Hyperbolic Algebraic and Analytic Curves

Jim Agler ^{∗} U.C. San Diego

La Jolla, California 92093 John E. McCarthy ^{†} Washington University St. Louis, Missouri 63130

November 7, 2006

Abstract

A hyperbolic algebraic curve is a bounded subset of an algebraic set. We study the function theory and functional analytic aspects of these sets. We show that their function theory can be described by finite codimensional subalgebras of the holomorphic functions on the desingularization. We show that classical analytic techniques, such as interpolation, can be used to answer geometric questions about the existence of biholomorphic maps. Conversely, we show that the algebraic-geometric viewpoint leads to interesting questions in classical analysis.

0 Introduction

By a hyperbolic algebraic curve we mean a set V that is the intersection of a one dimensional algebraic set C with a bounded open set in ℂ^{n}. We call them hyperbolic to emphasize that we wish to study the geometry and function theory on V, not the global theory on C, nor just the local theory on small neighborhoods of points of V. Note that by a holomorphic function on V we mean a function that in a neighborhood of every point of V is the restriction of a holomorphic function on a neighborhood in ℂ^{n}.

∗Partially supported by National Science Foundation Grant DMS 0400826 †Partially supported by National Science Foundation Grant DMS 0501079

Figure 1: V_{1}, a hyperbolic subset of Newton’s nodal cubic

The leimotif is the difference between studying the hyperbolic geometry and function theory on the unit disk 𝔻 in ℂ, as opposed to studying the “parabolic” theory on ℂ or the “spherical” theory on the Riemann sphere. Here are some simple examples of what we wish to consider.

Example 0.1 Let C be Newton’s nodal cubic, {(z, w) ∈ ℂ^{2} : z^{2} = w^{2}(1 − w)}, and let V_{1} = C ∩ {|1 − w| < 2} (see Figure 1). Now consider another hyperbolic algebraic curve V_{2} that is topologically the same as V_{1}, i.e. a disk that intersects itself once. When is there a biholomorphic bijection from V_{1} onto V_{2}?

The answer (see Section 7) is that the Kobayashi geodesic that goes from the multiple point across V_{2} and back to the multiple point must have the same length as the corresponding geodesic in V_{1}. If one cuts C very close to the multiple point, this length gets very large; if one cuts it very far away, the length is small. So there is a one parameter family of conformal structures on sets that are homeomorphic to V_{1}, and they can all be attained by taking the intersection of C with an appropriate bounded set U.

The set V_{1} can be obtained as the image of the unit disk 𝔻 under the map

h(ζ) = (\sqrt{2}(ζ − 2ζ^{3}), 1 − 2ζ^{2}).

The map h has nowhere vanishing derivative, and it is one-to-one except at the points ±1/\sqrt{2}.

The set {F ∘ h : F holomorphic on V_{1}} is the subalgebra of O(𝔻) (the functions analytic on 𝔻) that have the same value at ±1/\sqrt{2}. One could reverse the viewpoint, and ask:

Given a finite codimensional subalgebra A of O(𝔻), when does there exist a map h from 𝔻 to some hyperbolic algebraic curve V in ℂ^{n} such that

A = {F ∘ h : F holomorphic on V }?  (0.2)

If such an h exists, what is the smallest dimension n that suffices?

Example 0.3 Let A^{2} and A^{3} be the subalgebras of O(D) given by the following (the superscript is the codimension):

A^{2} = \{f ∈ O(𝔻) : f(0) = f(\frac{1}{2}) = f(−\frac{1}{2})\}
A^{3} = \{f ∈ A^{2} : f′(0) = −\frac{15}{64} [f′(\frac{1}{2}) + f′(−\frac{1}{2})]\}.

Can these algebras be realized as in (0.2) for some V in ℂ^{2}?

The algebra A^{2} cannot; because of the triple point, for any h mapping into ℂ^{2}, there must be some linear relation between h(0), h(\frac{1}{2}) and h(−\frac{1}{2}), and this relation will apply to the entire algebra (0.2) generated by h. One can realize A^{2} in ℂ^{3} — see Section 8.

Once one adds a linear relation on the derivatives, as in A^{3}, one can realize the algebra in ℂ^{2}. For the particular example of A^{3}, the function

h(z) = (z \frac{z^{2} − 1/4}{1 − z^{2}/4}, z^{2} \frac{z^{2} − 1/4}{1 − z^{2}/4})

works.

Examples 0.1 and 0.3 illustrate the first three of the general themes of this paper:

(i) deciding when two hyperbolic algebraic curves are isomorphic (in the sense of there being a biholomorphic homeomorphism between them);

(ii) studying the duality between hyperbolic algebraic curves and cofinite subalgebras of the holomorphic functions on the desingularization;

(iii) finding the minimum dimension into which a given hyperbolic algebraic curve can be isomorphically realised;

(iv) exploring the operator theoretic aspects of hyperbolic algebraic curves. There is in general a duality between hyperbolic algebraic curves and

maps like h that are one-to-one except on a finite set. To describe the duality formally, it is convenient to broaden our scope to the analytic analogue of a hyperbolic algebraic curve, which we call a hyperbolic analytic curve. It is a one dimensional analytic subvariety of a bounded open set.

Definition 0.4 A hyperbolic analytic curve V ⊆ ℂ^{n} is a set such that

(i) V is a relatively closed subset of some bounded open set U ⊆ ℂ^{n}. (ii) There is a finite set B ⊂ V such that for every P ∈ V \ B there exists a neighborhood W of P in ℂ^{n} and n−1 holomorphic functions F = (F_{2}, …, F_{n}) on W such that

(a) The rank of the derivative DF is n − 1 on W.

(b) V ∩ W = {z ∈ W : F (z) = 0}. (iii) For every P ∈ B there exists a neighborhood W of P in ℂ^{n} and finitely many holomorphic tuples of functions F^{j} = (F_{2}^{j}, …, F_{n_{j}}^{j}) on W such that

V ∩ W = ∪_{j}{z ∈ W : F^{j}(z) = 0}.

Definition 0.5 A holomap is a proper holomorphic map h from a Riemann surface S into a bounded open set U ⊆ ℂ^{n} such that there is a finite subset E of S with the property that h is non-singular and injective on S \ E.

All hyperbolic analytic curves arise as the images of Riemann surfaces under holomaps, and the surface is unique if it is unpunctured (see Section 1 for a precise definition of unpunctured).

Theorem 1.2 (i) If h is a holomap from a Riemann surface S into the bounded open set U ⊂ ℂ^{n}, then h(S) is a hyperbolic analytic curve .

(ii) Conversely, if V is a hyperbolic analytic curve relatively closed in the bounded open set U, then there exists a Riemann surface S and a holomap h : S → U ⊂⊂ ℂ^{n} such that V = h(S).

(iii) Moreover, if there is another Riemann surface S̃ and a holomap h̃ from S̃ onto V, and if S̃ is unpunctured on the pre-image of an open set containing all the singular points of V, then there is a biholomorphic bijection ψ : S → S̃ such that h = h̃ ∘ ψ.

One can also approach hyperbolic analytic curves algebraically. If h : S → V is a holomap, then any holomorphic function ϕ on V can be lifted to the function ϕ ∘ h on S. Not all functions on S arise in this way: for example, such functions must identify all points that h identifies. But the set of functions on S that do factor through V via h is an algebra, of finite codimension. Let us make two definitions.

Definition 0.6 If W is any subset of a set S with a conformal structure, then by a holomorphic function on W we mean a function f : W → ℂ with the property that for every point λ in W there is a neighborhood U of λ in S and a holomorphic function ϕ on U such that ϕ|_{U∩S} = f|_{U∩S}. We denote by O(W) the Fréchet space of all holomorphic functions on W, with the topology of uniform convergence on compacta.

Definition 0.7 A cofinite subalgebra A of O(W) is a unital closed subalgebra of O(W) whose codimension as a vector space is finite. When the set W is understood, we shall call A a cofinite algebra.

We prove in Theorem 2.9 that all cofinite subalgebras of O(S), where S is good Riemann surface, arise as {F ∘ h : F ∈ O(V)} for a holomap h : S → V, and conversely that every set of the form {F ∘ h : F ∈ O(V)} is a cofinite subalgebra. Moreover, if S is a reasonably nice hyperbolic algebraic curve, then V can also be chosen to be a hyperbolic algebraic curve . So all cofinite algebras of O(S) arise as in Example 0.1 by factoring through another hyperbolic algebraic curve that has a finite number of singularities.

We shall write

A_{h} := {F ∘ h : F ∈ O(V)}.

To state the theorem, we need one more definition.

Definition 0.8 A Riemann surface S is called good if there is a holomap h : S → U ⊆ ℂ^{n} such that h has no singular points and {p ∘ h : p ∈ ℂ[z_{1}, …, z_{n}]} is dense in O(S).

Remark 0.9 Every finite Riemann surface is good, where a finite Riemann surface is one obtained from a compact Riemann surface by deleting a positive finite number of smoothly bounded disjoint closed disks. Indeed, by a result of E.L. Stout [24], every finite Riemann surface S can be properly embedded in 𝔻^{3} by a triple of inner functions. The image h(S) is an analytic subvariety of 𝔻^{3} by Theorem 1.2, and has no singular points, so every analytic function on S is of the form F ∘ h for some analytic function F on a neighborhood of h(S). By a theorem of H. Cartan [7], F can be extended to an analytic function on the whole tridisk, and so can be approximated u.c.c. by polynomials. Therefore A_{h} = O(S).

We can now state the equivalence between cofinite algebras and realizations of them via holomaps.

Theorem 2.9 (i) If h : S → V ⊆ U is a holomap from an unpunctured Riemann surface onto a hyperbolic analytic curve V, then

A_{h} := {F ∘ h : F ∈ O(V)}

is a cofinite subalgebra of O(S).

(ii) Conversely, if A is a cofinite subalgebra of O(S) for some good Riemann surface S, then there exists a holomap h : S → ℂ^{n} such that A = A_{h}.

(iii) Let V be a hyperbolic algebraic curve, and assume that O(V) can be generated by a finite number of rational functions that are continuous on \overline{V}. If A is a cofinite subalgebra of O(V), then there is a hyperbolic algebraic curve W and a holomap h from V onto W such that A = A_{h}.

The purpose of proving these equivalences is to get the isomorphism theorem 3.1, which, roughly speaking, says that two hyperbolic algebraic curves (or hyperbolic analytic curves) are isomorphic (in the sense of there being a biholomorphic homeomorphism between them) if and only if there is an isomorphism of the corresponding Riemann surface that preserves the structure of the cofinite algebras.

