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

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