Explicit representation of membership in polynomial ideals

TL;DR

This paper presents a new division formula on projective space that offers explicit solutions to polynomial division problems, improving degree estimates and generalizing classical theorems like Macaulay and Noether's.

Contribution

It introduces a novel division formula on projective space that provides explicit solutions and sharp degree bounds, extending classical polynomial division results.

Findings

Provides explicit solutions to polynomial division problems

Achieves sharp degree estimates in division formulas

Generalizes classical theorems such as Macaulay and Noether's AF+BG theorem

Abstract

We introduce a new division formula on projective space which provides explicit solutions to various polynomial division problems with sharp degree estimates. We consider simple examples as the classical Macaulay theorem as well as a quite recent result by Hickel, related to the effective Nullstellensatz. We also obtain a related result that generalizes Max Noether's classical AF+BG theorem.