If A = A_{h} for some holomap h, we shall say that h holizes the algebra A, and that it holizes the set h(S).

In Sections 4-8, we consider cofinite subalgebras of O(𝔻), which we call petals. It is of particular interest to know when cofinite subalgebras of O(𝔻) can be holized by a pair of finite Blaschke products. The hyperbolic algebraic curve is then a special sort of subvariety V of the bidisk that has the property that \overline{V} ∩ ∂(𝔻^{2}) = \overline{V} ∩ (∂𝔻)^{2}. These are called distinguished varieties (since they exit the disk through the distinguished boundary) and have been studied in [3] and [27].

One difficulty in the theory is proving that a particular pair of functions in a cofinite algebra generate the whole algebra. Thanks to work of J. Wermer [29], petals are easier to understand than algebras on general surfaces. We explain this in Section 4. For a pair of Blaschke products, there is a convenient formula that gives the exact codimension of the algebra they generate.

Theorem 5.12 Let f and g be Blaschke products of degree m and n respectively. Suppose there are exactly r < ∞ unordered pairs of points on 𝕋 that are not separated by (f, g). Then the codimension of Alg(f, g) is \frac{(m−1)(n−1)−r}{2}.

In Section 6, we show how the question of whether there is a holomorphic map from one petal to another is reducible to a Pick interpolation problem. In Section 7, we describe all petals of codimension one. In Section 8, we discuss petals of codimension two, and our results here are not complete.

A detailed study of the bounded analytic functions on the unit disk, the space H^{∞}(𝔻), and the Hardy space H^{2} has had profound consequences in operator theory. The backward shift operator S^{∗} on H^{2} is in some sense universal for contractions (operators of norm less than or equal to 1). A celebrated theorem of B. Sz.-Nagy asserts that every completely non-unitary contraction can be realised as the restriction of a direct sum of copies of S^{∗} to an invariant subspace [25]. See e.g. [26], [19], or [2] for expositions.

In Section 9, we illustrate how every hyperbolic algebraic curve comes with its own model theory. We consider V = {(z, w) ∈ 𝔻^{2} : z^{2} = w^{2}}. We prove the following analogue of the von Neumann-Wold theorem. Theorem 9.11 Let T = (T_{1}, T_{2}) be a pair of commuting isometries on a Hilbert space H satisfying T_{1}^{2} = T_{2}^{2}. Then there is a decomposition H = M_{+}⊕M_{−}⊕K, isometries W_{±} on M_{±}, and a pair of operators E_{±} : K → M_{±} satisfying E_{+}^{∗}E_{+} + E_{−}^{∗}E_{−} = I, W_{+}^{∗}E_{+} = 0, and W_{−}^{∗}E_{−} = 0, so that

T_{1} ≅  ( \begin{matrix}& M_{+} & M_{−} & K \\ M_{+} & W_{+} & 0 & E_{+} \\ M_{−} & 0 & W_{−} & E_{−} \\ K & 0 & 0 & 0 \end{matrix} )

T_{2} ≅  ( \begin{matrix}& M_{+} & M_{−} & K \\ M_{+} & W_{+} & 0 & E_{+} \\ M_{−} & 0 & −W_{−} & −E_{−} \\ K & 0 & 0 & 0 \end{matrix} ).

In Section 10, we show how the geometry and analysis interact in finding the minimal inner functions on a hyperbolic algebraic curve. We show that on the Neil parabola, the minimal inner functions split into two sorts, one of degree 2 and the other of degree 3.

1 Lifting hyperbolic analytic curves

The following theorem is well-known, but we could not find a convenient proof in the literature, so we include one for completeness.

Definition 1.1 A puncture in a Riemann surface S is a domain D_{0} ⊆ S that is conformally equivalent via a map τ to {z ∈ ℂ : 0 < |z| < 1}, and such that every sequence {z_{n}} with τ(z_{n}) → 0 is discrete in S. We say that the surface is unpunctured if it has no punctures.

Theorem 1.2 (i) If h is a holomap from a Riemann surface S into the bounded open set U ⊂ ℂ^{n}, then h(S) is a hyperbolic analytic curve .

(ii) Conversely, if V is a hyperbolic analytic curve in the bounded open set U, then there exists a Riemann surface S and a holomap h : S → U ⊂⊂ ℂ^{n} such that V = h(S).

(iii) Moreover, if there is another Riemann surface S̃ and a holomap h̃ from S̃ onto V, and if S̃ is unpunctured on the pre-image of an open set containing all the singular points of V, then there is a biholomorphic bijection ψ : S → S̃ such that h = h̃ ∘ ψ.

Proof: (i) Because h is proper, h(S) must be relatively closed in U. Let V = h(S) and B = h(E). For every point P in V \ B, the inverse image Q = h^{−1}(P) is unique, and, after a change of basis if necessary, we can find a local coordinate ζ at Q so that locally

h(ζ) = (ζ, h_{2}(ζ), …, h_{n}(ζ)).

For 2 ≤ i ≤ n, define F_{i}(z) = z_{i} − h_{i}(z_{1}). Then the derivative of F has full rank.

Now let P ∈ B. Let Q be one of the (finitely many) pre-images of P under h. We will show that for some neighborhood D of Q, the set h(D) is an analytic subvariety of U.

Indeed, after a change of basis if necessary, we can assume that there is some local coordinate ζ at Q and some positive integer N such that, on some disk Δ = 𝔻(Q, ε), the function h is injective and of the form

h(ζ) = (ζ^{N}, h_{2}(ζ), …, h_{n}(ζ)).  (1.3)

(See [8, p.125].) Consider the function g : h(Δ) → D := 𝔻(0, ε^{N}) given by projection onto the first coordinate. Then g is an analytic branched cover of branching order N over D. (This means that g is proper, and that g is an N-sheeted covering map of D \ {0}). Therefore, by [16, Thm. III.B.19], h(Δ) is a one-dimensional analytic subvariety of a neigborhood of P, which means it satisfies condition (iii) of Definition 0.4. Taking the union over all pre-images of P, we get that h(S) is an analytic variety at P too.

(ii) Suppose V is a hyperbolic analytic curve . Let B be the set of singular points of V. Let S′ = V \ B with the conformal structure inherited from V. For each P in B, V is locally the union of finitely many irreducible analytic varieties. Each of these irreducible pieces is locally the image of a disk under a Puiseux series as in (1.3).

The disk Δ and the function h depend on the choice of point P and the choice of irreducible component. For each such choice, take the disjoint union of S′ with Δ, and then identify the points of Δ \ {0} with their image in S under h. This new Riemann surface is the desired S.

(iii) Define ψ = h̃^{−1} ∘ h. This is a well-defined biholomorphic bijection from S′ = S \ E onto S̃′ = S̃ \ Ẽ.

Now consider ψ on a punctured neighborhood of a point Q in E. In local coordinates, it is a bounded analytic function on a punctured disk, therefore it extends to be holomorphic at Q. Moreover, by continuity, as ζ tends to Q, h̃^{−1} ∘ h(ζ) will converge to some point in S̃, since S̃ is unpunctured.

Extend ψ in this manner to all of E. The new ψ will remain injective, since otherwise there would be points Q̃ in S̃ for which every neigborhood contained 2 (or more) disks intersecting only at Q̃. Moreover, by applying the same argument to ψ^{−1}, we see that ψ must also be surjective. □

2 Cofinite algebras

Cofinite sub-algebras of uniform algebras were studied in depth by T. Gamelin, who showed that they can be obtained inductively by imposing one linear restriction at a time on a cofinite algebra of lower codimension [14]. He stated his results in the context of uniform algebras, but his proofs of the parts that we shall use (Theorem 2.3 below), as he remarks in the paper, carry over to our setting.

Before giving his result, let us make two more definitions.

Definition 2.1 A linear functional on O(S) is called local if it comes from a finitely supported distribution, i.e. is of the form

Λ(f) = ∑_{i=1}^{m} ∑_{j=0}^{n_{i}} a_{ij}f^{(j)}(α_{i}).

Definition 2.2 A connection on {α_{1}, …, α_{m}} ⊂ S is a finite dimensional set Γ of local functionals Λ supported in {α_{1}, …, α_{m}}. We say Γ is algebraic if Γ^{⊥} := {f ∈ O(S) : Λ(f) = 0 ∀ Λ ∈ Γ} is an algebra. We say Γ is irreducible if every function f in Γ^{⊥} satisfies f(α_{1}) = f(α_{2}) = … = f(α_{m}).

Note that if the support of Γ is a singleton, it is by definition irreducible. In [6], E. Bishop conidered analogous objects to our Definitions 2.1 and 2.2. In his terminiology, each Λ in the algebraic connection Γ is a homogeneous linear operator of order no more than max(n_{i}) for the algebra Γ^{⊥}.

Theorem 2.3 [Gamelin] (i) If Γ is an algebraic connection on a Riemann surface S then Γ^{⊥} is a cofinite algebra.

(ii) Every cofinite subalgebra of O(S) is of the form Γ^{⊥} for some algebraic connection Γ.

(iii) Every algebraic connection is the union of a finite number of irreducible algebraic connections with disjoint supports.

(iv) Every cofinite algebra A of O(S) can be obtained by a chain A = A_{n} ⊊ A_{n−1} … ⊊ A_{1} = O(S), where each A_{j+1} is an algebra of codimension one in A_{j} and is obtained either as the kernel of a point derivation on A_{j} or by identifying two points of the maximal ideal space of A_{j}.

Proof: Part (i) is obvious. Parts (ii) - (iv) were proved by T. Gamelin [14, Lemma 9.7 and Thm. 9.8]. □

Before giving our holization theorem, we need some lemmata.

Lemma 2.4 Let N, K ≥ 1 be coprime. Let h_{1}(z) = z^{N} + ∑_{i=N+1}^{∞} a_{i}z^{i} and h_{2}(z) = z^{K} + ∑_{i=K+1}^{∞} a_{i}z^{i} be holomorphic on some disk centered at the origin of radius R > 0. Then there is some r > 0 such that, in the disk algebra A(𝔻_{r}), the algebra generated by h_{1} and h_{2} contains all holomorphic functions that vanish at the origin to a high order (that depends only on N and K). Moreover, for all large L, there is some function F_{L} holomorphic on 𝔻_{r} × 𝔻_{r}, such that F_{L}(h_{1}(z), h_{2}(z)) = z^{L}.

