Polydisc
Encyclopedia
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
the open
disc of center z and radius r in the complex plane
, then an open polydisc is a set of the form
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
, 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 group
s.
When
the term bidisc is sometimes used.
A polydisc is an example of logarithmically convex Reinhardt domain
.
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a polydisc is a Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of discs.
More specifically, if we denote by
the openOpen set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
disc of center z and radius r in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
, then an open polydisc is a set of the form
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
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
is the Euclidean distance
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...
in Cn.
When
, open balls and open polydiscs are not biholomorphically equivalent, that is, there is no biholomorphic mapping between the two. This was proven by PoincaréHenri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
in 1907 by showing that their automorphism groups have different dimensions as Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s.
When
the term bidisc is sometimes used.A polydisc is an example of logarithmically convex Reinhardt domain
Reinhardt domain
In mathematics, especially several complex variables, an open subset G of Cn is called Reinhardt domain if \in G implies \in G for all real numbers \theta_1, \dots, \theta_n....
.