# Analytic plane

complex-analytic plane

A non-empty set of points $z = ( z _ {1} \dots z _ {n} )$ in the complex vector space $\mathbf C ^ {n}$ that satisfies a system of equations

$$\sum _ {i = 1 } ^ { n } a _ {i j } z _ {i} = b _ {j} , \ j = 1 \dots k ; \ a _ {i j } , b _ {j} \in \mathbf C ;$$

$$\mathop{\rm rank} \| a _ {i j } \| = k < n .$$

The number $k$ is called the complex codimension, while $n - k$ is called the complex dimension of the analytic plane. The real dimension of the analytic plane equals $2 (n - k )$ and is even, but not all even-dimensional real planes in $\mathbf R ^ {2n} = \mathbf C ^ {n}$ are analytic planes. Complex one-dimensional analytic planes are sometimes called complex, or analytic, straight lines. See also Analytic surface.

How to Cite This Entry:
Analytic plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_plane&oldid=45178
This article was adapted from an original article by E.D. SolomentsevE.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article