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