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 $:

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

References

[a1]	W. Rudin, "Function theory in polydiscs" , Benjamin (1969)