Some results and problems on complex germs with definable Mittag-Leffler stars

A.J. Wilkie School of Mathematics The Alan Turing Building University of Manchester Manchester M13 9PL UK

June 20, 2012

Abstract

Working in an o-minimal expansion of the real field, we investigate when a germ (around 0 say) of a complex analytic function has a definable analytic continuation to its Mittag-Leffler star.

As an application we show that any algebro-logarithmic function that is complex analytic in a neighbourhood of the origin in ℂ has an analytic continuation to all but finitely many points in ℂ.

It is my pleasure to dedicate this paper to Anand Pillay on the occasion of his 60th birthday.

1 Introduction

This paper is motivated by the conjecture of Zilber stating that every ℂ_{exp}- definable subset of ℂ is either countable or co-countable. Here, ℂ_{exp} is the expansion of the ring of complex numbers by the complex exponential function. As far as I know, even sets of the form

{z ∈ ℂ : ∃w ∈ ℂ F (z, w) = 0} (∗)

where F (z, w) is a (two variable) term of the language L(ℂ_{exp}) have not been shown to satisfy Zilber’s conjecture. Our approach to this particular case is as follows. Let us suppose that

F (0, 0) = 0 ≠ \frac{∂F}{∂w} (0, 0).

Then by the implicit function theorem there exists ϵ > 0 and a complex analytic function ϕ : Δ(0; ϵ) → ℂ (where, in general, Δ(a; r) denotes the disk centred at a ∈ ℂ and having radius r) such that for all z ∈ Δ(0; ϵ), we have F (z, ϕ(z)) = 0. We must show that the set (∗) is co-countable and it seems reasonable to conjecture that the function element ϕ has an analytic continuation (which necessarily preserves the equation F (z, ϕ(z)) = 0) to all but countably many points in the complex plane. Indeed, one can fairly easily show that if one proves a suitably generalized version of this analytic continuation conjecture (in which w is allowed to be an n-tuple of variables and F an n-tuple of terms in the 1+n variables z, w, and where the countably many exceptional points have a certain specific form) then Zilber’s conjecture (even for subsets of ℂ defined by formulas of the language L_{ω_{1},ω}(ℂ_{exp})) would follow.

Let us now consider issues of definabilty. The approach to Zilber’s conjecture suggested above transcends L(ℂ_{exp})-definablility (at least, if Zilber’s conjecture is true!): one cannot define restricted functions ϕ : Δ(0; ϵ) → ℂ without the resource of the real line and the usual metric. So we follow the Peterzil-Starchenko idea of doing complex analysis definably in a suitable o-minimal structure via the usual identifications ℂ ∼ ℝ ⊕ iℝ ∼ ℝ × ℝ. Actually, we will only be considering a fixed o-minimal expansion ℝ̃ of the ordered field of real numbers \overline{ℝ}, so many of the subtleties of [4] will not be required here. But the uniform finiteness of the winding number for definable functions will be, and this was inspired by the Peterzil-Starchenko approach.

My aim in this paper, then, is to consider definable analytic continuation relative to an o-minimal expansion ℝ̃ of \overline{ℝ}. I shall only consider continuations along straight line paths emanating from the origin in ℂ, so let me discuss this now. The mathematical theory (i.e. without definability considerations) may be found in [1] and [3], but in very few modern texts as far as I can see.