Proof: Observe that the leading terms of the functions h_{2}^{j}, the monomials z^{jK}, have powers that are all different mod N for 1 ≤ j ≤ N. Let σ be a permutation of {0, 1, …, N − 1} such that the leading term of h_{2}^{σ(j)} has a power congruent to j mod N.

Define the functions f_{j} by

f_{j}(z) = (h_{1})^{n_{j}}h_{2}^{σ(j)},

where the non-negative integers n_{j} are chosen so that the powers of the leading terms of the functions f_{j} form the arithmetic progression {mN, mN + 1, …, (m + 1)N − 1}.

We wish to obtain the functions z^{mN} and z^{mN+1} in the algebra generated by h_{1} and h_{2}. Once we do this, the algebra generated by z^{mN} and z^{mN+1} contains each of the functions

(z^{mN})^{mN−j}(z^{mN+1})^{j} = z^{m^{2}N^{2}+j}

for 0 ≤ j ≤ mN, and so contains every monomial z^{k} for k ≥ m^{2}N^{2}.

Starting with f_{0}, one can subtract a multiple of f_{1} to cancel the coefficient of z^{mN+1}, and then multiples of f_{2}, …, f_{N−1} to cancel the next few coefficients. Then one subtracts multiples of h_{1}^{i}f_{j} to cancel out the coefficients one by one. If we can show that this process converges, we are done.

Without loss of generality, R can be taken greater than 1, and so there is some constant C that bounds the Taylor coefficients of each f_{j}. Let us relabel the functions

{f_{1}, f_{2}, …, f_{N−1}, h_{1}f_{0}, …, h_{1}f_{N−1}, h_{1}^{2}f_{0}, …}

as {g_{1}, g_{2}, …}. Then at the k^{th} stage of the cancellation, we have the function

f_{0} − ∑_{i=1}^{k} a_{i}g_{i},

and at the (k + 1)^{st} stage we have the function

f_{0} − ∑_{i=1}^{k} a_{i}g_{i} − (f̂_{0}(mN + k + 1) − ∑_{i=1}^{k} a_{i}ĝ_{i}(mN + k + 1)) g_{k+1}.

So we have the inequality

|a_{k+1}| ≤ C(1 + ∑_{i=1}^{k} |a_{k}|).  (2.5)

It follows from (2.5) by induction that

|a_{k}| ≤ C (C + 1)^{k−1} ∀ k.

On the disk of radius r, the modulus of g_{k} is bounded by a constant times r^{k/N}. Therefore if r is chosen so that

r < (C + 1)^{−N},

then the series ∑ a_{k}g_{k} will converge absolutely, and so the function z^{mN} will be in the closure in A(𝔻_{r}) of the algebra generated by h_{1} and h_{2}, as desired. Similarly, starting with g_{1}, one gets the function z^{mN+1}.

Finally, the function F_{L} is a finite product of functions of the form ∑ b_{ij}z_{1}^{i}z_{2}^{j}, where

|b_{ij}| ≤ (C + 1)^{iN}

and, for each i, there are at most N distinct j’s with b_{ij} ≠ 0. Therefore F_{L} is analytic on 𝔻_{r} × 𝔻_{r}, as desired. □

Lemma 2.6 Let T be an additive subsemigroup of ℕ, and suppose that gcd{t ∈ T } = k. Then kℕ \ T is finite.

Proof: By hypothesis, there exist t_{1}, …, t_{n} in T such that gcd(t_{1}, …, t_{n}) = k. Let τ = lcm(t_{1}, …, t_{n}). Then, since the subgroup generated by T is all of kℤ, we have integers m_{ij}, 1 ≤ i ≤ τ/k, 1 ≤ j ≤ n, so that

∑_{j=1}^{n} m_{ij}t_{j} = ik.

Let L ∈ ℕ be such that L + m_{ij} ≥ 0 for every i, j. Then T contains the set

\{∑_{j=1}^{n} t_{j}(m_{ij} + (L + r)\frac{τ}{t_{j}}) : 1 ≤ i ≤ τ/k, r ∈ ℕ\},

and this set equals

{s ∈ kℕ : s > Lτ}.
□

Definition 2.7 Let f be a holomorphic function in a neighborhood of 0. By the order of f, written ord(f), we mean the smallest number such that the corresponding Taylor coefficient for f is non-zero.

Lemma 2.8 Let h = (h_{1}, …, h_{n}) be an n-tuple of analytic functions on 𝔻, with h_{i}(0) = 0 for each i. Suppose h_{1}(z) = z^{N}, and that the n-tuple separates points on 𝔻. Let k > 1 be a divisor of N. Then the algebra generated by h_{1}, …, h_{n} contains a function f with ord(f) not a multiple of k.

Proof: Suppose not. Then every function in A, the algebra generated by h_{1}, …, h_{n}, has order a multiple of k. Some function must have some Taylor coefficient that is not a multiple of k, as the algebra separates points. Choose f in A such that the difference between the first power of z that is not a multiple of k and whose Taylor coefficient is non-zero and the order of f is minimal. So

f(z) = ∑_{i=i_{0}}^{∞} a_{ik}z^{ik} + a_{r}z^{r} + O(z^{r+1}),

where r is not a multiple of k and r − i_{0}k is minimal. Let N = nk. Now consider

g(z) = f^{n}(z) − z^{i_{0}N}.

Then ord(g) ≥ i_{0}N + 1, and it contains a term whose (n − 1)i_{0}k + r Taylor coefficient is non-zero. The difference between this and the order of g is less than or equal to r − i_{0}k − 1, a contradiction. □

It is a theorem of H. Rossi [21] that if X is a compact set in ℂ^{n}, then R(X) can be generated by n + 1 functions. So the condition in (iii) below is not too restrictive. It clearly applies, for example, to any smoothly bounded planar domain.

Theorem 2.9 (i) If h : S → V ⊆ U is a holomap from a Riemann surface onto a hyperbolic analytic curve V, then

A_{h} := {F ∘ h : F ∈ O(V)}

is a cofinite subalgebra of O(S).

(ii) Conversely, if A is a cofinite subalgebra of O(S) for some good Riemann surface S, then there exists a holomap h : S → ℂ^{n} such that A = A_{h}.

(iii) Let V be a hyperbolic algebraic curve, and assume that O(V) can be generated by a finite number of rational functions that are continuous on \overline{V}. If A is a cofinite subalgebra of O(V), then there is a hyperbolic algebraic curve W and a holomap h from V onto W such that A = A_{h}.

Proof: (i) (a) Let g ∈ O(S). Then g ∈ A_{h} if F = g ∘h^{−1} is in O(V). By Theorem 1.2 (iii), we can assume S is the surface constructed in the proof of Theorem 1.2 (ii). Then F is analytic on V \ B.

For F to be analytic at the point P in B, we need certain consistency conditions on g and its derivatives on the pre-images of P. Let Q be a preimage of P. We show that if g vanishes to high enough order at Q, then there will be an analytic function F on a neighborhood of P such that F ∘ h agrees with g on a neighborhood of Q.

Choose a disk Δ containing Q such that \overline{Δ} ∩ E = \{Q\}, and let z be a local coordinate at Q so that Δ = {z : |z| < 1}, and such that h_{1}(z) = z^{N} (after, perhaps, permuting the h_{i}’s).

Consider the subalgebra of O(Δ) generated by h_{1}, …, h_{n}. Let T be the set of natural numbers t such that there is a function in the algebra generated by h_{1}, …, h_{n} whose first non-zero Taylor coefficient is the t^{th}. Then T is a semigroup, and so by Lemma 2.6, is kℕ minus, perhaps, a finite set.

By Lemma 2.8, we must have k = 1. Therefore T contains numbers that are coprime, so by Lemma 2.4, {F ∘ h : F ∈ O(P)} will contain all functions analytic on a neigborhood of Q that vanish at Q to high order.

(b) In part (a), we showed that for each pre-image Q of P, any function g that vanishes at Q to high order can be locally represented as F_{Q} ∘ h for some F_{Q} analytic near P. To finish the proof, we need to show that we can isolate the (finitely many) different sheets at P. This means that if Q_{1} and Q_{2} are distinct preimages of P, and, as in the proof of part (a) we have disks Δ_{1} and Δ_{2} with maps

h^{i} : z_{i} ↦ (z_{i}^{N_{i}}, h_{2}^{i}(z_{i}), …) i = 1, 2,

then there is some function F analytic near P such that F ∘h^{1}(z_{1}) is a power of z_{1} and F ∘ h^{2} is identically 0. Indeed, by part (a) we could then find, for every set of functions g_{1}, …, g_{r} analytic on neighborhoods of Q_{1}, …, Q_{r} in S and vanishing to high enough order, functions F_{i} such that the function F_{i} ∘ h will be g_{i} near Q_{i} and 0 near all other Q_{j}’s. The sum of the F_{i}’s would then be the desired function.

By the proof of [16, Thm. III.B.19] (applied to the map that is projection onto the first coordinate), h^{2}(Δ_{2}) is a one dimensional analytic subvariety, and there exists some open set U′ containing P with the property that for every point R in U′ \ h^{2}(Δ_{2}) there is some function G in O(U′) with the property that G(R) ≠ 0 and G|_{h^{2}(Δ_{2})} ≡ 0.

Choose R in h^{1}(Δ_{1}) \ {P }. Then we get a G such that G ∘ h^{1} is nonconstant, and G ∘ h^{2} ≡ 0. By shrinking the neighborhood Δ_{1} further, if necessary, we can assume that the only zero of G∘h^{1} on \overline{Δ_{1}} is at 0. Therefore G∘h^{1}(z_{1}) is z_{1}^{N} times some unit in O(Δ_{1}). By part (a), {F ∘h^{1} : F ∈ O(P)} contains z_{1}^{k}O(Δ_{1}) for some K, so the set {(F · G) ∘ h : F ∈ O(P)} will contain all functions on Δ_{1} that vanish to order N + K at Q_{1}, and will be identically zero on a neighborhood of Q_{2}.

(ii) It follows from Theorem 2.3 that there is a natural number N with the property that every function in O(S) that vanishes at every point of the connection to order N is in A.

