View source for Analytic polyhedron

A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123801.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123803.png" />, which can be represented by inequalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123804.png" />, where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123806.png" />, are holomorphic in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123807.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123808.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a0123809.png" />. It is also assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238010.png" /> is compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238012.png" /> are polynomials, the analytic polyhedron is said to be a polynomial polyhedron. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238014.png" />, the analytic polyhedron is called a [[Polydisc|polydisc]]. The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238015.png" /> are called the faces of the analytic polyhedron. The intersection of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238016.png" /> different faces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238017.png" /> is said to be an edge of the analytic polyhedron. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238018.png" /> and all faces have dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238019.png" />, while no edge has dimension exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238020.png" />, the analytic polyhedron is a [[Weil domain|Weil domain]]. The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238021.png" />-dimensional edges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238022.png" /> forms the skeleton of the analytic polyhedron. The concept of an analytic polyhedron is important in problems of integral representations of analytic functions of several variables.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR></table>



====Comments====
The analytic polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238023.png" /> defined above is sometimes said to be an analytic polyhedron of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012380/a01238024.png" /> (cf. [[#References|[a1]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Hörmander,   "An introduction to complex analysis in several variables" , North-Holland  (1973)  pp. Chapt. 2.4</TD></TR></table>