a contradiction it is sufficient, by the previous lemma, to conclude that there are infinitely many points in π 1 (S) over which S is not open.
Note that since a is not isolated and {a}×G is finite, the set π 1 (B ∩S) is infinite for every open B ∋ a. By Lemma 6.2, we may find B ∋ a sufficiently small such that S is open over all but finitely many points in π 1 (B ∩ S). It follows that dim(B ∩ S) = dim(π 1 (B ∩ S)) = 2 for all such B.
By Theorem 4.9, we may find an open box B = B 1 × B 2 containing a such that S ∩ cl(B) is closed, and a 1 / ∈ int(π 1 (B ∩ S)). Let B S = B ∩ S and denoteB S = cl(B S ). Repeating the argument with a smaller box, we see that we also have a 1 / ∈ int(π 1 (B S )) = int(π 1 (S∩cl(B))).
Since S ∩ cl(B) is closed it follows that (a 1 , b) ∈ S, and therefore by our assumptions, b = a 2 . But then the curve γ(t) = (γ 1 (t), γ 2 (t)) tends to a, so for small enough t, it must belong to the open set B, and its projection is not in int(π 1 (B S )). Therefore S is not open over every γ 1 (t) for t small enough. This contradicts Lemma 6.2 and ends the proof of the first clause.
Assume now that in addition a is an injective point of S op . Then there exists a box B = B 1 × B 2 such that no point in B S is isolated, and the projection of B S on B 1 is at most one-to-one. By what we just saw, we may shrink B so that S is open over every point in π 1 (B S ) and therefore B S is the graph of an open map f : B 1 → B 2 . Since S is closed in B for a sufficiently small B, the function f is continuous on B 1 .
Notice that though, by o-minimality, the set of isolated points of any plane curve, S, is finite we still do not know at this point that it is contained in acl D ([S]). However, by Corollary 4.11, the non-injective points of S and of S op are in acl D ([S]) so we may conclude from the above: Corollary 6.4. Let S ⊆ G 2 be a plane curve whose projection on both coordinates is finiteto-one. If a ∈ S is a non-isolated point which is D-generic in S over

On D-functions
Every plane curve S ⊆ G 2 gives rise to a definable partial function from G into G, around almost every point in S (except when S is contained in finitely many fibers {a} × G). The goal of this section is to establish the basic theory of such functions. (1) We say that f : U → G is a D-function if there exists a plane curve S ⊆ G 2 such that Γ f ⊆ S. We say in this case that S represents f . (2) We say that f is D-represented over A is there exists S representing f which is D-definable over A.
Note that our definition does not require that S is, locally at (x 0 , f (x 0 )), the graph of a function, but only that it contains the graph of f . Indeed, at least for some of the D-functions which we need to consider we do not know if this stronger notion is true as well. Proof. Assume that f : U → G is D-represented over A by S. We let S = S 1 ∪ · · · ∪ S r be a decomposition of S into strongly minimal sets, definable in D over acl(A). By Theorem 4.9, we may assume, by adding finitely many points in acl D (A), that each S i is closed in G 2 , but now the intersection S i ∩ S j for i = j may be non-empty and finite. We claim that one of the S i must contain Γ f . Indeed, for each i = 1, . . . r, let C i = π(S i ∩ Γ f ) ⊆ U , where π : G 2 → G is the projection on the first coordinate. By the continuity of f , these are definable, relatively closed subsets of U , whose pair-wise intersection is at most finite. Let Because U is open and definably connected so is U ′ . For i = 1, . . . , r let C ′ i = C i ∩ U ′ . Now the C i 's are pairwise disjoint and still relatively closed in U ′ . But now each of the C ′ i is clopen in U ′ so for some j, C ′ j = U ′ . Because C j is closed in U it follows that C j = U . Proof. Let us see first that we may assume that each S t is a closed subset of G 2 . By Theorem 4.9, there exists a D-definable family of finite sets {R t : t ∈ T }, such that for every t ∈ T , cl(S t ) ⊆ S t ∪ R t . Since R t is finite it follows that S t ∪ R t is closed, so we can replace the original family with {S t ∪ R t : t ∈ T } (note that we do not claim that R t = fr(S t )).
By fixing a coordinate system near 0 we can identify some neighbourhood W ∋ 0 in G with an open subset of R 2 . For each r > 0, we consider the disc B r centered at 0, and let S r t = S t ∩ (B r × W ). By o-minimality, there exists a uniform cell decomposition of the sets {S r t : t ∈ T, r > 0}. In particular, there is a bound k ∈ N such that every such decomposition contains at most k cells. By allowing cells to be empty we obtain a definable collection of cells {C r t,i : t ∈ T , r > 0, i = 1, . . . , k}, such that for every t ∈ T , r > 0, Recall that the notion of a decomposition implies that for C r t,i , C r t,j , if π : G 2 → G is the projection onto the first coordinate then either π(C r t,i ) = π(C r j,j ) or π(C r t,i ) ∩ π(C r j,j ) = ∅.
Claim. For every t ∈ T , and a D-function f ∈ F 0 , the following are equivalent: (1) S t represents the germ of f at 0.

Proof of Claim.
(1) ⇒ (2). We assume that S t ∩(B r ×G) contains the graph of f |B r , for r > 0. To simplify notation we omit r and consider the cell decomposition S t = C t,1 ∪ · · · ∪ C t,k . We let A ⊆ {1, . . . , k} be all i such that C i ∩ Γ f = ∅. We fix a cell C = C i with i ∈ A and claim that C ⊆ Γ f . Without loss of generality dim C > 0, and since C ⊆ Γ f , the projection π : C → G is injective. Since C is definably connected it is sufficient to prove that C ∩ Γ f is clopen inside C. Because C is locally closed and f is continuous, it follows that C ∩ Γ f is closed in C, so we need to prove that it is also open in C.
Let U ⊆ B r be an open neighborhood of x 0 , and consider U ∩ π(C). Since Γ f ⊆ S t , there exists a cell C ′ in the decomposition of S t which contains Γ f ∩ [(U ∩ π(C)) × G] for some open set U ∋ x 0 . But then π(C) ∩ π(C ′ ) = ∅ and therefore π(C) = π(C ′ ). By the continuity of f it follows that (x 0 , f (x 0 )) ∈ C ′ , forcing C ′ = C. It follows that C ∩ Γ f is clopen in C, and therefore C ⊆ Γ f . We showed that for each i ∈ A, C i ⊆ Γ f and hence Γ f = i∈A C i .
We now return to the proof of Proposition 7.3 and consider the uniform decomposition For each A ⊆ {1, . . . , k}, we consider The family F = {G r t,A : G r t,A is the graph of a continuous function on B r } is definable in M, as t varies in T , A varies among subsets of {1, . . . , k} and r > 0. By the above claim, this family satisfies our requirements.

Remark.
(1) Note that in the above family F of D-functions, each germ of a function appears infinitely often since we allow arbitrarily small r. One can divide the family, definably in M, by the equivalence of germs at 0 and then, using Definable Choice in o-minimal structures, obtain a unique D-function in the family representing each germ. Thus, if f ∈ F 0 is represented by plane curve S t then there exists g ∈ F 0 which has the same germ as f at 0 and is definable in M over t.