Since S is good, there is a non-singular holomap f from S into some bounded set U ⊆ ℂ^{k} such that polynomials in f_{1}, …, f_{k} are dense in O(S). Let V = f(S), and let the image of the points of the connection under f be be {α_{1}, …, α_{m}}; without loss of generality assume no α_{i} is 0. Then the polynomials in ℂ[z_{1}, …, z_{k}] that vanish to order N at each α_{i} are finitely generated, as an algebra, by polynomials of the form

(z_{1} − α_{1}^{1})^{j_{11}}(z_{2} − α_{1}^{2})^{j_{12}} ⋯ (z_{k} − α_{m}^{k})^{j_{mk}} z_{1}^{i_{1}} ⋯ z_{k}^{i_{k}},  (2.10)

where j_{11} + j_{12} + ⋯ + j_{mk} = N and 0 ≤ i_{1}, …, i_{k} ≤ N − 1.

If necessary, add to the collection of functions given in (2.10) a finite number of additional polynomials that vanish to order less than N and satisfy the conditions of the connection. Then we will have a finite number of polynomials g_{1}, …, g_{n} with the property that the algebra generated by these functions will give all polynomials that satisfy the connection on V. Therefore the functions h_{1} := g_{1} ∘ f, …, h_{n} := g_{n} ∘ f are functions in A with the property that the algebra they generate is the intersection of A with the algebra of polynomials in f. Since polynomials in f are dense in O(S) by hypothesis, any function in A is the u.c.c. limit of a sequence of polynomials in h = (h_{1}, ⋯, h_{n}). Since V is a non-singular analytic subvariety of U, we have therefore that A = A_{h}.

Finally we must prove that h is proper, and therefore a holomap. But this follows because h is one-to-one off the set {α_{1}, …, α_{m}}, and h maps open subsets of S to relatively open subsets of h(S).

(iii) Let V be a hyperbolic algebraic curve as in the statement of the theorem, with V contained in the algebraic curve C. Assume that C has no irreducible components disjoint from V. We can choose f : V → U ⊆ ℂ^{k} as in part (ii), with the additional property that the f_{i}’s are rational and bounded on V. Moreover, f extends to be a regular rational function on C less, at most, a finite set. Choose the g_{j}’s as in part (ii), and let h = g ∘ f. By Chevalley’s constructibility theorem [17, Thm. 3.16], h(C) is contained in some algebraic curve D ⊆ ℂ^{n}. As h is a homeomorphism off the connection, W = h(V) is relatively open in D, so W is a hyperbolic algebraic curve, as required. □

3 Isomorphism theorem

We now reach the pay-off for setting things up carefully, the isomorphism theorem, which allows us to pass between algebraic and geometric information.

If Λ is a linear functional on O(S_{1}), and ψ : S_{1} → S_{2} is a holomoprphic map, we define the push-forward ψ_{∗}(Λ) by

ψ_{∗}(Λ)[f] = Λ[f ∘ ψ].

Applying this to each element of a connection Γ on S_{1}, one gets the pushforward ψ_{∗}(Γ) on S_{2}.

Theorem 3.1 Let S_{1} and S_{2} be Riemann surfaces, and, for r = 1, 2, let h_{r} : S_{r} → V_{r} ⊆ U_{r} be a holomap onto a hyperbolic analytic curve. Then the following are equivalent:

(i) There is a biholomorphic homeomorphism ϕ : V_{1} → V_{2}. (ii) There are biholomorphic homeomorphisms ψ : S_{1} → S_{2} and ϕ : V_{1} → V_{2} such that h_{2} ∘ ψ = ϕ ∘ h_{1}. (iii) There is a biholomorphic homeomorphism ψ : S_{1} → S_{2} such that

A_{h_{1}}(S_{1}) := {F ∘ h_{1} : F ∈ O(V_{1})}
= {G ∘ h_{2} ∘ ψ : G ∈ O(V_{2})}.

(iv) There is a biholomorphic homeomorphism ψ : S_{1} → S_{2} that pushes forward the connection on S_{1} induced by h_{1} to the connection on S_{2} induced by h_{2}.

Proof: (i) ⇒ (ii): Let E_{1} and E_{2} be the finite subsets of S_{1} and S_{2} respectively, off of which h_{1} and h_{2} are injective and non-singular. Let B_{1} = h(E_{1}) and B_{2} = h(E_{2}). Define ψ by

ψ = h_{2}^{−1} ∘ ϕ ∘ h_{1}.

Then ψ is holomorphic on S_{1}\(ϕ∘h_{1})^{−1}(B_{2}). For any Q in (ϕ∘h_{1})^{−1}(B_{2}), the function ψ maps a small punctured neighborhood of Q to a small punctured neighborhood in S_{2} that is disjoint from E_{2}. So the singularity at Q is removable, and ψ extends to be holomorphic from S_{1} to S_{2}. The same argument shows that ψ^{−1} is holomorphic.

(ii) ⇒ (iii): Define G = F ∘ ϕ^{−1}.

(iii) ⇒ (iv): Let the connections Γ_{1}, Γ_{2} be given by

Γ_{r} = A_{h_{r}}(S_{r})^{⊥}, r = 1, 2.

Define τ_{ψ} : O(S_{2}) → O(S_{1}) by τ_{ψ}(g) = g ∘ ψ. Then (iii) says

Γ_{1} = {Λ ∈ O(S_{1})^{∗} : Λ(F ∘ h_{1}) = 0 ∀ F ∈ O(V_{1})}
= {Λ ∈ O(S_{1})^{∗} : Λ ∘ τ_{ψ}(G ∘ h_{2}) = 0 ∀ G ∈ O(V_{2})}
= Γ_{2} ∘ τ_{ψ}^{−1}.

Therefore

Γ_{1} ∘ τ_{ψ} = Γ_{2}.

(iv) ⇒ (i): For every G in O(V_{2}), the function G ∘ h_{2} ∘ ψ is in O(S_{1}). Moreover, if Λ is in Γ_{1}, then

Λ(G ∘ h_{2} ∘ ψ) = (Λ ∘ τ_{ψ})(G ∘ h_{2}) = 0.

Therefore G ∘ h_{2} ∘ ψ is in A_{h_{1}} for every G. By Theorem 2.9, this means there always exists some F in O(V_{1}) so that G ∘ h_{2} ∘ ψ = F ∘ h_{1}.

Letting G run over the coordinate functions, we get that there is a holomorphic mapping ϕ on V_{1} such that

h_{2} ∘ ψ = ϕ ∘ h_{1}.

As ϕ is one-to-one by construction, the same argument applied to ψ^{−1} yields that ϕ is a biholomorphic equivalence. □

We shall say that two hyperbolic analytic curves V_{1} and V_{2} are isomorphic if there is a biholomorphic homeomorphism between them. Note that even if they are both hyperbolic algebraic curves, one cannot assume that the map is algebraic; for example, a disk and a square in the plane are isomorphic by our definition. We shall say that two cofinite algebras are isomorphic if they are related as in (iii) of Theorem 3.1.

A very similar argument to the proof of Theorem 3.1 allows us to tell when there is any holomorphic map between two hyperbolic analytic curves. Using the same notation, we have:

Theorem 3.2 Let h_{r} : S_{r} → V_{r} be holomaps onto hyperbolic analytic curves. Then the following are equivalent:

(i) There is a holomorphic map ϕ : V_{1} → V_{2}. (ii) There are holomorphic maps ψ : S_{1} → S_{2} and ϕ : V_{1} → V_{2} such that h_{2} ∘ ψ = ϕ ∘ h_{1}. (iii) There is a holomorphic map ψ : S_{1} → S_{2} such that

A_{h_{1}}(S_{1}) ⊇ {A_{h_{2}}(S_{2}) ∘ ψ}.

(iv) There is a holomorphic map ψ : S_{1} → S_{2} such that

Γ_{1} ∘ τ_{ψ} ⊆ Γ_{2}.

4 Generators

If A is a cofinite subalgebra of O(S), and f_{1}, …, f_{n} are functions in A, it is of interest to know when they generate the whole algebra. Let Alg(f_{1}, …, f_{n}) denote the closure of the polynomials in f_{1}, …, f_{n} (in the Fréchet topology).

In order for a unital sub-algebra A to equal O(S), the following two conditions are clearly necessary:

(SP): The functions in A separate points of S.

(ND): At every point of S there is a function in A whose derivative does not vanish there.

We shall show that these conditions are sufficient in two particular cases. The first condition, named after J. Wermer, is topological; the second, named after E. Bishop, is a Mergelyan-like theorem, in which the algebra A must take the topology of S into account.

(W): Suppose that S is the interior of a compact set D in a Riemann surface S, and the boundary of S is a simple closed analytic curve.

(B): Suppose that there is a sequence of compact sets K_{n} such that each K_{n} is contained in the interior of K_{n+1}, the union of all the K_{n} is S, and there exists n_{0} such that, for all n ≥ n_{0}, for all p in S \ K_{n}, there exists some f in A with |f(p)| > ∥f∥_{K_{n}}.

Theorem 4.1 Let A be a unital sub-algebra of O(S) satisfying (SP) and (ND). If either (W) or (B) is also satisfied, then A is dense in O(S).

Proof: First assume (W). Then one can find an increasing sequence of compact sets D_{n} whose union is S and such that each D_{n} has a boundary that is a simple closed analytic curve γ_{n}. Moreover, one can find a function ϕ in A whose derivative is non-zero anywhere on γ_{n}. (Indeed, choose ϕ to be any function whose derivative does not vanish identically. Then its zeroes are isolated, and one can perturb γ_{n} to miss them all).

By a result of Wermer [29, Appendix], the algebra A is then uniformly dense in A(D_{n}), the functions analytic on the interior of D_{n} and continuous up to the boundary. Now suppose that A were not dense in O(S). Then by the Hahn-Banach theorem, there would be a non-zero linear functional Λ on O(S) that annihilated A. Every element of the dual of O(S) comes from integration against a compactly supported measure μ [10, Thm. 4.10.1]. For n large enough, μ will be supported on D_{n}. But as integration against μ annihilates a dense subspace of A(D_{n}), it must annihilate all holomorphic functions on S restricted to D_{n}, a contradiction.

Now assume (B). Again, argue by contradiction, and assume that there is a compactly supported measure μ that annihilates A but not all of O(S). Let the support of μ be K. Let K̂ be {p ∈ S : ∀ f ∈ A, |f(p)| ≤ ∥f∥_{K}}. By assumption (B), K̂ is compact. By a result of Bishop [6], A is dense in A(K̂), and again we achieve a contradiction. □

It is not true that every subalgebra of O(S) is contained in a proper cofinite subalgebra: consider, for example, the algebra of polynomials inside the algebra of all holomorphic functions on an annulus. Theorem 4.1 shows, however, that if the set S satisfies (W), or the algebra A satisifies (B), then it is either dense, or contained in a codimension one subalgebra of O(S).

