We consider simple cusp algebras, that is certain subalgebras of the algebra
of holomorphic functions on a disk that are annihilated by some distributions
living on a singleton. We determine when these algebras can be holized in two
dimensions, and when these holizations are globally biholomorphic

Agler, Jim

McCarthy, John E.

English

arXiv

A cusp is the image of the unit disk under a proper holomorphic map into Cn that is one-to-one and whose derivative vanishes at exactly one point. It is simple if not all the second derivatives vanish. We characterize when two simple cusps are isomorphic, and show that they can all be realized in C²

Diposit Digital de Documents de la UAB

Publ. Mat. 53 (2009), 111–118CUSP ALGEBRASJim Agler∗ and John E. McCarthy†AbstractA cusp is the image of the unit disk under a proper holomorphicmap into Cn that is one-to-one and whose derivative vanishes atexactly one point. It is simple if not all the second derivativesvanish. We characterize when two simple cusps are isomorphic,and show that they can all be realized in C2.1. IntroductionBy a cusp V we shall mean the image of the unit disk D under abounded injective holomorphic map h into Cn whose derivative vanishesat exactly one point. The simplest example is the Neil parabola, givenby h(ζ) = (ζ2, ζ3). See [3], [4] for background and theory on the Neilparabola, which is pictured in Figure 1.A generalization of a cusp is a petal. A petal is the image of the unitdisk D under a proper holomorphic map h from D into some boundedopen set Ω, where, except for a finite set Eh, h is one-to-one and non-singular. If Eh is a singleton, then V is a cusp. Since the automorphismgroup of D is transitive, we may assume then that Eh = {0}.The function h : D→ Ω induces a finite codimensional subalgebra Ahof the algebra O(D) of all analytic functions on D, namelyAh = {F ◦ h : F ∈ O(V )},where O(V ) is the algebra of functions on V that, in the neighborhoodof every point P of V , coincides with the restriction to V of a functionholomorphic in a neighborhood of Cn containing P . When V is a cusp,the algebraAh has the property that it contains zmO(D) for somem ≥ 2.Conversely, if a cofinite subalgebra of O(D) contains zmO(D), then itarises in this way.2000 Mathematics Subject Classification. 14H10, 30F20.Key words. Petal, cusp, holomap.∗Partially supported by National Science Foundation Grant DMS 0400826.†Partially supported by National Science Foundation Grant DMS 0501079.112 J. Agler, J. E. McCarthyFigure 1. A simple cusp - the Neil parabola z3 = w2.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, zmO(D) ⊆ A.We shall say a map h : D→ Ω ( Cn is a holomap if it is a proper mapthat is one-to-one and non-singular except on a finite set. We shall saythat a holomap h is a holization of the cofinite algebra A if A = Ah.If A is a cusp algebra, then we define three integers attached to A.The codimension of A is defined bycod(A) = dim (O(D)/A) .The order of A is defined byord(A) = min{k : zk+1O(D) ⊆ A}.The contact of A is defined bycon(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 quan-tities for the corresponding algebra Ah.By a simple cusp algebra, we shall mean a cusp algebra A such thatcon(A) = 1. There is a unique cusp algebra of codimension 1, whichcorresponds via a holization to the holomorphic functions on the Neilparabola {(z, w) ∈ D2 : z3 = w2} (this result can be seen directly, or willfollow from Section 4). This can be thought of as a Riemann mappingtheorem for cusps of codimension 1. In Section 4, we generalize this“Riemann mapping theorem” to arbitrary simple cusps. We show thatCusp Algebras 113locally the only invariant of simple cusps is the codimension, but globallycusps of codimension n+ 1 have a 2n− 1 real parameter moduli space.This paper is a continuation of the authors’ earlier work in [1], wherethe ideas are put in a more general context. A principal concern in thatpaper is when a given finite codimensional subalgebra can be holizedin 2 dimensions, i.e. by a map into C2. In Section 5, we show that allsimple cusp algebras can be holized in C2.2. PreliminariesThroughout this section, we shall assume that A is a simple cuspalgebra, i.e. of contact one. By a primitive for A we mean a function pithat lies in A, and satisfies pi(0) = 0, pi′′(0) = 2; so pi has a Taylorexpansion pi(z) = z2 + · · · .For