(2) It follows from the above that if S t represents f ∈ F 0 then J 0 (f ) is in dcl(t).
Notation. For a D-function f , we reserve the notation S f for a strongly minimal set representing f . Definition 7.4. We say that a D-function f : U → G is G-affine if there exist non-empty open sets V ⊆ U and W ∋ 0 such that for every x 1 , x 2 ∈ V and x ∈ W , As we already noted for G-affine subsets of G 2 , we have: Remark 7.6. If f is G-affine and f (0) = 0 then f is a partial group homomorphism, in a neighborhood of 0. Hence, if f is G-affine and J 0 f = 0 then f vanishes at 0.
As we already saw in Fact 3.9, since D is not locally modular there exists at least one D-function which is not G-affine. 8. The ring of Jacobian matrices 8.1. The ring R. Our next goal is to show that if f is a D-function then its Jacobian matrix vanishes at 0 if and only if f is not locally invertible at 0. This will be done in this and the next section. In the present section we prove that similarly to a complex analytic function, the Jacobian matrix is non-zero if and only if it is an invertible matrix.
Throughout this section we fix a definable local coordinate system for G near 0 G , identifying 0 G with 0 ∈ R 2 . From now on we identify G locally with an open subset of R 2 . For a differentiable D-function f in a neighborhood of 0, with f (0) = 0, the Jacobian matrix at x, denoted by J x f , is computed with respect to this fixed coordinate system, and we denote by |J x f | its determinant. We use d x f to denote the differential of f , viewed as a map from the tangent space of G at x, denoted by T x (G), to T f (x) (G). As we soon observe, the collection of all matrices J 0 f forms a ring, and the main goal of this section is to show that it is in fact a field (thus every nonzero matrix is invertible).
We first observe the following: It is important here to distinguish between the group operation in G and the usual ring operations in M 2 (R). Thus we reserve the additive notation ± for matrix addition, and let ⊕, ⊖ denote the group operations in G.
Lemma 8.3. The set R is a subring of M 2 (R) and for every A ∈ R which is invertible, Proof. We first note that the collection of germs of functions in F 0 is closed under ⊕ and functional composition. Indeed, if S f and S g represent D-functions f and g in F 0 then the plane curve S f • S g represents f • g and the plane curve Using the chain rule it is easy to verify that for f, g ∈ F 0 , J 0 (f ⊕ g) = J 0 f + J 0 g, and J 0 (f • g) = J 0 f · J 0 g. Since the germs in F 0 are closed under ⊕ and functional composition, it follows that R is a ring. If J 0 f is invertible then f is a locally invertible function in which case it is clear that f −1 is also in F 0 , and therefore (J 0 f ) −1 ∈ R.
Note. Even if S f and S g are locally at (0, 0) the graphs of functions, the sets S f • S g and S f ⊞ S g need not be such, but they still represent f • g and f ⊕ g, respectively. It is in fact possible that no strongly minimal set representing f • g will be locally at (0, 0) the graph of a function.

Definability and dimension of R.
Our aim is to show that R is a definable field isomorphic to R( √ −1). This is achieved in several steps. We first show (Theorem 8.11) that R is a definable ring of one of two kinds, and then -by eliminating one of these possibilities -we deduce the desired result.
We are going to use extensively the following operation: Definition 8.4. For a D-function f which is C 1 in a neighborhood of some a ∈ G, we let .
If we let ℓ a (x) = x ⊕ a then we have: Proof.
(1) is easy to verify and (2) follows, so we prove (3): Note that As we noted in the proof of Lemma 8.3, J 0 (h 1 ⊖ h 2 ) = J 0 h 1 − J 0 h 2 , therefore the above equals We are going to need the following: Lemma 8.6. There are invertible matrices in R arbitrarily close to the 0 matrix.
Proof. We go via the following claim which is also used later in the text.
Claim 8.7. There exists g ∈ F 0 which is not G-affine, with J 0 g = 0.
Proof. For y ∈ G and n ∈ N we write ny := n-times y ⊕ · · · ⊕ y . Fix f : U → G in F 0 which is not G-affine, and for n ∈ N, let g n (x) = f (nx) − nf (x). It is easy to see that g n ∈ F 0 and J 0 g n = nJ 0 f − nJ 0 f = 0. We want to show that for some n ∈ N, the function g n is not G-affine, so gives the desired g.
Notice that if g n is G-affine, then since J 0 g n = 0, the function g n must vanish on its domain. Assume towards contradiction that for every n ∈ N, the function g n vanishes on its domain, namely f (nx) = nf (x) whenever nx ∈ U . Pick a D-generic x 0 ∈ U sufficiently close to 0 so that for all n, nx ∈ U and nf (x) ∈ U (we can do it by saturation). For all n we have Thus, there are infinitely many y ∈ G such that f (x 0 + y) = f (x 0 ) + f (y). Because f is a D-function it follows that for almost all y with x 0 + y ∈ U , f (x 0 + y) = f (x 0 ) + f (y). Since x 0 is D-generic, the function f must be G-affine, contradiction.
We return to Lemma 8.6, take the function g : V → G from the above claim. By Lemma 8.1, for every x ∈ V generic, J x g and henceJ x g is invertible. Because g is smooth and J 0 g = 0 there are invertible matrices of the formJ x g ∈ R arbitrarily close to the 0 matrix.
Proposition 8.9. The ring R is a definable subset of M 2 (R).
Proof. Let us see first that R can be viewed as a locally definable subring of M 2 (R), namely a bounded union of definable subsets of M 2 (R). By Proposition 7.3 and the remarks which follow, any D-definable over ∅, family {S t : t ∈ T } of plane curves all containing (0, 0) gives rise to a ∅-definable family of D-functions in F 0 , F 0 = {f t : t ∈ T 0 } containing precisely all functions in F 0 whose germ at 0 is represented by S t for some t ∈ T . The family F 0 realizes in the above sense the definable set J = {J 0 f t : t ∈ T 0 } ⊆ R.
As pointed out in the proof of Lemma 8.3, the families are also a families of D-functions. Hence, the sets of matrices J + J, J · J, and J −1 = {A −1 : A ∈ J invertible} can be realized by definable families of D-functions. Thus R can be first identified as a bounded -definable subring of M 2 (R), where the set of disjuncts is bounded in size by the cardinality of the language of D. It follows that there is a definable open neighborhood U ⊆ M 2 (R) of the zero matrix, such that U ∩ R is definable (for more on locally definable groups and rings see [20]). More precisely, there exists a ∅-definable family of D-functions which realizes U . We now proceed to show that R is actually a definable subset of M 2 (R). Let U ∋ 0 be a neighborhood of 0 in M 2 (R) such that U ∩ R is definable as above. We claim that Indeed, for every C ∈ R we can find, by Lemma 8.6, an invertible matrix B ∈ U ∩ R, sufficiently close to 0, such that CB ∈ U ∩ R. It follows that R is definable. Proof. Since dim U = 2 we have dimJ (U ) ≤ 2. Assume towards contradiction, that dimJ(U ) ≤ 1.
