07-19-2026, 10:41 PM
Analytic polyhedron
Summary
An analytic polyhedron is a concept from the field of several complex variables, where geometry is described using complex-valued analytic (holomorphic) functions rather than only linear or algebraic boundaries. It is a subset of complex space defined by conditions of the form ($ |f_j(z)| < 1 $), where the functions ($ f_j $) are holomorphic and the region is bounded inside a larger complex domain. If these defining functions are polynomials, the object is called a polynomial polyhedron.
Analytic polyhedra are important because they are examples of domains of holomorphy, meaning that holomorphic functions defined on them cannot generally be extended beyond their boundaries, and they possess strong connections with complex analysis, pseudoconvexity, and the theory of several complex variables. Their boundaries are formed by portions of hypersurfaces where ($ |f_j(z)| = 1 $), and they provide useful models for studying the behavior of analytic functions in higher-dimensional complex spaces.
ARTICLE
Summary
An analytic polyhedron is a concept from the field of several complex variables, where geometry is described using complex-valued analytic (holomorphic) functions rather than only linear or algebraic boundaries. It is a subset of complex space defined by conditions of the form ($ |f_j(z)| < 1 $), where the functions ($ f_j $) are holomorphic and the region is bounded inside a larger complex domain. If these defining functions are polynomials, the object is called a polynomial polyhedron.
Analytic polyhedra are important because they are examples of domains of holomorphy, meaning that holomorphic functions defined on them cannot generally be extended beyond their boundaries, and they possess strong connections with complex analysis, pseudoconvexity, and the theory of several complex variables. Their boundaries are formed by portions of hypersurfaces where ($ |f_j(z)| = 1 $), and they provide useful models for studying the behavior of analytic functions in higher-dimensional complex spaces.
ARTICLE
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