each n, let An be the spaceAn =[A ∩ z2nO(D)],and let En be the quotientEn = An/An+1.Observe that multiplication by any primitive pi is always one-to-onefrom En to En+1. Therefore, there is some integer n0 such that dim(En)is 1 for n ≤ n0, and 2 for n > n0. Since dim(En) = 2 if and only ifboth z2n and z2n+1 are in A, it follows thatord(A) = 2n0 + 1.As pin is in An for each n, if f is in An for some n ≤ n0, then there is aconstant cn such that f − cnpin is in An+1. Therefore every function fin A has the representation(2.1) f = c0 + c1pi + · · ·+ cn0pin0 + z2n0+2g,where c0, . . . , cn0 are in C and g ∈ O(D). From (2.1), we see thatcod(A) = n0 + 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 n0 = cod(A) − 1, and pi is a primitive for A,then every function f in A has a unique representation(2.3) f(z) = p ◦ pi(z) + z2n0+2g(z)for some polynomial p of degree at most n0 and some g ∈ O(D).114 J. Agler, J. E. McCarthy3. Connections and local theoryLet U be an open set in C. A linear functional on O(U) is called localif it comes from a finitely supported distribution, i.e. is of the formΛ(f) =m∑i=1ni∑j=0aijf(j)(αi).Definition 3.1. A connection on {α1, . . . , αm} ⊂ U is a finite dimen-sional set Γ of local functionals Λ supported by {α1, . . . , αm}. We sayΓ is algebraic if Γ⊥ := {f ∈ O(U) : Λ(f) = 0, ∀Λ ∈ Γ} is an algebra.It was proved by T. W. Gamelin [2] that every finite codimensionalsubalgebra A of O(D) is Γ⊥ for some algebraic connection on D. More-over, 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 isa one-to-one proper map h from D onto a neigborhood U of P in Vsuch that the derivative of h vanishes only at the pre-image of P . Thealgebra Ah 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 P1 and P2 are cusp points in cusps V1 and V2, andthere is an injective holomorphic map φ : V1 → V2. For r = 1, 2 thereare holizations hr : D → Vr with hr(0) = Pr. By [1, Theorem 3.2], themap φ from V1 to V2 induces a one-to-one map ψ : D → D such thath2 ◦ ψ = φ ◦ h1, and the connections Γ1 and Γ2 induced by h1 and h2are related byΓ2 ⊇ ψ∗ Γ1:= {f 7→ Λ1 (f ◦ ψ) : Λ1 ∈ Γ1}.If(3.2) Λ1 : f 7→n∑j=0ajf(j)(0),then(3.3) ψ∗ Λ1 : g 7→n∑j=0aj(g ◦ ψ)(j)(0).We can use Faa´ di Bruno’s formula to evaluate the derivatives of g ◦ψ interms of those of g and ψ. As ψ(0) = 0 and ψ′(0) 6= 0, note in particularthat the coefficient of g(n)(0) on the right-hand side of (3.3) is an[ψ′(0)]n.Therefore ψ∗ is injective, so cod(A1) ≤ cod(A2).If con(A1) = k, it means that for each 1 ≤ j ≤ k, the functionalf 7→ f (j)(0) is in Γ1. Therefore each functional g 7→ g(j)(0) is in Γ2, andCusp Algebras 115con(A1) ≤ con(A2). Finally if ord(A1) = n, it means there is some Λ1in Γ1 of the form (3.2) with an 6= 0; it follows that ord(A1) ≤ ord(A2).Summarizing, we have shown:Proposition 3.4. If φ : V1 → V2 is an injective holomorphic map fromone cusp to another, then cod(V1) ≤ cod(V2), con(V1) ≤ con(V2) andord(V1) ≤ ord(V2).Now, if P is a cusp point in a petal V1, and one surrounds P bya decreasing sequence of open sets Un, then with each Un there is acusp algebra with a codimension, contact and order, and there is aninclusion map from each Un+1 to Un. By Proposition 3.4, the positiveintegers cod(Un), con(Un) and ord(Un) must decrease as Un shrinks.Therefore, at some point they must stabilize, and for all sufficientlysmall 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 cuspsWe shall say that two points P1 and P2 in the petals V1 and V2are locally equivalent if there is a neighborhood U1 of P1 in V1, and aneighborhood U2 of P2 in V2, and a biholomorphic homeomorphism Ffrom U1 onto U2 that maps P1 to P2.Theorem 4.1. Two simple cusp points are locally equivalent if and onlythey have the same codimension.Proof: Necessity follows from Proposition 3.4. To prove sufficiency, letpi1 be a primitive of the first algebra, and pi2 a primitive of the secondalgebra. By Proposition 2.2, it is sufficient to prove that there are neigh-borhoods W1 and W2 of the origin, and a univalent map φ : W1 → W2that maps 0 to 0 and such that(4.2) pi2 ◦ φ = pi1.Each pir has a square root χr in a neighborhood of 0, and each χr islocally univalent (because pir is of order 2). So defineφ := χ−12 ◦ χ1on a suitable neighborhood W1, and (4.2) is satisfied.How can we globally parametrize simple cusp algebras? We shallshow that there is an essentially unique primitive with all its even Taylorcoefficients (except for the second) zero. By pˆi(k) we mean the kth Taylorcoefficient at 0.116 J. Agler, J. E. McCarthyLemma 4.3. Every simple cusp algebra of codimension n + 1 has aprimitive pi such that pˆi(2k) = 0 for all 2 ≤ k ≤ n. Moreover, pi isunique up to O(z2n+2).Proof: (Existence): Let χ1 be any primitive. Defineχ2 := χ1 − χˆ1(4)(χ1)2.Proceed inductively, withχk := χk−1 − χˆk−1(2k)(χk−1)k.Then let pi = χn.