Proof of Claim. For every matrix A ∈J(U ) let C A := {x ∈ U :J x f = A}. By our assumptions, there exists A ∈J(U ) such that dim C A ≥ 1, and by possibly shrinking it, we assume that C A is definably connected. Consider B A = C A ⊖ C A ⊆ G. There are two cases to consider: We may apply [17,Lemma 2.7] and conclude that the set C A consists of a subset of a coset of a locally definable one dimensional subgroup H of G. It follows that for g 0 ∈ H sufficiently small there are infinitely many a ∈ C A such that a ⊕ g 0 ∈ C A , and thus

Case 2 For all A ∈J (U ), dim B A = 2, so B A contains an open subset of G.
Given A generic inJ(U ) we may find an open set W ⊆ G in B A such that A is still generic inJ(U ) over the parameters defining W . Thus there are infinitely many A ∈J(U ) for which W ⊆ B A . Pick g 0 generic in W and then for each A such that g 0 ∈ B A there are a, b ∈ C A such that a ⊖ b = g 0 , so a = b ⊕ g 0 . By definition of C A we know that for every such pair (a, b) we haveJ a f =J b f , soJ b⊕g 0 f =J b f . We get: Since there are infinitely many such pairs b, b ⊕ g 0 , as A varies, we are done.
To conclude the proof, fix g 0 as provided by the Claim and infinitely many a such that J a f =J a f (x ⊕ g 0 ). By Lemma 8.5 (3), for each such a,d a (f (x ⊕ g 0 ) ⊖ f (x)) = 0. But then, by Lemma 8.5 (2), the D-function k(x) = f (x ⊕ g 0 ) ⊖ f (x), has infinitely many a where J a k = 0, so k(x) is constant on that set, say of value d. By strong minimality of D, (g 0 , d) is in Stab * (S f ). Since g 0 is not in dcl(∅), it is not a torsion-element so Stab * (S f ) is infinite and therefore f is G-affine, contradiction.
8.3. The structure of R. The main result of this section is: Theorem 8.11. There exists a fixed invertible matrix M ∈ M 2 (R) such that one of the following two holds: (1) In particular, R is a field isomorphic to R( √ −1). Or, We need some preliminaries. Then |J x f | has constant sign at all x ∈ U where f is differentiable and J x f is invertible.
Proof. By Corollary 4.11, we may assume -possibly removing finitely many points from U -that f is locally injective. The result now follows from [22,Theorem 3.2].
Now, for f ∈ F 0 non-constant we denote by σ(f ) the sign of |J x (f )| for all x sufficiently close to 0 at which J x f is invertible. Proof. Fix A 0 ∈ R generic over ∅, and W ⊆ R an open neighborhood of A 0 . Fix also a definable family of D-functions, {f t : t ∈ T } realizing W . Let a 0 be generic in T , such that J 0 f a 0 = A 0 . We may assume that T is a cell in some M k , and by definable choice in o-minimal structures further assume that the map t → J 0 f t is a homeomorphism of T and W . By Proposition 8.10, dim T = dim W = dim R ≥ 2.
For every t ∈ T , let U t ⊆ G be the domain of f t (containing 0). We may find now a definably connected neighborhood U 0 ∋ 0 and a definably connected neighborhood T 0 ∋ a 0 in T , such that for every t ∈ T 0 , U 0 ⊆ U t . The sets U 0 and T 0 might use additional parameters but we may choose these so that a 0 is still generic in T 0 over these parameters.
Consider now the set of matricesŴ It is an open neighborhood of 0 in R, which is realized by the family {f t ⊖ f a 0 : t ∈ T 0 }.
Our goal is to show that every invertible matrix inŴ 0 has positive determinant. Let us first see that they all have the same determinant sign. Note that for all t ∈ T 0 \ {a 0 }, the function f t ⊖ f a 0 is non-constant on U 0 , thus by Theorem 7.7 it is an open map. We now show, using our above notation, that σ(f t ⊖ f a 0 ) is constant as t varies in a punctured neighborhood of a 0 . Fix is still definably connected, and for each t ∈T 0 , the function f t ⊖ f a 0 is open onÛ 0 . Given, t 1 = t 2 ∈T 0 , there exists a definable path p : [0, 1] →T 0 connecting t 1 and t 2 , and by possibly shrinkingÛ 0 , the induced map (s, x) → F (p(s), x) is a proper homomotopy of f t 1 ⊖ f a 0 and f t 2 ⊖ f a 0 , hence by [22,Theorem 3.19], for every x generic inÛ 0 , |J x (f t 1 ⊖ f a 0 )| and |J x (f t 2 ⊖ f a 0 )| have the same sign. It follows that Thus, every invertible matrix inŴ 0 has the same determinant sign.
Next, note that for every invertible A ∈Ŵ 0 sufficiently close to 0, the matrix A 2 is also in W 0 and clearly has positive determinant. Thus all invertible matrices inŴ 0 have positive determinant.
Finally, as we saw in Proposition 8.9, R = {AB −1 : A, B ∈Ŵ 0 , B invertible}, and hence all invertible matrices in R have positive determinant.
Proof of Theorem 8.11. Assume first that every non-zero A ∈ R is invertible, namely that R is a definable division ring. It follows from [24, Theorem 4.1] that R is definably isomorphic to either R or R( √ −1) or the ring of quaternions over R. Because dim R ≥ 2, we are left with the last two possibilities. The ring of quaternions is not isomorphic to a definable subring of M 2 (R), so R is necessarily isomorphic to R( √ −1). It is now not difficult to see that there exists an invertible M ∈ M 2 (R) such that R is of the form (1).
We thus assume that there exists at least one matrix A which is not invertible, of rank 1. We want to show that there exists an invertible M ∈ M 2 (R) such that R has form as in (2).
We conjugate R by some fixed matrix so that A, written in columns, has the form (w, 0) for some w ∈ R 2 . We now show that every matrix in R is of the form a 0 b a for some a, b ∈ R. Consider the set It is a definable subring of R of positive dimension (since H is a non-trivial subring of R) with dim H ≤ 2.
Proof of Claim 1. Write the matrices in H in the form B = a 0 b 0 , and note that for Assume towards a contradiction that dim H = 2, and then H consists of all matrices of the form a 0 b 0 . We may now take C = c d e f ∈ R invertible, sufficiently close to 0, and since d, f cannot be both 0, it easy to see that by choosing a, b appropriately, we may obtain a matrix B + C ∈ R whose determinant is negative, a contradiction.
Thus, H is a 1-dimensional ring.
Claim 2. The matrices in H are not of the form B = a 0 αa 0 , for some fixed α ∈ R.
Proof of Claim 2. For every invertible C = c d e f ∈ R, and B = a 0 αa 0 ∈ H, we have |B + C| = |C| + (af − αad). By choosing a appropriately, we obtain |B + C| < 0 unless f = αd. We now consider Since this matrix must be in H, we must have α(c + αd) = e + α 2 d, which implies that e = αc. However, this would make C non-invertible, a contradiction.
