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

Notes

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

References

• Åhag, Per; Czyż, Rafał; Lodin, Sam; Wikström, Frank (2007), "Plurisubharmonic extension in non-degenerate analytic polyhedra" (PDF), Universitatis Iagellonicae Acta Mathematica, 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, 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 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".