Local geometry of zero sets of holomorphic functions near the torus
Let \(\mathbb D\) denote the unit disk in the complex plane, \(\mathbb T\) be a unit circle, \(\mathbb E\) be the complement of the closed unit disk in \(\mathbb C\) and let \(A(\mathbb D^n)\) denote the polydisk algebra, the algebra of functions that are continuous on the closure of \(\mathbb D^n\) and holomorphic on the interior. When studying function theory on the polydisk \(\mathbb D^n\), it is often useful to focus on the torus \(\mathbb T^n\), which is the distinguished boundary of \(\mathbb D^n\). In several important ways, the behavior of a function in \(A(\mathbb D^n)\) is controlled by its behavior on \(\mathbb T^n\): not only is \(\mathbb T^n\) a set of uniqueness, but every function in the algebra attains its maximum modulus on \(\mathbb T^n\). We say that a variety (an irreducible algebraic set) \(V\) is toral if its intersection with \(\mathbb T^n\) is fat enough to be a determining set for holomorphic functions on \(V\). Otherwise we call the variety atoral. A polynomial in \(\mathbb C[z_1,\dots,z_n]\) is toral (respectively, atoral) if the zero set of every irreducible factor is toral (respectively, atoral). Earlier the authors shown that knowing an algebraic set is toral has interesting consequences in function theory and operator theory. In this paper, they study the localized versions of this and other related geometric properties of zero sets. Let \(f\) be a holomorphic function defined on an open subset \(G\) of \(\mathbb C^n\) and suppose \(\tau\in\mathbb T^n\cap G\). We say \(f\) is locally toral at \(\tau\) if \(f(\tau) = 0\) and, for every neighborhood \(U\) of \(\tau \in G\), there is a neighborhood \(V\) of \(\tau \in U\) such that \(Z_f\cap V\cap \mathbb T^n\) is determining for \(Z_f\cap V\). 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 \(\tau\), then we say \(f\) is locally atoral at \(\tau\) if \(f(\tau) = 0\) and \(f\) is not locally toral at \(\tau\). 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. Let \(n = 2\), \(f\) be a holomorphic function defined on an open subset \(G\) of \(\mathbb C^2\) and suppose \(\tau\in\mathbb T^2\cap G\). A function \(\varphi\) which is holomorphic on an open neighborhood of the closed unit disk is inner if \(|\varphi(\alpha)| = 1\) for all \(\alpha\in\mathbb D\) and therefore the zero set of the function \(w - \varphi(z)\) lies in \((\mathbb D\times\mathbb D)\cup (\mathbb E \times \mathbb E)\cup\mathbb T^2\). Define \(f\) to be locally inner at \(\tau\) if \(f(\tau) = 0\) and its zero set near \(\tau\) lies in \((\mathbb D\times\mathbb D)\cup (\mathbb E \times \mathbb E)\cup\mathbb T^2\) and locally outer at \(\tau\) if \(f(\tau) = 0\) and its zero sets near \(\tau\) lies in \((\mathbb D\times\mathbb E)\cup (\mathbb E\times\mathbb D)\cup\mathbb T^2\). Note that if \(f\) is locally inner (or outer) at \(\tau\), then the zero set of \(f\) does not intersect \(\mathbb S\), where \(\mathbb S\) is the set of points \((z,w)\in\mathbb C^2\) where either \(z\) or \(w\) (but not both) is unimodular. With this notation, \(\mathbb C^2\) is the disjoint union of \(\mathbb T^2\), \(\mathbb S\) and the four open ``quadrants'' in \(\mathbb C^2\) determined by the sets \(\mathbb D\) and \(\mathbb E\), namely, \(\mathbb D\times\mathbb D\), \(\mathbb E \times \mathbb D\), \(\mathbb D \times\mathbb E\), and \(\mathbb E\times\mathbb E\). Any curve is \(\mathbb C^2\) which intersects two or more of the above quadrants does so by passing through either \(\mathbb S\) or \(\mathbb T^2\). Replacing \(\mathbb T^2\) with \(\mathbb S\) in the definitions of toral, atoral, locally toral and locally atoral results in these definitions of sidal, asidal, locally sidal and locally asidal, respectively. If \(f\) is locally inner (or locally outer) at \(\tau\), then \(f\) is locally asidal at \(\tau\). 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''), the authors find that every nonzero atoral polynomial \(p(z,w)\) is a product of an inner polynomial and an outer polynomial. A summary of the types of irreducible polynomials is given.
polydisk algebra of holomorphic functions
local geometry of analytic sets near torus
scientific article; zbMATH DE number 5377300