We are thus left with the case that matrices in H are of the form 0 0 a 0 . If we now take an arbitrary c d e f ∈ R and multiply it on the right by a non-zero element of H, we obtain another element of H, forcing d to be 0. Thus all matrices in R are lower triangular.
Because H contains all matrices of the form 0 0 a 0 , every matrix in R can be written as the sum of a diagonal matrix in R and a matrix in H. Finally, we note that the diagonal matrices in R form a sub-ring, and since the determinant of each such non-zero matrix must be positive, it follows that they are of the form a 0 0 a . Thus, the matrices in R are of the form a 0 b a , as required. This ends the proof of Theorem 8.11.
8.4. From ring to field.
Definition 8.14. We say that R is of analytic form if it satisfies (1) of Theorem 8.11.
Our goal in this section is to prove that Case (2) of Theorem 8.11 contradicts the strong minimality of D. Thus, our negation assumption is that there exists a matrix M ∈ GL(2, R) such that all matrices in M −1 RM are of the form Let us first note that we may assume that all matrices in R itself are in form (2). Indeed, we fix a small definable open chart U ∋ 0 in G and identify it with an open subset of R 2 . We may also assume that M U = U . Now consider the definable bijection h : G → G which is the identity outside U and h(x) = M x on U . The push-forward of the structure D under h is an isomorphic, definable, strongly minimal structure D ′ , and it is easy to verify that for any D-function f ∈ F 0 , its image in D ′ is a function whose differential at 0 is M −1 J 0 f M . Thus the ring of Jacobians at 0 of all smooth D ′ -functions f with f (0) = 0 consists of matrices as in (2). We now replace D with D ′ .
In the case where G = R 2 , + then [4, Corollary 2.18] would immediately yield the desired result. The goal of this sub-section is to prove an analogue of that result in the context of an arbitrary group G.
We first need the following version of the uniqueness of definable solutions to definable ODEs. It can be easily deduced from [18, Theorem 2.3]: Proposition 8.15. Let Gr(k, n) be the space of all k-dimensional linear subspaces of R n . Let U ⊆ R n be an open set and assume that L : U → Gr(k, n) is a definable C 3 -function assigning to each p ∈ U a k-linear space L p . Assume that C 1 , C 2 ⊆ U are definable kdimensional smooth manifolds such that for every p ∈ C 1 ∩ C 2 , the tangent space of C i at p equals L p . Then for every p ∈ C 1 ∩ C 2 there exists a neighbourhood V ∋ p such that Definition 8. 16. A definable vector field on an open U ⊆ G, is given by a definable partial function F : U → T (U ) from U to its tangent bundle T (U ), such for every g ∈ G, F (g) ∈ T g (G).
Every definable non-vanishing vector field F on U gives rise to a a definable line field, still denoted by F , where to each g ∈ U we assign the 1-dimensional subspace of T g (U ) spanned F (g).
We say that a line field F is (left) G-invariant if if for every g, h ∈ U , Given a line field F , we say that a definable smooth 1-dimensional set C ⊆ U is a trajectory of F if for every g ∈ U , the tangent space to C at g is F (g).
Lemma 8.17. Let F be a definable non-vanishing G-invariant line field. Assume that C ⊆ G is a definably connected smooth 1-dimensional trajectory of F . Then C is a coset of a definable local subgroup of G.
Proof. Recall that we identify an open neighborhood U of 0 with an open subset of R 2 , and T (U ) is identified with U × R 2 . The line field can be viewed as a map F : U → Gr(1, 2).
Without loss of generality 0 ∈ C. Since F is left-invariant, for any g ∈ G, g ⊕ C is also a solution to F . By Proposition 8.15, C and g ⊕ C coincide on some neighborhood of g. It follows that every x ∈ C and g ∈ C sufficiently small, we also have x ⊕ g ∈ C. Thus C is a local subgroup of G.
We can now return to our main goal: proving that R is of analytic form. Recall that we assume that for a D- When f is clear from the context we omit the subscript f .
Let v 0 = 0 1 and consider the non-vanishing G-invariant vector field F given by namely the line-field induced by F is invariant under df . If in addition α(b) = 0 then Proof. By assumption on the form of matrices in R, Writing d b f explicitly (and composing on the left with d 0 ℓ f (b) ), we obtain which implies the first clause. For the second clause, notice the special form ofd b f implies that when α(b) = 0, then for every v ∈ T 0 (G), we haved b f · v ∈ Rv 0 . The result easily follows. Lemma 8.19. Assume that f : U → G is a D-function, and that C ⊆ G is a definable smooth curve which is a trajectory of F . Then so is f (C).
Proof. By the first clause of Lemma 8.18, the image of C under f is also a solution to F , hence a trajectory. Lemma 8.20. Assume that f is a D-function, and C ⊆ G is a definable smooth curve, such that at every b ∈ C we have α(b) = 0 (in the above notation). Then for every generic b ∈ C, the tangent space of Proof. Consider the restriction of f to C, and pick a generic b in C. Since b is generic, the map f |C : Lemma 8.21. There exists a D-function h and a definable curve C ⊆ G such that h(C) is a trajectory of F .
Proof. This is similar to the proof of the claim in Proposition 8.10. Fix any D-function f , with α, β as above. We claim that there is a 0 ∈ G such that for infinitely many Indeed, for r ∈ R, let C r = {b ∈ G : α f (b) = r}. Pick r generic in the image of α f so that C r is 1-dimensional, and consider D r = C r ⊖ C r . If D r is still 1-dimensional then as we already saw several times, C r is contained in a coset of a locally definable subgroup H r and then picking a 0 ∈ H r small enough will work with any b ∈ C r .
Otherwise, D r is 2-dimensional. We may now pick a 0 ∈ D r generic over r. Since r is still generic over a 0 there are infinitely many r ′ such that a 0 ∈ D r ′ . For each such r ′ , there Fix a 0 as above, and consider the D-function h( . It is easily verified that for each b ∈ G we havẽ It follows that α h (b) = 0 for every b ∈ G such that α g (b ⊕ a 0 ) = α g (b). Let C be the collection of all those elements b. By Lemma 8.20, the curve h(C) is a trajectory of F near b.
We can now conclude: The ring R is of analytic form.
Proof. We still work under the negation assumption that we are in Case 2 of Theorem 8.11. Using Lemma 8.21 and Lemma 8.17 we obtain a definable local subgroup H which is a solution to the vector field F , and thus all of its cosets are also trajectories of F . Let U be a neighbourhood of 0 which can be covered by cosets of H, all solutions to F . Fix any D-function f ∈ F 0 which is not G-affine. By Lemma 8.19, for every a ∈ U such that f (a) ∈ U , the image f (H ⊕ a) is also a coset of H. Fix a 0 ∈ H close enough to 0 and consider the D-function k(x) = f (x) ⊖ f (x ⊕ a 0 ). Since f is not G-affine, the function k is not constant.
