polydisk

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 form

:$D(z_1,r_1) \times \dots \times D(z_n,r_n).$

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 Lie groups.Poincare, H,Les fonctions analytiques de deux variables et la r?epresentation 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.

References

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{{PlanetMath attribution, urlname=polydisc, title=polydisc

Several complex variables