Polynomial Convexity and Polynomial Approximations of Certain Sets in $$\mathbb {C}^{2n}$$ with Non-isolated CR-Singularities

Abstract

In this paper, we first consider the graph of $$(F_1,F_{2},\ldots ,F_{n})$$ on $$\overline{\mathbb {D}}^{n},$$ where $$F_{j}(z)=\bar{z}^{m_{j}}_{j}+R_{j}(z),j=1,2,\ldots ,n,$$ which has non-isolated CR-singularities if $$m_{j}>1$$ for some $$j\in \{1,2,\ldots ,n\}.$$ We show that under the certain condition on $$R_{j},$$ the graph is polynomially convex, and holomorphic polynomials on the graph approximate all continuous functions. We also show that there exists an open polydisk D centered at the origin such that the set $$\{(z^{m_{1}}_{1},\ldots , z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}} + R_{1}(z),\ldots , \bar{z_{n}}^{m_{2n}} + R_{n}(z)):z\in \overline{D},m_{j}\in \mathbb {N}, j=1,\ldots ,2n\}$$ is polynomially convex, and if $$\gcd (m_{j},m_{k})=1~~\forall j\not =k,$$ the algebra generated by the functions $$z^{m_{1}}_{1},\ldots , z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}} + R_{1},\ldots , \bar{z_{n}}^{m_{2n}} + R_{n}$$ is dense in $$\mathcal {C}(\overline{D}).$$ We prove an analogue of Minsker’s theorem over the closed unit polydisk, i.e., if $$\gcd (m_{j},m_{k})=1~~\forall j\not =k,$$ the algebra $$[z^{m_{1}}_{1},\ldots , z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}},\ldots , \bar{z_{n}}^{m_{2n}};{\overline{\mathbb {D}}^{n}} ]=\mathcal {C}(\overline{\mathbb {D}}^{n}).$$