An elementary proof of the Briancon-Skoda theorem

TL;DR

This paper presents a new elementary proof of the Briançon-Skoda theorem, establishing a key inclusion relation between the integral closure powers and the ideal itself in the context of analytic function germs.

Contribution

It provides a novel, elementary proof of the Briançon-Skoda theorem, simplifying the understanding of the integral closure and ideal containment in analytic function germs.

Findings

Proof simplifies the understanding of the Briançon-Skoda theorem

Establishes the inclusion of the u-th power of the integral closure in the ideal

Applicable to ideals generated by m elements in complex analytic germs

Abstract

We give a new elementary proof of the Brian\c{c}on-Skoda theorem, which states that for an $m$-generated ideal $\mathfrak{a}$ in the ring of germs of analytic functions at $0\in \C^n$, the $\nu$:th power of its integral closure is contained in $\mathfrak{a}$, where $\nu = \min(m,n)$.