New York Journal of Mathematics

New York J. Math. 14 (2008) 517–538.

Local geometry of zero sets of holomorphic functions near the torus

Jim Agler, John E. McCarthy and Mark Stankus

Abstract. We call a holomorphic function f on a domain in ℂ^{n} locally toral at the point P in the n-torus if the intersection of the zero set of f with the n-torus has dimension n − 1 at P. We study the interplay between the structure of locally toral functions and the geometry of their zero sets.

Contents

0. Introduction 517 1. Preliminaries 519 2. The local geometry of analytic sets near the torus 522 3. The global geometry of algebraic sets 534 References 538

0. Introduction

Throughout this paper, we shall let 𝔻 denote the unit disk in the complex plane, 𝕋 be a unit circle, 𝔼 be the complement of the closed unit disk in ℂ and let A(𝔻^{n}) denote the polydisk algebra, the algebra of functions that are continuous on the closure of 𝔻^{n} and holomorphic on the interior.

When studying function theory on the polydisk 𝔻^{n}, it is often useful to focus on the torus 𝕋^{n}, which is the distinguished boundary of 𝔻^{n}. In several important ways, the behavior of a function in A(𝔻^{n}) is controlled by its

517^{ISSN 1076-9803/08}

Received October 30, 2007.

Mathematics Subject Classification. 14J989, 32B99.

Key words and phrases. Toral polynomial; locally toral; sidal; locally sidal.

The first author was partially supported by National Science Foundation Grant DMS 0400826.

The second author was partially supported by National Science Foundation Grant DMS 0501079.

behavior on 𝕋^{n}: not only is 𝕋^{n} a set of uniqueness, but every function in the algebra attains its maximum modulus on 𝕋^{n}.

We shall say that a variety (by which we always mean an irreducible algebraic set) V is toral if its intersection with 𝕋^{n} is fat enough to be a determining set for holomorphic functions on V (see Section 2 for a precise definition). Otherwise we shall call the variety atoral. We shall say that a polynomial in ℂ[z_{1}, …, z_{n}] is toral (respectively, atoral) if the zero set of every irreducible factor is toral (respectively, atoral).

In [2] it was shown that knowing an algebraic set is toral has interesting consequences in function theory and operator theory. In this paper we study the localized versions of this and other related geometric properties of zero sets. In Section 1 we collect standard basic tools for studying the local geometry of an analytic set. In Section 2 we define and analyze the local properties of interest to us. In Section 3 we develop the relationship between the local properties of Section 2 and the global properties studied in [2]. The results in Sections 2 and 3 include constraints, both positive and negative, on which combinations of properties can occur simultaneously. In those sections we also include a number of specific examples showing that various combinations of properties can occur.

Let f be a holomorphic function defined on an open subset G of ℂ^{n} and suppose τ ∈ 𝕋^{n} ∩ G. To localize the notion of torality, we say f is locally toral at τ if f(τ) = 0 and, for every neighborhood U of τ in G, there is a neighborhood V of τ in U such that Z_{f} ∩ V ∩ 𝕋^{n} is determining for Z_{f} ∩ V. An irreducible holomorphic function may no longer be irreducible if its domain is restricted to a smaller set, however, there is a well-defined notion of locally irreducible at a point. If f is locally irreducible at τ, then we say f is locally atoral at τ if f(τ) = 0 and f is not locally toral at τ.

Every toral polynomial is locally toral at some point and vice versa. A polynomial which is toral, locally toral at exactly one point and locally atoral on an arc is presented.

Now let n = 2, f be a holomorphic function defined on an open subset G of ℂ^{2} and suppose τ ∈ 𝕋^{2}∩G. A function φ which is holomorphic on an open neighborhood of the closed unit disk is inner if |φ(α)| = 1 for all α ∈ 𝔻 and therefore the zero set of the function w − φ(z) lies in (𝔻 × 𝔻) ∪ (𝔼 × 𝔼) ∪ 𝕋^{2}. We define f to be locally inner at τ if f(τ) = 0 and its zero set near τ lies in (𝔻×𝔻)∪(𝔼×𝔼)∪𝕋^{2} and locally outer at τ if f(τ) = 0 and its zero sets near τ lies in (𝔻 × 𝔼) ∪ (𝔼 × 𝔻) ∪ 𝕋^{2}. Notice that if f is locally inner (or outer) at τ, then the zero set of f does not intersect 𝕊 where 𝕊 is the set of points (z, w) ∈ ℂ^{2} where either z or w (but not both) is unimodular. With this notation, ℂ^{2} is the disjoint union of 𝕋^{2}, 𝕊 and the four open “quadrants” in ℂ^{2} determined by the sets 𝔻 and 𝔼, namely, 𝔻×𝔻, 𝔼×𝔻, 𝔻×𝔼, and 𝔼×𝔼. Any curve is ℂ^{2} which intersects two or more of the above quadrants does so by passing through either 𝕊 or 𝕋^{2}. Replacing 𝕋^{2} with 𝕊 in the definitions of toral, atoral, locally toral and locally atoral results in our definitions of sidal, asidal, locally sidal and locally asidal, respectively.

If f is locally inner (or locally outer) at τ, then f is locally asidal at τ. Moreover, the product of locally inner and locally outer functions is locally asidal. Every locally asidal function can be decomposed (locally) as a product of a local inner times a local outer.

After defining inner and outer for holomorphic functions of 2 variables (the nonlocalized versions of “locally inner” and “locally outer”), we find that every nonzero atoral polynomial p(z, w) is a product of an inner polynomial and an outer polynomial.

While an inner polynomial is locally inner at each point in its zero set intersect 𝕋^{2}, the converse is not true. The counterexample to the converse is a polynomial p which is locally inner at each point which is both in the zero set of p and in 𝕋^{2} (and, therefore, locally asidal at each of these points), but is not inner since it is not (globally) sidal.

1. Preliminaries

For the convenience of the reader, in this section we compile a number of elementary results from the literature ([4],[3]) that describe the local properties of 0-sets of holomorphic functions of several variables. In addition, we will indicate, in a general sense, how these results will be used in the subsequent sections of the paper. If n ≥ 1 and G is an open set in ℂ^{n}, let Hol(G) denote the algebra of holomorphic functions on G. If d ≥ 2 and G_{0} is an open set in ℂ^{d−1}, let Hol(G_{0})[w] denote the ring of polynomials in w with coefficients in Hol(G_{0}). Let P(G_{0}) denote the collection of pseudopolynomials over G_{0}, i.e., the monic elements of Hol(G_{0})[w]. Thus, P ∈ P(G_{0}) if and only if there exists an integer n ≥ 1 and s_{1}, …, s_{n} ∈ Hol(G_{0}) such that

(1.1)  P (z, w) = w^{n} − s_{1}(z)w^{n−1} + ⋯ + (−1)^{n}s_{n}(z).

If G_{1} is an open subset of G_{0}, then we let P |G_{1} denote the pseudopolynomial over G_{1} obtained by restricting the coefficients of P to G_{1}. Throughout this section, we denote points in ℂ^{d} as ordered pairs ζ = (z, w) with z ∈ ℂ^{d−1} and w ∈ ℂ. If f ∈ Hol(G), then we let Z_{f} = {ζ | f(ζ) = 0 and ζ ∈ G}. If P ∈ P(G_{0}), then we shall view P as an element of Hol(G_{0} × ℂ). Thus,

Z_{P} = {(z, w) ∈ G_{0} × ℂ | P (z, w) = 0}.

If ζ ∈ ℂ^{d}, then we will say that G is a neighborhood of ζ if G is an open subset which contains ζ and say that U is a neighborhood of ζ in G if U is an open subset of G which contains ζ.

Theorem 1.2 (Weierstrass Preparation Theorem). If G is an open subset of ℂ^{d}, (z_{0}, w_{0}) ∈ G, f ∈ Hol(G) and f(z_{0}, w), the holomorphic function of the single variable w, has a 0 of order n at w = w_{0}, then there exist (i) a connected neighborhood G_{0} of z_{0} in ℂ^{d−1}, (ii) a neighborhood D_{0} of w_{0} in ℂ, (iii) an h ∈ Hol(G_{0} × D_{0}) such that Z_{h} = ∅, and (iv) a pseudopolynomial P over G_{0} such that G_{0} × D_{0} ⊆ G,

(1.3)  P (z_{0}, w) = (w − w_{0})^{n},

and

(1.4)  f(z, w) = P (z, w)h(z, w)

for all z ∈ G_{0} and all w ∈ D_{0}. Moreover, the representation in (1.4) is unique in the following sense: if Q is a pseudopolynomial over G_{0}, k ∈ Hol(G_{0} × D_{0}), Z_{k} = ∅, Q(z_{0}, w) = (w − w_{0})^{m} for some m ≥ 1, and f = Qk on G_{0} × D_{0}, then P = Q and h = k.

In light of the uniqueness assertion in Theorem 1.2, the pseudopolynomial P of Theorem 1.2 is referred to as the Weierstrass polynomial of f at (z_{0}, w_{0}) over G_{0}.

Evidently, if f ∈ Hol(G), (z_{0}, w_{0}) ∈ G, f(z_{0}, w_{0}) = 0 and the function f(z_{0}, w) has a zero of finite order at w = w_{0}, then Theorem 1.2 can be used to obtain sets G_{0} and D_{0} and P ∈ P(G_{0}) such that Z_{f} ∩ (G_{0} × D_{0}) = Z_{P} ∩ (G_{0} × D_{0}).

