Homogenization of a nonsymmetric embedding-dimension-three numerical semigroup

Abstract

Let $n_1,n_2,n_3$ be positive integers with $gcd\left(n_1,n_2,n_3\right)=1$. For $S=\langle n_1,n_2,n_3\rangle$ nonsymmetric, we give an alternative description, using elementary techniques, of a minimal presentation of its homogenization $\bar{S}=\langle \left(1,0\right),\left(1,n_1\right),\left(1,n_2\right),\left(1,n_3\right)\rangle$. As a consequence, we show that this minimal presentation is unique. We recover Bresinsky’s characterization of the Cohen–Macaulay property of $\bar{S}$ and present a procedure to compute all possible catenary degrees of the elements of $\bar{S}$.