---
tag: A6HN207Y2T
kind: concept
name: Distinguished boundary of a polydisc
msc: [32A10, 32A26]
specializes: [JHQNT58FEH]
sources:
  - url: https://en.wikipedia.org/wiki/Polydisc
---

## Summary

The distinguished boundary of a [[polydisc|DJBWP7D3M8]] $P(a, r)$ in $\C^n$ is the torus of points with $\abs{z_j - a_j} = r_j$ for every $j$.
It is the set over which the [[Cauchy integral formula on a polydisc|FG5448NZ18]] integrates.
A function holomorphic on the polydisc and continuous on its closure attains its largest modulus on this torus.
It is the [[Shilov boundary|JHQNT58FEH]] of the algebra of such functions, and for $n \ge 2$ it is a small part of the topological boundary.

## Explanations

### L1 human

The boundary of the unit polydisc in $\C^2$ is the set where $\abs{z_1} = 1$ or $\abs{z_2} = 1$.
The distinguished boundary is the smaller set where both $\abs{z_1} = 1$ and $\abs{z_2} = 1$.
It is a product of two circles, a torus of real dimension $2$, while the full boundary has real dimension $3$.
For example, $f(z_1, z_2) = z_1 z_2$ has modulus at most $1$ on the closed polydisc, and it reaches $1$ exactly on the distinguished boundary.
So, for holomorphic functions, this thin torus already sees the maximum.