Analytic polyhedron

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

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

where D is a bounded connected open subset of Cn and f_j are holomorphic on D.[1] 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 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 k hypersurfaces has dimension no greater than 2n-k.[2]

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Notes[edit]

  1. ^ See (Åhag et al. 2007, p. 139) and (Khenkin 1990, p. 35).
  2. ^ (Khenkin 1990, pp. 35-36).

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