Now, one of the many hurdles to overcome in the quest to understand the 0-sets of holomorphic functions in several variables results from the fact that the ring Hol(G_{0}), while an integral domain (when G_{0} is connected), fails to be a unique factorization domain. This has the unpleasant consequence that Hol(G_{0})[w] is not a unique factorization domain. It is indeed both a fundamental and fortuitous event that nevertheless P(G_{0}), while not a ring, behaves like a unique factorization domain in the following sense.

Theorem 1.5. If d ≥ 2, G_{0} is an open connected set in ℂ^{d−1} and P ∈ P(G_{0}), then P can be written, uniquely up to order, as a finite product of irreducible elements of P(G_{0}).

In many proofs using Theorem 1.5, it would be desirable to have that if Q is an irreducible factor of P and G_{1} ⊂ G_{0}, then Q|G_{1} is irreducible. Unfortunately, this is not true. For example, if Q is the pseudopolynomial over 𝔻 defined by Q(z, w) = w^{2}−z, U is an open disk in 𝔻, and 0 ∉ U, then Q is irreducible, but Q|U is reducible since \sqrt{z} ∈ Hol(U). These considerations motivate the following robust localized notions of irreducibility.

Definition 1.6. Let d ≥ 2, z_{0} ∈ ℂ^{d−1}, G_{0} be a neighborhood of z_{0}, and P ∈ P(G_{0}). Let us agree to say that P is locally irreducible at z_{0} if, there exists a neighborhood U of z_{0} in G_{0} such that for every connected neighborhood V of z_{0} in U, P |V is irreducible.

Definition 1.7. If d ≥ 2, ζ ∈ ℂ^{d}, G is a neighborhood of ζ and f ∈ Hol(G), we will say that f is locally irreducible at ζ if there exists a neighborhood U of ζ in G such that for every connected neighborhood V of ζ in U, for every g ∈ Hol(V) and every h ∈ Hol(V), f|V = gh implies either Z_{g} or Z_{h} equals the empty set.

Clearly, if G is a connected neighborhood of (z_{0}, w_{0}), f ∈ Hol(G), and P is a Weierstrass polynomial for f at (z_{0}, w_{0}), then f is locally irreducible at (z_{0}, w_{0}) if and only if P is locally irreducible at z_{0}.

An alternate approach to handling considerations pertaining to local irreducibility is to employ the theory of germs of analytic varities. We will not use this theory.

Since the ring of power series which converge on some neighborhood of a given point z_{0} ∈ ℂ^{d−1} is a unique factorization domain, the following theorems follow from Theorem 1.5.

Theorem 1.8. If d ≥ 2, z_{0} ∈ ℂ^{d−1}, G_{0} is a neighborhood of z_{0}, and P ∈ P(G_{0}), then there exists a connected neighborhood G_{1} of z_{0} in G_{0} such that every irreducible factor of P |G_{1} is locally irreducible at z_{0}.

Theorem 1.9. If ζ ∈ ℂ^{d}, G is a connected neighborhood of ζ, f ∈ Hol(G) and f is not identically zero, then there exists a connected neighborhood U of ζ and f_{1}, …, f_{N} ∈ Hol(U) such that f|U = f_{1}f_{2} ⋯ f_{N} and each f_{j} is locally irreducible at ζ.

The singular points of a pseudopolynomial without multiple factors can be described via a discriminant. Recall that if p = w^{n}−s_{1}w^{n−1}+⋯+(−1)^{n}s_{n} = (w − r_{1}) ⋯ (w − r_{n}) is a polynomial with complex coefficients, then the coefficients s_{1}, …, s_{n} equal symmetric polynomials evaluated at the roots r_{1}, …, r_{n} and △_{p}, the discriminant of p, is defined by writing the symmetric function ∏_{i<j}(r_{i} − r_{j})^{2} as a polynomial △_{p} in the elementary symmetric

functions of r_{1}, …, r_{n}. Thus,

△_{p} = △_{p}(s_{1}, …, s_{n}) = ∏_{i<j}(r_{i} − r_{j})^{2} .

As a consequence, p has a multiple factor if and only if △_{p} = 0. Now, if P ∈ P(G_{0}), then △_{P} is naturally an element of Hol(G_{0}). Furthermore, the following theorem is obtained.

Theorem 1.10. If G_{0} is a connected set in ℂ^{d−1} and P ∈ P(G_{0}), then P has a multiple factor if and only if △_{P} is identically 0 on G_{0}.

The next theorem shows that if △_{P} does not vanish at z_{0}, then, for a sufficiently small neighborhood U of z_{0} in G_{0}, P factors as a product of pseudopolynomials of degree 1.

Theorem 1.11. Let z_{0} ∈ ℂ^{d−1}, G_{0} be a neighborhood of z_{0} and P be a pseudopolynomial over G_{0} of degree n. If △_{P} (z_{0}) ≠ 0, then there exists a neighborhood U of z_{0} in G_{0} and n functions r_{1}, … r_{n} ∈ Hol(U) so that

P (z, w) = (w − r_{1}(z)) ⋯ (w − r_{n}(z))

for all z ∈ U and w ∈ ℂ.

For P ∈ P(G_{0}) define S_{P} (G_{0}, D_{0}) = {(z_{0}, w_{0}) ∈ G_{0} × D_{0}|△_{P} (z_{0}) = 0 and w = w_{0} is a repeated root of the one variable polynomial P (z_{0}, w)}. Thus, if P has no multiple factors, then S_{P} (G_{0}, D_{0}) consists of the singular points of the analytic set Z_{P} ∩ (G_{0} × D_{0}).

Proposition 1.12. If G_{0} is an open connected set in ℂ^{d−1}, D_{0} is an open set in ℂ, P ∈ P(G_{0}), P is irreducible and Z_{P} ⊆ G_{0} × D_{0}, then Z_{P} \S_{P} (G_{0}, D_{0}) is connected.

In applying Proposition 1.12 the following observation is often useful. Notice that if a pseudopolynomial P arises as the Weierstrass polynomial of a function f at (z_{0}, w_{0}) over G_{0}, then condition (1.3) of Theorem 1.2 implies that for a sufficiently small connected neighborhood U of z_{0} in G_{0},

(1.13)  P (z, w) = 0 and z ∈ U imply w ∈ D_{0} .

If Q is an irreducible factor of P |U, then Proposition 1.12 implies that

(1.14)  Z_{Q}\S_{Q}(U, D_{0})

is connected.

Since every pseudopolynomial is monic, the following theorem holds.

Theorem 1.15. If G_{0} is an open set in ℂ^{d−1}, and both P and Q are pseudopolynomials over G_{0}, then there exist a quotient R ∈ P(G_{0}) and a remainder S ∈ Hol(G_{0})[w] such that P = RQ + S and either S is identically zero or the degree of S is less than the degree of Q.

Combining Theorems 1.5, 1.10 and 1.15, it is easy to see the following Nullstellensatz result for pseudopolynomials: if G_{0} is an open connected set in ℂ^{d−1}, P ∈ P(G_{0}), Q ∈ P(G_{0}), and Z_{Q} ⊆ Z_{P}, then there exists a positive integer n such that Q divides P^{n}. Therefore, if (z_{0}, w_{0}) ∈ 𝕋^{d}, G is a connected neighborhood of (z_{0}, w_{0}), f ∈ Hol(G), and g ∈ Hol(G), then there exists a neighborhood of the form G_{0} × D_{0} of (z_{0}, w_{0}) in G such that

(1.16)

Z_{g} ⊆ Z_{f} implies there exists a positive integer n such that g|(G_{0} × D_{0}) divides (f|(G_{0} × D_{0}))^{n} .

2. The local geometry of analytic sets near the torus

We shall employ standard notations by letting 𝔻 denote the open unit disc in ℂ centered at the origin and 𝕋 denote the boundary of 𝔻. Thus, for d ≥ 1, 𝔻^{d} is the standard unit polyidisc centered at the origin in ℂ^{d} and 𝕋^{d} is the distinguished boundary of 𝔻^{d}. In addition we set 𝔼 = ℂ\(𝔻 ∪ 𝕋). For the special case when d = 2, note that there are four open “quadrants” in ℂ^{2} determined by the sets 𝔻 and 𝔼, namely, 𝔻 × 𝔻, 𝔼 × 𝔻, 𝔻 × 𝔼, and 𝔼 × 𝔼. We shall let 𝕊 denote the union of the “sides” of these quadrants. Thus,

𝕊 = [(𝕋 × ℂ) ∪ (ℂ × 𝕋)]\𝕋^{2}.

Recall from [2] that a polynomial p ∈ ℂ[z_{1}, …, z_{n}] is said to be toral if Z_{p} ∩𝕋^{n} is determining for Z_{p} and p is said to be atoral if there does not exist a nonconstant q ∈ ℂ[z_{1}, …, z_{n}] such that q is toral and q divides p. Here, Z_{p} denotes the 0-set of p and for a set X ⊆ ℂ^{n}, we say X is determining for Z_{p} if X ∩ Z_{p} is a set of uniqueness for Hol(Z_{p}), the algebra of holomorphic functions on Z_{p}.