Notice that for every x sufficiently close to 0, the elements x and x ⊕ a 0 belong to the coset x ⊕ H, and therefore as we just noted, f (x) and f (x ⊕ a 0 ) belong to the same coset of H. It follows that k(x) ∈ H and therefore k sends an open subset of G into H, contradicting strong minimality (the pre-image of some point will be infinite).
Note that the above argument does not really use the definability of the trajectory C but merely its existence. Thus, if we worked over the reals then we could have used the usual existence theorem for solutions to differential equations in order to derive a contradiction.

Some intersection theory for D-curves
Our ultimate goal is to show, under suitable assumptions, that if two plane curves C, D ⊆ G 2 are tangent at some point p, and C belongs to a D-definable family F of plane curves then by varying C within F one gains additional intersection points with D, near the point p (see Proposition 9.12 (2)). This will allow us to detect tangency D-definably.
The main tool towards this end is the following theorem, whose proof will be carried out in this section via a sequence of lemmas.
We now digress to report on an unsuccessful strategy, which nevertheless may be of some interest.
9.1. Digression: on almost complex structures. Let K = R( √ −1). In analogy to the notion of an almost complex structure on a real manifold, we call a definable almost Kstructure on a definable R-manifold M , a definable smooth linear J : Note that every definable K-manifold admits a natural almost K-structure, induced by multiplication of each T x (M ) by i = √ −1. It is known that when K = C any 2dimensional almost complex structure is isomorphic, as an almost complex structure, to a complex manifold. The proof of this result seems to be using integration and thus we do not expect it to hold for almost K-structures in arbitrary o-minimal expansions of real close fields.
Returning now to our 2-dimensional group G, we can endow G with a definable almost K-structure in the following way: Just as we did at the beginning of Section 8.4, we may first assume that every matrix in R has the form a −b b a .
Next, we identify naturally T 0 (G) with R 2 ∼ K and let J : T 0 (G) → T 0 (G) be defined by J(x, y) = (x, −y). Next, use the differential of ℓ a to obtain J : T G → T G as required. Note that since T G is a trivial tangent bundle, this step can be carried out for any definable group of even dimension. However in the case of G, our choice of J and the fact that for each D-function f ,J a f has analytic form, implies that f is so-called J-holomorphic, namely that each each a ∈ dom f we have Now, if our underlying real closed field R were the field of real numbers then G would be isomorphic as an almost complex structure to a complex manifoldĜ, and this isomorphism would send every J-holomorphic function from G to G to a holomorphic function fromĜ tô G. In particular, by our above observation every D-function would be sent to a holomorphic function. This would give an immediate proof of Theorem 9.1, due to the fact that the result is true for holomorphic maps.
Unfortunately, we do not know how to prove for arbitrary K that every 2-dimensional almost K-manifold is (definably) isomorphic to a K-manifold, and hence we cannot use the theory of K-holomorphic maps in order to deduce Theorem 9.1. We thus use a different strategy. We now consider the complex function M (z, w) = z · w, and for a, b ∈ C near 0, let Let deg 0 (f ) be the local degree of f at 0 (see details below). Since the local degree is preserved under definable homotopy (see Fact 9.2 below), it follows from the general theory that deg 0 (M a,b ) = deg 0 (f ) for sufficiently small a, b. Because each M a,b is holomorphic, the sign of |J z M a,b | is positive at a generic z in int(C), and therefore deg 0 (M a,b ) ≥ |M −1 a,b (w)|, for all w close to 0.
This implies that f is not locally injective near 0.
Our objective is to imitate the above proof, using D-functions instead of holomorphic ones. The main obstacle is the fact that we do not have multiplication or division in D, so we want to produce a D-function which sufficiently resembles the multiplication function M . 9.3. Topological preliminaries. Throughout this section we will be using implicitly the o-minimal version of Jordan's plane curve theorem (see [31]) . We recall some definitions and results (see [21, Section 2.2-2.3]): Given a circle C ⊆ R 2 , a definable continuous f : R 2 → R 2 and w / ∈ f (C), we let W C (f, w) denote the winding number of f along C around w. If f −1 (w) is finite, p ∈ R 2 and f (p) = w then deg p (f ) is defined to be W C (f, f (p)) for all sufficiently small C around p. We need the following results: (1) If {f t : t ∈ T } is a definable continuous family of functions with w / ∈ f t (C) for any t ∈ T and T definably connected, then W C (f t 1 , w) = W C (f t 2 , w) for all t 1 , t 2 ∈ T .
(2) Assume that C is a circle around p, f : C → R 2 definable and continuous, and w 1 , w 2 are in the same component of (3) If f is definable and R-differentiable at p and J p (f ) is invertible, then deg p (f ) is either 1 or −1, depending on whether |J p (f )| is positive or negative. (4) Assume that f is a definable R-smooth, open map, finite-to-one in a neighbourhood U of p and that f (z) = f (p) for all z = p in U . Assume also that J z (f ) is invertible of positive determinant for all generic z ∈ U . Let C ⊆ U be a circle around p. Then for all w ∈ f (int(C)), if w and f (p) are in the same component of Proof. (1) follows from [21,Lemma 2.13(4)].
(2) is just [21,Lemma 2.15]. The proof of (3) is the same as the classical one, so we omit it.
(4) It follows from (2) and then fix a generic w 0 ∈ f (int(C i )) sufficiently close to w, so in particular, the Jacobian of f at each pre-image of w 0 is invertible of positive determinant. By [21,Lemma 2.25], The same argument shows that for generic w 0 near p, we have W C (f, f (p)) = |f −1 (w 0 )|. (2) f and h are, respectively, generically k-to-one and m-to-one near 0, Then g(x) = M (f (x), h(x)) is, generically, at least k + m-to-one near 0.
Proof. By Corollary 4.11 each g a,b is open, thus, since it is a D-function, it is finite-to-one near 0. Also, it follows from Corollary 8.13 and Proposition 8.22 that g a,b has positive determinant of the Jacobian at every point where the Jacobian matrix does not vanish, which by Lemma 8.1 is a co-finite set.
We now fix C around 0 such that 0 / ∈ g(C) = g 0,0 (C) and deg 0 (g) = W C (g, 0). By continuity of M and g we can find an open U 1 ∋ 0, and an open disc U 2 ∋ 0, such that for all a, b ∈ U 1 , g a,b (0) ∈ U 2 and g a,b (C) ∩ U 2 = ∅. It follows that g a,b (0) and 0 are in the same component of R 2 \ g a,b (C).
By Fact 9.2, , g a,b (0)) ≥ |g −1 a,b (0)|. If we take a, b generic and independent near 0, then f −1 (a) ∩ h −1 (b) = ∅. Also, by our assumptions on M and the definition of g a,b , we have It follows from Fact 9.2 (4) that deg 0 (g) ≥ m + k and that g is generically, at least, k + m-to-one near 0. 9.5. Producing the function M . We now proceed to construct the desired D-function M as in Lemma 9.3. We start with a D-function k(x) which is not G-affine and fix a generic a 0 ∈ dom k. Define We write M a (y) = M (a, y).
