Uniform approximation on certain polynomial polyhedra in $\mathbb{C}^2$

TL;DR

The paper addresses extending Wermer’s maximality dichotomy to uniform algebras generated by polynomials and pluriharmonic boundary data on complex non-degenerate polynomial polyhedra in $\mathbb{C}^2$. It develops a framework based on the polynomial convexity of the graph of pluriharmonic data over the distinguished boundary, yielding a dichotomy: either the resulting uniform algebra equals the full boundary algebra $\mathcal{C}(\Gamma_{\mathfrak{D}_2})$ or there exists a nontrivial algebraic variety $V$ inside $\overline{\mathfrak{D}}_2$ where the pluriharmonic extensions become holomorphic. The authors prove this dichotomy using graph-geometry (totally real graphs), Tornehave-type dimension reductions to algebraic varieties, and boundary-regularity arguments, and they describe the polynomial hulls of graphs in cases where the pluriharmonic data are conjugates of holomorphic polynomials. They also extend the results to domains with low boundary regularity via segment-property arguments and provide explicit hull descriptions for graphs under proper polynomial mappings. Overall, the work broadens Wermer-type dichotomies to a natural class of two-variable polynomial polyhedra and supplies concrete hull descriptions and examples that illustrate the interplay between uniform approximation, polynomial convexity, and pluriharmonic extension.

Abstract

In this paper we extend the dichotomy given by Samuelsson and Wold that can be thought of as an analogue of the Wermer maximality theorem in $\mathbb{C}^2$ for certain polynomial polyhedra. We consider complex non-degenerate simply connected polynomial polyhedra of the form $Ω:=\{z\in\mathbb{C}^2: |p_1(z)|<1, |p_2(z)|<1\}$ such that $\overlineΩ$ is compact. Under a mild condition of the polynomials $p_1$ and $p_2$, we prove that either the uniform algebra, generated by polynomials and some continuous functions $f_1,\dots, f_N$ on the distinguished boundary that extends as pluriharmonic functions on $Ω$, is all continuous functions on the distinguished boundary or there exists an algebraic variety in $\overlineΩ$ on which each $f_j$ is holomorphic. We also compute the polynomial hull of the graph of pluriharmonic functions in some cases where the pluriharmonic functions are conjugates of holomorphic polynomials. We also give a couple of general theorem about uniform approximation on the domains with low boundary regularity.

Theorems & Definitions (67)

• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Remark 1.9
• Remark 1.10
• Remark 1.11