In the theory of functions of

several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a polydisc is a

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

of discs. More specifically, if we denote by

D(z,r)

the

open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979 * ''Open'' (Go ...

disc of center ''z'' and radius ''r'' in the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

, 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 mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...

in Cⁿ, which is defined as :

\.

Here, the

norm Norm, the Norm or NORM may refer to: In academic disciplines * Normativity, phenomenon of designating things as good or bad * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative e ...

is the

Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...

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 Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philos ...

in 1907 by showing that their

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

s have different dimensions as

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

s.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.

References

* * {{PlanetMath attribution, urlname=polydisc, title=polydisc Several complex variables