Division formulas on projective varieties

TL;DR

This paper develops a division formula for possibly singular projective varieties, enabling explicit solutions to polynomial division problems and providing a global effective version of the Briançon-Skoda-Huneke theorem.

Contribution

It introduces a division formula applicable to singular projective varieties, extending polynomial division techniques and the Briançon-Skoda-Huneke theorem to a global setting.

Findings

Provides explicit polynomial division solutions on singular projective varieties.

Extends the Briançon-Skoda-Huneke theorem globally and effectively.

Offers new tools for algebraic geometry and polynomial ideal theory.

Abstract

We introduce a division formula on a possibly singular projective subvariety $X$ of complex projective space $\Pk^N$, which, e.g., provides explicit representations of solutions to various polynomial division problems on the affine part of $X$. Especially we consider a global effective version of the Brian\c con-Skoda-Huneke theorem.