I have trouble defining the biggest centered polydisc of holomorphy (where I can apply cauchy's inequality) of a multivariate complex holomorphic function. As an example, suppose a function of 2 variables with 3 poles:
$$f(z_1,z_2) = \frac{1}{\left(1-\frac{z_1}{2}\right)\left(1-\frac{z_2}{2}\right)\left(1-z_1z_2\right)}$$
The function has singularities:
- On $\left\{(z_1,z_2) \in \mathbb C^2:\, z_1 = 2\right\}$
- On $\left\{(z_1,z_2) \in \mathbb C^2:\, z_2 = 2\right\}$
- On $\left\{(z_1,z_2) \in \mathbb C^2:\, z_1 = \frac{1}{z_2}\right\}$
I have two questions:
- For each of the three singularities, if only this one occured, what would be the polyradius of the biggest polydisc of holomorphy ?
- What is the polyradius of the biggest polydisc of holomorphy for the full function ?
The OP asks for the largest polydisk contained in a certain set, as though the existence of a largest polydisk was clear. It's been commented that the existence of a largest polydisk is not at all clear to at least one reader, while of course a maximal polydisk is much more plausible. When the OP replies "if you prefer to call [largest] [maximal] be my guest" it seems like an example clarifying the difference may be a good idea. $\newcommand{\R}{\mathcal R}$ Say $D=\{(x,y):x,y\ge0,xy\le1\}$. Say a rectangle is a set of the form $[a,b]\times[c,d]$, let $\R$ be the set of all rectangles contained in $D$, and let $R_0=[0,1]\times[0,1]$.
Then $R_0$ is not the largest element of $\R$, because for example if $R_1=[0,2]\times[0,1/2]$ then $R_1\in\R$ but $R_1\not\subset R_0$.
But $R_0$ is a maximal element of $\R$; this means that, as you can easily verify, if $R_1\in\R$ and $R_0\subset R_1$ then $R_0=R_1$.