I am reading some lecture notes on holomorphic functions of several complex variables, see page 105. The part I am struggling with is a proof by Wermer
I have asked about runge domains, and polynimial convex domains earlier. See this link for reference. I am strugling with the following parts of the proof:
- Is $\mathbb{T}$ the boundary of the disk, or is it the torus? How can i visualize $(\mathbb{T}) \times \{ (0,\ldots\,0)\}\subset D$?
- Why does it follow from the maximum principle that $\overline{\mathbb{D}} \times \{ 0 , \ldots , 0 )\} \subset \hat{K}_n$? (Does this define the boundary of the surface?
- Lastly how does it follow that $(1/2,0,\ldots,0)$ lies in $\overline{\mathbb{D}} \times \{ 0 , \ldots , 0 )\}$?
Also it seems this is an example to biholomorphic image to polydisc, which fails to be polynomically convex. However I do not quite see how it follows that it is not polynomically convex, even though it is not a runge domain.
Check the book for notation, but from the above, it's clear that $\mathbb{T} = \partial\mathbb{D}$
If $p$ is a polynomial in $n $ variables, then $q(z_1) = p(z_1,0,0,\ldots,0)$ is a polynomial in $z_1$, so if $w\in \bar{\mathbb{D}}$, then $$|q(w)| \le \max_{\mathbb{T}} |q| = \max_{\mathbb{T}\times(0,\ldots,0)} |p|$$ Hence $\bar{\mathbb{D}}\times(0,\ldots,0) \subset \hat K$.
$1/2 \in \bar{\mathbb{D}}$