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

TL;DR

This paper extends a dichotomy in polynomial approximation within certain complex polyhedra in a2^2, showing conditions under which uniform approximation either recovers all continuous functions or is constrained by algebraic varieties.

Contribution

It generalizes the Wermer maximality theorem analogue for polynomial polyhedra in a2^2, providing new criteria for uniform approximation and computing polynomial hulls in specific cases.

Findings

Either the uniform algebra equals all continuous functions on the boundary

Existence of algebraic varieties where functions are holomorphic

Computed polynomial hulls for graphs of pluriharmonic functions

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 $\Omega:=\{z\in\mathbb{C}^2: |p_1(z)|<1, |p_2(z)|<1\}$ such that $\overline{\Omega}$ 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 $\Omega$, is all continuous functions on the distinguished boundary or there exists an algebraic variety in $\overline{\Omega}$ 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.