analytic polyhedron

Definition.

Suppose $G\subset {ℂ}^n$ is a domain and let $W\subset G$be an open set. Let $f_1,\mathrm{\dots },f_k:W\to ℂ$ be holomorphicfunctions. Then if the set

is relatively compact in $W$, we say that $\mathrm{\Omega }$ is ananalytic polyhedron in $G$. Sometimes it is denoted$\mathrm{\Omega }(f_1,\mathrm{\dots },f_k)$. Further $(W,f_1,\mathrm{\dots },f_k)$ is called the of the analytic polyhedron.

An analytic polyhedron is automatically a domain of holomorphy by usingthe functions that define it as $g(z):=\frac{1}{e^{i\theta }-f_j(z)}$ toshow that $g$ cannot be extended beyond a point where $f_j(z)=e^{i\theta }$.Every boundary point of $\mathrm{\Omega }$ is of that form for some $f_j$.

Furthermore every domain of holomorphy can be exhausted by analytic polyhedra (that is, every compact subset is contained in an analytic polyhedron)and in fact only domains of holomorphy can be exhausted by analyticpolyhedra, see the Behnke-Stein theorem.

Note that sometimes $W$ is required to be homeomorphic to the unit ball.