Uniform approximation on certain polynomial polyhedra in $\mathbb{C}^2$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| author | Gorai, Sushil Mondal, Golam Mostafa |
| author_facet | Gorai, Sushil Mondal, Golam Mostafa |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06195 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform approximation on certain polynomial polyhedra in $\mathbb{C}^2$ Gorai, Sushil Mondal, Golam Mostafa Complex Variables 32E30, 32E20, 46J10, 32A65 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. |
| title | Uniform approximation on certain polynomial polyhedra in $\mathbb{C}^2$ |
| topic | Complex Variables 32E30, 32E20, 46J10, 32A65 |
| url | https://arxiv.org/abs/2404.06195 |