Analytic polyhedron

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

${\displaystyle P=\{z\in D:|f_{j}(z)|<1,\;\;1\leq j\leq N\}}$

where D is a bounded connected open subset of Cⁿ, ${\displaystyle f_{j}}$ are holomorphic on D and P is assumed to be relatively compact in D.^[1] If ${\displaystyle f_{j}}$ 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

${\displaystyle \sigma _{j}=\{z\in D:|f_{j}(z)|=1\},\;1\leq j\leq N.}$

An analytic polyhedron is a Weil polyhedron, or Weil domain if the intersection of any k of the above hypersurfaces has dimension no greater than 2n-k.^[2]

See also[]

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

Notes[]

References[]

• Åhag, Per; Czyż, Rafał; Lodin, Sam; Wikström, Frank (2007), "Plurisubharmonic extension in non-degenerate analytic polyhedra" (PDF), , Fasciculus XLV: 139–145, MR 2453953, Zbl 1176.31010.
• Khenkin, G. M. (1990), "The Method of Complex Integral Representations in Complex Analysis", in Vitushkin, A. G. (ed.), Several Complex Variables I, Encyclopaedia of Mathematical Sciences, 7, Berlin–Heidelberg–New York: Springer-Verlag, pp. 19–116, ISBN 3-540-17004-9, MR 0850491, Zbl 0781.32007 (also available as ISBN 0-387-17004-9).
• Gunning, Robert C.; Rossi, Hugo (1965), Analytic Functions of Several Complex Variables, Prentice–Hall series in Modern Analysis, Englewood Cliffs, N.J.: Prentice-Hall, pp. xiv+317, ISBN 9780821869536, MR 0180696, Zbl 0141.08601.
• Gunning, Robert C. (1990), Introduction to Holomorphic Functions of Several Variables. Volume I: Function Theory, Wadsworth & Brooks/Cole Mathematics Series, Belmont, California: Wadsworth & Brooks/Cole, pp. xx+203, ISBN 0-534-13308-8, MR 1052649, Zbl 0699.32001.
• Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland, ISBN 0-444-88446-7, MR 1045639, Zbl 0685.32001.
• Kaup, Ludger; Kaup, Burchard (1983), Holomorphic functions of several variables, de Gruyter Studies in Mathematics, 3, Berlin–New York: Walter de Gruyter, pp. XV+349, ISBN 978-3-11-004150-7, MR 0716497, Zbl 0528.32001.
• Severi, Francesco (1958), Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma (in Italian), Padova: CEDAM – Casa Editrice Dott. Antonio Milani, pp. XIV+255, Zbl 0094.28002. 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 . 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".

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