Essential and inessential elements of a standard basis

TL;DR

This paper introduces the concept of inessential elements in a standard basis of a homogeneous ideal, analyzing their properties and implications for ideal saturation and dehomogenization.

Contribution

It defines inessential elements within standard bases and explores their properties, providing new insights into the structure of homogeneous ideals.

Findings

Inessential elements do not affect the saturation of the ideal.

Omission of inessential elements leaves the saturation unchanged.

Inessential elements are irrelevant in any dehomogenization process.

Abstract

In this paper we introduce the concept of inessential element of a standard basis of I, where I is any homogeneous ideal of a polynomial ring. An inessential element is, roughly speaking, a form of the basis whose omission produces an ideal having the same saturation of I; it becomes useless in any dehomogenization of I with respect to a linear form. We study the properties of the basis linked to the presence of inessential elements and give some examples.