2 The Mittag-Leffler star

So consider any complex analytic function ϕ : Δ(0; ϵ) → ℂ. The Mittag-Leffler star of ϕ (henceforth just the star of ϕ), denoted S_{ϕ}, is defined to be the set of all z ∈ ℂ such that there exists an open set U_{z} ⊆ ℂ with Δ(0; r) ∪ [0, z] ⊆ U_{z} and a complex analytic function ψ : U_{z} → ℂ with ψ ↾ Δ(0; r) = ϕ. (Here, [0, z] denotes the straight line segment in ℂ from 0 to z, i.e. [0, z] := {tz : 0 ≤ t ≤ 1}.) It can be shown (see [3] Volume 3) that S_{ϕ} is an open, connected and simply connected set containing Δ(0; r), and that there exists a unique complex analytic function ⋆ϕ : S_{ϕ} → ℂ with ⋆ϕ ↾ Δ(0; r) = ϕ. I call ⋆ϕ the star function of ϕ. Also, a point z such that z ∉ S_{ϕ} but satisfying [0, w] ⊆ S_{ϕ} for all w ∈ [0, z] ∖ {z} will be called a singular point of ϕ. In general one can say very little else about the nature of the set S_{ϕ} (it could, for example, be bounded). However, we have the following result in the definable situation.

Theorem 1. Let ℝ̃ be an o-minimal expansion of the ordered field of real numbers \overline{ℝ} and assume that ℝ̃ has analytic cell decomposition. Let ϕ : Δ(0; r) → ℂ be a complex analytic function and suppose that its star function ⋆ϕ : S_{ϕ} → ℂ (and hence its star S_{ϕ}) is definable (in ℝ̃). Then ϕ has only finitely many singular points.

Proof. Let ϕ : Δ(0; r) → ℂ and ⋆ϕ : S_{ϕ} → ℂ be as in the hypotheses of the theorem and suppose that ϕ has infinitely many singular points. Then by analytic cell decomposition there would exist (possibly after rotating ℂ about 0) a 2-cell C of the form

C = {x + iy : a < x < b, f(x) < y < g(x)},

where f, g : (a, b) → ℝ are definable real analytic functions, such that C ⊆ S_{ϕ} and such that ⋆ϕ ↾ C has no analytic continuation to any open set in ℂ containing a point of graph(g). We may further assume (by refining the original cell decomposition) that either for all z ∈ C, | ⋆ ϕ(z)| < 1 or for all z ∈ C, | ⋆ ϕ(z)| > 1. Let us consider the first case. By o-minimality there is a finite set s ⊆ graph(g) such that ⋆ϕ has a definable continuous extension (which we also denote by ⋆ϕ) to (C ∪ (graph(g))) \ s. By analytic cell decomposition again, there exists a′, b′ with a < a′ < b′ < b such that ⋆ϕ ∘ g ↾ (a′, b′) is a definable real analytic function. We now obtain a contradiction by a using a classical argument (as described in, for example, [2], Chapter IX). Namely, fix x_{0} ∈ (a′, b′) and let ϵ > 0 be chosen small enough so that a′ < x_{0} − ϵ < x_{0} < x_{0} + ϵ < b′ and also so that the (real) Taylor series of both g and ⋆ϕ ∘ g extend (via the same power series) to (not necessarily definable) complex analytic functions G : Δ(x_{0}; ϵ) → ℂ and Φ : Δ(x_{0}; ϵ) → ℂ respectively. Define the complex analytic function H : Δ(x_{0}; ϵ) → ℂ by H(z) := z+iG(z). Since the Taylor coefficients of G are real it follows that H′(x_{0}) ≠ 0 and hence (by reducing ϵ if necessary) that H is a holomorphic homeomorphism from Δ(x_{0}; ϵ) onto an open set, U say. Further, H maps the interval (x_{0} −ϵ, x_{0} +ϵ) onto graph(g ↾ (x_{0} − ϵ, x_{0} + ϵ)). Now consider the function

Ψ := ⋆ϕ − Φ ∘ H^{−1} : (C ∪ graph(g ↾ (x_{0} − ϵ, x_{0} + ϵ))) ∩ U → ℂ.

