Polydisc

 ( Several Complex Variables ) 

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 set disc of center z and radius r in the complex plane, then an open polydisc is a set of the form

$D(z\_1,r\_1)\; \backslash times\; \backslash dots\; \backslash times\; D(z\_n,r\_n).$

It can be equivalently written as

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

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

When $n=2$ the term bidisc is sometimes used.

A polydisc is an example of logarithmically convex Reinhardt domain.

• Steven G Krantz (2002). 9780821827246, American Mathematical Society. ISBN 9780821827246
• John P D'Angelo, D'Angelo P D'Angelo (1993). 9780849382727, CRC Press. ISBN 9780849382727