A global Brian\c{c} con-Skoda-Huneke-Sznajdman theorem

TL;DR

This paper establishes a global effective membership theorem for polynomials on non-reduced algebraic subvarieties in complex space, extending local results to a broader global context.

Contribution

It generalizes the Briançon-Skoda-Huneke theorem to non-reduced algebraic subvarieties, providing a new global version of a local analytic result.

Findings

Proves a global effective membership result for polynomials.

Extends local Briançon-Skoda-Huneke theorem to non-reduced varieties.

Provides tools for polynomial membership problems in complex algebraic geometry.

Abstract

We prove a global effective membership result for polynomials on a non-reduced algebraic subvariety of $\C^N$. It can be seen as a global version of a recent local result of Sznajdman, generalizing the Brian\c{c}on-Skoda-Huneke theorem for the local ring of holomorphic functions at a point on a reduced analytic space.