By our construction Ψ is continuous, holomorphic on C ∩ U, and identically zero on the analytic curve graph(g ↾ (x_{0}−ϵ, x_{0}+ϵ)) which forms (a nontrivial) part of the boundary of C ∩ U. This implies (see [2], page 303, exercise 6) that Ψ is identically zero throughout (C ∪ graph(g ↾ (x_{0} − ϵ, x_{0} + ϵ))) ∩ U. It follows that Φ ∘ H^{−1} provides an analytic continuation of ⋆ϕ to the open set C ∪ U. But this is a contradiction since U contains the point x_{0} + ig(x_{0}) of graph(g). The case that | ⋆ ϕ(z)| > 1 for all z ∈ C is dealt with by applying the above argument to the function \frac{1}{⋆ϕ} and then inverting the analytic continuation. (The proof actually shows that ⋆ϕ is necessarily locally bounded at all but finitely many points of graph(g).) This completes the proof of Theorem 1.

Later I shall show that the collection of all those complex analytic germs having a definable Mittag-Leffler star has a reasonably rich structure, at least if ℝ̃ does. This is in contrast to those germs having definable entire, or definable meromorphic, extensions which, as one can easily show, are (for any o-minimal ℝ̃) just polynomials or rational functions respectively.

The proof of Theorem 1 shows that any definable complex analytic function whose domain is an open cell C in ℂ has an analytic continuation (though not necessarily a definable one) across the boundary of C at all but finitely many points. I leave the reader to combine this remark with Theorem 1 itself to give a proof of the following result.

Theorem 2. Let ℝ̃ be an o-minimal expansion of the ordered field of real numbers \overline{ℝ} and assume that ℝ̃ has analytic cell decomposition. Let ϕ : Δ(0; r) → ℂ be a complex analytic function and suppose that its star function ⋆ϕ : S_{ϕ} → ℂ (and hence its star S_{ϕ}) is definable (in ℝ̃). Then for all but finitely many points z ∈ ℂ there exists a continuous, piecewise linear path begining at 0 and terminating at z (and, in fact, consisting of at most two line segments) along which ϕ has an analytic continuation.

However, I am unable to settle the following question.

Open Problem 1 Let ϕ and ⋆ϕ be as in the hypotheses of Theorems 1 and 2. Does there exist a finite set s ⊆ ℂ \ {0} such that ϕ has an analytic continuation along all continuous, definable paths that begin at 0 and avoid s?

Before finishing this section I should mention the Mittag-Leffler Star Theorem. This provides a remarkable series expansion for ⋆ϕ which converges uniformly to ⋆ϕ on compact subsets of S_{ϕ}. It is completely analagous to the Taylor expansion on the disk of convergence of ϕ in the sense that the only dependence of the series on the germ ϕ is a fixed (i.e. independent of ϕ) linear one on the numbers ϕ(0), ϕ′(0), …, ϕ^{(n)}(0), . . .. I will not need this result and so I will not expand on this remark. The interested reader may consult [3] for further information and proofs.

3 The ring of definable star functions

I now fix an o-minimal expasion ℝ̃ of \overline{ℝ} and I assume that ℝ̃ has analytic cell decomposition. Notions of definablility are relative to ℝ̃ and are without parameters. I denote by G̃ the collection of all definable, complex analytic germs at 0, i.e. the collection of definable, complex analytic functions f : U → ℂ (where U is a (definable) open neighbourhood of 0), where two such functions are identified if there is some open neighbourhood of 0 on which they agree. I will, however, not distinguish notationally betwen functions and their germs.

It is clear that G̃ is an integral domain (under pointwise operations) and a differential ring (under the usual derivative \frac{d}{dz}). We are interested in its subset consisting of those ϕ ∈ G̃ having a definable star function ⋆ϕ : S_{ϕ} → ℂ.

It is not immediately obvious this is a subring of G̃: it could be the case, for example, that ⋆ϕ : S_{ϕ} → ℂ and ⋆ψ : S_{ψ} → ℂ are definable but that the domain of ⋆(ϕ + ψ) (i.e. S_{(ϕ+ψ)}) is strictly larger than S_{ϕ} ∩ S_{ψ}. So we would need to show that the extension of the (obviously definable) function ⋆ϕ + ⋆ψ : S_{ϕ} ∩ S_{ψ} → ℂ to S_{(ϕ+ψ)} is definable.

