Finite Type Points On Subsets Of $\mathbb C^n$

TL;DR

This paper extends the concept of finite type points from real hypersurfaces to arbitrary subsets of complex space, proving the openness of such points in a more general setting.

Contribution

It generalizes D'Angelo's notion of finite type points and proves their openness for any subset of rac{ ext{C}^n}{}, expanding previous results.

Findings

The set of finite type points is open in any subset of rac{ ext{C}^n}{}.

Extension of finite type point theory from hypersurfaces to arbitrary sets.

Generalization of previous openness results to broader contexts.

Abstract

In [4], D'Angelo introduced the notion of points of finite type for a real hypersurface $M$ in $\mathbb C^n$ and showed that the set of points of finite type in $M$ is open. Later, Lamel-Mir [8] considered a natural extension of D'Angelo's definition for an arbitrary set $M$ in $\mathbb C^n$. Building on D'Angelo's work, we prove the openness of the set of points of finite type for any subset $M$ in $ \mathbb C^n.$