Analytic polyhedron

Template:Short description In mathematics, especially several complex variables, an analytic polyhedron is a subset of the complex space Template:Math of the form

$P=\{z\in D:|f_j(z)|<1,1\le j\le N\}$

where Template:Math is a bounded connected open subset of Template:Math, $f_j$ are holomorphic on Template:Math and Template:Math is assumed to be relatively compact in Template:Math. If $f_j$ above are polynomials, then the set is called a polynomial polyhedron. Every analytic polyhedron is a domain of holomorphy and it is thus pseudo-convex.

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

${\sigma }_j=\{z\in D:|f_j(z)|=1\},1\le j\le N.$

An analytic polyhedron is a Weil polyhedron, or Weil domain if the intersection of any Template:Math of the above hypersurfaces has dimension no greater than Template:Math.

See also

• Behnke–Stein theorem
• Bergman–Weil formula
• Oka–Weil theorem

References

• Template:Citation.
• Template:Citation (also available as Template:ISBN).
• Template:Citation.
• Template:Citation.
• Template:Citation.
• Template:Citation.
• Template:Citation. 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".

Template:Mathanalysis-stub