# Polydisc

polycylinder

A region

$$\Delta = \Delta ( a = ( a _ {1} \dots a _ {n} ),\ r = ( r _ {1} \dots r _ {n} )) =$$

$$= \ \{ z = ( z _ {1} \dots z _ {n} ) \in \mathbf C ^ {n} : | z _ \nu - a _ \nu | < r _ \nu , \nu = 1 \dots n \}$$

in a complex space $\mathbf C ^ {n}$, $n \geq 1$, which is the topological product of $n$ discs

$$\Delta = \Delta _ {1} \times \dots \times \Delta _ {n} ,$$

$$\Delta _ \nu = \{ z _ \nu \in \mathbf C : | z _ \nu - a _ \nu | < r _ \nu \} ,\ \nu = 1 \dots n .$$

The point $a = ( a _ {1} \dots a _ {n} ) \in \mathbf C ^ {n}$ is the centre of the polydisc $\Delta$, $r = ( r _ {1} \dots r _ {n} )$, $r _ \nu > 0$, $\nu = 1 \dots n$, is its polyradius. With $a = 0$, $r = ( 1 \dots 1 )$ one obtains the unit polydisc. The distinguished boundary of $\Delta$ is the set

$$T = T( a, r) = \{ {z \in \mathbf C ^ {n} } : { | z _ \nu - a _ \nu | = r _ \nu , \nu = 1 \dots n } \} ,$$

which is a part of its complete topological boundary $\partial \Delta$. A polydisc is a complete Reinhardt domain.

A natural generalization of the concept of a polydisc is that of a polyregion (polycircular region, generalized polycylinder) $D = D _ {1} \times \dots \times D _ {n}$, which is the topological product of, in general multiply-connected, regions $D _ \nu \subset \mathbf C$, $\nu = 1 \dots n$. The boundary $\Gamma = \partial D$ of a polyregion $D$ consists of $n$ sets of dimension $2n - 1$:

$$\Gamma _ \nu = \ \{ {z \in \mathbf C ^ {n} } : {z _ \nu \in \partial D _ \nu ,\ z _ \mu \in \overline{D}\; _ \mu , \mu \neq \nu } \} ,\ \ \nu = 1 \dots n,$$

the common part of which is the $n$- dimensional distinguished boundary of $D$:

$$T = \partial D _ {1} \times \dots \times \partial D _ {n} = \ \{ {z \in \mathbf C ^ {n} } : { z _ \nu \in \partial D _ \nu , \nu = 1 \dots n } \} .$$