By definition, we have (A): For x, y near 0, M (0, y) = M (x, 0) = 0. Our next goal is to show that M can be used, similarly to multiplication, to "divide (an appropriate) function f by x". Namely, that we can implicitly solve M (x, y) = f (x) in some neighborhood of x = 0. This is the purpose of the next few results.
By Theorem 8.22 and the discussion in Section 8.4, we may assume that for a smooth f ∈ F 0 , the matrix J 0 (f ) has the form with c, e ∈ R. We consider the partial definable map d on G, given by d(a) = J 0 (M a ). It is convenient to identify every d(a) with the first row of the above matrix, so we view d as a map from G into R 2 . By Claim 2 in the proof of Proposition 8.10 (and using the fact thatJ 0 (f ) = J 0 (f )), we get that d(a) is equal to: By Proposition 8.10, applied to k(x), the image of every open U ∋ 0 under x →J x (k) is a 2-dimensional subset of R, hence by o-minimality this map is locally injective near the generic a 0 . Equivalently, the map x →J a 0 ⊕x (k) is locally injective near 0. SinceJ a 0 (k) is constant, it follows that d(x) is locally injective at 0. In particular, we have We are going to use several different norms in the next argument, so we set and for a linear map T we denote the operator norm by It is well-known (and easy to see) that if we identify every linear map with a 2 × 2 matrix then ||T || op and ||T || are equivalent norms. We need an additional property of M . Given two functions α, β : U * → R ≥0 on a punctured neighborhood U * ⊆ R 2 of 0, we write α ∼ β if lim t→0 α(t)/β(t) is a positive element of R. We will show: (C): There are definable R >0 -valued functions e(a) and δ(a), in some punctured neighborhood U * of 0, with e(a) ∼ ||d(a)|| and δ(a) ∼ ||d(a)|| 2 , such that for every a ∈ U * , the function M a = M (a, −) is invertible on the disc B e(a) and its image contains the disc B δ(a) (recall that for a = 0 we have M a (x) ≡ 0 near 0). In order to prove (C), we use an effective version of the inverse function theorem, as appearing in [30, §7.2]. We give the details, with references to [30].
Proposition 9.4. There exist definable functions e(a) and δ(a) from a punctured neighborhood of 0 into R >0 , such that for a = 0 in a small neighborhood of 0, the function M a (y) is injective on B(0; e(a)) and the its image contains a ball of radius δ(a) around 0. Furthermore, there is a constant C > 0 such that e(a) = ||d(a)||/4C and δ(a) = e(a) 2 /2.
Proof. We start with some observations. If For each a, y, we view D(a, y) both as a linear operator and a vector in R 4 . Since M is a C 2 -function, ||J (a,y) D|| op is bounded by some constant C, as (a, y) varies in a neighborhood B 1 × B 2 of (0, 0), and by normalizing M we may assume that C > 1. By [30,Lemma 7.2.8] applied to D, for every (a 1 , y 1 ), (a 2 , y 2 ) ∈ B 1 × B 2 we have Note also that D(0, 0) = J 0 M 0 = 0, so restricting further B 1 , B 2 we may also assume that ||D(a, y)|| < 1 for all (a, y) ∈ B 1 × B 2 .
We now need a version of [30, Lemma 7.2.10]: Lemma 9.5. For every a ∈ B 1 such that J 0 M a is invertible, and for all y 1 , y 2 ∈ B 2 , there exists e(a) ∈ R >0 , such that if ||y 1 ||, ||y 2 || ≤ e(a) then (1) the matrices J y 1 M a , J y 2 M a are invertible.
Proof. We fix a with J 0 (M a ) invertible and we write J 0 M a = c e −e c . By (*), for every y ∈ B 2 and for every E > 0, if ||y|| < E/2C then In particular, since J 0 M a = 0, we may take E < ||d(a)|| = ||J 0 M a || op and then J y M a must be non-zero. Because M a is a D-function it follows that J y M a is invertible. Let c ′ = 1/||J 0 (M a ) −1 || op . As we pointed out earlier, in our case Hence, for any two y 1 , y 2 ∈ R 2 : (**) ||J 0 M a · (y 1 − y 2 )|| ≥ ||d(a)|| · ||y 1 − y 2 ||.
Using the fact that f (0) = 0, we may apply [30, Lemma 7.2.8] to f and conclude that ||f (x)|| ≤ C||x||, for all x in an open disc U centered at 0, where C is a bound on ||J a (f )|| op in U . By the equivalence of norms we may assume that C is also a bound on ||J a (f )|| as a varies in U .
Consider now the map a → J a (f ), as a map from U into R 4 , and let C ′ be a bound on the norm of the differential of this map in U . We apply again [30, Lemma 7.2.8] to this map, and using the fact that J 0 (f ) = 0, we conclude that ||J a (f )|| < C ′ ||a||, for x in some U ′ ∋ 0, and a ∈ B(0; ||x||) Thus, for all x ∈ U , It follows that ||f • f (x)|| ≤ C ′ ||x|| 4 and hence lim x→0 ||f • f (x)||/||x|| 2 = 0.
We also need: Lemma 9.7. If x(t) : (a, ǫ) → R 2 is a definable curve tending to 0 as t → 0, then lim Proof. Recall that d is a map from U into R 2 , defined by d(a) = (c, e), where J 0 (M a ) = c −e e c . Notice that since R is 2-dimensional we may also view d as a map from G into R. We claim that the Jacobian at 0 of d is invertible. Indeed, we have seen in ( †) above (Subsection 9.5), J 0 (M a ) =J a 0 ⊕a k −J a 0 k. By Proposition 8.10, the function a →J a f is a diffeomorphism in a small neighbourhood of the generic point a 0 onto an open subset of R. Since x → a 0 ⊕ x is a diffeomorphism (between open subsets of G) in a neighbourhood of 0 we get that a →J a 0 ⊕a k is a diffeomorphism near 0 between an open subset of G and R. SinceJ a 0 k is a constant matrix it follows that a → J 0 (M a ) is a diffeomorphism near 0.
It follows from the definition of the differential that Since J 0 (d) is invertible the limit of J 0 (d)·x(t) ||x(t)|| is a non-zero vector, and hence lim Proof. We first prove (i) for some neighborhood U , so we assume that (i) fails. Then there exists an definable function x(t) tending to 0 in G, such that for all t, ||g(x(t))|| ≥ δ(x(t)) = ||d(x(t))|| 2 /32C 2 .
Because J 0 f = 0, Fact 9.6 implies that lim ||g(x(t))||/||x(t)|| 2 = 0. Combined with the above inequality we get lim ||d(x(t))|| 2 /||x(t)|| 2 = 0, hence lim  Proof. Clause (i) is just Corollary 9.8. To see that h is differentiable everywhere we apply the Implicit Function Theorem to M (x, y) − g(x). By Lemma 9.5, J y M x is invertible for every x ∈ U * and |y| < e(a), so indeed h(x), the solution to M (x, y) − g(x) = 0, is differentiable at x. To see that the limit of h at 0 is 0, we compute the limit along an arbitrary curve x(t) tending to 0. By definition, |h(x(t))| < ||e(t)|| ∼ ||d(t)||, so since d(0) = 0, also h(x(t)) must tend to 0. The second clause of (ii) follows since if h were constant with h(0) = 0 necessarily h would vanish on its domain, implying that g was identically 0.