Given a particular cofinite subalgebra, we know from Theorem 2.9 that it can be holized into ℂ^{n} for some n. To prove that n can be chosen to be small, we need to show that some choice of f_{1}, …, f_{n} in A do not generate a proper subalgebra. The next theorem says that, in certain important circumstances (such as trying to holize an algebra on the disk into ℂ^{2}) you only need to worry about the obvious obstructions.

Theorem 4.2 Let S be a set satisfying (W). Let Γ be a connection on S, and let A = Γ^{⊥}. Let F = {α_{1}, …, α_{m}} be the support of Γ. Let f = (f_{1}, f_{2}) be a pair of functions in A. Assume that

(i) If f identifies a pair of points in S, then both points are in F. (ii) The derivative of f is non-vanishing on S \ F.

Then Alg(f_{1}, f_{2}) is a cofinite algebra given by a connection that is supported on F.

Proof: Choose an increasing sequence D_{n} as in the proof of Theorem 4.1, and moreover assume that F ⊂ int(D_{1}). If necessary, perturb the boundaries of the D_{n}’s so that either df_{1} or df_{2} is never zero on ∂D_{n}. Then by the proof of Theorem 1.2 in [29], there is an integer k that is independent of n and such that every function in A(D_{n}) that vanishes to order k on F is uniformly approximable by functions in Alg(f_{1}, f_{2}). Therefore Alg(f_{1}, f_{2}) contains all functions on S that vanish to order k on F, so by Theorem 2.3 we are done. □

Because of the topology we are using, many subtleties of boundary behavior disappear. In [23], Stessin and Thomas consider the problem of determining the H^{p}-closure of the algebra generated by a pair of functions.

If an algebra is generated by an n-tuple h = (h_{1}, …, h_{n}), then h is not necessarily a holomap: if h(∂S) is not disjoint from h(S), then one cannot find an open set U containing h(S) such that h : S → U is proper. Conversely, if A = A_{h} for some holomap h : S → V ⊂ U, then the polynomials in h need not be all the functions in A, if there are holomorphic functions on V that are not approximable by polynomials. If all the functions h_{i} are inner functions that are continuous up to the boundary of S, however, then everything is nice. Indeed, the map h is then a a holomap into the polydisk 𝔻^{n}. By Theorem 1.2, the set V = h(S) is a hyperbolic analytic curve. By Cartan’s theorem [7], any holomorphic function F on V extends to a holomorphic function on 𝔻^{n}, and is therefore uniformly approximable on compact sets by polynomials. Thus we have:

Proposition 4.3 Let S be a smoothly bounded Riemann surface. Let h = (h_{1}, …, h_{n}) be an n-tuple of holomorphic functions that are continuous up to ∂S and have modulus one everywhere on ∂S. Then Alg(h_{1}, …, h_{n}) = A_{h}.

5 Pairs of Blaschke products

It is natural to ask if a given cofinite subalgebra of O(𝔻) can be holized by a pair of Blaschke products. S.I. Fedorov proved that on every C^{ω}-bounded planar domain Ω there are a pair of continuous inner functions ϕ_{1} and ϕ_{2} that continue analytically across the boundary and such that the pair Φ = (ϕ_{1}, ϕ_{2}) separate points of Ω and dΦ ≠ 0 on Ω [11]. This means that Φ holizes Ω as a distinguished variety. (The fact that ϕ_{1} and ϕ_{2} satisfy an algebraic equation was proved by Fedorov by extending them to the Schottky double, but also follows from general arguments [3].) However, W. Rudin showed that if the connectivity of Ω is greater than one, then Φ cannot be one-to-one on ∂Ω [22].

We shall want to know how many multiple points a pair (f, g) of finite Blaschke products (each of degree greater than one) has on ℂ^{2}, i.e. how many solutions (λ, μ) ∈ ℂ^{2} there are to the simultaneous equations

f(λ) = f(μ)
g(λ) = g(μ).  (5.1)

When counting these, we shall consider (λ, μ) and (μ, λ) to be distinct solutions, and we shall need to count solutions with multiplicity. To this end, write

f(z) = \frac{p^{∼}(z)}{p(z)}  (5.2)
g(z) = \frac{q^{∼}(z)}{q(z)}  (5.3)

where p(z) = ∏_{i=1}^{m}(1 − \overline{α_{i}}z), p^{∼}(z) = ∏_{i=1}^{m}(z − α_{i}) and q(z) = ∏_{i=1}^{n}(1 − \overline{β_{i}}z), q^{∼}(z) = ∏_{i=1}^{n}(z−β_{i}). Let C_{F} and C_{G} be the zero sets of the polynomials

F (z, w) := \frac{p(z)p^{∼}(w) − p(w)p^{∼}(z)}{z − w}  (5.4)
G(z, w) := \frac{q(z)q^{∼}(w) − q(w)q^{∼}(z)}{z − w}  (5.5)

respectively.

Note that generically F in (5.4) is a polynomial of degree 2(m − 1); the coefficient of z^{m−1}w^{m−1} is

(−1)^{m+1} [∑_{i=1}^{m} α_{i} ∏_{j=1}^{m} ᾱ_{j} − ∑_{i=1}^{m} ∏_{j≠i} ᾱ_{j}] = (−1)^{m+1}\overline{f′(0)}.  (5.6)

Let F̂(t : z : w) and Ĝ(t : z : w) be the homogeneous polynomials of degrees exactly 2(m − 1) and 2(n − 1) so that

F̂(1 : z : w) = F (z, w)
Ĝ(1 : z : w) = G(z, w).

Let CF̂ and CĜ be the zero sets of F̂ and Ĝ in ℂℙ^{2}. Define the multiplicity of the solution to (5.1) at the point (λ, μ) as the intersection multiplicity of the curves CF̂ and CĜ at the point (1 : λ : μ).

Equivalently one can define the multiplicity of the solution to (5.1) at the point (λ, μ) as the intersection multiplicity of the curves C_{F} and C_{G} at the point (λ, μ).

Define F^{∼} by

F^{∼}(z, w) := z^{k}w^{ℓ}\overline{F (1/z̄, 1/w̄)},

where k is the highest power of z in F and ℓ is the highest power of w. Then both F and G are symmetric, in the sense that F^{∼} equals F and G^{∼} equals G.

See [4] for a discussion of polynomials that are symmetric with respect to the torus. Moreover, since the functions f and g have modulus less than one in 𝔻, equal to one on 𝕋 := ∂𝔻, and greater than one on 𝔼 := ℂ \ \overline{𝔻}, both C_{F} and C_{G} are contained in 𝔻 × 𝔻 ∪ 𝕋 × 𝕋 ∪ 𝔼 ∪ 𝔼.

Theorem 5.7 Let f and g be Blaschke products of degree m and n respectively, m, n ≥ 2. Using the notation (5.2 - 5.5), let N_{f,g} be the number of points in C_{F} ∩ C_{G} in 𝔻 × 𝔻 plus half the number of points in 𝕋 × 𝕋. Then N_{f,g} is either infinite or (m − 1)(n − 1). Moreover, if neither λ nor μ is zero and (λ, μ) is a solution to (5.1), so is (1/\overline{λ}, 1/\overline{μ}).

Proof: Let us prove the last assertion first. It is an immediate consequence of the symmetry of C_{F} and C_{G} with respect to 𝕋^{2}. To see it directly,

f(1/\overline{λ}) = \frac{p^{∼}(1/\overline{λ})}{p(1/\overline{λ})}
= \frac{\overline{p(λ)}}{\overline{p^{∼}(λ)}}
= \frac{1}{\overline{f(1/\overline{λ})}}.

So if f(λ) = f(μ), then f(1/\overline{λ}) = f(1/\overline{μ}), and similarly for g.

We can assume without loss of generality that (5.6) is non-zero. Indeed, otherwise we can choose some Möbius map

m_{γ}(z) := \frac{γ − z}{1 − γ̄z}  (5.8)

and consider the Blaschke products f ∘ m_{γ} and g ∘ m_{γ} instead, where γ is chosen as a non-critical point of f. To count the intersections, first assume that g(z) = z^{n}, i.e. that all the β_{i} are 0. For ω an n^{th} root of unity, g(λ) = g(ωλ). For μ = ωλ to also satisfy (5.1), we need

\frac{p(λ)p^{∼}(ωλ) − p(ωλ)p^{∼}(λ)}{λ − ωλ} = 0.  (5.9)

The left-hand side of (5.9) is a polynomial of degree exactly 2m − 2. Indeed, the coefficient of λ^{2m−2} is ω^{m−1} times (5.6), which we have assumed to be non-zero. So for n − 1 choices of ω, there are 2m − 2 different λ’s that solve (5.9), giving 2(m − 1)(n − 1) solutions to (5.1) in ℂ × ℂ.

By the symmetry F = F^{∼}, or by direct calculation, the constant term in F (λ, ωλ) is the complex conjugate of the highest order coefficient, and this is non-zero by hypothesis. So none of the points (λ, ωλ) on C_{F} have λ = 0. Therefore each point with λ ∈ 𝔻 is matched by a point with λ ∈ 𝔼, by the first part of the proof. Therefore we have

N_{f,z^{n}} = \frac{1}{2}2(m − 1)(n − 1) = (m − 1)(n − 1).

Now, move the zeroes of g away from 0 to the points β_{i}. The function G|_{C_{F}} is changing continuously, so the number of zeroes of G in C_{F} ∩ 𝔻^{2} can only change either if a zero moves to C_{F} ∩ 𝕋^{2}, in which case its reflection also moves to C_{F} ∩ 𝕋^{2} creating a double point there, or if a double point on C_{F} ∩ 𝕋^{2} splits, and one moves in and the other moves out. In either case, N_{f,g} remains constant.

The only other possibility is that suddenly G|_{C_{F}} become identically zero on some component. This might happen at the end-point of the path, but can be avoided at all intermediate stages. (Indeed, otherwise there would be a branch of an analytic function, ω(z) ≢ z, such that f(z) ≡ f(ω(z)) and g(z) ≡ g(ω(z). Perturbing just one of g’s zeroes destroys this symmetry).

Therefore, at all points along the path, N_{f,g} = (m − 1)(n − 1), and at the end-point it either has this value or is infinite. □

Bézout’s theorem tells us that the number of intersection points in ℂℙ^{2} of F̂ and Ĝ is precisely deg(F̂) · deg(Ĝ). If one knew the multiplicities at infinity, one could recover the number of intersections in ℂ^{2}. Unfortunately, this is hard to do; but one can use Theorem 5.7 to calculate the multiplicity at infinity.

