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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 ?
lrnv
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  • Why do you believe there is such a thing as the biggest polydisk of holomorphy??? – David C. Ullrich Mar 17 '21 at 09:50
  • Well, I do not know. Assuming the function is holomorphic in a neighbourhood of the origin, there should be a polydisc centered at the origin that is contained in this neighbourhood, no ? – lrnv Mar 17 '21 at 12:08
  • yes of course. I don't see what that has to do with a biggest polydisk. A maximal polydisk, certainly. – David C. Ullrich Mar 17 '21 at 19:03
  • I never claimed that my terminology was the good one. If you prefer to call that a maximal polydisc, be my gest, as long as we have the same object in mind ;) However google searchs on maximal polydisk instead of biggest polydisc still doesnt return anything relevant to my question... – lrnv Mar 18 '21 at 08:12
  • ??? The difference between "biggest" and "maximal" is perfectly standard; when I say that although I don't see where there should be a biggest polydisk but there may well be a maximal one that's not because I "prefer to call" it that. I don't see what a google search has to do with this. "as long as we have te same object in mind": Huh? If you have the standard definition in mind when you say "biggest polydisk" then what you have in mind is definitely not the same as what one means by "maximal polydisk". – David C. Ullrich Mar 18 '21 at 09:58
  • The problem is the following to me: I am not skilled enough to defined properly the maximum and/or biggest polydisc. What i wanted was the disc of polyradius $r_1,r_2$ that had the biggest polyradius possible, say w.r.t a given norm in $L2(R^2)$. What do you call a 'maximal' polydisc ? – lrnv Mar 18 '21 at 12:57
  • Btw you just did this thing you've been insisting you're unable to do. The sentence "What I wanted..." is a perfectly precise and clear statement of what you meant by "largest". (yes, there certainly will be largest polydiscs in that sense) – David C. Ullrich Mar 18 '21 at 19:42
  • Thanks for helping me clearing it out :) – lrnv Mar 18 '21 at 22:33

1 Answers1

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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 $R_0=R_1$.

supinf
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David C. Ullrich
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  • Thanks for the definition of 'maximal' that you used. Indeed, it was very unclear to me. Both of these difinitions will work out for my initial problem: behind this question, i want to show that i can apply Cauchy's inequality with radii > 1 for certain functions, but i have trouble defining, from the set of singularities, what are the biggest (and/or maximal) radii that i can use. I hope i'm clearer. – lrnv Mar 18 '21 at 13:00