For (iii), note that the graph of h is contained in the plane curve B = {(x, y) : (x, y, g(x)) ∈ M (x, y)} whereM is the D-definable set of Morley rank 2 containing the graph of M .
We note that locally near the point (0, 0) itself, the D-definable set B need not be the graph of a function, but this does not come up in the argument.
Proof of Theorem 9.1: Assume that f ∈ F 0 and J 0 (f ) = 0. We will show that f is not injective near 0.
Proof. Consider g(x) = f (f (x)), and assume towards contradiction that f and thus also g is injective. near 0. By Corollary 9.9, there exists a D-function h in a neighborhood U of 0, with h(0) = 0 such that for all x ∈ U , We now wish to apply Lemma 9.3 to the functions x and h(x). For that we just need to note that for a and b near 0 the function g a,b (x) = M (x ⊖ a, h(x) ⊖ b) is non-constant near 0. Indeed, we can find a fixed definably connected open W ∋ 0, such that for a, b close to 0, W ⊆ dom(g a,b ). Since each g a,b is a D-function, its graph is contained in a strongly minimal set, and hence if it were constant near 0 then it would have to be constant on the whole of W . But then, by the continuity of M , the function g = g 0,0 must also be constant on W , contradiction.
By applying Lemma 9.3 we conclude that g(x) is at least 1+k-to-one near 0, where k ≥ 1. This contradicts the assumption that f and thus g were locally injective. Contradiction.
The following example shows that the proof of Theorem 9.1 uses more than just the basic geometric properties of the function f . Example 9.10. A crucial point in our above argument was that f (z)/z, or in the language of our proof, the implicitly defined function h, is an open map. This followed from the fact that it was a D-function.
Consider the function f (z) = |z| 2 z from C to C. The function is smooth everywhere, J 0 f = 0, and yet it is injective everywhere. However, the function f (z)/|z| is clearly not an open map. 9.7. Intersection theory in families. Based on the topological properties we established thus far we can develop some intersection theory resembling that of complex analytic curves. Definition 9.11. Given two plane curves X, Y , and p = (p 1 , p 2 ) ∈ X ∩ Y . We say that X and Y are tangent at p if there are D-functions f, g which are C 1 in a neighborhood of p 1 , with Γ f ⊆ X and Γ g ⊆ Y , such that f (p 1 ) = p 2 = g(p 1 ) and J p 1 f = J p 1 g. Proposition 9.12. Let F = {E a : a ∈ T } be a D-definable almost faithful family of plane curves, D-definable over ∅, and let X be a strongly minimal plane curve whose projections on both coordinates are finite-to-one.
Assume that a is generic in T over ∅, E a strongly minimal, X ∩ E a is finite and p = (x 0 , y 0 ) ∈ E a ∩ X.
(1) If p is generic in E a over a, not topologically isolated in X and also D-generic in X over [X] then for every neighborhood U ∋ p, there is a neighborhood V ∋ a in T , such that for every a ′ ∈ V , E a ′ intersects X in U . (2) (Here we do not make any genericity assumptions on p). Assume that for some open V ∋ a whenever a ′ ∈ V the set E a ′ represents a D-function f a ′ in a neighborhood of x 0 and that the map (a ′ , x) → f a ′ (x) is continuous at (a, x 0 ). Assume also that X represents a function g at p and that J x 0 (f a ) = J x 0 (g). Then for every neighborhood U ∋ p there is a neighborhood V ∋ a in T , such that for every a ′ ∈ V , either E a ′ and X are tangent at some point in U or |E a ′ ∩ X ∩ U | > 1.
Proof. (1) Fix an open U = U 1 × U 2 ∋ p definably connected. Since p is generic in E a over a, we may choose U so that E a is locally the graph of a continuous function f a : U 1 → U 2 or f a : U 2 → U 1 . Since our assumptions are symmetric with respect to both coordinates we may assume the former.
Since, in addition, a is generic in T over ∅ we may shrink U and find an open definably connected V 0 ∋ a in T such that for every a ′ ∈ V , the set E a ′ ∩ U 1 × U 2 is the graph of a D-function f a ′ : U 1 → U 2 and furthermore, the map (a ′ , x) → f a ′ (x) is continuous on V 0 × U 1 .
Since p is not isolated in X, D-generic in X over [X], and the projections of X on both coordinates are finite-to-one, it follows from Corollary 6.4 that, after possibly shrinking U further, the set X ∩ U is the graph of an open continuous map g : U 1 → U 2 .
Notice that for every a ′ ∈ V 0 , and (x, y) ∈ U , Because X ∩ E a is finite, the function f a ⊖ g is not constant on its domain, so by Theorem 7.7, f a ⊖ g is open on U 1 .
Claim There exists V ∋ a such that for every a ′ ∈ V \ {a}, the function f a ′ ⊖ g is an open map on U 1 .
Indeed, assume towards a contradiction that for a ′ ∈ V 0 arbitrarily close to a the map f a ′ ⊖ g is not an open map. Thus, by Theorem 7.7, it is constant on U 1 . It follows from continuity that f a ⊖ g is constant on U 1 , contradicting out assumption.
Thus, we showed that there exists V ∋ a such that for all a ′ ∈ V , the function f a ′ ⊖ g is open and finite-to-one on U 1 . In addition, the the map (a ′ , x) → (f a ′ ⊖ g)(x) is continuous in a neighborhood (a, x 0 ). Because 0 ∈ (f a ⊖ g)(U ) it follows from Fact 9.2(1), (4) that for some open V 0 ∋ a small enough and for all a ′ ∈ V 0 , the set (f a ′ ⊖ g)(U ) contains 0, namely X ∩ E a ′ ∩ U = ∅. This ends the proof of (1).
(2) Let g be a D-function with g(x 0 ) = y 0 such that Γ g ⊆ X, and J x 0 f a = J x 0 g. Note that (f a ⊖g)(x 0 ) = 0 and f a = g. So, for C ⊆ G a sufficiently small circle around x 0 the only zero of f a ⊖ g in the closed ball B determined by C is x 0 . By continuity of (x, a ′ ) → f a ′ (x), we may find some neighborhood V ⊆ V 0 of a such that for every a ′ ∈ V , 0 / ∈ (f a ′ ⊖ g)(C).
It follows from Fact 9.2(1) that W C (f a ′ ⊖ g, 0) = W C (f a ⊖ g, 0) for every a ′ ∈ V . By our assumptions, J x 0 (f a ⊖ g) = 0 and therefore by Theorem 9.1, f a ⊖ g is not injective in any neighborhood of x 0 , that is, for every generic y near 0, |(f a ⊖ g) −1 (y)| > 1. It follows from Fact 9.2(4) that W C (f a ⊖ g, 0) > 1. Thus, for every a ′ ∈ V , W C (f a ′ ⊖ g, 0) > 1.
