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

In this paper a new classification of monomial curves in ${\mathbb{A}}^4(\text{mathbbmss}k)$ 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 ${\mathbb{A}}^4(\text{mathbbmss}k)$ 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.