(Uniqueness): Suppose pi and χ are both primitives with their evenTaylor coefficients, starting at 4, vanishing. By Proposition 2.2, we haveχ = c1pi + c2pi2 + · · ·+ cnpin +O(z2n+2).By looking at the coefficient of z2 we see c1 = 1. Now looking at thecoefficients of z4, z6, . . . , z2n in order, we see that c2 = 0 = c4 = · · · =cn.So every algebraic connection of dimension n + 1 supported at theorigin is the annihilator of an algebra generated locally by a uniqueprimitive(4.4) pi(z) = z2 + α1z3 + α2z5 + · · ·+ αnz2n+1.Let us denote the codimension n + 1 subalgebra of O(D) with prim-itive (4.4) by A(α1, . . . , αn), and let V (α1, . . . , αn) denote the corre-sponding petal h(D), where h holizes A(α1, . . . , αn).Two cusps V1 and V2 are globally equivalent if there is a biholomorphichomeomorphism φ from V1 onto V2. If each Vr is holized by hr : D→ Vr,then by [1, Theorem 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 israther restrictive.Theorem 4.5. The cusps V (α1, . . . , αn) and V (β1, . . . , βn) are isomor-phic if and only if there is a unimodular constant τ such that βj =τ2j−1αj for 1 ≤ j ≤ n.As a corollary we haveCorollary 4.6. The moduli space of all simple cusps of codimen-sion n+ 1 is R+ × Cn−1.Cusp Algebras 1175. EmbeddingThe purpose of this section is to prove that every simple cusp alge-bra A contains a pair of functions h1, h2 such that the pair holizes thealgebra (which is equivalent to saying that polynomials in h1 and h2 aredense in A in the topology of uniform convergence on compacta).For the rest of this section, A will be a fixed simple cusp algebra ofcodimension n+ 1.Lemma 5.1. For every α on D \ {0}, there is a function ψα in A thathas a single simple zero at α, and no other zeroes on D.Proof: Consider functions of the form (z − α)eh(z). In order to be in A,h must satisfy n+ 1 equations on its first 2n+ 1 derivatives at 0. Thissystem 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/f2, . . . , 1/fkto A. Since A is of finite codimension in O(D), for some k there is a linearrelation1/fk =k−1∑j=1cj1/fj +A.Multiplying both sides by fk−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/ψα isin A.Proof: Let Γ be the connection at 0 such that A = Γ⊥. Consider thealgebraA′ :={f ∈ O(|α|2D): Λ(f) = 0, ∀Λ ∈ Γ}.Then ψα is in A′, so by Lemma 5.2, 1/ψα is in A′. Therefore f/ψα isin A′, and is therefore annihilated by every functional in Γ. But f/ψα isalso in O(D), so it is in A as required.Theorem 5.4. There exists a map h : D→ C2 that holizes A.Proof: By applying Lemma 5.3 repeatedly, one can find a primitive h1of A that has no zeroes in D \ {0}. Defineh2(z) := z (h1(z))n+1.118 J. Agler, J. E. McCarthyConsider the algebra A′ that is the closure (in the topology of uniformconvergence on compacta) of polynomials in h1 and h2. By [1, Theo-rem 4.2], the algebra A′ is of finite codimension, and it is a cusp alge-bra, because h(z) = (h1(z), h2(z)) is one-to-one away from the origin.If A′ 6= 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 h2 and hn+11 , and these functions have order 2n + 2and 2n+ 3 respectively at the origin. So there can be no linear relationin 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 analyticcurves, Indiana Univ. Math. J. 56(6) (2007), 2899–2933.[2] T. W. Gamelin, Embedding Riemann surfaces in maximal idealspaces, J. Functional Analysis 2 (1968), 123–146.[3] G. Knese, Function theory on the Neil parabola, Michigan Math.J. 55(1) (2007), 139–154.[4] V. I. Paulsen an D. Singh, Modules over subalgebras of the diskalgebra, Indiana Univ. Math. J. 55(5) (2006), 1751–1766.Jim Agler:Department of MathematicsUniversity of California, San Diego (UCSD)La Jolla, California 92093-0112USAE-mail address: jagler@ucsd.eduJohn E. McCarthy:Mathematics DepartmentWashington UniversitySt. Louis, Missouri 63130USAE-mail address: mccarthy@math.wustl.eduRebut el 14 de setembre de 2007.