We wish here to localize the notions of torality and atorality to a point τ on the torus. Accordingly, if G is an open set in ℂ^{d} and f ∈ Hol(G), let Z_{f} = {ζ ∈ G | f(ζ) = 0}. Further, if X ⊆ G, we say X is determining for Z_{f} if X ∩ Z_{f} is a set of uniqueness for Hol(Z_{f}), the algebra of holomorphic functions on Z_{f}.

Definition 2.1. If G is an open set in ℂ^{d}, f ∈ Hol(G) and τ ∈ 𝕋^{d}, then we say f is locally toral at τ if f(τ) = 0 and, for every neighborhood U of τ in G, there is a neighborhood V of τ in U such that Z_{f} ∩ V ∩ 𝕋^{d} is determining for Z_{f} ∩ V. We say f is locally atoral at τ if f(τ) = 0 and there does not exist a neighborhood W of τ in G together with a g ∈ Hol(W) such that g is locally toral at τ and Z_{g} ⊆ Z_{f}.

It easily follows from these definitions that if f ∈ Hol(G), τ ∈ 𝕋^{d} and U is a neighborhood of τ in G, then f is locally toral (respectively, atoral) at τ if and only if f|U is locally toral (respectively, atoral) at τ. In addition, if g ∈ Hol(G) and g(τ) ≠ 0, then f is locally toral (respectively, atoral) at τ if and only if fg is locally toral (respectively, atoral) at τ.

In [2] it was an immediate consequence of the definitions that any irreducible polynomial in ℂ[z, w] is either toral or atoral. We now prove the localized version of this result. Let τ = (τ_{1}, τ_{2}) ∈ 𝕋^{2}, G be a neighborhood of τ and f ∈ Hol(G). We will say that f is nondegerate at τ if there exists a neighborhood U of τ in G such that Z_{z−τ_{1}} ∩ Z_{f} ∩ U = {τ} and say that f is degenerate at τ if f(τ) = 0 and f is not nondegenerate at τ. Clearly, for any f, if f(τ) = 0, then f can be written uniquely in the form f = (z − τ_{1})^{m}g where m ≥ 0 and either g(τ) ≠ 0 or g is nondegenerate. Notice also that if f is nondegenerate at τ, then f(τ_{1}, w) has a zero of finite order at w = τ_{2}.

Proposition 2.2. Let τ ∈ 𝕋^{2}, G be a neighborhood of τ, and f ∈ Hol(G) with f(τ) = 0. If f is locally irreducible at τ, then either f is locally toral at τ or f is locally atoral at τ.

Proof. Suppose f is locally irreducible at τ so that either Z_{z−τ_{1}}∩U = Z_{f} ∩U for some neighborhood U of τ in G or f is nondegenerate at τ. In the first case, clearly, f is locally toral at τ. In the second case, we let P be a Weierstrass polynomial of f at τ over G_{0} so that Z_{f} ∩ (G_{0} × D_{0}) = Z_{P} ∩(G_{0} ×D_{0}). Suppose that f is not locally atoral at τ. Thus, there exists a neighborhood W of τ in G_{0} × D_{0} and g ∈ Hol(W) such that g is locally toral at τ and Z_{g} ⊆ Z_{f}. The local irreducibility of P, the torality of g and (1.16) imply that P is locally toral at τ. Thus, f is locally toral at τ. □

If f is locally atoral at τ, g is a divisor of f and g(τ) = 0, then, by the definition of locally atoral, g is locally atoral at τ. A straightforward modification of the proof of Proposition 1.3 of [2] shows that if f is locally toral at τ, g is a divisor of f and g(τ) = 0, then g is locally toral at τ.

Now, the product of toral (respectively, atoral) polynomials is clearly toral (respectively, atoral). Clearly, the product of functions which are locally atoral is locally atoral. The following proposition shows that the product of functions which are locally toral is locally toral.

Proposition 2.3. Let τ ∈ 𝕋^{2}, G be a neighborhood of τ and f, g ∈ Hol(G). If f and g are locally toral at τ, then the product fg is locally toral at τ.

Proof. Without loss of generality, we may assume that both f and g are nondegenerate at τ. Let h = fg. By Theorems 1.2 and 1.8, there exist connected neighborhoods G_{0} and G_{1} of τ_{1}, neighborhoods D_{0} and D_{1} of τ_{2}, and pseudopolynomials P_{f} and P_{g} over G_{0} such that G_{0} × D_{0} ⊂ G, G_{1} × D_{1} ⊂ G,

Z_{f} ∩ (G_{0} × D_{0}) = Z_{P_{f}} ∩ (G_{0} × D_{0}),
Z_{g} ∩ (G_{1} × D_{1}) = Z_{P_{g}} ∩ (G_{1} × D_{1}),

each irreducible factor of P_{f} is locally irreducible at τ and each irreducible factor of P_{g} is locally irreducible at τ. Let R_{1}, …, R_{m}, S_{1}, …, S_{n} be pseudopolynomials over G_{0} ∩ G_{1} such that

(2.4) P_{f}|(G_{0} ∩ G_{1}) = R_{1} ⋯ R_{m} and P_{g}|(G_{0} ∩ G_{1}) = S_{1}S_{2} ⋯ S_{n}

are the decompositions of P_{f} and P_{g} into a product of irreducible pseudopolynomials.

To show h is locally toral at τ, let U be a neighborhood of τ in G. There exists a sufficiently small connected neighborhood V of τ in

U ∩ ((G_{0} ∩ G_{1}) × (D_{0} ∩ D_{1}))

such that for 1 ≤ j ≤ m and 1 ≤ k ≤ n,

(2.5)  (Z_{R_{j}} ∩ V)\{τ} and (Z_{S_{k}} ∩ V)\{τ} are connected

and

(2.6) (Z_{R_{j}} ∩ V)\{τ} and (Z_{S_{k}} ∩ V)\{τ} do not have any singular points.

To show that Z_{h} ∩ V ∩ 𝕋^{2} is determining for Z_{h} ∩ V, fix F ∈ Hol(Z_{h} ∩ V) with F |(Z_{h} ∩ V ∩ 𝕋^{2}) = 0. We shall show that F = 0 on Z_{f} ∩ V by showing that F = 0 on Z_{R_{j}} ∩ V for each j.

Fix j such that 1 ≤ j ≤ m. R_{j} is locally toral at τ and so there exists a neighborhood W of τ in V such that Z_{R_{j}} ∩W ∩𝕋^{2} is determining for Z_{R_{j}} ∩W. Since Z_{R_{j}} ∩ W ∩ 𝕋^{2} ⊂ Z_{f} ∩ W ∩ 𝕋^{2} ⊂ Z_{h} ∩ W ∩ 𝕋^{2}, F |(Z_{R_{j}} ∩ W ∩ 𝕋^{2}) = 0, and, since Z_{R_{j}} ∩ W ∩ 𝕋^{2} is determining for Z_{R_{j}} ∩ W, F |(Z_{R_{j}} ∩ W) = 0. Now (Z_{R_{j}} ∩W)\{τ} is a relatively open subset of the connected set (Z_{R_{j}} ∩V)\{τ} and each point of (Z_{R_{j}} ∩ V)\{τ} is not a singular point of Z_{R_{j}}. Thus, the use of Theorem 1.11 and analytic continuation yields F |(Z_{R_{j}} ∩V) = 0. Since F |(Z_{R_{j}} ∩ V) = 0 for each j, (2.4) shows that F |(Z_{f} ∩ V) = 0.

An analogous argument show that F |(Z_{g} ∩ V) = 0. Thus, F is identically zero. In summary, therefore, Z_{h} ∩ V ∩ 𝕋^{2} is determining for Z_{h} ∩ V and h is locally toral at τ. □

Corollary 2.2 of [2] shows that every polynomial can be written uniquely, up to multiplicitive constants, as a product of a toral and an atoral polynomial. Proposition 2.2, Proposition 2.3 and Theorem 1.9 imply the following localized version of that result.

Corollary 2.7. Let τ ∈ 𝕋^{2}, G be a neighborhood of τ, and f ∈ Hol(G) with f(τ) = 0. If f is neither locally toral nor locally atoral at τ, then there exists a neighborhood U of τ, and an essentially unique factorization f|U = qr for some q ∈ Hol(U) which is locally toral at τ and r ∈ Hol(U) which is locally atoral at τ.

Theorem 2.8 below generalizes and proves the following: if τ = (τ_{1}, τ_{2}) ∈ 𝕋^{2}, G_{0} is a neighborhood of τ_{1}, r ∈ Hol(G_{0}) and r(τ_{1}) = τ_{2}, then the pseudopolynomial w − r(z) is locally toral at τ if and only if there exists a connected neighborhood N_{1} of τ_{1} in G_{0} such that, for all z ∈ N_{1} ∩ 𝕋, r(z) ∈ 𝕋.

Theorem 2.8. Let ζ = (z_{0}, w_{0}) ∈ 𝕋^{2}, G be a neighborhood of ζ and f ∈ Hol(G) be such that ζ is not a singular point of Z_{f}, f(ζ) = 0, and f is nondegenerate at ζ. f is locally toral at ζ if and only if there exist a neighborhood N_{1} of z_{0}, a neighborhood N_{2} of w_{0} and a function r ∈ Hol(N_{1}) such that

(2.9)  Z_{f} ∩ (N_{1} × N_{2}) = {(z, r(z)) : z ∈ N_{1}}

and

(2.10)  r(z) ∈ 𝕋 whenever z ∈ N_{1} ∩ 𝕋 .

