# Weil domain

A special case of an analytic polyhedron. A bounded domain $ D $
in $ n $-
dimensional space $ \mathbf C ^ {n} $
is said to be a Weil domain if there exist $ N \geq n $
functions $ f _ {i} ( z) $,
$ i= 1 \dots N $,
holomorphic in a fixed neighbourhood $ U ( \overline{D}\; ) $
of the closure $ \overline{D}\; $,
such that

1) $ D= \{ {z } : {| f _ {i} ( z) | < 1, i = 1 \dots N, z \in U ( \overline{D}\; ) } \} $;

2) the faces of the Weil domain $ D $, i.e. the sets

$$ \sigma _ {i} = \{ {z \in D } : { | f _ {i} ( z) | = 1 ,\ | f _ {j} ( z) | \leq 1 ,\ j \neq i } \} , $$

have dimension $ 2n - 1 $;

3) the edges of the Weil domain $ D $, i.e. the intersections of any $ k $( $ 2 \leq k \leq n $) different faces, have dimension $ \leq 2n - k $.

The totality of all $ n $- dimensional edges of a Weil domain is called the skeleton of the domain. The Bergman–Weil representation applies to Weil domains. These domains are named for A. Weil [1], who obtained the first important results for these domains.

#### References

[1] | A. Weil, "L'intégrale de Cauchy et les fonctions de plusieurs variables" Math. Ann. , 111 (1935) pp. 178–182 |

[2] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) |

[3] | V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian) |

#### Comments

#### References

[a1] | B.A. Fuks, "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1963) (Translated from Russian) |

[a2] | G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1984) |

**How to Cite This Entry:**

Weil domain.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Weil_domain&oldid=49200