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A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123801.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123803.png" />, which can be represented by inequalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123804.png" />, where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123806.png" />, are holomorphic in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123807.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123808.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123809.png" />. It is also assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238010.png" /> is compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238012.png" /> are polynomials, the analytic polyhedron is said to be a polynomial polyhedron. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238014.png" />, the analytic polyhedron is called a [[Polydisc|polydisc]]. The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238015.png" /> are called the faces of the analytic polyhedron. The intersection of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238016.png" /> different faces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238017.png" /> is said to be an edge of the analytic polyhedron. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238018.png" /> and all faces have dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238019.png" />, while no edge has dimension exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238020.png" />, the analytic polyhedron is a [[Weil domain|Weil domain]]. The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238021.png" />-dimensional edges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238022.png" /> forms the skeleton of the analytic polyhedron. The concept of an analytic polyhedron is important in problems of integral representations of analytic functions of several variables.
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A domain  $  \Pi $
 +
of the complex space  $  \mathbf C  ^ {n} $,  
 +
$  n \geq  1 $,
 +
which can be represented by inequalities $  | f _ {i} (z) | < 1 $,  
 +
where the functions $  f _ {i} (z) $,  
 +
$  i = 1 \dots m $,  
 +
are holomorphic in some domain $  D \subset  \mathbf C  ^ {n} $
 +
containing $  \Pi $,  
 +
i.e. $  \Pi = \{ {z \in D } : {| f _ {i} (z) | < 1,  i = 1 \dots m } \} $.  
 +
It is also assumed that $  \Pi $
 +
is compact in $  D $.  
 +
If $  f _ {i} ( z ) $
 +
are polynomials, the analytic polyhedron is said to be a polynomial polyhedron. If $  m = n $
 +
and $  f _ {i} ( z ) = a _ {i} z _ {i} $,  
 +
the analytic polyhedron is called a [[Polydisc|polydisc]]. The sets $  \sigma _ {i} = \{ {z \in D } : {| f _ {i} ( z) | = 1;  | f _ {j} ( z ) | < 1,  j \neq i } \} $
 +
are called the faces of the analytic polyhedron. The intersection of any $  k $
 +
different faces $  (2 \leq  k \leq  n ) $
 +
is said to be an edge of the analytic polyhedron. If $  m \geq  n $
 +
and all faces have dimension $  2n - 1 $,  
 +
while no edge has dimension exceeding $  2n - k $,  
 +
the analytic polyhedron is a [[Weil domain|Weil domain]]. The set of $  n $-
 +
dimensional edges $  \sigma _ {i _ {1}  \dots i _ {n} } = \sigma _ {i _ {1}  } \cap \dots \cap \sigma _ {i _ {n}  } $
 +
forms the skeleton of the analytic polyhedron. The concept of an analytic polyhedron is important in problems of integral representations of analytic functions of several variables.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The analytic polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238023.png" /> defined above is sometimes said to be an analytic polyhedron of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238024.png" /> (cf. [[#References|[a1]]]).
+
The analytic polyhedron $  \Pi $
 +
defined above is sometimes said to be an analytic polyhedron of order $  m $(
 +
cf. [[#References|[a1]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Hörmander,  "An introduction to complex analysis in several variables" , North-Holland  (1973)  pp. Chapt. 2.4</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Hörmander,  "An introduction to complex analysis in several variables" , North-Holland  (1973)  pp. Chapt. 2.4</TD></TR></table>

Latest revision as of 18:47, 5 April 2020


A domain $ \Pi $ of the complex space $ \mathbf C ^ {n} $, $ n \geq 1 $, which can be represented by inequalities $ | f _ {i} (z) | < 1 $, where the functions $ f _ {i} (z) $, $ i = 1 \dots m $, are holomorphic in some domain $ D \subset \mathbf C ^ {n} $ containing $ \Pi $, i.e. $ \Pi = \{ {z \in D } : {| f _ {i} (z) | < 1, i = 1 \dots m } \} $. It is also assumed that $ \Pi $ is compact in $ D $. If $ f _ {i} ( z ) $ are polynomials, the analytic polyhedron is said to be a polynomial polyhedron. If $ m = n $ and $ f _ {i} ( z ) = a _ {i} z _ {i} $, the analytic polyhedron is called a polydisc. The sets $ \sigma _ {i} = \{ {z \in D } : {| f _ {i} ( z) | = 1; | f _ {j} ( z ) | < 1, j \neq i } \} $ are called the faces of the analytic polyhedron. The intersection of any $ k $ different faces $ (2 \leq k \leq n ) $ is said to be an edge of the analytic polyhedron. If $ m \geq n $ and all faces have dimension $ 2n - 1 $, while no edge has dimension exceeding $ 2n - k $, the analytic polyhedron is a Weil domain. The set of $ n $- dimensional edges $ \sigma _ {i _ {1} \dots i _ {n} } = \sigma _ {i _ {1} } \cap \dots \cap \sigma _ {i _ {n} } $ forms the skeleton of the analytic polyhedron. The concept of an analytic polyhedron is important in problems of integral representations of analytic functions of several variables.

References

[1] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)

Comments

The analytic polyhedron $ \Pi $ defined above is sometimes said to be an analytic polyhedron of order $ m $( cf. [a1]).

References

[a1] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4
How to Cite This Entry:
Analytic polyhedron. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_polyhedron&oldid=12335
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article