Proof. Suppose that f is locally toral at ζ. By shrinking G if necessary, we may assume that Z_{f} does not have any singular points. By Theorem 1.2, there exists a neighborhood G_{0} ×D_{0} of ζ in G and P a Weierstruass polynomial of f at ζ over G_{0} such that Z_{f} ∩ (G_{0} × D_{0}) = Z_{P} ∩ (G_{0} × D_{0}). Without loss of generality, we may assume that 0 ∉ G_{0}. By the discussion preceeding (1.13), we may assume Z_{P} ∩ (G_{0} × ℂ) = Z_{P} ∩ (G_{0} × D_{0}). Let Q be a pseudopolynomial over G_{0} without multiple factors such that Z_{P} = Z_{Q}. Since Z_{f} ∩ (G_{0} × D_{0}) = Z_{Q} ∩ (G_{0} × D_{0}) and Z_{f} does not have any singular points, S_{Q}(G_{0}, D_{0}) = ∅ and △_{Q}(z) ≠ 0 for all z ∈ G_{0}. Since P is a Weierstrass polynomial at ζ and Z_{Q} = Z_{P}, Q(z_{0}, w) = (w − w_{0})^{m} where m is the degree of Q. To show that m = 1, note that since ζ is not a singular point of Z_{f} and Z_{f} ∩ (G_{0} × D_{0}) = Z_{Q} ∩ (G_{0} × D_{0}), ζ is not a singular point of Z_{Q} and △_{Q}(z_{0}) ≠ 0. Thus, by Theorem 1.11, m = 1 and there exists r ∈ Hol(G_{0}) such that Q(z, w) = w − r(z). For a sufficiently small neighborhood U of z_{0} in G_{0}, \overline{r(1/z)} is well-defined and nonzero and since Q is locally toral at ζ by construction, Z_{Q} ∩ U ∩ 𝕋^{2} is determining for Z_{Q} ∩ U. Therefore, since w − 1/\overline{r(1/z)} vanishes on Z_{Q} ∩ U ∩ 𝕋^{2}, w = 1/\overline{r(1/z)} for z ∈ Z_{Q} ∩ U and so r(z) ∈ 𝕋 whenever z ∈ U ∩ 𝕋. Thus, (2.9) and (2.10) hold with N_{1} = U and N_{2} = D_{0}.

Now suppose that N_{1} × N_{2} is a neighborhood of ζ and (2.9) and (2.10) hold. Let Q be the pseudopolynomial over N_{1} defined by Q(z, w) = w − r(z). To show Q is locally toral at (z_{0}, w_{0}), let U be a neighborhood of (z_{0}, w_{0}) in G. By the discussion preceeding (1.13), there exist a connected neighborhood V_{1} of z_{0} and a neighborhood V_{2} of w_{0} such that V_{1} × V_{2} ⊆ U, V_{1} × V_{2} ⊆ G_{0} × D_{0} and Z_{Q} ∩ (V_{1} × ℂ) = Z_{Q} ∩ (V_{1} × V_{2}). With this choice of V_{1} and V_{2},

(2.11)  Z_{Q} ∩ (V_{1} × V_{2}) = {(z, r(z)) : z ∈ V_{1}}.

To show Z_{Q} ∩ (V_{1} × V_{2}) ∩ 𝕋^{2} is determining for Z_{Q} ∩ (V_{1} × V_{2}), fix F ∈ Hol(Z_{Q}∩(V_{1}×V_{2})) with F |(Z_{Q}∩(V_{1}×V_{2})∩𝕋^{2}) = 0. By (2.9), (z, r(z)) ∈ 𝕋^{2} whenever z ∈ V_{1} ∩𝕋 and so F (z, r(z)) = 0 whenever z ∈ V_{1} ∩𝕋. Since V_{1} ∩𝕋 is determining for V_{1}, (2.11) implies that F vanishes on Z_{Q} ∩ (V_{1} × V_{2}). Hence, Z_{Q} ∩ (V_{1} × V_{2}) ∩ 𝕋^{2} is determining for Z_{Q} ∩ (V_{1} × V_{2}), Q is locally toral at τ and, consequently, f is locally toral at τ. □

Recall from [2] that p ∈ ℂ[z_{1}, …, z_{n}] is atoral if and only if there exists an algebraic set A such that dim A ≤ n − 2 and Z_{p} ∩ 𝕋^{2} ⊆ A. When n = 2, this result simply says that p is atoral if and only if Z_{p} ∩ 𝕋^{2} is finite. Thus, Proposition 2.12 below, which gives a characterization of local atorality, does not come as a great surprise.

Proposition 2.12. Let τ ∈ 𝕋^{2}, G be a neighborhood of τ and f ∈ Hol(G) with f(τ) = 0. f is locally atoral at τ if and only if τ is an isolated point of Z_{f} ∩ 𝕋^{2}.

Proof. First assume that τ is an isolated point of Z_{f} ∩ 𝕋^{2}. We argue by contradiction. If f is not locally atoral at τ, then there exist a neighborhood W of τ and g ∈ Hol(W) such that g is locally toral at τ and Z_{g} ⊆ Z_{f}. Since τ is isolated, there exists an open set U ⊆ ℂ^{2} such that Z_{g} ∩ U ∩ 𝕋^{2} = {τ}. Since g is locally toral at τ, there exists a neighborhood V of τ in U such that Z_{g} ∩V ∩𝕋^{2} is determining for Z_{f} ∩V. Since Z_{g} ∩V ∩𝕋^{2} ⊆ Z_{g} ∩U ∩𝕋^{2} = {τ}, {τ} is determining for Z_{g} ∩ V, an impossibility.

Now assume that τ = (τ_{1}, τ_{2}) is not an isolated point of Z_{f} ∩ 𝕋^{2}. If f(τ_{1}, w) is identically 0, then f is degenerate at τ and hence is not locally atoral at τ. Therefore, we may assume that f(τ_{1}, w) has a zero of finite order n at w = τ_{2}. Let G_{0}, D_{0}, P and h be the sets and functions guaranteed by Theorem 1.2 with P the Weierstrass polynomial for f at τ over G_{0}. Without loss of generality we may assume that 0 ∉ D_{0} and D_{0} is symmetric, i.e., if z ∈ D_{0}, then 1/z ∈ D_{0}. Similarly, we may assume that 0 ∉ G_{0} and G_{0} is symmetric. Since Theorem 1.2 asserts that P (τ_{1}, 0) = (−τ_{2})^{n}, P (z, 0) ≠ 0 on a neighborhood of τ_{1}. As s_{n}(z) = (−1)^{n}P (z, 0), we may choose G_{0} to be sufficiently small so that

(2.13)  s_{n}(z) ≠ 0 for z ∈ G_{0}.

Now since τ is not isolated, P has an irreducible factor Q ∈ P(G_{0}) such that there is a sequence of distinct points τ^{ℓ} ∈ 𝕋^{2} ∩ (G_{0} × ℂ) such that τ^{ℓ} → τ and

(2.14)  Q(τ^{ℓ}) = 0 for all ℓ.

Furthermore, since f(τ_{1}, w) is not identically 0, we may assume that τ_{1}^{ℓ} ≠ τ_{1}^{j} if ℓ ≠ j. Let m be the degree of Q and t_{1}, …, t_{m} ∈ Hol(G_{0}) be such that

Q(z, w) = w^{m} − t_{1}(z)w^{m−1} + ⋯ + (−1)^{m} t_{m}(z) .

Now, recalling that 0 ∉ G_{0}, 0 ∉ D_{0}, and that both G_{0} and D_{0} are symmetric, we define R ∈ P(G_{0}) by

(2.15)  R(z, w) = \frac{(−1)^{m}}{\overline{t_{m}(1/z)}}w^{m}\overline{Q (\frac{1}{z}, \frac{1}{w} )} .

Here, t_{m} ≠ 0 on G_{0} by (2.13).

Now consider QR ∈ P(G_{0}). By (2.14), Q(τ^{ℓ}) = 0 for ℓ ≥ 1 and by (2.15) R(τ^{ℓ}) = 0 for ℓ ≥ 1. Hence △_{QR} vanishes on the sequence {τ^{ℓ}}, a set of uniqueness for G_{0}, and we see that △_{QR} is identically 0 on G_{0}. Since Q and R are irreducible pseudopolynomials and △_{QR} is identically zero, Q = R by Theorem 1.10.

Now notice that by construction Z_{Q} ∩ (G_{0} × D_{0}) ⊆ Z_{f} ∩ (G_{0} × D_{0}). Thus, as we need to show that f is not locally atoral, the proof of the proposition will be complete if we can show Q is locally toral at (z_{0}, w_{0}). Accordingly, assume that U is a neighborhood of τ in G_{0}×D_{0}. By the remark following Proposition 1.12, there exists a neighborhood V_{0} of z_{0} in G_{0} and a neighborhood D_{1} of w_{0} in D_{0} such that V_{0} × D_{1} ⊆ U, S_{Q}(V_{0}, D_{1}) = {τ}, and

(Z_{Q} ∩ (V_{0} × D_{1}))\{τ}

is connected. We claim that Z_{Q} ∩ (V_{0} × D_{1}) ∩ 𝕋^{2} is determining for Z_{Q} ∩ (V_{0} × D_{1}). Since τ^{ℓ} → τ, there exists ℓ ≥ 1 such that τ^{ℓ} ∈ V_{0} × D_{1}.