We can now conclude that for every a ′ , either 0 is a regular value of the function f a ′ ⊖ g on int(C), in which case it has more than one pre-image and then E a ′ and X intersect more than once in int(C), or 0 is a singular value, in which case the curves E a ′ and X are tangent at some point in int(C).

The main theorem
We are now ready to prove our main result. Our proof follows closely that of [4,Theorem 7.3]. As some of the technicalities of that proof were dealt with in earlier stages of the present paper, the proof will be somewhat simpler. We begin with a series of useful technical facts. Throughout this section we let K := R. (1) For i = 0,1, T i is strongly minimal and C i is almost faithful.
(2) Every generic curve in C i , i = 0,1, is closed, strongly minimal and has no isolated points. } are open subsets of K, with 0 ∈ cl(W 0 ) and 1 ∈ cl(W 1 ). (5) For each i = 0,1, the map (a, x) → f i a (x) is continuous on T ′ i × U . Proof. By Claim 8.7, there exists a D-function f : U → G which is not G-affine, such that J 0 f = 0. Let S ⊆ G 2 be a strongly minimal set representing f . By Theorem 4.9, we may assume that S is closed, and by allowing parameters we may assume that S has no isolated points. Let C 0 = {S ⊖ p : p ∈ S}. Let T 0 := S and for a ∈ S let E 0 a := S ⊖ a. For every a = (x 0 , f (x 0 )) ∈ S, the curve S a represents the D-function f (x ⊕ x 0 ) ⊖ f (x 0 ). By Proposition 8.10, the set of elements of K W = {J 0 (f (x ⊕ x 0 ) ⊖ f (x 0 )) : x 0 ∈ U } has dimension 2, and by applying the same proposition to a smaller U , we see that J 0 f = 0 is in the closure of a 2-dimensional component of W . By o-minimality, we may find an open U ′ ⊆ U such that the set W 0 = {J 0 (f (x ⊕ x 0 ) ⊖ f (x 0 )) : x 0 ∈ U ′ } is an open subset of K with the 0 matrix in its closure. We let T ′ 0 := {(x 0 , f (x 0 )) : x 0 ∈ U ′ }. By its definition, the sets U ′ , T ′ 0 and W 0 satisfy all clauses of the lemma. In order to obtain C 1 , we replace f with the function h(x) = f (x) ⊕ x. It is a D-function which is not G-affine, with J 0 h = 1 ∈ K. We repeat the above process and obtain the rest of the lemma.
Recall the following definition.   Let G m and G a denote the multiplicative and additive groups of the algebraically closed field K. The action of G m ⋉ G a on G a (defined by (a, c) · b = ab + c) gives rise, naturally, to a field configuration on the structure (K, +, ·) as follows: take g, h ∈ G m ⋉ G a independent generics (in M), and b ∈ G a generic over g, h (in M). Then where · denotes the action of G m ⋉ G a on G a is readily verified to be a field configuration in the field K.
Our aim is to pull this field configuration back into the structure D. First note that although W 0 and W 1 from Lemma 10.1, are not neighborhoods of 0 and 1, respectively, it is still the case that for every B ∈ W 0 , if A ∈ W 1 and C ∈ W 0 are sufficiently close to 1 and 0, respectively, then AB + C is still in W 0 (since W 0 is open). Similarly, for every A ∈ W 1 , if C ∈ W 1 is sufficiently close to 1 then AC ∈ W 1 .
Let e = (1, 0) be the identity of G m ⋉ G a , and choose b ∈ W 0 and h, g in W 1 × W 0 ⊆ G m ⋉ G a sufficiently close to e so that gh ∈ W 1 × W 0 , and h · b and hg · b are in W 0 . Note that we may choose g, h, b to be independent generics in the sense of M (and then also independent in the sense of K).
Abusing notation, we sometimes write f ∈ C i for a D-function f which is represented by a curve in C i . In particular, let us denote, for i = 1, 2, We are going to reconstruct F as a set of jacobian matrices of D-functions in C ′ 0 and C ′ 1 , and show that it is, in fact, a field configuration in D.
(2) hg ∈ W 1 × W 0 , and there are f 3 ∈ C ′ 0 and g 3 ∈ C ′ 1 with (J 0 g 3 , J 0 f 3 ) = hg. (3) There are k 2 , k 3 ∈ C ′ 0 such that J 0 k 2 = h · b and J 0 k 3 = hg · b. For a D-function Ψ, we denote by [Ψ] the D-canonical parameter of some fixed strongly minimal set representing it. Our goal is to prove the following proposition. . We will show that a ∈ F , thus reaching a contradiction. By our choice of a, dim(J 0 f a /∅) = 2 = dim G and since J 0 f a ∈ dcl(a) we also have dim(a/∅) = 2. Thus we also have a ∈ dcl(J 0 f a ) = dcl(J 0 X). Claim 2. Let {x 1 , . . . , x k } := X ∩ E a . Then for every i = 1, . . . , k, either dim(x i /a) = 2, or x i ∈ acl D (∅).
Proof of Claim 2. We consider the family and for simplicity write the members of F ′ as {X t : t ∈ T }. By our choice of X, there is t 0 generic in T such that X is a strongly minimal subset of X t 0 , so definable over acl D (t 0 ). We may now replace F ′ by another family of the same dimension, defined over ∅, such that the generic member of F ′ is strongly minimal and X belongs to the family. Without loss of generality, we still call this new family F.
Thus X = X t 0 , with F = {X t : t ∈ T } a D-definable almost faithful family of plane curves, and t 0 generic in T over ∅. So, if we let d = RM(T ) then dim(t 0 /∅) = 2d. Because a ∈ dcl(J 0 X), we have a ∈ dcl(t 0 ). Also, by our underlying negation assumption, RM(a t 0 /∅) = d + 1, hence RM(t 0 /a) = RM(t 0 /∅) = d. Since x i ∈ E a , we have dim(x i /a) ≤ 2, so we assume that dim(x i /a) ≤ 1 and show that a ∈ acl D (∅).
We now consider the D-type of t 0 over x i . Since RM(t 0 /x i ) = RM(t 0 /∅) it follows that almost all curves X t contain x i . Because the family {X t : t ∈ T } is almost faithful, it follows that there are only finitely many points x ∈ G 2 which belong to almost all curves X t , and therefore x i ∈ acl D (∅). This ends the proof of Claim 10.
We now return to the proof of Lemma 10.6. By Claim 2 we may assume that for i = 1, . . . , r, we have dim(x i /a) = 2 and for i = r + 1, . . . , k, we have x i ∈ acl D (∅). Without loss of generality, x k = 0.
In order to show that a ∈ F , we have to show that k < n. Towards that end, we will show that there are infinitely many a ′ ∈ T 0 such that n = |X ∩ E a ′ | ≥ k + 1.
the more general conjecture, allowing underlying sets of arbitrary dimension is open even for reducts of the complex field.