multidisk

Multidisk, in mathematics, is commonly known as the polydisk. It denotes the Cartesian product of n copies of the unit disk in the complex plane. Formally, D^n = D × D × ... × D, where D = { z ∈ C : |z| < 1 }. More generally, one can consider scaled polydisks D_r^n = { z ∈ C^n : |zi| < ri for i = 1,...,n }.

As a subset of complex n-space, D^n is a bounded, open domain in C^n. It is a standard example in several complex variables and complex geometry, and it is a Reinhardt domain, meaning it is invariant under separate rotations in each coordinate. It is not biholomorphic to the unit ball when n > 1, reflecting distinct geometric and function-theoretic properties from the ball.

The automorphism group of the polydisk is generated by independent automorphisms of each unit disk (Möbius transformations preserving D) together with the possibility of permuting coordinates when appropriate. Function theory on the polydisk often reduces to considering each variable separately, though the boundary geometry and holomorphic function behavior still present rich structures.

Boundary and convexity: The topological boundary ∂D^n consists of points where at least one coordinate lies on the unit circle. The distinguished boundary is the torus T^n. The polydisk is pseudoconvex and serves as a prototypical example of domains of holomorphy in several complex variables.

Other uses: the term multidisk can occasionally refer to multiple physical disks or a disk array in computing contexts. This article focuses on the mathematical concept commonly called the polydisk.