Since △_{Q}(τ_{1}^{ℓ}) ≠ 0, Theorem 1.11 implies that there exists a symmetric neighborhood N_{1} of τ_{1}^{ℓ}, a symmetric neighborhood N_{2} of τ_{2}^{ℓ} and a function r ∈ Hol(N_{1}) such that r(τ_{1}^{ℓ}) = τ_{2}^{ℓ} and

(2.16)  Z_{Q} ∩ (N_{1} × N_{2}) = {(z, r(z))|z ∈ N_{1}} .

Recalling (2.15) and that Q = R, we see immediately, that for z ∈ N_{1}, r(z) = _{\frac{1}{\overline{r( \frac{1}{z} )}}}, i.e.,

(2.17)  |r(z)| = 1 whenever z ∈ N_{1} ∩ 𝕋.

To show Z_{Q} ∩ (V_{0} × D_{1}) ∩ 𝕋^{2} is determining for Z_{Q} ∩ (V_{0} × D_{1}), fix F ∈ Hol(Z_{Q} ∩ (V_{0} × D_{1})) with F |(Z_{Q} ∩ (V_{0} × D_{1}) ∩ 𝕋^{2}) = 0. Since

F |(Z_{Q} ∩ (N_{1} × N_{2}) ∩ 𝕋^{2})

is identically zero, (2.15) and (2.17) and Theorem 2.8 imply

F |(Z_{Q} ∩ (N_{1} × N_{2}))

is identically zero. Now (Z_{Q} ∩ (N_{1} × N_{2}))\{τ} is a relatively open subset of the connected set (Z_{Q}∩(V_{0}×D_{1}))\{τ} and each point of (Z_{Q}∩(V_{0}×D_{1}))\{τ} is not a singular point of Z_{Q} by (2.16). Thus the use of Theorem 1.11 and analytic continuation yields F |(Z_{Q}∩(V_{1}×D_{0})) = 0. Thus, Z_{Q}∩(V_{1}×D_{0})∩𝕋^{2} is determining for Z_{Q} ∩ (V_{1} × D_{0}), Q is locally toral at τ and, consequently, f is locally toral at τ. Since Z_{Q} ⊂ Z_{f} and Q is locally toral at τ, f is not locally atoral at τ. □

Before continuing to the geometry of toral points we wish to formalize an additional fact about the geometry of atoral points (Proposition 2.19 below). Note the following definition is exactly the same as Definition 2.1 with 𝕋^{2} replaced by 𝕊.

Definition 2.18. Let τ ∈ 𝕋^{2}, G be a neighborhood of τ, and f ∈ Hol(G). Let us agree to say f is locally sidal at τ if f(τ) = 0 and for every neighborhood U of τ in G there exists a neighborhood V of τ in U such that Z_{f} ∩ V ∩ 𝕊 is determining for Z_{f} ∩ V. We say f is locally asidal at τ if f(τ) = 0 and there does not exist a neighborhood W of τ in G together with a g ∈ Hol(W) such that g is locally sidal at τ and Z_{g} ⊆ Z_{f}.

The following facts follow in a direct way using the ideas in the proof of Proposition 2.12.

Proposition 2.19. Let τ ∈ 𝕋^{2}, G be a neighborhood of τ, and f ∈ Hol(G). f is locally asidal at τ if and only if f(τ) = 0 and there exists a neighborhood U of τ in G such that Z_{f} ∩ U ∩ 𝕊 = ∅.

Corollary 2.20. Let τ ∈ 𝕋^{2}, G be a neighborhood of τ and f ∈ Hol(G). If f(τ) = 0 and f is neither locally sidal nor locally asidal at τ, then there exists a neighborhood U of τ in G and functions f_{1}, f_{2} ∈ Hol(U) such that f_{1} is locally sidal at τ, f_{2} is locally asidal at τ and f|U = f_{1}f_{2}. Furthermore, f_{1} and f_{2} are essentially unique on U (i.e., if f|U = g_{1}g_{2} is any other factorization of f with g_{1} locally sidal at τ and g_{2} locally asidal at τ, then there exists a nonvanishing u ∈ Hol(U) such that g_{1} = uf_{1}).

Proposition 2.21. Let τ ∈ 𝕋^{2}, G a neighborhood of τ, and f ∈ Hol(G). If f is locally atoral at τ, then f is locally sidal at τ.

While the converse of Proposition 2.21 is false, nevertheless, it is true that in a certain generic sense, locally sidal points are locally atoral. We make this assertion precise later in this section.

Before continuing, note that Propositions 2.21, 2.2 and 2.3 imply that if f(τ) = 0 and f is locally asidal at τ, then f is locally toral at τ. If τ is a nonsingular point of Z_{f}, then the following local conditions which are weaker than local asidality guarantee local torality. In the following corollary, the first conclusion follows from Theorem 2.8 and the second follows from the first by considering g(z, w) = f(w, z).

Corollary 2.22. Let τ ∈ 𝕋^{2}, G be a neighborhood of τ and f ∈ Hol(G) with f(τ) = 0. Suppose τ is not a singular point of Z_{f}. If W is a neighborhood of τ in G and Z_{f} ∩ W ∩ (𝕋 × ℂ) ⊆ 𝕋^{2}, then f is locally toral at τ. If W is a neighborhood of τ in G and Z_{f} ∩ W ∩ (ℂ × 𝕋) ⊆ 𝕋^{2}, then f is locally toral at τ.

We now turn to the geometry of toral points. Recall from [2] that if p is a toral polynomial of degree (m, n), then each of the following equivalent statements is true.

(2.23) Z_{p} is symmetric, (i.e., (z, w) ∈ Z_{p} ∩ (ℂ^{∗})^{2} ⇒ (1/z, 1/w) ∈ Z_{p}) .

(2.24) p is essentially symmetric,

(i.e., z^{m}w^{n}\overline{p (\frac{1}{z} , \frac{1}{w} )} = σp(z, w) for some nonzero constant σ) .

We localize the notions in (2.23) and (2.24) to a point on the torus in the following definition.

Definition 2.25. Let us agree to say a set S ⊆ ℂ^{2} is symmetric if S ⊆ (ℂ\{0})^{2} and 1/ζ ∈ S whenever ζ ∈ S. Here, if ζ = (z, w), then 1/ζ = (1/z, 1/w). If U ⊆ ℂ^{2} is an open symmetric set and f ∈ Hol(U), we say f is essentially symmetric if there exists a nonvanishing σ ∈ Hol(U) such that \overline{f (1/ζ)} = σ(ζ)f(ζ) for all ζ ∈ U. If τ ∈ 𝕋^{2}, G is a neighborhood of τ and f ∈ Hol(G), we say Z_{f} is locally symmetric at τ if there exists a neighborhood U of τ in G such that Z_{f} ∩ U is symmetric. In addition, we say f is locally essentially symmetric at τ if there exists a symmetric neighborhood U of τ in G such that f|U is essentially symmetric.

Proposition 2.26. Let τ ∈ 𝕋^{2}, G a neighborhood of τ, and f ∈ Hol(G). If f is locally toral at τ, then f is locally essentially symmetric at τ.

Proof. By Theorem 1.9, it suffices to show that if f is locally toral at τ and f is locally irreducible at τ, then f is locally essentially symmetric at τ. Suppose f is locally toral at τ and f is locally irreducible at τ. By Proposition 2.12, τ is not an isolated point of Z_{f} ∩ 𝕋^{2}. Thus, the proof of Proposition 2.12 shows that for sufficiently small neighborhood G_{0} and D_{0},

P (z, w) = σ(z, w)\overline{P (1/z, 1/w)} for (z, w) ∈ G_{0} × D_{0}

for a nonvanishing σ ∈ Hol(G_{0} × D_{0}). Thus, P, and therefore f, is locally essentially symmetric at τ. □

Now recall that atoral points necessarily are sidal. This is not the case for toral points. Indeed, toral points come in 3 types: sidal, asidal, and neither sidal nor asidal. However, toral points that are neither sidal nor asidal arise from sidal toral points and asidal toral points in a particularly simple manner, as the following proposition asserts.

Proposition 2.27. Let τ ∈ 𝕋^{2}, G a neighborhood of τ, and f ∈ Hol(G). If f is locally toral at τ, then one of the following holds:

(i) f is locally sidal at τ. (ii) f is locally asidal at τ. (iii) There exists a neighborhood U of τ in G and essentially unique f_{1} and

f_{2} ∈ Hol(U) such that f_{1} is locally toral and sidal at τ, f_{2} is locally toral and asidal at τ, and f|U = f_{1}f_{2}.

As the following examples show, all three cases mentioned in Proposition 2.27 can occur.

Example 2.28. If β ∈ 𝕋, p(z, w) = w − βz or p(z, w) = zw − β, then p is both locally toral and locally asidal at each point in Z_{p} ∩ 𝕋^{2}.

