Polydisc Explained

See also: Duocylinder.

In the theory of functions of several complex variables, a branch of mathematics, a polydisc is a Cartesian product of discs.

More specifically, if we denote by

D(z,r) the open disc of center z and radius r in the complex plane, then an open polydisc is a set of the formD(z _{1,r} _{1)} x ... x D(z _{n,r} _{n).}

It can be equivalently written as

\{ w = (w _{1,} w _{2,} ... , w _{n)} \in { C }^n : \vert z_k - w_k \vert < r_k, \mbox k = 1,\dots,n \}.

One should not confuse the polydisc with the open ball in Cⁿ, which is defined as

\{ w \in C ^{n} : \lVert z - w \rVert < r \}.

Here, the norm is the Euclidean distance in Cⁿ.

When

n > 1 , open balls and open polydiscs are not biholomorphically equivalent, that is, there is no biholomorphic mapping between the two. This was proven by Poincaré in 1907 by showing that their automorphism groups have different dimensions as Lie groups.^[1]

When

n =2 the term bidisc is sometimes used.

A polydisc is an example of logarithmically convex Reinhardt domain.

References

• Book: Steven G Krantz . Function Theory of Several Complex Variables . American Mathematical Society . Jan 1, 2002 . 0-8218-2724-3.
• Book: John P D'Angelo, D'Angelo P D'Angelo . Several Complex Variables and the Geometry of Real Hypersurfaces . CRC Press . Jan 6, 1993 . 0-8493-8272-6.

Notes and References

1. Poincare, H, Les fonctions analytiques de deux variables et la representation conforme, Rend. Circ. Mat. Palermo23 (1907), 185-220