An indispensable classification of monomial curves in $\mathbb{A}^4(\mathbbmss{k}) $

TL;DR

This paper introduces a new classification method for monomial curves in four-dimensional affine space based on indispensable binomials, providing criteria for unique minimal generating systems and extending to broader binomial ideals.

Contribution

It offers a novel classification approach for monomial curves in $ ext{A}^4$ using indispensable binomials, with criteria for unique minimal generating sets and generalizations to other binomial ideals.

Findings

Characterization of monomial curves with unique minimal binomial generators

Necessary and sufficient conditions for the defining ideal's minimal generating set

Extension of results to broader classes of binomial ideals

Abstract

In this paper a new classification of monomial curves in $\mathbb{A}^4(\mathbbmss{k})$ is given. Our classification relies on the detection of those binomials and monomials that have to appear in every system of binomial generators of the defining ideal of the monomial curve; these special binomials and monomials are called indispensable in the literature. This way to proceed has the advantage of producing a natural necessary and sufficient condition for the definining ideal of a monomial curve in $\mathbb{A}^4(\mathbbmss{k})$ to have a unique minimal system of binomial generators. Furthermore, some other interesting results on more general classes of binomial ideals with unique minimal system of binomial generators are obtained.