Example 2.29. Let α ∈ 𝔻\{0}, q_{α}(z, w) = (1 − αz)zw − (z − α) and r_{α}(z) = \frac{z−α}{z(1−αz)}. Clearly, q_{α} is irreducible,

Z_{q_{α}} = {(z, r_{α}(z)) : z ∈ ℂ\{0, 1/α}} ,

and Z_{q_{α}} does not have any singular points. Since r_{α}(z) ∈ 𝕋 whenever z ∈ 𝕋, Theorem 2.8 implies q_{α} is locally toral at each point in Z_{q_{α}} ∩ 𝕋^{2}. Now if a_{α} = \frac{1}{α} [|α|^{2} + i|α|\sqrt{1 − |α^{2}|}] and b_{α} = \frac{1}{α} [|α|^{2} − i|α|\sqrt{1 − |α^{2}|}], then a_{α}, b_{α} ∈ 𝕋 and for τ_{1} ∈ 𝕋, r′_{α}(τ_{1}) = 0 if and only if τ_{1} = a_{α} or τ_{1} = b_{α}. In these cases, r_{α} is not one-to-one on any neighborhood of τ_{1} and so p is locally toral and locally sidal at (a_{α}, r_{α}(a_{α})) and at (b_{α}, r_{α}(b_{α})).

Example 2.30. Let α ∈ 𝔻\{0} and q_{α}, r_{α} and a_{α} be as in Example 2.29. For an appropriate choice at β ∈ 𝕋, (a_{α}, r_{α}(a_{α})) is a zero of w − βz. It is easy to see that if ζ = (a_{α}, r_{α}(a_{α})), then w − τ_{1}z is locally asidal at ζ and q_{α} is locally sidal at ζ and (w − βz)q_{α}(z, w) is neither locally sidal nor locally asidal at ζ.

It would appear that the local geometry of sidal toral points can be quite complex. However, much can be said about the geometry of asidal points. If f is locally asidal at τ, then Proposition 2.19 implies that there exists a neighborhood U of τ in G such that Z_{f} ∩ U ∩ 𝕊 = ∅, i.e.,

(2.31)  Z_{f} ∩ U ⊆ (𝔻 × 𝔻) ∪ (𝔻 × 𝔼) ∪ (𝔼 × 𝔻) ∪ (𝔼 × 𝔼) ∪ 𝕋^{2}.

Two special cases of (2.31) would be:

(2.32)  Z_{f} ∩ U ⊆ (𝔻 × 𝔻) ∪ (𝔼 × 𝔼) ∪ 𝕋^{2}

and

(2.33)  Z_{f} ∩ U ⊆ (𝔻 × 𝔼) ∪ (𝔼 × 𝔻) ∪ 𝕋^{2}.

In light of these two special possibilities we make the following definition.

Definition 2.34. Let τ ∈ 𝕋^{2}, G a neighborhood of τ and f ∈ Hol(G) with f(τ) = 0. We say f is locally inner (respectively, locally outer) at τ if there exists a neighborhood U of τ in G such that (2.32) (respectively, (2.33)) holds.

The following theorem together with Theorems 1.2 and 1.11 show that if f is locally toral at τ, f is nondegenerate at τ and τ is not a singular point of Z_{f}, then either f is locally inner at τ, f is locally outer at τ or f is locally sidal at τ. Moreover, we can determine which it is using the angular derivative, the definition of which we now recall. Let τ = (τ_{1}, τ_{2}) ∈ 𝕋^{2}, G_{0} a neighborhood of τ_{1} and r ∈ Hol(G_{0}) be such that r(z) ∈ 𝕋 whenever z ∈ G_{0} ∩ 𝕋. Since

i\frac{d}{dθ} Arg(r(e^{iθ})) = \frac{d}{dθ} log(r(e^{iθ})) = \frac{1}{r(e^{iθ})} \frac{d}{dθ} r(e^{iθ}) = \frac{r′(e^{iθ})e^{iθ}i}{r(e^{iθ})} ,

the angular derivative of r at τ is

A_{r}(τ) = \frac{τ_{1}r′(τ_{1})}{r(τ_{1})} .

Now, if f is a holomorphic function, f is locally toral at τ, and τ is not a singular point of Z_{f}, then using Theorems 1.2 and 1.11 and the following lemma can be used to determine if f is locally inner at τ, is locally outer at τ or is locally sidal at τ.

Lemma 2.35. Let τ_{1} ∈ 𝕋, G_{0} a neighborhood of τ_{1}, and r ∈ Hol(G_{0}) be such that r(z) ∈ 𝕋 whenever z ∈ 𝕋 ∩ G_{0} and P (z, w) = (z, w) = w − r(z). Then the following hold.

(i) If A_{r}(τ_{1}) > 0, then P is locally inner at (τ_{1}, r(τ_{1})). (ii) If A_{r}(τ_{1}) < 0, then P is locally outer at (τ_{1}, r(τ_{1})). (iii) If A_{r}(τ_{1}) = 0, then r is not one-to-one on any neighborhood of τ_{1} and

so P is locally sidal at (τ_{1}, r(τ_{1})).

Example 2.36. If p(z, w) = w − z, then p is locally inner at each point of Z_{p} ∩ 𝕋^{2}. If q(z, w) = zw − 1, then q is locally outer at each ponit of Z_{q} ∩ 𝕋^{2}.

Example 2.37. Let α ∈ 𝔻\{0} and q_{α}, r_{α}, a_{α} and b_{α} be as in Example 2.29. If z_{0} ∈ ℂ\{0, 1/α} and G_{0} is a connected neighborhood of z_{0} in ℂ\{0, 1/α}, then Q(z, w) = w − r_{α}(z) is the Weierstrass polynomial of q_{α} at (z_{0}, r_{α}(z_{0})) over G_{0}. Since A_{r_{α}}(\frac{|α|}{α}) > 0, A_{r_{α}}( − \frac{|α|}{α}) < 0, A_{r_{α}}(a_{α}) = A_{r_{α}}(b_{α}) = 0 and A_{r_{α}}(τ_{1}) ≠ 0 whenever τ ∈ 𝕋\{a_{α}, b_{α}}, Lemma 2.35 implies q_{α} is locally inner at each point in the arc of 𝕋\\{a_{α}, b_{α}\} which contains \frac{|α|}{α} and q_{α} is locally outer at each point in the arc of 𝕋\\{a_{α}, b_{α}\} which contains \frac{|α|}{α}.

Proposition 2.38. Let τ ∈ 𝕋^{2}, G a neighborhood of τ and f ∈ Hol(G) with f(τ) = 0. If f is locally inner at τ or f is locally outer at τ, then f is locally toral at τ.

Proof. If f is either locally inner or locally outer at τ, then f is locally asidal at τ and, therefore, f is locally toral at τ. □

Theorem 2.39. Let τ ∈ 𝕋^{2}, G a neighborhood of τ and f ∈ Hol(G). If f is locally asidal at τ, then there exist a neighborhood U of τ in G and f_{1}, f_{2} ∈ Hol(U) such that f|U = f_{1}f_{2}, f_{1} is locally inner at τ and f_{2} is locally outer at τ. The factorization is unique in the following sense: if g_{1}, g_{2} ∈ Hol(U), f|U = g_{1}g_{2}, g_{1} is locally inner at τ and g_{2} is locally outer at τ, then there exists u ∈ Hol(U) such that g_{1} = uf and Z_{u} = ∅.

Proof. The uniqueness assertion follows from Definition 2.34 and Theorem 1.5.

It suffices to show that if f is locally asidal at τ and locally irreducible at τ, then f is either locally inner at τ or locally outer at τ. Suppose f is locally asidal at τ and locally irreducible at τ. By Proposition 2.2, the definition of locally irreducible, the comments preceding (1.13) and the fact that each singular point of Z_{f} is isolated in Z_{f}, there exists a connected neighborhood G_{0} of τ_{1}, a neighborhood D_{0} of τ_{2}, and P ∈ P(G_{0}) such that G_{0} × D_{0} ⊆ G and P is the Weierstrass polynomial of f at τ over G_{0}, Z_{P} ∩ (G_{0} × D_{0}) = Z_{P} ∩ (G_{0} × ℂ), S_{P} (G_{0}, D_{0}) ⊂ {τ}, and P |U is irreducible whenever U is a connected neighborhood of τ in G_{0}. Thus, by Proposition 1.12, Z_{P} \{τ} is connected.

Since S_{P} (G_{0}, D_{0}) ⊂ {τ}, △_{P} (z) ≠ 0 whenever z ∈ G_{0}\{τ_{1}}. Since △_{P} ∈ Hol(G_{0}) and G_{0} ⊆ ℂ, Z_{△_{P}} is finite and there exists a neighborhood G_{1} of τ_{1} in G_{0} such that Z_{△_{P} |G_{1}} ⊆ {τ_{1}}.

To show that P is either locally inner or locally outer at τ, fix z_{0} ∈ G_{1}\{τ_{1}} and let w_{0} be such that P (z_{0}, w_{0}) = 0. By Theorem 1.11, there exists a neighborhood V of τ_{1} in G_{1} and r ∈ Hol(V) such that r(z_{0}) = w_{0} and P (z, r(z)) = 0 for z ∈ V. Since f is locally asidal at τ, r(z) ∈ 𝕋 for z ∈ V ∩ 𝕋. Furthermore, if r′(z_{1}) = 0 for some z_{1} ∈ V ∩ 𝕋, then there would necessarily exist a point z_{2} near z_{1} in V \𝕋 such that |r(z_{2})| = 1. Since f is locally asidal at τ, no such z_{1} exists and we see that r′ ≠ 0 on V ∩ 𝕋.