To resolve this rather annoying difficulty, we first let S̃ be the collection of all definable, open subsets of ℂ of the form ℂ \ ⋃_{j=1}^{n}[a_{j}, ∞), where a_{1}, …, a_{n} are (necessarily definable) nonzero complex numbers (and where, for a ∈ ℂ, [a, ∞) := {ta : 1 ≤ t}). Now let

M̃:= {ϕ ∈ G̃: ϕ has a (definable) representative ϕ̄: U → ℂ for some U ∈ S̃}.

Now it is certainly clear that M̃ is a subring of G̃ since S̃ is closed under intersection. We would like to show that M̃ may be identified with the collection of those ϕ ∈ G̃ having a definable star function. Such a result is in the spirit of those in section 2.7 of [4] but does not seem to follow directly from them. So instead we argue as follows. Consider, more generally, any definable complex analytic function F : U → ℂ where U is a (definable) open subset of ℂ of co-dimension at most 1 (in the sense of the o-minimal structure ℝ̃). Let E(F) denote the collection of all (not necessarily definable) complex analytic functions G : V → ℂ with U ⊆ V ⊆ ℂ, V open, and G ↾ U = F. Now if G_{i} : V_{i} → ℂ are in E(F) for i = 1, 2, and z ∈ V_{1} ∩ V_{2} then G_{1}(z) = G_{2}(z). This is because for some ϵ > 0, Δ(z; ϵ) ⊆ V_{1} ∩ V_{2} and Δ(z; ϵ) ∩ U is a nonempty open set (as U has codimension 1) on which G_{1} and G_{2} agree (with F). It follows that G_{1} and G_{2}, being complex analytic, agree throughout Δ(z; ϵ) and hence in particular that G_{1}(z) = G_{2}(z). It now follows that all functions in E(F) have a common extension, H : W → ℂ say, which also lies in E(F). Further, H is definable. To see this let A ⊆ ℂ × ℂ be the closure of the graph of F : U → ℂ. Then one easily shows that for all z, w ∈ ℂ, H(z) = w if and only if ⟨z, w⟩ ∈ A and for some ϵ > 0, A ∩ (Δ(z; ϵ) × ℂ) is the graph of a continuously (complex) differentiable function with domain Δ(z; ϵ), and this is a definable condition.

Now suppose that ϕ ∈ M̃, represented by ϕ̄ : U → ℂ with U ∈ S̃. Since sets in S̃ obviously have co-dimension at most 1, we may apply the argument above to F = ϕ̄ and let H : W → ℂ be the resulting maximal extension. Then as the function ⋆ϕ : S_{ϕ} → ℂ lies in E(ϕ̄) it follows that S_{ϕ} ⊆ W and H ↾ S_{ϕ} = ⋆ϕ. Now it may be the case that the inclusion here is proper (e.g. if ϕ(z) = (1 − z)^{−1}, then S_{ϕ} = ℂ \ [1, ∞) whereas W = ℂ \ {1}), but, given W it is very easy to define the singular points of ϕ, and hence also the set S_{ϕ}. Since ⋆ϕ is just the restriction of H to S_{ϕ} its definabilty also follows, as required.

The rest of this paper is devoted to proving that M̃ is closed under various operations. Our first observation is now clear.

Theorem 3. M̃ is a subring (in fact, a differential subring) of G̃.

We also have the following.

Theorem 4. M̃ is algebraically closed in G̃.

Proof. Firstly, if f ∈ M̃ is invertible in G̃ (i.e. if f(0) ≠ 0) then it is invertible in M̃. For if domain(f) = U ∈ S̃, let Z_{f} := {a ∈ U : f(a) = 0}. Since Z_{f} is a discrete set, it is finite (by o-minimality). So if we set V := U \ ⋃_{a∈Z_{f}} [a, ∞) then V ∈ S̃ and \frac{1}{f} : V → ℂ is a definable, complex analytic function. Hence \frac{1}{f} ∈ M̃.