Corollary 5.10 With notation as above, assume that F and G have degrees exactly 2m−2 and 2n−2, respectively, and that C_{F} ∩C_{G} is finite. Let r be the number of points λ in 𝔻\{0} that satisfy f(λ) = f(0) and g(λ) = g(0). Let s be the intersection multiplicity of C_{F} and C_{G} at (0, 0). Then the intersection multiplicity of the zero sets of F̂ and Ĝ at both (0 : 1 : 0) and (0 : 0 : 1) is exactly

(m − 1)(n − 1) + r + \frac{1}{2}s.

Proof: The number of solutions at points of the form (1 : z : w), by Theorem 5.7, is twice N_{f,g} minus the number of zeroes with either z or w equal to 0. So the number at points of the form (0 : z : w) is, by Bézout’s theorem,

4(m − 1)(n − 1) − 2(m − 1)(n − 1) + 2r + s.  (5.11)

At a point (0 : z : w), the homogeneous polynomial F̂ is (5.6) times z^{m−1}w^{m−1}, so the only roots at infinity occur when either z = 0 or w = 0. By symmetry, the intersection multiplicities at (0 : 1 : 0) and (0 : 0 : 1) are the same, so each must be exactly half (5.11). □

We shall be interested in proving that a pair of Blaschke products (f, g) generates a specific cofinite algebra (see Section 8). Recall that Alg(f, g) denotes the algebra generated by f and g (i.e. the closure in O(𝔻) of the set of polynomials in f and g). The codimension of Alg(f, g) is exactly half the number of points in C_{F} ∩ C_{G} ∩ 𝔻^{2} (because both (λ, μ) and (μ, λ) are counted). The following theorem is the principal result in this section.

Theorem 5.12 Let f and g be Blaschke products of degree m and n respectively. Suppose there are exactly r < ∞ unordered pairs of points on 𝕋 that are not separated by (f, g). Then the codimension of Alg(f, g) is \frac{(m−1)(n−1)−r}{2}.

Proof: If the codimension were infinite, then there would be some distinguished variety on which both F and G vanished, and hence there would be an infinite number of pairs of points in 𝕋 × 𝕋 that were identified by (f, g). When the codimension is finite, the result follows from Theorem 5.7. □

Sometimes one can just prove that there are at least r pairs of points on 𝕋 that are identified by (f, g). To apply Theorem 5.12 to conclude that the codimension of Alg(f, g) is at most ⌊\frac{(m−1)(n−1)−r}{2}⌋, one needs to know that Alg(f, g) is cofinite. Here is a sufficient condition.

Proposition 5.13 Let f and g be Blaschke products of degree m and n respectively, with 2 ≤ m ≤ n < ∞. Using the notation (5.2 - 5.5), assume that F is irreducible. If n is not a multiple of m, then Alg(f, g) is cofinite. If n is a multiple of m, then either Alg(f, g) is cofinite or g is in Alg(f).

Proof: If C_{F} ∩ C_{G} is infinite, then G vanishes on all of C_{F}, and F divides G since F is assumed irreducible. Therefore, whenever f(λ) = f(μ), we also have g(λ) = g(μ), so the function h := g ∘ f^{−1} is well-defined and analytic, and g = h ∘ f is in Alg(f). Since f and g are Blaschke products, so is h, and so the degree of g is a multiple of the degree of f. □

In practice, it may be hard to verify if F is irreducible, so for later use we shall refine the proposition in the case that m is 2 or 3.

Proposition 5.14 Let f and g be Blaschke products of degree m and n respectively, with 2 ≤ m ≤ n < ∞. Assume that m is either 2 or 3. If n is not a multiple of m, then Alg(f, g) is cofinite. If n is a multiple of m, then either Alg(f, g) is cofinite or g is in Alg(f).

Proof: By composing f and g with some m_{γ}, as in the proof of Theorem 5.7, we can assume that F is of degree exactly 2(m − 1) with the only term of that order being z^{m−1}w^{m−1}, and that G is of degree exactly 2(n − 1) with the only term of that order being z^{n−1}w^{n−1}.

If F is reducible, it cannot have a factor that is a polynomial in just z (or just w), as this would force p to be constant. Moreover, it cannot have a linear factor, because if a, b are non-zero and

z^{m−1}w^{m−1} + lower order = (az + bw + c) (stuff),

the second factor would have to have a term in either z^{m−2}w^{m−1} or z^{m−1}w^{m−2} to get a term in z^{m−1}w^{m−1}. Either way, one would then also get a term in either z^{m−2}w^{m} or z^{m}w^{m−2}, which is not allowed.

So if m = 2, we must have that F is irreducible. If m = 3, the only possible factoring is into a pair of quadratic factors. If either factor has a term in say z^{2}, the other can only have powers of w, and this has been ruled out. So the only possible factoring is of the form

F (z, w) = (azw + bz + cw + d) (a′zw + b′z + c′w + d′),

with aa′ ≠ 0. This means C_{F} decomposes into {(z, ϕ_{1}(z))} ∪ {(z, ϕ_{2}(z))} for linear fractional transformations ϕ_{1} and ϕ_{2}. As

f(z) = f(ϕ_{1}(z)) = f(ϕ_{2}(z)),

and these are the only points where this value is attained, we have ϕ_{2} = ϕ_{1}∘ϕ_{1} and ϕ_{1} ∘ ϕ_{1} ∘ ϕ_{1} = id. Since C_{F} is contained in 𝔻^{2} ∪ 𝕋^{2} ∪ 𝔼^{2}, it follows that ϕ_{1} and ϕ_{2} are disk automorphisms.

For definiteness, assume that G vanishes on the sheet {(z, ϕ_{1}(z))}. This means that

g(z) = g(ϕ_{1}(z)).

But then

g(z) = g(ϕ_{1}(z)) = g(ϕ_{1} ∘ ϕ_{1}(z)),

so G actually vanishes on all of C_{F}. Then we are back in the situation of the proof of Proposition 5.13, and we get that g is on Alg(f). □

6 Petals

A petal is a hyperbolic analytic curve that is holized by holomaps from 𝔻, such as the set V_{1} in Example 0.1 of Section 0. By Theorem 3.1, two petals V_{1} and V_{2} are isomorphic if and only if there is a Möbius map m : 𝔻 → 𝔻 that pushes forward the first connection onto the second.

To determine if there is any holomorphic map from V_{1} to V_{2}, again by Theorem 3.2 we can pass to their desingularizations, and ask if there is a holomorphic map ψ from 𝔻 to 𝔻 that pushes forward the connection Γ_{1} corresponding to V_{1} into the connection Γ_{2} corresponding to V_{2}. The simplest case is when the only singularities are double points. Then Γ_{1} identifies the values at the points α_{2i−1} and α_{2i}, for i = 1, …, n, and Γ_{2} identifies β_{2j−1} and β_{2j} for j = 1, …, m, with m ≥ n. After reindexing the β’s, the problem becomes whether there is a map ψ : 𝔻 → 𝔻 such that α_{i} goes to β_{i} for every 1 ≤ i ≤ 2n. By Pick’s theorem [20], (see e.g. [15], [12] or [2] for an exposition), this can be done if and only if the matrix

(\frac{1 − \overline{β_{i}}β_{j}}{1 − \overline{α_{i}}α_{j}} )_{i,j=1}^{2n}

is positive semi-definite.

For more general connections, the question becomes whether there is a holomoprhic function ψ : 𝔻 → 𝔻 that satisfies a finite number of conditions on its values and its derivatives’ values on the support of Γ_{1}. Again, this is answerable using a modification of Pick’s theorem [12] [2]. In particular, if the condition is ever satisfied, it is always satisfied by a finite Blaschke product. So we obtain:

Proposition 6.1 Let V_{1} and V_{2} be petals. If there exists any holomorphic map ϕ from V_{1} to V_{2}, then there exists a proper holomorphic map of constant finite valence.

7 Petals of codimension one

In order to prove that certain holomaps actually realize particular algebras, the following lemma is useful (it can also be derived from Theorem 2.3).

Lemma 7.1 Suppose Γ is an algebraic connection on S, and that h is a holomap that holizes the algebra Γ^{⊥}. Let Dh denote the derivative of h.

(i) If {α} = {h^{−1}(h(α))} and Dh(α) ≠ 0, then α ∉ suppt(Γ). (ii) If {h^{−1}(h(α_{1}))} = {α_{1}, …, α_{m}} and there are functions f_{1}, …, f_{m} in A_{h} such that Df_{j}(α_{i}) = δ_{ij}, then the irreducible component of Γ with support {α_{1}, …, α_{m}} is just the set of equations

f(α_{1}) = f(α_{2}) = … = f(α_{m}).

Proof: (i) Suppose

Λ(f) = ∑_{i=1}^{m} ∑_{j=0}^{n_{i}} a_{ij}f^{(j)}(α_{i})  (7.2)

is a functional in Γ, with α = α_{1}. Choose a polynomial p so that p ∘ h(α) = 1, D(p ∘ h)(α) = 1 and p ∘ h(α_{i}) = 0 for all i > 1. Then for every k, ℓ ≥ 1, the functions [(p ∘ h)^{k} − 1]^{ℓ} are in A_{h}, and the only way they can all be in the kernel of (7.2) is if all the a_{ij}’s are zero.

(ii) The functions [f_{j}^{k} − 1]^{ℓ} give functions that vanish to high order at each α_{i}, for i ≠ j, and the first ℓ − 1 derivatives vanish at α_{j}. So if Λ is as in (7.2), then every a_{ij} must be zero if j > 0, and Λ cannot actually depend on the values of the derivatives. □

The simplest petals that are not disks correspond to codimension one algebras. There are two kinds. The first comes from a simple cusp. It can be described as the algebra

{f ∈ O(𝔻) : f′(α) = 0},  (7.3)

for some α ∈ 𝔻. As a hyperbolic algebraic curve, it can be holized by the map

h(ζ) = (m_{α}(ζ)^{2}, m_{α}(ζ)^{3}),  (7.4)

where m_{α} is the Möbius map

m_{α}(ζ) := \frac{α − ζ}{1 − \overline{α}ζ} .

All of these algebras are isomorphic, and in fact each of the holizations in (7.4) actually holize the algebra into exactly the same set,

{(z, w) ∈ ℂ^{2} : z^{3} = w^{2}, |z| < 1}.

