Row-Factorization Matrices in Arf Numerical Semigroups and Defining Ideals
Meral Süer, Mehmet Yeşil
TL;DR
The article advances the understanding of Arf numerical semigroups by developing and cataloging row-factorization matrices (RF-matrices) and connecting them to the structure of defining ideals in semigroup rings. It provides complete RF-matrix classifications for multiplicities $m\leq 5$ and parametrizations for cases where the conductor is a multiple of the multiplicity, establishing that the Frobenius number $F(S)$ always admits an RF-matrix with $|\det RF(F)|=F$. Using these RF matrices, the authors characterize when the defining ideal $I_S$ is generated by RF$(F)$-relations, showing genericity precisely in the low-multiplicity regime and detailing explicit generators in concrete examples. These results illuminate the interplay between the Arf property, RF-matrix structure, and toric defining ideals, with implications for Arf rings and the study of plane curve singularities.
Abstract
In this paper, we investigate the row-factorization matrices of Arf numerical semigroups, and we provide the full list of such matrices of certain Arf numerical semigroups. We use the information of row-factorization matrices to detect the generic nature and to find generators of the defining ideals.