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 $(F_1,F_2,\cdots ,F_n)$ on ${\overline{\mathbb{D}}}^n,$ where $F_j(z)={\bar{z}}_j^{m_j}+R_j(z),j=1,2,\cdots ,n,$ which has non-isolated CR-singularities if $m_j>1$ for some $j\in \{1,2,\cdots ,n\}.$ We show that under certain condition on $R_j,$ the graph is polynomially convex and holomorphic polynomials on the graph approximates all continuous functions. We also show that there exists an open polydisc $D$ centred at the origin such that the set $\{(z_1^{m_1},\cdots ,z_n^{m_n},{\overline{z_1}}^{m_{n+1}}+R_1(z),\cdots ,{\overline{z_n}}^{m_{2n}}+R_n(z)):z\in \overline{D},m_j\in \mathbb{N},j=1,\cdots ,2n\}$ is polynomially convex; and if $\gcd (m_j,m_k)=1\forall j\ne k,$ the algebra generated by the functions $z_1^{m_1},\cdots ,z_n^{m_n},{\overline{z_1}}^{m_{n+1}}+R_1,\cdots ,{\overline{z_n}}^{m_{2n}}+R_n$ is dense in $\mathcal{C}(\overline{D}).$ We prove an analogue of Minsker's theorem over the closed unit polydisc, i.e, if $\gcd (m_j,m_k)=1\forall j\ne k,$ the algebra $[z_1^{m_1},\cdots ,z_n^{m_n},{\overline{z_1}}^{m_{n+1}},\cdots ,{\overline{z_n}}^{m_{2n}};{\overline{\mathbb{D}}}^n]=\mathcal{C}({\overline{\mathbb{D}}}^n).$ In the process of proving the above results, we also studied the polynomial convexity and approximation of certain graphs.