So to prove the theorem it is now sufficient to consider a monic polynomial

P (w) = w^{n} + f_{1} · w^{n−1} + ⋯ + f_{n}

where f_{1}, …, f_{n} ∈ M̃, which has a root, ϕ say, in G̃. We may assume that P is irreducible over (the field of fractions of) M̃ and, in particular, that its discriminant, D say, is a nonzero element of M̃. It follows, as above, that we can find a set U ∈ S̃ such that both U ⊆ ⋂_{j=1}^{n} domain(f_{j}) and D(z) ≠ 0 for all z ∈ U \ {0}. It now follows from classical theory that ϕ has an analytic continuation to all points of U. (The continuation is single valued since U is simply connected.) In particular, U ⊆ S_{ϕ} and this continuation is necessarily equal to ⋆ϕ ↾ U (as both functions agree on an open neighbourhood of zero) and, further, P (⋆ϕ) = 0 (in the ring M̃). It remains to show that ⋆ϕ ↾ U is definable. However, this follows easily by considering a cell decomposition of ℝ^{3} compatible with the (definable) set consisting of all ⟨x, y, t⟩ ∈ ℝ^{3} such that x + iy ∈ U and for some u ∈ ℝ,

(t + iu)^{n} + f_{1}(x + iy) · (t + iu)^{n−1} + ⋯ + f_{0}(x + iy) = 0.

Then the graph of the real part of ⋆ϕ ↾ U is given by piecing together certain 0, 1 and 2 cells of this decomposition. The imaginary part of ⋆ϕ ↾ U is dealt with similarly and this completes the proof of the theorem.

It follows from Theorem 4 that if ℝ̃ = \overline{ℝ} then M̃ = G̃.

Open Problem 2 Does there exist an o-minimal expansion ℝ̃ of \overline{ℝ} in which some nonalgebraic, complex analytic germ f : Δ(0; r) → ℂ is definable, but is such that M̃ = G̃?

Certainly the complex exponential function restricted to a disk Δ(0; r) could not be definable in such an ℝ̃ since its star function is the entire exponential function which is not definable in any o-minimal structure.

From now on we assume that, for any R > 0, exp ↾ {x + iy : −R < y < R} is definable in ℝ̃. This is equivalent to both the real exponential function exp ↾ ℝ and the restricted sine function sin : [0, 2π) → ℝ being definable in ℝ̃. The structure ℝ_{an,exp} is an example.

Theorem 5. Let f ∈ M̃ and assume that f(0) ≠ 0. Then any branch of log f (restricted to some set in S̃) is in M̃.

Proof. Clearly we may choose r > 0 small enough so that all determinations of log f ↾ Δ(0; r) are in G̃. Let L_{f} : Δ(0; r) → ℂ be such a determination. Let U = domain(f), so that U ∈ M̃. Now arguing as before we may assume that f(z) ≠ 0 for all z ∈ U. Thus, since U is simply connected, L_{f} extends to a single valued logarithm of f on all of U via the usual formula

L_{f}(z) = L_{f}(0) + ∫_{0}^{z} \frac{f′(w)}{f(w)}dw (z ∈ U)

where the integration is along the straight line segment [0, z] ⊆ U. To see that L_{f} : U → ℂ is definable let us first note that the functions |f| : U → ℝ_{>0} and \frac{f}{|f|} : U → \{w ∈ ℂ : |w| = 1\} are definable, continuous functions. Since the real logarithm function from ℝ_{>0} to ℝ is definable, we obtain immediately that the real part of L_{f} (= log |f|) is definable. To deal with the imaginary part we note that as L_{f} is definable in some neighbourhood of 0, the number L_{f}(0) is definable and hence so is its imaginary part, θ_{0} say. Then for z ∈ U, the imaginary part of L_{f}(z) is given by θ_{z}(z), where θ_{z} : [0, z] → ℝ is the unique continuous function satisfying (a) θ_{z}(0) = θ_{0}, and (b) \frac{f}{|f|}(w) = e^{iθ_{z}(w)} for w ∈ [0, z].

