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

TL;DR

This paper investigates polynomial convexity and approximation properties of certain complex sets with non-isolated CR-singularities, establishing conditions under which these sets are polynomially convex and dense in continuous functions.

Contribution

It introduces new conditions ensuring polynomial convexity and approximation for graphs with non-isolated CR-singularities, extending classical results to more complex sets.

Findings

Graphs are polynomially convex under certain conditions.

Holomorphic polynomials approximate all continuous functions on these graphs.

The algebra generated by specific functions is dense in continuous functions on the polydisc.

Abstract

In this paper, we first consider the graph of $(F_1,F_{2},\cdots,F_{n})$ on $\overline{\mathbb{D}}^{n},$ where $F_{j}(z)=\bar{z}^{m_{j}}_{j}+R_{j}(z),j=1,2,\cdots,n,$ 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.