Cusp algebras

J. Agler

J. E. McCarthy

Crossref

Publicacions Matemàtiques

Agler, J.

McCarthy, J. E.

Revistes Catalanes amb Accés Obert

Publ. Mat. 53 (2009), 111–118
CUSP ALGEBRAS
Jim Agler∗ and John E. McCarthy†
Abstract
A cusp is the image of the unit disk under a proper holomorphic
map into Cn that is one-to-one and whose derivative vanishes at
exactly one point. It is simple if not all the second derivatives
vanish. We characterize when two simple cusps are isomorphic,
and show that they can all be realized in C2.
1. Introduction
By a cusp V we shall mean the image of the unit disk D under a
bounded injective holomorphic map h into Cn whose derivative vanishes
at exactly one point. The simplest example is the Neil parabola, given
by h(ζ) = (ζ2, ζ3). 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 Eh, h is one-to-one and non-
singular. If Eh is a singleton, then V is a cusp. Since the automorphism
group of D is transitive, we may assume then that Eh = {0}.
The function h : D→ Ω induces a finite codimensional subalgebra Ah
of the algebra O(D) of all analytic functions on D, namely
Ah = {F ◦ h : F ∈ O(V )},
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 Cn containing P . When V is a cusp,
the algebraAh has the property that it contains z
mO(D) for somem ≥ 2.
Conversely, if a cofinite subalgebra of O(D) contains zmO(D), then it
arises in this way.
2000 Mathematics Subject Classification. 14H10, 30F20.
Key words. Petal, cusp, holomap.
∗Partially supported by National Science Foundation Grant DMS 0400826.
†Partially supported by National Science Foundation Grant DMS 0501079.
112 J. Agler, J. E. McCarthy
Figure 1. A simple cusp - the Neil parabola z3 = w2.
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, zmO(D) ⊆ A.
We shall say a map h : D→ Ω ( Cn 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 = Ah.
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 : zk+1O(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 quan-
tities for the corresponding algebra Ah.
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) ∈ D2 : z3 = w2} (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
Cusp Algebras 113
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 C2. In Section 5, we show that all
simple cusp algebras can be holized in C2.
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 pi
that lies in A, and satisfies pi(0) = 0, pi′′(0) = 2; so pi has a Taylor
expansion pi(z) = z2 + · · · .
For each n, let An be the space
An =
[
A ∩ z2nO(D)
]
,
and let En be the quotient
En = An/An+1.
Observe that multiplication by any primitive pi is always one-to-one
from En to En+1. Therefore, there is some integer n0 such that dim(En)
is 1 for n ≤ n0, and 2 for n > n0. Since dim(En) = 2 if and only if
both z2n and z2n+1 are in A, it follows that
ord(A) = 2n0 + 1.
As pin is in An for each n, if f is in An for some n ≤ n0, then there is a
constant cn such that f − cnpi
n is in An+1. Therefore every function f
in A has the representation
(2.1) f = c0 + c1pi + · · ·+ cn0pi
n0 + z2n0+2g,
where c0, . . . , cn0 are in C and g ∈ O(D). From (2.1), we see that
cod(A) = n0 + 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 n0 = cod(A) − 1, and pi is a primitive for A,
then every function f in A has a unique representation
(2.3) f(z) = p ◦ pi(z) + z2n0+2g(z)
for some polynomial p of degree at most n0 and some g ∈ O(D).
114 J. Agler, J. E. McCarthy