Accordingly, either

(2.40)  r(V ∩ 𝔻) ⊆ 𝔻 and r(V ∩ 𝔼) ⊆ 𝔼

or

(2.41)  r(V ∩ 𝔻) ⊆ 𝔼 and r(V ∩ 𝔼) ⊆ 𝔻.

Noting that (2.40) implies that

{(z, r(z)) : z ∈ V } ⊆ (𝔻 × 𝔻) ∪ (𝔼 × 𝔼) ∪ 𝕋^{2}

and that (2.41) implies that

{(z, r(z)) : z ∈ V } ⊆ (𝔻 × 𝔼) ∪ (𝔼 × 𝔻) ∪ 𝕋^{2}

we have shown that if (z_{0}, w_{0}) ∈ (Z_{P} ∩𝕋^{2})\{τ}, then there exists a relatively open neighborhood U_{(z_{0},w_{0})} of (z_{0}, w_{0}) in Z_{P} \{τ} such that exactly one of the following inclusions is obtained:

(2.42)  U_{(z_{0},w_{0})} ⊆ (𝔻 × 𝔻) ∪ (𝔼 × 𝔼) ∪ 𝕋^{2},
(2.43)  U_{(z_{0},w_{0})} ⊆ (𝔻 × 𝔼) ∪ (𝔼 × 𝔻) ∪ 𝕋^{2}.

Now, since the asidality of P at τ implies that if (z_{0}, w_{0}) ∈ Z_{P} \𝕋^{2}, then (z_{0}, w_{0}) is an element of one of the open sets (𝔻 × 𝔻) ∪ (𝔼 × 𝔼) or (𝔻 × 𝔼) ∪ (𝔼 × 𝔻), we see that in fact for every point (z_{0}, w_{0}) in Z_{P} \{τ}, there exists a relatively open neighborhood of (z_{0}, w_{0}) in Z_{P} \{τ} such that either (2.42) or (2.43) holds, but not both. Consequently, since Z_{P} \{τ} is connected, either Z_{P} \{τ} ⊆ (𝔻 × 𝔻) ∪ (𝔼 × 𝔼) ∪ 𝕋^{2} and P is locally inner at τ or Z_{P} \{τ} ⊆ (𝔻 × 𝔼) ∪ (𝔼 × 𝔻) ∪ 𝕋^{2} and P is locally outer at τ. □

Before we consider the global properties of zero sets of globally defined polynomials, we summarize our classification of points τ in the 0-sets of locally irreducible holomorphic functions. Indeed, if τ ∈ 𝕋^{2}, f is holomorphic on a neighborhood G of τ, and f is locally irreducible at τ, then one of the following occurs.

(i) f could be locally toral and locally sidal at τ. (ii) f could be locally inner at τ. (iii) f could be locally outer at τ. (iv) f could be locally atoral and locally sidal at τ.

If τ is not a singular point of Z_{f} and f is locally toral at τ, then we can determine whether (i), (ii) or (iii) above occur via Lemma 2.35.

Example 2.36 gives examples of (ii) and (iii) above.

Example 2.29 gives an example of (i) above.

Proposition 2.21 states that (iv) occurs if f is locally atoral at τ. The polynomial 2 − τ_{1}z − τ_{2}w is locally atoral at τ and therefore is an example of (iv) above.

For α ∈ 𝔻, let B_{α}(z) = \frac{z−α}{1−αz}. If τ_{1} ∈ 𝕋, then B_{α}(τ_{1}) ∈ 𝕋 and

(2.44)  A_{B_{α}}(τ_{1}) = \frac{1 − |α|^{2}}{|1 − αz|^{2}} .

Example 2.45. Let r(z) = \frac{zB_{1/3}(z)}{B_{1/2}(z)},

U = {z ∈ ℂ : |z| < 3/2 and z ≠ 1/2},

and f(z, w) = w − r(z) for (z, w) ∈ U × ℂ. For τ_{1} = cos(t) + i sin(t), t ∈ ℝ,

A_{r}(τ_{1}) = \frac{12(5 − 2 cos(t)) sin^{2} (\frac{t}{2})}{(5 − 3 cos(t))(5 − 4 cos(t))}.

Thus, A_{r}(1) = 0 and A_{r}(τ_{1}) > 0 whenever τ_{1} ∈ 𝕋\{1}. Thus, f is locally sidal at (1, 1) and f is locally inner at each point of (Z_{f} ∩ 𝕋^{2})\{(1, 1)}.

Example 2.46. Let r(z) = \frac{zB_{1/4}(z)}{B_{1/2}(z)},

U = {z ∈ ℂ : |z| < 3/2 and z ≠ 1/2},

and f(z, w) = w − r(z) for (z, w) ∈ U × ℂ. For τ_{1} = cos(t) + i sin(t), t ∈ ℝ,

A_{r}(τ_{1}) = \frac{32 cos^{2}(t) − 144 cos(t) + 109}{(5 − 4 cos(t))(17 − 8 cos(t))}.

Thus, if we set a = arccos (\frac{18−\sqrt{106}}{8}), then f is locally outer at each point

in the set {(e^{it}, r(e^{it})) : −a < t < a}, f is locally inner at each point in the set {(e^{it}, r(e^{it})) : a < t < 2π − a}, and f is locally sidal at (e^{ia}, r(e^{ia})) and (e^{−ia}, r(e^{−ia})).

3. The global geometry of algebraic sets

In this section we shall discuss the various connections between the concepts of torality and atorality for polynomials in ℂ^{2} introduced in [2] and the concepts of local torality and local atorality for holomorphic functions introduced in the previous section. We then shall extend the local concepts of sidality, asidality, inner and outer from the previous section to the context of globally defined algebraic sets in ℂ^{2}.

Theorem 3.1. Let p be an irreducible polynomial. The following are equivalent.

(i) p is toral. (ii) There exists τ ∈ 𝕋^{2} such that p is locally toral at τ. (iii) There exists τ ∈ Z_{p} ∩ 𝕋^{2} such that ▽p(τ) ≠ 0 and Z_{p} is locally sym-

metric at τ.

Proof. We shall show that (i)→(iii)→(ii)→(i). First assume that (i) holds. Since p is toral, Theorem 2.4 of [2] implies that Z_{p} ∩ 𝕋^{2} is infinite. Since p is irreducible, Z_{▽p} is finite. Hence there exists τ ∈ Z_{p} ∩ 𝕋^{2} such that ▽p(τ) ≠ 0. Now, by Proposition 1.5 of [2], Z_{p} is symmetric. Hence Z_{p} is locally symmetric at τ and (iii) holds.

Now assume that (iii) holds. Since ▽p(τ) ≠ 0, τ is not a singular point of Z_{p} and, by Theorems 1.11 and 1.2, there exists a neighborhood G_{0} of τ_{1}, a neighborhood D_{0} of τ_{2} and r ∈ Hol(G_{0}) such that

Z_{f} ∩ (G_{0} × D_{0}) = {(z, r(z)) : z ∈ G_{0}}.

Thus, since Z_{f} is locally symmetric, r(z) ∈ 𝕋 whenever z ∈ 𝕋. By Theorem 2.8, f is locally toral at τ and (ii) holds.

Finally assume that (ii) holds. By Proposition 2.12, Z_{p} ∩ 𝕋^{2} is an infinite set. By Theorem 2.4 of [2], p is not atoral and therefore, since p is irreducible, p is toral. □

Theorem 3.1 cannot be much improved since the following examples illustrate the facts that the zeros of toral polynomials on the torus need not be locally toral (i.e., (i) does not imply (ii) for every τ) and that local symmetry at a singular point need not imply local torality (i.e., (iii) does not imply (ii) without the assumption that ▽p(τ) ≠ 0). To work efficiently with pseudopolynomials of degree 2 in w, we state the following lemma.

Lemma 3.2. Let s, p, r_{1}, r_{2} ∈ ℂ. If P (w) = w^{2} − sw + p = (w − r_{1})(w − r_{2}), then the following hold.

(i) P is a symmetric polynomial if and only if |p| = 1 and sp = s. (ii) If |p| = 1 and |s| < 2, then P has two distinct unimodular roots. (iii) If |p| = 1 and |s| = 2, then P has a double unimodular root. (iv) If |p| = 1 and |s| > 2, then P has two roots r_{1} and r_{2} neither of which

is unimodular and |r_{1}||r_{2}| = 1.

Example 3.3. Let h(x) = ∑_{n=0}^{2} a_{n}e^{inx} be a trigonometric polynomial of

degree 2 such that h(x) > 0 for all real numbers x and there exist numbers a, b, c and d such that 0 < a < b < c < d, h > 4 on [0, a), h < 4 on (b, c), h > 4 on (c, d), h(d) = 4 and h > 4 on (d, 2π]. Let τ_{1} = e^{id}. By Fejer’s Theorem, there exists a polynomial s(z) of degree two such that s(e^{ix}) = h(x). Let p(z) = s(z)/\overline{s(1/z)} and P (z, w) = w^{2} − s(z)w + p(z). Now |s(τ_{1})| = 2 and |s(z)| > 2 whenever z ∈ 𝕋, z is near τ_{1} and z ≠ τ_{1}. Thus, by Lemma 3.2, there exists τ_{2} ∈ 𝕋 such that P (τ_{1}, τ_{2}) = 0 and for a sufficiently small neighborhood G_{0} of τ_{1}, Z_{P} ∩𝕋^{2} ∩(G_{0} ×ℂ) = {τ}. Thus, P is locally atoral at τ. By clearing denominators, we obtain a polynomial of degree (4,2) which is irreducible, toral and is locally atoral at a single point.

