Cusp algebras

Jim Agler ^{∗} U.C. San Diego

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

April 5, 2007

Abstract

1 Introduction

By a cusp V we shall mean the image of the unit disk 𝔻 under a bounded injective holomorphic map h into ℂ^{n} whose derivative vanishes at exactly one point. The simplest example is the Neil parabola, given by h(ζ) = (ζ^{2}, ζ^{3}). See [3, 4] for background and theory on the Neil parabola, which is pictured in Figure 1.

A generalization of a cusp is a petal. A petal is the image of the unit disk 𝔻 under a proper holomorphic map h from 𝔻 into some bounded open set Ω, where, except for a finite set E_{h}, h is one-to-one and non-singular. If E_{h} is a singleton, then V is a cusp. Since the automorphism group of 𝔻 is transitive, we may assume then that E_{h} = {0}.

The function h : 𝔻 → Ω induces a finite codimensional subalgebra A_{h} of the algebra O(𝔻) of all analytic functions on 𝔻, namely

A_{h} = {F ∘ h : F ∈ O(V)},

∗Partially supported by National Science Foundation Grant DMS 0400826

†Partially supported by National Science Foundation Grant DMS 0501079

Figure 1: A simple cusp - the Neil parabola z^{3} = w^{2}

where O(V) is the algebra of functions on V that, in the neighborhood of every point P of V, coincides with the restriction to V of a function holomorphic in a neighborhood of ℂ^{n} containing P. When V is a cusp, the algebra A_{h} has the property that it contains z^{m}O(𝔻) for some m ≥ 2. Conversely, if a cofinite subalgebra of O(𝔻) contains z^{m}O(𝔻), then it arises in this way.

In light of these remarks, let us agree to say that a unital algebra A ⊆ O(𝔻) is a cusp algebra if

(i) dim(O(𝔻)/A) is finite, and

(ii) For some m ≥ 2, z^{m}O(𝔻) ⊆ A.

We shall say a map h : 𝔻 → Ω ⊊ ℂ^{n} is a holomap if it is a proper map that is one-to-one and non-singular except on a finite set. We shall say that a holomap h is a holization of the cofinite algebra A if A = A_{h}.

If A is a cusp algebra, then we define three integers attached to A. The codimension of A is defined by

cod(A) = dim (O(𝔻)/A) .

The order of A is defined by

ord(A) = min{k : z^{k+1}O(𝔻) ⊆ A}.

The contact of A is defined by

con(A) = max{n : f^{(j)}(0) = 0, ∀ f ∈ A, 1 ≤ j ≤ n}.

If V is a cusp, then we define cod(V), ord(V) and con(V) as the quantities for the corresponding algebra A_{h}.

By a simple cusp algebra, we shall mean a cusp algebra A such that con(A) = 1. There is a unique cusp algebra of codimension 1, which corresponds via a holization to the holomorphic functions on the Neil parabola {(z, w) ∈ 𝔻^{2} : z^{3} = w^{2}} (this result can be seen directly, or will follow from Section 4.) This can be thought of as a Riemann mapping theorem for cusps of codimension 1. In Section 4, we generalize this “Riemann mapping theorem” to arbitrary simple cusps. We show that locally the only invariant of simple cusps is the codimension, but globally cusps of codimension n + 1 have a 2n − 1 real parameter moduli space.

This paper is a continuation of the authors’ earlier work in [1], where the ideas are put in a more general context. A principal concern in that paper is when a given finite codimensional subalgebra can be holized in 2 dimensions, i.e. by a map into ℂ^{2}. In Section 5, we show that all simple cusp algebras can be holized in ℂ^{2}.

2 Preliminaries

Throughout this section, we shall assume that A is a simple cusp algebra, i.e. of contact one. By a primitive for A we mean a function π that lies in A, and satisfies π(0) = 0, π′′(0) = 2; so π has a Taylor expansion π(z) = z^{2} + . . ..

For each n, let A_{n} be the space

A_{n} = [A ∩ z^{2n}O(𝔻)] ,

and let E_{n} be the quotient

E_{n} = A_{n}/A_{n+1}.