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) =
m∑
i=1
ni∑
j=0
aijf
(j)(αi).
Definition 3.1. A connection on {α1, . . . , αm} ⊂ U is a finite dimen-
sional set Γ of local functionals Λ supported by {α1, . . . , αm}. We say
Γ is algebraic if Γ⊥ := {f ∈ O(U) : Λ(f) = 0, ∀Λ ∈ Γ} is an algebra.
It was proved by T. W. Gamelin [2] that every finite codimensional
subalgebra A of O(D) is Γ⊥ for some algebraic connection on D. More-
over, 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 one-to-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 Ah 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 P1 and P2 are cusp points in cusps V1 and V2, and
there is an injective holomorphic map φ : V1 → V2. For r = 1, 2 there
are holizations hr : D → Vr with hr(0) = Pr. By [1, Theorem 3.2], the
map φ from V1 to V2 induces a one-to-one map ψ : D → D such that
h2 ◦ ψ = φ ◦ h1, and the connections Γ1 and Γ2 induced by h1 and h2
are related by
Γ2 ⊇ ψ∗ Γ1
:= {f 7→ Λ1 (f ◦ ψ) : Λ1 ∈ Γ1}.
If
(3.2) Λ1 : f 7→
n∑
j=0
ajf
(j)(0),
then
(3.3) ψ∗ Λ1 : g 7→
n∑
j=0
aj(g ◦ ψ)
(j)(0).
We can use Faa´ 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) 6= 0, note in particular
that the coefficient of g(n)(0) on the right-hand side of (3.3) is an[ψ
′(0)]n.
Therefore ψ∗ is injective, so cod(A1) ≤ cod(A2).
If con(A1) = k, it means that for each 1 ≤ j ≤ k, the functional
f 7→ f (j)(0) is in Γ1. Therefore each functional g 7→ g
(j)(0) is in Γ2, and
Cusp Algebras 115
con(A1) ≤ con(A2). Finally if ord(A1) = n, it means there is some Λ1
in Γ1 of the form (3.2) with an 6= 0; it follows that ord(A1) ≤ ord(A2).
Summarizing, we have shown:
Proposition 3.4. If φ : V1 → V2 is an injective holomorphic map from
one cusp to another, then cod(V1) ≤ cod(V2), con(V1) ≤ con(V2) and
ord(V1) ≤ ord(V2).
Now, if P is a cusp point in a petal V1, and one surrounds P by
a decreasing sequence of open sets Un, then with each Un there is a
cusp algebra with a codimension, contact and order, and there is an
inclusion map from each Un+1 to Un. By Proposition 3.4, the positive
integers cod(Un), con(Un) and ord(Un) must decrease as Un 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 P1 and P2 in the petals V1 and V2
are locally equivalent if there is a neighborhood U1 of P1 in V1, and a
neighborhood U2 of P2 in V2, and a biholomorphic homeomorphism F
from U1 onto U2 that maps P1 to P2.
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
pi1 be a primitive of the first algebra, and pi2 a primitive of the second
algebra. By Proposition 2.2, it is sufficient to prove that there are neigh-
borhoods W1 and W2 of the origin, and a univalent map φ : W1 → W2
that maps 0 to 0 and such that
(4.2) pi2 ◦ φ = pi1.
Each pir has a square root χr in a neighborhood of 0, and each χr is
locally univalent (because pir is of order 2). So define
φ := χ−12 ◦ χ1
on a suitable neighborhood W1, 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 pˆi(k) we mean the kth Taylor
coefficient at 0.
116 J. Agler, J. E. McCarthy
Lemma 4.3. Every simple cusp algebra of codimension n + 1 has a
primitive pi such that pˆi(2k) = 0 for all 2 ≤ k ≤ n. Moreover, pi is
unique up to O(z2n+2).
Proof: (Existence): Let χ1 be any primitive. Define
χ2 := χ1 − χˆ1(4)(χ1)
2.
Proceed inductively, with
χk := χk−1 − χˆk−1(2k)(χk−1)
k.
Then let pi = χn.
(Uniqueness): Suppose pi and χ are both primitives with their even
Taylor coefficients, starting at 4, vanishing. By Proposition 2.2, we have
χ = c1pi + c2pi
