Cusp algebras

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.

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 The order of A is defined by The contact of A is defined by 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 .

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 and let E n be the quotient 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 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: 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 for some polynomial p of degree at most n 0 and some g ∈ O(D).

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 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 , and the connections Γ 1 and Γ 2 induced by h 1 and h 2 are related by 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: 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 de- creasing 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.

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 . 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 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 on a suitable neighborhood W 1 , and (4.2) is satisfied. 2 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. Byπ(k) we mean the k th Taylor coefficient at 0.
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 .
As a corollary we have Corollary 4.6 The moduli space of all simple cusps of codimension n + 1 is R + × C n−1 .

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. 2 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 Multiplying both sides by f k−1 , we get that 1/f is in A. Proof: Let Γ be the connection at 0 such that A = Γ ⊥ . Consider the algebra 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. 2 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 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.