Observe that multiplication by any primitive π is always one-to-one from E_{n} to E_{n+1}. Therefore, there is some integer n_{0} such that dim(E_{n}) is 1 for n ≤ n_{0}, and 2 for n > n_{0}. Since dim(E_{n}) = 2 if and only if both z^{2n} and z^{2n+1} are in A, it follows that

ord(A) = 2n_{0} + 1.

As π^{n} is in A_{n} for each n, if f is in A_{n} for some n ≤ n_{0}, then there is a constant c_{n} such that f − c_{n}π^{n} is in A_{n+1}. Therefore every function f in A has the representation

f = c_{0} + c_{1}π + … + c_{n_{0}}π^{n_{0}} + z^{2n_{0}+2}g,  (2.1)

where c_{0}, …, c_{n_{0}} are in ℂ and g ∈ O(𝔻). From (2.1), we see that cod(A) = n_{0} + 1. Summarizing, we have:

Proposition 2.2 If A is a simple cusp algebra, then ord(A) = 2 cod(A) − 1. Furthermore, if we set n_{0} = cod(A) − 1, and π is a primitive for A, then every function f in A has a unique representation

f(z) = p ∘ π(z) + z^{2n_{0}+2}g(z)  (2.3)

for some polynomial p of degree at most n_{0} and some g ∈ O(𝔻).

3 Connections and local theory

Let U be an open set in ℂ. A linear functional on O(U) 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 3.1 A connection on {α_{1}, …, α_{m}} ⊂ U is a finite dimensional set Γ of local functionals Λ supported by {α_{1}, …, α_{m}}. We say Γ is algebraic if Γ^{⊥} := {f ∈ O(U) : Λ(f) = 0 ∀ Λ ∈ Γ} is an algebra.

It was proved by T. Gamelin [2] that every finite codimensional subalgebra A of O(𝔻) is Γ^{⊥} for some algebraic connection on 𝔻. Moreover, A will be a cusp algebra iff the support of Γ is {0}.

We shall say that a point P in a petal V is a cusp point if there is a oneto-one proper map h from 𝔻 onto a neigborhood U of P in V such that the derivative of h vanishes only at the pre-image of P. The algebra A_{h} is then a cusp algebra. We wish to show that its codimension, order and contact do not depend on the choice of U.

Suppose that P_{1} and P_{2} are cusp points in cusps V_{1} and V_{2}, and there is an injective holomorphic map ϕ : V_{1} → V_{2}. For r = 1, 2 there are holizations h_{r} : 𝔻 → V_{r} with h_{r}(0) = P_{r}. By [1, Thm. 3.2], the map ϕ from V_{1} to V_{2} induces a one-to-one map ψ : 𝔻 → 𝔻 such that h_{2} ∘ ψ = ϕ ∘ h_{1}, and the connections Γ_{1} and Γ_{2} induced by h_{1} and h_{2} are related by

Γ_{2} ⊇ ψ_{∗} Γ_{1}
:= {f ↦ Λ_{1}(f ∘ ψ) : Λ_{1} ∈ Γ_{1}}.

If

Λ_{1} : f ↦ ∑_{j=0}^{n} a_{j}f^{(j)}(0),  (3.2)

then

ψ_{∗} Λ_{1} : g ↦ ∑_{j=0}^{n} a_{j}(g ∘ ψ)^{(j)}(0),  (3.3)

We can use Faá di Bruno’s formula to evaluate the derivatives of g ∘ ψ in terms of those of g and ψ. As ψ(0) = 0 and ψ′(0) ≠ 0, note in particular that the coefficient of g^{(n)}(0) on the right-hand side of (3.3) is a_{n}[ψ′(0)]^{n}. Therefore ψ_{∗} is injective, so cod(A_{1}) ≤ cod(A_{2}).

If con(A_{1}) = k, it means that for each 1 ≤ j ≤ k, the functional f ↦ f^{(j)}(0) is in Γ_{1}. Therefore each functional g ↦ g^{(j)}(0) is in Γ_{2}, and con(A_{1}) ≤ con(A_{2}). Finally if ord(A_{1}) = n, it means there is some Λ_{1} in Γ_{1} of the form (3.2) with a_{n} ≠ 0; it follows that ord(A_{1}) ≤ ord(A_{2}).