Example 3.4. Let q(z, w) = (3z + 1)w^{2} − (z + 3)(3z + 1)w + z(z + 3), s(z) = z + 3 and p(z) = \frac{z(z+3)}{3z+1}. It is easy to show that q is irreducible, q is essentially symmetric (see (2.24)) and

Z_{q} = {(z, w) ∈ ℂ^{2} | z ≠ −1/3 and w^{2} − s(z)w + p(z) = 0}.

If z ∈ 𝕋\{−1}, then |s(z)| > 2. Thus, by Lemma 3.2, Z_{q}∩𝕋^{2} = {(−1, 1)}. In summary, q is irreducible, q is locally atoral at (−1, 1) and Z_{q} is locally symmetric at (−1, 1). Also ▽q(−1, 1) = 0. Thus, condition (iii) of Theorem 3.1. without the ‘▽p(τ) ≠ 0” does not imply condition (i) of Theorem 3.1.

Corollary 3.5. If p ∈ ℂ[z, w], then p is atoral if and only if p is locally atoral at each point in Z_{p} ∩ 𝕋^{2}.

We now globalize Definition 2.18.

Definition 3.6. Let us agree to say that a polynomial p = p(z, w) ∈ ℂ[z, w] is sidal if 𝕊 is determining for Z_{p} and that p is asidal if no nonconstant divisor of p is sidal.

Noting that the logic of Definition 3.6 is parallel to that of Definition 1.2 of [2] (with 𝕋^{2} replaced by 𝕊, we see that the following analogs of Corollaries 2.1 and 2.2 from [2] are obtained.

Proposition 3.7. Let p be a nonzero polynomial in ℂ[z, w]. The following are equivalent.

(i) p is sidal (respectively, asidal). (ii) Each irreducible factor of p is sidal (respectively, asidal). (iii) Every divisor of p is sidal (respectively, asidal).

Proposition 3.8. Let p be a nonzero polynomial in ℂ[z, w]. There exist an essentially unique factorization p = qr with q sidal and r asidal.

Notice that Proposition 3.8 is a global analog of Corollary 2.20 from the previous section.

Now, while it is true that one can prove Propositions 3.7 and 3.8 above by following the arguments from [2], it is also true that the propositions can be deduced from the following simple geometric characterizations on sidal and asidal polynomials.

Theorem 3.9 (cf. Proposition 2.19). If p ∈ ℂ[z, w], then p is asidal if and only if Z_{p} ∩ 𝕊 = ∅.

Theorem 3.10. If p ∈ ℂ[z, w] and p is irreducible, then p is sidal if and only if Z_{p} ∩ 𝕊 ≠ ∅.

The reason that Theorems 3.9 and 3.10 are so much simpler than Theorems 2.4 and 2.7 from [2] is due to the fact that if p ∈ ℂ[z, w], then Z_{p} ∩ 𝕊 does not have isolated points. This fact also yields immediate proofs of the theorems.

Now recall Theorem 2.39 which asserted that if f is locally asidal at τ ∈ 𝕋^{2}, then f has an essentially unique factorization f = f_{1}f_{2} with f_{1} locally inner at τ and f_{2} locally outer at τ. Following Definition 2.34, we introduce the following definition.

Definition 3.11. If p ∈ ℂ[z, w], we say p is inner if Z_{p} ⊂ (𝔻×𝔻)∩(𝔼×𝔼)∩𝕋^{2} and we say p is outer if Z_{p} ⊂ (𝔻 × 𝔼) ∩ (𝔼 × 𝔻) ∩ 𝕋^{2}.

Every polynomial p ∈ ℂ[z, w] which is inner or outer is toral as can be seen by the following proposition.

Proposition 3.12. If p ∈ ℂ[z, w] and p is asidal, then p is toral.

Notice that the product of an inner polynomial and outer polynomial is asidal. The converse is true and is shown in the following theorem.

Theorem 3.13. Let p(z, w) ∈ ℂ[z, w]. If Z_{p} ∩ 𝕊 = ∅, then there exists p_{1}, p_{2} ∈ ℂ[z, w] such that p = p_{1}p_{2}, p_{1} is inner and p_{2} is outer.

Proof. We need only show that if p is irreducible and Z_{p} ∩ 𝕊 = ∅, then p is either inner or outer. By Corollary 2.1 in [2], it suffices to show that if p is irreducible and Z_{p} ∩ 𝕊 = ∅, then p is toral. Suppose p is irreducible and Z_{p} ∩ 𝕊 = ∅. Since p is irreducible, △_{p} is not identically zero. Since △_{p} ∈ ℂ[z], Z_{△_{p}} is finite and Z_{p}\S_{p}(ℂ, ℂ) is connected. Let U_{1} and U_{2} be the sets

U_{1} = {(z, w) ∈ Z_{p} : there exists a neighborhood V of (z, w)
such that Z_{p} ∩ V ⊆ (𝔻 × 𝔻) ∪ (𝔼 × 𝔼) ∪ 𝕋^{2}}

and

U_{2} = {(z, w) ∈ Z_{p} : there exists a neighborhood V of (z, w)
such that Z_{p} ∩ V ⊆ (𝔻 × 𝔼) ∪ (𝔼 × 𝔻) ∪ 𝕋^{2}}.

By Theorem 2.39, Z_{p} ∩ 𝕋^{2} ⊆ U_{1} ∪ U_{2}. Clearly,

Z_{p} ∩ (𝔻 × 𝔻) ⊆ U_{1},  Z_{p} ∩ (𝔻 × 𝔼) ⊆ U_{2},
Z_{p} ∩ (𝔼 × 𝔻) ⊆ U_{2}  Z_{p} ∩ (𝔼 × 𝔼) ⊆ U_{1}.

Since Z_{p} ∩ 𝕊 = ∅,

Z_{p} = (Z_{p}∩(𝔻×𝔻))∪(Z_{p}∩(𝔻×𝔼))∪(Z_{p}∩(𝔼×𝔻))∪(Z_{p}∩(𝔼×𝔼))∪(Z_{p}∩𝕋^{2}).

Thus Z_{p} = U_{1} ∪ U_{2}, U_{1} ∩ U_{2} = ∅ and both U_{1} and U_{2} are open. Thus U_{1} = ∅ and p is outer or U_{2} = ∅ and p is inner. □

We close this section by remarking that every polynomial in ℂ[z, w] is a product of irreducible polynomials in ℂ[z, w]. Here is a summary of the types of irreducible polynomials which we have encountered.

Example 3.14. Example 3.3 gives an example of a toral polynomial which is locally atoral at some point.

Example 3.15. Let q(z, w) = (az + 1)w^{2} − (z + a)(az + 1)w + z(z + a). In the case that a = 3, Example 3.4 shows that Z_{q} ∩ 𝕋^{2} is a singleton set, q is atoral and q is symmetric.

Example 3.16. If a > 3 and q(z, w) = (az+1)w^{2}−(z+a)(az+1)w+z(z+a), then Z_{q} ∩ 𝕋^{2} = ∅, q is atoral and q is symmetric.

538 Jim Agler, John E. McCarthy and Mark Stankus

Example 3.17. Recall that for α ∈ 𝔻, we set B_{α}(z) = \frac{z−α}{1−αz}. Let

r(z) = \frac{zB_{2/5}(z)}{B_{1/2}(z)},

U = {z ∈ ℂ : |z| < 3/2 and z ≠ 1/2}, and f(z, w) = w − r(z) for (z, w) ∈ U × ℂ. For t ∈ ℝ and τ_{1} = cos(t) + i sin(t),

A_{r}(τ_{1}) = \frac{80 cos^{2}(t) − 240 cos(t) + 163}{(5 − 4 cos(t))(29 − 20 cos(t))}.

Thus, A_{r}(τ_{1}) > 0 for all τ_{1} ∈ 𝕋 and, by Lemma 2.35, f is locally inner and locally asidal at each point in Z_{f} ∩ 𝕋^{2}. For z near 1/2, r(z) > 1 and therefore, f is not inner since f is sidal. Hence, locally inner does not imply inner and locally asidal does not imply asidal.

References

[1] Agler, Jim; McCarthy, John E. Distinguished varieties. Acta Math. 194(2005) 133–153. MR2231339 (2007c:47006).

[2] Agler, Jim; McCarthy, John E.; Stankus, Mark. Toral algebraic sets and function theory on polydisks. J. Geometric Anal. 16 (2006) 551–562. MR2271943 (2007j:32002).

[3] Gunning, Robert C.; Rossi, Hugo. Analytic functions of several complex variables. Prentice-Hall Series in Modern Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1965. xiv+317 pp. MR0180696 (31 #4927), Zbl 0141.08601.

[4] Rudin, Walter. Function theory in polydiscs. Mathematics Lecture Note Series. W. A. Benjamin, Inc., New York, 1969. vii+188 pp. MR0255841 (41 #501), Zbl 0177.34101.

U.C. San Diego, La Jolla, CA 92093 jagler@ucsd.edu

Washington University, St. Louis, MO 63130 mccarthy@math.wustl.edu

California Polytechnic State University, San Luis Obispo, CA 93407 mstankus@calpoly.edu

This paper is available via http://nyjm.albany.edu/j/2008/14-24.html.