It follows therefore that if one takes the Neil parabola, that is the curve {z^{3} = w^{2}}, and cuts it with any simple closed curve that contains (0, 0) in its interior, the corresponding hyperbolic algebraic curves are isomorphic.

The second type of codimension one petal is the single-crossing algebra defined, for each pair (α_{1}, α_{2}) of distinct points in 𝔻, as

A_{α_{1},α_{2}}^{1} := {f ∈ O(𝔻) : f(α_{1}) = f(α_{2})}.  (7.5)

Two such algebras A_{α_{1},α_{2}}^{1} and A_{β_{1},β_{2}}^{1} are isomorphic if and only if there is an automorphism of the disk that takes the pair (α_{1}, α_{2}) to the pair (β_{1}, β_{2}), and this in turn happens if and only if the hyperbolic distance from α_{1} to α_{2} equals the hyperbolic distance from β_{1} to β_{2}.

If one holizes the algebra A_{α_{1},α_{2}}^{1}, for example by the map

h(ζ) = (m_{α_{1}}(ζ)m_{α_{2}}(ζ), ζm_{α_{1}}(ζ)m_{α_{2}}(ζ) ),

the hyperbolic geodesic connecting α_{1} to α_{2} gets mapped to the Kobayashi geodesic that connects the multiple point to itself by going around the curve, and these two geodesics have the same length. This length therefore forms a complete isomorphism invariant for hyperbolic analytic curves with just a single crossing and no cusps, as in Example 0.1 of Section 0.

8 Petals of codimension two

There are six different types of petals of codimension two.

8.1 Triple Points

The first consists of those functions that identify a triple (α_{1}, α_{2}, α_{3}) of distinct points:

A_{α_{1},α_{2},α_{3}}^{2} := {f ∈ O(𝔻) : f(α_{1}) = f(α_{2}) = f(α_{3})}.  (8.1)

The isomorphism problem for two such algebras A_{α_{1},α_{2},α_{3}}^{2} and A_{β_{1},β_{2},β_{3}}^{2} is easy: as explained in Section 6, there is a disk automorphism that takes the first triple to the second if and only if one of the 3-by-3 matrices

(\frac{1 − \overline{σ(β_{i})}σ(β_{j})}{1 − \overline{α_{i}}α_{j}})_{i,j=1}^{3}

is rank one, where σ ranges over the permutation group S_{3}.

For the embedding dimension, note first that the local embedding dimension must exceed two, because of the triple point. Indeed, let h be any holomap into ℂ^{2} that sends α_{1}, α_{2} and α_{3} to the same point, γ say. Then {Dh(α_{i}) : 1 ≤ i ≤ 3} is a set of three vectors in ℂ^{2}, so must be linearly dependent. If F is holomorphic in a neighborhood of γ, then {D(F ∘ h)(α_{i})} can be at most two dimensional, and therefore any algebra holized by h must have a linear relation on the derivatives at α_{1}, α_{2} and α_{3}.

A global embedding dimension of three can be attained. To see this, assume without loss of generality that α_{1} = 0, and define f(ζ) = ζm_{α_{2}}(ζ)m_{α_{3}}(ζ). Let h be given by

h(ζ) = (f(ζ), ζf(ζ), ζ^{2}f(ζ)).

Then h is one-to-one except on {α_{1}, α_{2}, α_{3}}. Its derivative never vanishes. We just have to rule out the possibility of linear dependence between the

Figure 2: Petal with a triple point

vectors Dh(α_{i}), and then we will be done by Lemma 7.1. A calculation yields that the matrix whose rows are Dh(α_{i}) is

( \begin{matrix}α_{2}α_{3} & 0 & 0 \\ \frac{−α_{2}m_{α_{3}}(α_{2})}{1−|α_{2}|^{2}} & \frac{−α_{2}^{2}m_{α_{3}}(α_{2})}{1−|α_{2}|^{2}} & \frac{−α_{2}^{3}m_{α_{3}}(α_{2})}{1−|α_{2}|^{2}} \\ \frac{−α_{3}m_{α_{2}}(α_{3})}{1−|α_{3}|^{2}} & \frac{−α_{3}^{2}m_{α_{2}}(α_{3})}{1−|α_{3}|^{2}} & \frac{−α_{3}^{3}m_{α_{2}}(α_{3})}{1−|α_{3}|^{2}} \end{matrix} ) .

This has non-zero determinant, so has full rank.

8.2 Cusps of Order 2

Consider now a connection supported on one point, which we may as well take to be the origin. A cofinite algebra of codimension 2 coming from such a connection must be contained in an algebra of codimension 1 by Theorem 2.3, so one of the linear functionals must be

f ↦ f′(0).

The other must be of the form

Λ : f ↦ ∑_{j=2}^{n} a_{j}f^{(j)}(0).  (8.2)

If a_{2} = 0, then z^{2} is in the algebra, and all the non-zero a_{j} must have j odd. If only one a_{j} ≠ 0, it must be a_{3}, and we get the algebra

{f ∈ O(D) : f′(0) = f′′′(0) = 0}.

This algebra is holized by the functions (z^{2}, z^{5}).

Otherwise, let j_{0} be the index of the first non-zero coefficient, and n ≥ j_{0} + 2 be the index of the last. The algebra contains some polynomial of the form

p(z) = z^{j_{0}} + c_{j_{0}+2}z^{j_{0}+2} + … + c_{n}z^{n}

Subtracting appropriate linear multiples of z^{2}p(z), z^{4}p(z), ... from p, one gets that z^{j_{0}} is in the algebra, and so Λ has to be the zero functional, a contradiction.

So assume a_{2} ≠ 0. Suppose k is the smallest integer larger than 1 such that z^{k} is not in the algebra. If k > 2, then

p(z) := z^{2} − \frac{2 a_{2}}{k! a_{k}} z^{k}

is in the algebra, and so is p^{2}, and hence so is z^{4}. Therefore k is either two or three.

Putting all this together, we conclude that the algebra must be of the form

B_{0,c}^{2} := {f ∈ O(𝔻) : f′(0) = 0, f′′(0) + cf′′′(0) = 0}  (8.3)

for some c in ℂ. Two such algebras are isomorphic iff |c| is the same.

If c = 0, then the algebra is not locally embeddable in ℂ^{2}. Indeed, however the generators are chosen, there must be a linear relationship between f′′′(0), f^{(iv)}(0) and f^{(v)}(0).

If c ≠ 0, the algebra can be holized in ℂ^{2}, as we shall show in a subsequent paper [1]. For |c| large enough, the holization has a particularly simple form. Let f(z) = z^{4} and g(z) = z^{2}m_{α}(z), where α is chosen so that

g(1) = g(i).  (8.4)

A calculation shows that (8.4) holds iff α lies on the circle centered at 1 + i and radius 1. Another calculation shows that g lies in B_{0,c}^{2} for

c = \frac{α}{3(1 − |α|^{2})}.  (8.5)

So for this value of c, both f and g lie in B_{0,c}^{2}, and they identify the points 1 and i. Therefore, by Theorem 5.12 and Proposition 5.14, they generate the algebra B_{0,c}^{2}. As α moves along the admissible circular arc, we get that every algebra B_{0,c}^{2} is holized by z^{4} and a suitable Blascke product of degree 3 provided that

|c| ≥ \frac{2 − \sqrt{2}}{3(4\sqrt{2} − 5)} = 0.297 · · · .

8.3 Other codimension 2 petals

There are four other types of codimension 2 petals: two crossings, two cusps, and a crossing and a cusp which may or may not be at one of the points on the crossing. The generic such petal is two crossings, and for four distinct points α_{1}, α_{2}, β_{1}, β_{2} we shall let

C_{α_{1},α_{2};β_{1},β_{2}}^{2} := {f ∈ O(𝔻) : f(α_{1}) = f(α_{2}), f(β_{1}) = f(β_{2})}.

If one wishes to map the closed disk \overline{𝔻} into ℂ^{2} with a pair of Blaschke products so that there are precisely two crossings, then the points α_{1}, α_{2}, β_{1}, β_{2} must satisfy a polynomial equation.

Theorem 8.6 The algebra C_{α_{1},α_{2};β_{1},β_{2}}^{2} can be holized by a pair of finite Blaschke products (f, g) that are also one-to-one on 𝕋 if and only if

\frac{(α_{1} − β_{1})(α_{2} − β_{1})}{(1 − \overline{α_{1}}β_{1})(1 − \overline{α_{2}}β_{1})} = \frac{(α_{1} − β_{2})(α_{2} − β_{2})}{(1 − \overline{α_{1}}β_{2})(1 − \overline{α_{2}}β_{2})}.  (8.7)

Proof: Let f = m_{α_{1}}m_{α_{2}}. Condition (8.7) is the requirement that f(β_{1}) = f(β_{2}). If this holds, then f is a Blaschke product of degree two that lies in the algebra. Let g be any Blaschke product of degree 5 in the algebra, for example

g(z) = zm_{α_{1}}(z)m_{α_{2}}(z)m_{β_{1}}(z)m_{β_{2}}(z).

Then by Theorem 5.12 and Proposition 5.14, the pair f and g holize the algebra and do not identify any points on 𝕋.

Conversely, suppose f and g are Blaschke products that holize the algebra and are one-to-one on 𝕋. By Theorem 5.12, they either have degrees 2 and 5 or degrees 3 and 3. In the first case, if f is in the algebra and has degree 2, let f(α_{1}) = γ. Then m_{γ} ∘ f is a constant times m_{α_{1}}m_{α_{2}}, and has the same values at β_{1} and β_{2}, so (8.7) holds.

The second case cannot hold. Indeed, suppose f and g are both Blaschke products of degree 3 in the algebra. After composing with the Möbius transformations that send their values at α_{1} to 0, we can take

f(z) = m_{α_{1}}(z)m_{α_{2}}(z)m_{γ}(z)
g(z) = m_{α_{1}}(z)m_{α_{2}}(z)m_{δ}(z).

Let m_{α_{1}}(β_{1})m_{α_{2}}(β_{1}) = C_{1} and m_{α_{1}}(β_{2})m_{α_{2}}(β_{2}) = C_{2}. Then

C_{1}m_{γ}(β_{1}) = C_{2}m_{γ}(β_{2})  (8.8)
C_{1}m_{δ}(β_{1}) = C_{2}m_{δ}(β_{2}).

