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Difference between revisions of "Analytic plane"

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''complex-analytic plane''
 
''complex-analytic plane''
  
A non-empty set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012370/a0123701.png" /> in the complex vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012370/a0123702.png" /> that satisfies a system of equations
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A non-empty set of points $  z = ( z _ {1} \dots z _ {n} ) $
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in the complex vector space $  \mathbf C  ^ {n} $
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that satisfies a system of equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012370/a0123703.png" /></td> </tr></table>
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$$
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\sum _ {i = 1 } ^ { n }
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a _ {i j }  z _ {i}  = b _ {j} ,
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\  j = 1 \dots k ;
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\  a _ {i j }  , b _ {j} \in \mathbf C ;
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012370/a0123704.png" /></td> </tr></table>
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$$
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\mathop{\rm rank}  \| a _ {i j }  \|  = < n .
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$$
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012370/a0123705.png" /> is called the complex codimension, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012370/a0123706.png" /> is called the complex dimension of the analytic plane. The real dimension of the analytic plane equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012370/a0123707.png" /> and is even, but not all even-dimensional real planes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012370/a0123708.png" /> are analytic planes. Complex one-dimensional analytic planes are sometimes called complex, or analytic, straight lines. See also [[Analytic surface|Analytic surface]].
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The number $  k $
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is called the complex codimension, while $  n - k $
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is called the complex dimension of the analytic plane. The real dimension of the analytic plane equals $  2 (n - k ) $
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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|Analytic surface]].

Latest revision as of 18:47, 5 April 2020


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=16200
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