On the effective membership problem for polynomial ideals

TL;DR

This paper explores optimal degree bounds for representing elements in polynomial ideals in complex space, extending classical theorems to subvarieties and presenting new variants and generalizations.

Contribution

It introduces new variants and generalizations of classical theorems on polynomial ideal representations, focusing on optimal degree bounds in complex varieties.

Findings

Classical theorems are extended to subvarieties of ^N.

New variants of ideal representation theorems are presented.

Results provide conditions for optimal degree bounds in polynomial ideal representations.

Abstract

We discuss the possibility of representing elements in polynomial ideals in $\C^N$ with optimal degree bounds. Classical theorems due to Macaulay and Max Noether say that such a representation is possible under certain conditions on the variety of the associated homogeneous ideal. We present some variants of these results, as well as generalizations to subvarieties of $\C^N$.