But there is a unique γ satisfying (8.8), as can be seen either by direct calculation, or observing that m_{γ} is the unique solution to the extremal problem of finding a holomorphic map from 𝔻 to 𝔻 that sends β_{1} to r and β_{2} to \frac{C_{2}}{C_{1}} r, with r as large as possible. So g would be a function of f, and the pair would not holize the algebra. □

The algebra that has a single crossing and a cusp at the crossing, such as

{f ∈ O(𝔻) : f(0) = f(α), f′(0) = 0}

can not be locally holized in two dimensions. The other three can be.

We do not know the answer to either of the following questions, even in the case of codimension 2.

Question 8.9 If a petal can be locally holized in dimension 2, can it be globally holized by a pair of functions?

Question 8.10 If a petal can be holized by a pair of functions, can it be holized by a pair of Blaschke products?

9 Model theory

In this section, we propose to study what happens to the Hardy space theory if the disk is replaced by a nice hyperbolic algebraic curve. When there are no singularities, the Hardy space theory for bordered Riemann surfaces has been developed by J. Ball and V. Vinnikov [5]. Throughout this section, we shall take C to be the algebraic set

C = {(z, w) ∈ ℂ^{2} : z^{2} = w^{2}},

and V to be C ∩ 𝔻^{2}. The set V consists of two disks:

𝔻^{+} := {(ζ, ζ) : ζ ∈ 𝔻}
𝔻^{−} := {(ζ, −ζ) : ζ ∈ 𝔻}.

If we let 0^{+} and 0^{−} denote the centers of 𝔻^{+} and 𝔻^{−}, respectively, we can think of O(V) as the set of holomorphic functions f on two disjoint disks with the connection f(0^{+}) = f(0^{−}).

Let σ^{+} and σ^{−} denote Lebesgue measure on ∂𝔻^{+} and ∂𝔻^{−} respectively. If one replaces the defining equation for C by z^{2} = m_{α_{1}}(w)m_{α_{2}}(w) for distinct points α_{1} and α_{2}, then C ∩ 𝔻^{2} is an annulus. In the limiting case of V, it behaves like an annulus function theoretically, in the sense that the obstruction to solving certain problems is give by one real parameter.

Proposition 9.1 (i) Let u be a real-valued function on ∂V. Then u is the real part of a holomorphic function on V iff

∫_{∂𝔻^{+}} udσ^{+} = ∫_{∂𝔻^{−}} udσ^{−}.

(ii) The points {(α_{1}, α_{1}), …, (α_{m}, α_{m}), (β_{1}, −β_{1}), …, (β_{n}, −β_{n})} in V \ {(0, 0)} form the zero set of a rational inner function iff

∏_{i=1}^{m} |α_{i}| = ∏_{j=1}^{n} |β_{j}|.

The proof is straightforward, and we omit it.

Let T = (T_{1}, T_{2}) be a pair of commuting operators on a Hilbert space. We say that T has V as a spectral set if, for every polynomial p in two variables,

∥p(T_{1}, T_{2})∥ ≤ ∥p∥_{V} .  (9.2)

Von Neumann’s inequality [28] says that a single operator has the disk as a spectral set iff it is a contraction. The analogous theorem for V has a parameter in it. Note first that if T has V as a spectral set, then in particular any polynomial that vanishes on V must vanish on T, so

T_{1}^{2} = T_{2}^{2}.  (9.3)

If (9.3) holds, we shall write T ≺ V.

Theorem 9.4 Suppose T ≺ V. Then T has V as a spectral set iff

∥T_{1} + T_{2} + e^{iθ}(T_{1} − T_{2})∥ ≤ 2  ∀ θ ∈ ℝ.  (9.5)

Proof: By taking the Cayley transforms of the functions in (9.2), proving that V is a spectral set is the same as proving that whenever a holomorphic function has positive real part on V, then it has positive real part on T. To analyze this condition, it is sufficient to look at the extreme points of the maps ϕ from V to the right-half plane, normalized to take the origin to 1. These extreme points are the functions

ϕ|_{𝔻^{+}} : (ζ, ζ) ↦ \frac{τ^{+} + ζ}{τ^{+} − ζ}
ϕ|_{𝔻^{−}} : (ζ, −ζ) ↦ \frac{τ^{−} + ζ}{τ^{−} − ζ} ,

where τ^{+} and τ^{−} are unimodular.

Now,

ϕ(T) = ϕ|_{𝔻^{+}}(\frac{1}{2}(T_{1} + T_{2})) + ϕ|_{𝔻^{−}}(\frac{1}{2}(T_{1} − T_{2})) − ϕ(0).  (9.6)

Then (9.6) has positive real part iff (9.5) holds with e^{iθ} = τ^{+}/τ^{−}. □

Remark 9.7 Condition (9.5) can be rephrased in terms of the numerical radius of a single operator, because of the following lemma whose proof we omit (if the first operator in (9.10) is not invertible, take an appropriate limit).

Lemma 9.8 Operators A and B satisfy

∥A + e^{iθ}B∥ ≤ 1  ∀ θ ∈ ℝ  (9.9)

iff the numerical radius of

(1 − A^{∗}A − B^{∗}B)^{−1/2} A^{∗}B (1 − A^{∗}A − B^{∗}B)^{−1/2}  (9.10)

is less than or equal to \frac{1}{2}.

The von Neumann-Wold theorem states that all isometries are direct sums of unitaries and copies of the forward shift. What do pairs of isometries that live on V look like?

Theorem 9.11 Let T = (T_{1}, T_{2}) be a pair of commuting isometries on a Hilbert space H satisfying T_{1}^{2} = T_{2}^{2}. Then there is a decomposition H = M_{+}⊕M_{−}⊕K, isometries W_{±} on M_{±}, and a pair of operators E_{±} : K → M_{±} satisfying E_{+}^{∗}E_{+} + E_{−}^{∗}E_{−} = I, W_{+}^{∗}E_{+} = 0, and W_{−}^{∗}E_{−} = 0, so that

T_{1} ≅  ( \begin{matrix}& M_{+} & M_{−} & K \\ M_{+} & W_{+} & 0 & E_{+} \\ M_{−} & 0 & W_{−} & E_{−} \\ K & 0 & 0 & 0 \end{matrix} )

T_{2} ≅  ( \begin{matrix}& M_{+} & M_{−} & K \\ M_{+} & W_{+} & 0 & E_{+} \\ M_{−} & 0 & −W_{−} & −E_{−} \\ K & 0 & 0 & 0 \end{matrix} ).  (9.12)

Proof: Let M_{+} be the range of T_{1} + T_{2}, and M_{−} be the range of T_{1} − T_{2}. Claim: M_{+} and M_{−} are orthogonal. Proof of Claim: Suppose first that T_{1} and T_{2} are unitary.

(T_{1} − T_{2})^{∗}(T_{1} + T_{2})T_{1} = (T_{1}^{∗}T_{2} − T_{2}^{∗}T_{1})T_{1}
= T_{1}^{∗}T_{1}T_{2} − T_{2}^{∗}T_{2}T_{2}
= 0.

As T_{1} is unitary, it follows that

(T_{1} − T_{2})^{∗}(T_{1} + T_{2}) = 0,

so the range of T_{1} − T_{2} is orthogonal to the range of T_{1} + T_{2}.

For general isometries T_{1} and T_{2} as in the statement of the theorem, there are commuting unitaries U_{1} and U_{2} that extend them and satisfy U_{1}^{2} = U_{2}^{2} (as can be seen, for example, from the Sz.-Nagy-Foiaş model for commuting isometries [26]). From the previous argument, the range of U_{1} − U_{2} is orthogonal to the range of U_{1} + U_{2}, so the range of T_{1} − T_{2} is orthogonal to the range of T_{1} + T_{2}.

Let K := H ⊖ (M_{+} ⊕ M_{−}). Then K is orthogonal to the range of T_{1} and T_{2}, so the third rows of (9.12) will be zero. Using the fact that T ≺ V and that T_{1} and T_{2} are isometries, one gets the rest of (9.12). □

10 Neil parabola

We shall give another example of how the function theory interacts with the geometry of a hyperbolic algebraic curve. Let

V_{2} = {(z, w) ∈ 𝔻^{2} : z^{3} = w^{2}}.

In a recent paper [18], G. Knese succeeded in finding an exact formula for the Carathéodry metric on V_{2}; this was the first example of an explicit calculation of the Carathéodory metric on a singular hyperbolic algebraic curve.

What are the extreme points of the set of holomorphic functions on V_{2} with positive real part, normalized to take (0, 0) to 1? As V_{2} is a petal, holized by the map

h_{2} : z ↦ (z^{2}, z^{3}),

the question is equivalent to asking for the extreme points of the set P of functions on the disk that have positive real part, send 0 to 1, and whose derivative vanishes at the origin. Every function ϕ in P has a Herglotz representation as

ϕ(z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)

for some probability measure μ on the circle. Being in P means in addition that

∫ e^{−iθ}dμ(θ) = 0.  (10.1)

Absent (10.1), the extreme points just correspond to measures that have a single atom. With the condition, there are two sorts of extreme point: the first is two atoms of mass 1/2 that are diametrically opposite. The second is three atoms that form the vertices of a triangle with 0 in the interior, and such that their masses sum to 1 and (10.1) is satisfied. One never needs more than three atoms, since by a theorem of Carathéodory if a point in ℝ^{d} is in a convex set, it is in the convex hull of d + 1 extreme points (see e.g. [13] for a proof).

After taking the Cayley transform, we see that the minimal inner functions on V_{2} either have two zeroes (which must be (α^{2}, α^{3}) and (α^{2}, −α^{3})), or three.

11 Conclusion

The purpose of this paper is to show how studying hyperbolic algebraic curves gives rise to a host of problems in classical function theory, in Riemann surfaces, in function algebras and in operator theory.

A cofinite algebra has local restrictions on the minimal dimension of the space in which it can be holized, as in Example 0.3. There are also global restrictions – the desingularizing Riemann surface S will not in general be embeddable in ℂ^{1}. It is still an open question whether all finite Riemann surfaces are embeddable in ℂ^{2} [9, p. 347]. Let the global embedding dimension of a cofinite algebra A be the smallest n such that A can be holized in ℂ^{n}. Let the local embedding dimension at any point λ in the maximal ideal space be the smallest n such that some neighborhood of λ can be embedded in ℂ^{n}. Perhaps the most significant open problem arising from our approach is the following.

Question 11.1 When is the global embedding dimension of a cofinite algebra on S equal to the maximum of the local embedding dimensions and the global embedding dimension of S?

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