论文标题
$ \ mathbb {c}^{2n} $中的某些集合的多项式凸度和多项式近似值与非偏度CR-SINGULULITIS
Polynomial Convexity and Polynomial approximations of certain sets in $\mathbb{C}^{2n}$ with non-isolated CR-singularities
论文作者
论文摘要
在本文中，我们首先考虑$（f_1，f_ {2}，\ cdots，f_ {n}）$的图表上的图形$ f_ {j}（z）= \ bar {z}^{m_ {m_ {j}} _ {j}+r_ {j}+r_ {j}（z），j = 1,2，\ cdots，n，$，如果$ m_ {j} $ $ j} $ n，$ n，$ j \ we \ c cd c.1 $ n.在$ r_ {j}的某些条件下，$该图是图形上的多项式凸，图上的多态多项式近似于所有连续函数。我们还表明，存在一个以起源为中心的开放式polydisc $ d $，使得$ \ {（z^{m_ {1}} _ {1} _ {1}，\ cdots，z^{m_ {n}}} _ {n} _ {n} \ bar {z_ {n}}^{m_ {2n}} + r_ {n}（z））：z \ in \ In \ In \ overline {d}，m_ {j} \ in \ mathbb {n}，j = 1，j = 1，j = 1，\ cdots，\ cdots，\ cdn \} $ polynomeline convex;如果$ \ gcd（m_ {j}，m_ {k}）= 1 ~~ \ forall j \ not = k，$ the Algebra由函数$ z^{m_ {1}} _ {1} _ {1}，\ cdots，\ cdots，z^{m_ {m_ {n}} _ {n} _ {n} _ {n}， \ bar {z_1}^{m_ {n + 1}} + r_ {1}，\ cdots，\ bar {z__ {n}}^{m_ {2n}} + r_ {n} $在$ \ mathcal {c}（c} c}（c}（\ edime）中，封闭的单位polydisc，即，如果$ \ gcd（m_ {j}，m_ {k}）= 1 ~~ \ forall j \ not = k，$ the algebra $ [z^{m_ {1}}} _ {1} _ {1} \ bar {z_1}^{m_ {n+1}}}，\ cdots，\ bar {z__ {n}}}^{m_ {2n}}; {\ overline {\ mathbb {d} ] = \ Mathcal {c}（\ overline {\ mathbb {d}}}^{n}）。$在证明上述结果的过程中，我们还研究了某些图的多项式凸性和近似值。
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.
