View source for Analytic polyhedron

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

====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>