In
mathematics, especially
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
, an analytic polyhedron is a subset of the
complex space of the form
:
where is a bounded connected open subset of ,
are
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
on and is assumed to be
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
in .
[See and .] If
above are polynomials, then the set is called a polynomial polyhedron. Every analytic polyhedron is a
domain of holomorphy
In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain.
Formal ...
and it is thus
pseudo-convex.
The boundary of an analytic polyhedron is contained in the union of the set of hypersurfaces
:
An analytic polyhedron is a ''Weil polyhedron'', or
Weil domain if the intersection of any of the above hypersurfaces has dimension no greater than .
[.]
See also
*
Behnke–Stein theorem
In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence G_k \subset \mathbb^n (i.e., G_k \subset G_) of domains of holomorphy is again a domain of holomorphy. It was proved by ...
*
Bergman–Weil formula
*
Oka–Weil theorem
Notes
References
*.
* (also available as ).
*.
*.
*.
*.
*. Notes from a course held by Francesco Severi at the
Istituto Nazionale di Alta Matematica (which at present bears his name), containing appendices of Enzo Martinelli, Giovanni Battista Rizza and
Mario Benedicty. An English translation of the title reads as:-"''Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome''".
Several complex variables
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