Polynomial Convexity and Polynomial approximations of certain sets in $\mathbb{C}^{2n}$ with non-isolated CR-singularities

In this paper, we first consider the graph of $(F_1,F_{2},\cdots,F_{n})$ on $\overline{\mathbb{D}}^{n},$ where $F_{j}(z)=\bar{z}^{m_{j}}_{j}+R_{j}(z),j=1,2,\cdots,n,$ 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 $\{(z^{m_{1}}_{1},\cdots, z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}} + R_{1}(z),\cdots, \bar{z_{n}}^{m_{2n}} + R_{n}(z)):z\in \overline{D},m_{j}\in \mathbb{N}, j=1,\cdots,2n\}$ is polynomially convex; and if $\gcd(m_{j},m_{k})=1~~\forall j\not=k,$ the algebra generated by the functions $z^{m_{1}}_{1},\cdots, z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}} + R_{1},\cdots, \bar{z_{n}}^{m_{2n}} + R_{n}$ is dense in $\mathcal{C}(\overline{D}).$ We prove an analogue of Minsker's theorem over the closed unit polydisc, i.e, if $\gcd(m_{j},m_{k})=1~~\forall j\not=k,$ the algebra $[z^{m_{1}}_{1},\cdots, z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}},\cdots , \bar{z_{n}}^{m_{2n}};{\overline{\mathbb{D}}^{n}} ]=\mathcal{C}(\overline{\mathbb{D}}^{n}).$ In the process of proving the above results, we also studied the polynomial convexity and approximation of certain graphs.