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In [[mathematics]], especially [[several complex variables]], an '''analytic polyhedron''' is a subset of the [[complex space]] <math>\mathbf{C}^n</math> of the form
 
:<math>\{ z \in D : |f_j(z)| < 1, 1 \le j \le N \}\,</math>
 
where <math>D</math> is a bounded connected open subset of <math>\mathbf{C}^n</math> and <math>f_j</math> are [[holomorphic function|holomorphic]] on ''D''.<ref>http://www.emis.de/journals/UIAM/PDF/45-139-145.pdf</ref> If <math>f_j</math> above are polynomials, then the set is called a '''polynomial polyhedron'''. Every analytic polyhedron is a [[domain of holomorphy]] (thus, [[pseudoconvex set|pseudo-convex]].)
 
The boundary of an analytic polyhedron is the union of the set of hypersurfaces
 
: <math> \sigma_j = \{ z \in D : |f_j(z)| = 1 \}, 1 \le j \le N. </math>  
 
An analytic polyhedron is a ''Weil polyhedron'', or [[Weil domain]] if the intersection of <math>k</math> hypersurfaces has dimension no greater than <math>2n-k</math>.<ref name=Chirka1997>{{cite book|last=E. M. Chirka, A. G. Vitushkin|title=Introduction to Complex Analysis|year=1997|publisher=Springer|isbn=9783540630050|pages=35-36}}</ref>
 
See also: the [[Behnke–Stein theorem]].
 
== References ==
{{reflist}}
*Lars Hörmander. ''An Introduction to Complex Analysis in Several Variables,'' North-Holland Publishing Company, New York, New York, 1973.
 
{{mathanalysis-stub}}
[[Category:Several complex variables]]

Latest revision as of 20:54, 7 December 2012

In mathematics, especially several complex variables, an analytic polyhedron is a subset of the complex space of the form

where is a bounded connected open subset of and are holomorphic on D.[1] If above are polynomials, then the set is called a polynomial polyhedron. Every analytic polyhedron is a domain of holomorphy (thus, pseudo-convex.)

The boundary of an analytic polyhedron is the union of the set of hypersurfaces

An analytic polyhedron is a Weil polyhedron, or Weil domain if the intersection of hypersurfaces has dimension no greater than .[2]

See also: the Behnke–Stein theorem.

References

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  • Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.

Template:Mathanalysis-stub

  1. ↑ http://www.emis.de/journals/UIAM/PDF/45-139-145.pdf
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