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Erschienen in:
19.06.2019 | Original Paper
Arf good semigroups with fixed genus
Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 1/2020
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The concept of Arf numerical semigroup plays a significant role in the study of the equivalence between algebroid branches. Namely, two algebroid branches are said to be equivalent if their Arf closures have the same value semigroup, that is an Arf numerical semigroup. By introducing the concept of Arf good semigroups of \(\mathbb {N}^r\), it is possible to extend the aforementioned equivalence to the more general context of algebroid curves with \(r>1\) branches. Arf good semigroups can be completely described by their multiplicity trees that are combinatorial objects whose study is independent from the ring theory context. In this paper we give an algorithm for the computation of all Arf numerical semigroups with a given genus. Moreover, we generalize the concept of genus of a numerical semigroup to good semigroups of \(\mathbb {N}^r\) and we give a procedure to calculate all Arf good semigroups of \(\mathbb {N}^r\) with a given genus.