Domain of finite type

Definition

Let $\Omega \subseteq \C^n$ be a domain with $C^\infty$ boundary. Let $\rho$ be a smooth defining function of $\Omega$, as in the definition of the Levi form. Let $D \subseteq \C$ be the unit disc. For a function $g$ of class $C^\infty$ near $0 \in \C$, let $\nu(g)$ be its order of vanishing at $0$. This is the smallest $k$ such that some partial derivative of $g$ of order $k$ is nonzero at $0$, and $\nu(g) = \infty$ if there is none. For $p \in \partial\Omega$, the type of $\partial\Omega$ at $p$ is $$\Delta(p) = \sup_{\gamma} \frac{\nu(\rho \circ \gamma)}{\nu(\gamma - p)},$$ where $\gamma \colon D \to \C^n$ runs over the nonconstant holomorphic maps with $\gamma(0) = p$. Here $\nu(\gamma - p)$ is the least order of vanishing of the components of $\gamma - p$. The domain $\Omega$ is of finite type if $\Delta(p) < \infty$ for every $p \in \partial\Omega$.