Summarizing, we have shown:

Proposition 3.4 If ϕ : V_{1} → V_{2} is an injective holomorphic map from one cusp to another, then cod(V_{1}) ≤ cod(V_{2}), con(V_{1}) ≤ con(V_{2}) and ord(V_{1}) ≤ ord(V_{2}).

Now, if P is a cusp point in a petal V_{1}, and one surrounds P by a decreasing sequence of open sets U_{n}, then with each U_{n} there is a cusp algebra with a codimension, contact and order, and there is an inclusion map from each U_{n+1} to U_{n}. By Proposition 3.4, the positive integers cod(U_{n}), con(U_{n}) and ord(U_{n}) must decrease as U_{n} shrinks. Therefore, at some point they must stabilize, and for all sufficiently small neighborhoods of P, the cusp algebras have the same codimension, contact and order. We shall call these (cod)(P), (con)(P) and (ord)(P) respectively, and say P is a simple cusp point if con(P) = 1.

4 Equivalence of simple cusps

We shall say that two points P_{1} and P_{2} in the petals V_{1} and V_{2} are locally equivalent if there is a neighborhood U_{1} of P_{1} in V_{1}, and a neighborhood U_{2} of P_{2} in V_{2}, and a biholomorphic homeomorphism F from U_{1} onto U_{2} that maps P_{1} to P_{2}.

Theorem 4.1 Two simple cusp points are locally equivalent if and only they have the same codimension.

Proof: Necessity follows from Proposition 3.4. To prove sufficiency, let π_{1} be a primitive of the first algebra, and π_{2} a primitive of the second algebra. By Proposition 2.2, it is sufficient to prove that there are neighborhoods W_{1} and W_{2} of the origin, and a univalent map ϕ : W_{1} → W_{2} that maps 0 to 0 and such that

π_{2} ∘ ϕ = π_{1}.  (4.2)

Each π_{r} has a square root χ_{r} in a neighborhood of 0, and each χ_{r} is locally univalent (because π_{r} is of order 2). So define

ϕ := χ_{2}^{−1} ∘ χ_{1}

on a suitable neighborhood W_{1}, and (4.2) is satisfied. □

How can we globally parametrize simple cusp algebras? We shall show that there is an essentially unique primitive with all its even Taylor coefficients (except for the second) zero. By π̂(k) we mean the k^{th} Taylor coefficient at 0.

Lemma 4.3 Every simple cusp algebra of codimension n+1 has a primitive π such that π̂(2k) = 0 for all 2 ≤ k ≤ n. Moreover, π is unique up to O(z^{2n+2}).

Proof: (Existence) Let χ_{1} be any primitive. Define

χ_{2} := χ_{1} − χ̂_{1}(4)(χ_{1})^{2}.

Proceed inductively, with

χ_{k} := χ_{k−1} − χ̂_{k−1}(2k)(χ_{k−1})^{k}.

Then let π = χ_{n}.

(Uniqueness) Suppose π and χ are both primitives with their even Taylor coefficients, starting at 4, vanishing. By Proposition 2.2, we have

χ = c_{1}π + c_{2}π^{2} + … + c_{n}π^{n} + O(z^{2n+2}).

By looking at the coefficient of z^{2} we see c_{1} = 1. Now looking at the coefficients of z^{4}, z^{6}, …, z^{2n} in order, we see that c_{2} = 0 = c_{4} = … = c_{n}. □

So every algebraic connection of dimension n + 1 supported at the origin is the annihilator of an algebra generated locally by a unique primitive

π(z) = z^{2} + α_{1}z^{3} + α_{2}z^{5} + … + α_{n}z^{2n+1}.  (4.4)

Let us denote the codimension n + 1 subalgebra of O(𝔻) with primitive (4.4) by A(α_{1}, …, α_{n}), and let V (α_{1}, …, α_{n}) denote the corresponding petal h(𝔻), where h holizes A(α_{1}, …, α_{n}).

