---
tag: DJBWP7D3M8
kind: concept
name: Polydisc
msc: [32A10, 32Q02]
sources:
  - url: https://en.wikipedia.org/wiki/Polydisc
same_as:
  wikidata: Q2103196
  wikipedia: Polydisc
---

## Summary

A polydisc in $\C^n$ is a product of $n$ open discs of the complex plane, one in each coordinate.
It is the basic domain of [[several complex variables|BH57D2FBNR]], playing the role that the disc plays in one variable.
The [[Cauchy integral formula on a polydisc|FG5448NZ18]] follows from the one-variable formula applied in each coordinate, and it leads to power series expansions.
Its [[distinguished boundary|A6HN207Y2T]] is a torus, much smaller than its topological boundary.

## Explanations

### L1 human

A polydisc is what one gets by letting each coordinate $z_j$ range over its own disc, independently of the others.
In $\C^2$, the unit polydisc is the set of pairs $(z_1, z_2)$ with $\abs{z_1} < 1$ and $\abs{z_2} < 1$.
It is a complex analogue of a box, or of a square in the plane.
The point $(0.9, 0.9)$ lies in the unit polydisc of $\C^2$, but not in the [[unit ball|PDQ9KW9FAP]], since $0.81 + 0.81 > 1$.