D\'etermination finie sur un espace de Stein

TL;DR

This paper generalizes the finite determinacy theorem for holomorphic functions with isolated critical points to functions defined near Stein compact subsets and arbitrary ideals, expanding its applicability.

Contribution

It extends the classical finite determinacy theorem to a broader setting involving Stein spaces and arbitrary ideals.

Findings

Generalization of the finite determinacy theorem to Stein neighborhoods

Applicable to functions with arbitrary ideals, not just isolated critical points

Provides new tools for local holomorphic function classification

Abstract

Consider the ring of holomorphic function germs in $C^n$ and denote by $M$ the maximal ideal of this ring. For any a holomorphic function germ $f$ with an isolated critical point, the finite determinacy theorem (Mather-Tougeron) asserts that there exists some $k$, such that $f+g$ can be brought back to $f$, via a holomorphic change of variables, for any $g \in M^k$. In this paper, a generalisation of this theorem for functions defined in a neighbourhood of a Stein compact subset and for an arbitrary ideal is given.