Two cusps V_{1} and V_{2} are globally equivalent if there is a biholomorphic homeomorphism ϕ from V_{1} onto V_{2}. If each V_{r} is holized by h_{r} : 𝔻 → V_{r}, then by [1, Thm. 3.1], this occurs if and only if there is a map ψ : 𝔻 → 𝔻 such that ψ_{∗} maps the first connection to the second. As the only self-maps of the disk that leave the origin invariant are rotations, this is rather restrictive.

Theorem 4.5 The cusps V (α_{1}, …, α_{n}) and V (β_{1}, …, β_{n}) are isomporphic if and only if there is a unimodular constant τ such that β_{j} = τ^{2j−1}α_{j} for 1 ≤ j ≤ n.

As a corollary we have

Corollary 4.6 The moduli space of all simple cusps of codimension n + 1 is ℝ^{+} × ℂ^{n−1}.

5 Embedding

The purpose of this section is to prove that every simple cusp algebra A contains a pair of functions h_{1}, h_{2} such that the pair holizes the algebra (which is equivalent to saying that polynomials in h_{1} and h_{2} are dense in A in the topology of uniform convergence on compacta).

For the rest of this section, A will be a fixed simple cusp algebra of codimension n + 1.

Lemma 5.1 For every α on 𝔻 \ {0}, there is a function ψ_{α} in A that has a single simple zero at α, and no other zeroes on 𝔻.

Proof: Consider functions of the form (z − α)e^{h(z)}. In order to be in A, h must satisfy n + 1 equations on its first 2n + 1 derivatives at 0. This system is triangular, so can be solved with h a polynomial. □

Lemma 5.2 If f is in A and f has no zeroes on 𝔻, then 1/f is in A.

Proof: Consider the vector space obtained by adjoining 1/f, 1/f^{2}, …, 1/f^{k} to A. Since A is of finite codimension in O(𝔻), for some k there is a linear relation

1/f^{k} = ∑_{j=1}^{k−1} c_{j}1/f^{j} + A.

Multiplying both sides by f^{k−1}, we get that 1/f is in A. □

Lemma 5.3 If f ∈ A and f(α) = 0 for some α ∈ 𝔻 \ {0}, then f/ψ_{α} is in A.

Proof: Let Γ be the connection at 0 such that A = Γ^{⊥}. Consider the algebra

A′ := \{f ∈ O(\frac{|α|}{2}𝔻) : Λ(f) = 0 ∀ Λ ∈ Γ\}.

Then ψ_{α} is in A′, so by Lemma 5.2, 1/ψ_{α} is in A′. Therefore f/ψ_{α} is in A′, and is therefore annihilated by every functional in Γ. But f/ψ_{α} is also in O(𝔻), so it is in A as required. □

Theorem 5.4 There exists a map h : 𝔻 → ℂ^{2} that holizes A.

Proof: By applying Lemma 5.3 repeatedly, one can find a primitive h_{1} of A that has no zeroes in 𝔻 \ {0}. Define

h_{2}(z) := z (h_{1}(z))^{n+1}.

Consider the algebra A′ that is the closure (in the topology of uniform convergence on compacta) of polynomials in h_{1} and h_{2}. By [1, Thm. 4.2], the algebra A′ is of finite codimension, and it is a cusp algebra, because h(z) = (h_{1}(z), h_{2}(z)) is one-to-one away from the origin. If A′ ≠ A, then it is contained in a maximal proper subalgebra A′′ of A, which must have codimension n + 2, and therefore order 2n + 3. But A′′ contains h_{2} and h_{1}^{n+1}, and these functions have order 2n + 2 and 2n + 3 respectively at the origin. So there can be no linear relation in A′′ between the derivatives at the origin of order (2n + 2) and (2n + 3), and so the order of A′′ must actually be at most 2n + 1. □

References

[1] J. Agler and J.E. McCarthy. Hyperbolic algebraic and analytic curves. To appear.

[2] T.W. Gamelin. Embedding Riemann surfaces in maximal ideal spaces. J. Funct. Anal., 2:123–146, 1968.

[3] G. Knese. Function theory on the Neil parabola. Michigan Math. J. To appear.

[4] V.I. Paulsen and D. Singh. Modules over subalgebras of the disk algebra. Indiana Math. J., 55(5):1751–1766, 2006.