2 + · · ·+ cnpi
n +O(z2n+2).
By looking at the coefficient of z2 we see c1 = 1. Now looking at the
coefficients of z4, z6, . . . , z2n in order, we see that c2 = 0 = c4 = · · · =
cn.
So every algebraic connection of dimension n + 1 supported at the
origin is the annihilator of an algebra generated locally by a unique
primitive
(4.4) pi(z) = z2 + α1z
3 + α2z
5 + · · ·+ αnz
2n+1.
Let us denote the codimension n + 1 subalgebra of O(D) with prim-
itive (4.4) by A(α1, . . . , αn), and let V (α1, . . . , αn) denote the corre-
sponding petal h(D), where h holizes A(α1, . . . , αn).
Two cusps V1 and V2 are globally equivalent if there is a biholomorphic
homeomorphism φ from V1 onto V2. If each Vr is holized by hr : D→ Vr,
then by [1, Theorem 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 (α1, . . . , αn) and V (β1, . . . , βn) are isomor-
phic if and only if there is a unimodular constant τ such that βj =
τ2j−1αj for 1 ≤ j ≤ n.
As a corollary we have
Corollary 4.6. The moduli space of all simple cusps of codimen-
sion n+ 1 is R+ × Cn−1.
Cusp Algebras 117
5. Embedding
The purpose of this section is to prove that every simple cusp alge-
bra A contains a pair of functions h1, h2 such that the pair holizes the
algebra (which is equivalent to saying that polynomials in h1 and h2 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 ψα in A that
has a single simple zero at α, and no other zeroes on D.
Proof: Consider functions of the form (z − α)eh(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/f2, . . . , 1/fk
to A. Since A is of finite codimension in O(D), for some k there is a linear
relation
1/fk =
k−1∑
j=1
cj1/f
j +A.
Multiplying both sides by fk−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/ψα is
in A.
Proof: Let Γ be the connection at 0 such that A = Γ⊥. Consider the
algebra
A′ :=
{
f ∈ O
(
|α|
2
D
)
: Λ(f) = 0, ∀Λ ∈ Γ
}
.
Then ψα is in A
′, so by Lemma 5.2, 1/ψα is in A
′. Therefore f/ψα is
in A′, and is therefore annihilated by every functional in Γ. But f/ψα is
also in O(D), so it is in A as required.
Theorem 5.4. There exists a map h : D→ C2 that holizes A.
Proof: By applying Lemma 5.3 repeatedly, one can find a primitive h1
of A that has no zeroes in D \ {0}. Define
h2(z) := z (h1(z))
n+1.
118 J. Agler, J. E. McCarthy
Consider the algebra A′ that is the closure (in the topology of uniform
convergence on compacta) of polynomials in h1 and h2. By [1, Theo-
rem 4.2], the algebra A′ is of finite codimension, and it is a cusp alge-
bra, because h(z) = (h1(z), h2(z)) is one-to-one away from the origin.
If A′ 6= 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 h2 and h
n+1
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, Indiana Univ. Math. J. 56(6) (2007), 2899–2933.
[2] T. W. Gamelin, Embedding Riemann surfaces in maximal ideal
spaces, J. Functional Analysis 2 (1968), 123–146.
[3] G. Knese, Function theory on the Neil parabola, Michigan Math.
J. 55(1) (2007), 139–154.
[4] V. I. Paulsen an D. Singh, Modules over subalgebras of the disk
algebra, Indiana Univ. Math. J. 55(5) (2006), 1751–1766.
Jim Agler:
Department of Mathematics
University of California, San Diego (UCSD)
La Jolla, California 92093-0112
USA
E-mail address: jagler@ucsd.edu
John E. McCarthy:
Mathematics Department
Washington University
St. Louis, Missouri 63130
USA
E-mail address: mccarthy@math.wustl.edu
Rebut el 14 de setembre de 2007.


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