Analytic polyhedron

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