So we must show that θ_{z}(z) is a definable function. For z ∈ U, let

A_{z} = \{t ∈ ℝ : 0 ≤ t ≤ 1 and \frac{f}{|f|}(tz) = 1\}

Then A_{z} is, uniformly in z, a definable subset of [0, 1]. It follows by ominimality that there exists N > 0 such that for all z ∈ U, either A_{z} contains at most N points, or else A_{z} contains an open interval. In the former case we clearly have that

θ_{0} − 2π(N + 1) ≤ θ_{z}(w) ≤ θ_{0} + 2π(N + 1)

for all w ∈ [0, z]. This holds in the latter case too since then, by analyticity, f is real (and positive) throughout [0, z], and hence θ_{z} is constant with value θ_{0}. We now consider a cell decomposition of ℝ^{3} compatible with the set

{⟨x, y, θ⟩ ∈ ℝ^{3} : x + iy ∈ U, θ_{0} − 2π(N + 1) ≤ θ ≤ θ_{0} + 2π(N + 1) and cos θ + i sin θ = \frac{f(x+iy)}{|f(x+iy)|}\}.

(Notice that this set is definable by our assumptions on ℝ̃.) The graph of the function z ⟼ θ_{z}(z) (z ∈ U) is now obtained by piecing together certain 0, 1 and 2 cells of this decomposition.

By a similar argument one can also establish the following result.

Theorem 6. If f ∈ M̃, f(0) ≠ 0, and α is an exponent of ℝ̃, then (any branch of) f^{α} also lies in M̃.

I am reasonably confident that Theorems 4 and 5 have a common generalization as suggested by the following problem.

Open Problem 3 Let f_{1}, …, f_{n} ∈ M̃ \ {0} and let τ_{1}(w), …, τ_{n}(w) be one variable terms of L(ℂ_{exp}). Assume that ϕ ∈ G̃ satisfies

∑_{j=1}^{n} f_{j}(z) · τ_{j}(ϕ(z)) = 0

for all z ∈ domain(ϕ). Is it the case that ϕ ∈ M̃?

As for our original motivation, I conjecture a positive answer to the following.

Open Problem 4 Let τ(z, w) be a two variable term of L(ℂ_{exp}) and let ϕ ∈ G̃ be such that

τ(z, ϕ(z)) = 0

for all z ∈ domain(ϕ). Then does the star of ϕ have at most countably many singular points? If so, is each such point definable?

Finally, I state a result that makes no reference to definablity. It follows immediately from Theorems 4, 5, 6 and 2. However, I see no way of proving it without using the o-minimality of, say, ℝ_{an,exp}.

Theorem 7. Let O denote the ring of complex analytic germs at the origin in ℂ. Let H be the smallest subset of O containing the polynomial ring ℂ[z] and satisfying the following closure conditions:

(i) H is a subring of O and is algebraically closed in O; (ii) if f ∈ H and f(0) ≠ 0, then log f ∈ H; (iii) if f ∈ H, f(0) ≠ 0 and α ∈ ℝ, then f^{α} ∈ H. Then for every germ f ∈ H there exists a finite set s_{f} ⊆ ℂ such that for all z ∈ ℂ \ s_{f} there exists a piecewise linear path starting at 0 and terminating at z along which f can be analytically continued.

References

[1] G. H. Hardy, Divergent Series, Oxford (Édition Jacques Gabay), (1992).

[2] Serge Lang, Complex Analysis, Springer (Fourth edition), (1999).

[3] A.I. Markushevich, Theory of Functions of a Complex Variable, (Translated from Russian) Chelsea (1977).

[4] Y. Peterzil, S. Starchenko, Expansions of algebraically closed fields in o-minimal structures, Selecta Math. (N.S.) 7, no.3 (2000), pp.409-445.