A proof of A. Gabrielov's rank Theorem

TL;DR

This paper provides a complete proof of Gabrielov's rank Theorem, introduces formal-geometric techniques for clarity, and extends the theorem to related areas in commutative algebra.

Contribution

It offers a comprehensive proof of Gabrielov's rank Theorem and develops new formal-geometric methods that also lead to extensions in algebraic contexts.

Findings

Complete proof of Gabrielov's rank Theorem

New formal-geometric techniques for analytic map germs

Extensions to Zariski main Theorem and elimination theory

Abstract

This article contains a complete proof of Gabrielov's rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung Theorem. We include, furthermore, new extensions of the rank Theorem (concerning the Zariski main Theorem and elimination theory) to commutative algebra.