Cyclotomic exponent sequences of numerical semigroups

Ciolan, Alexandru and García-Sánchez, Pedro A. and Herrera-Poyatos, Andrés and Moree, Pieter (2021) Cyclotomic exponent sequences of numerical semigroups. MPIM Preprint Series 2021 (5).

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We study the cyclotomic exponent sequence of a numerical semigroup $S,$ and we compute its values at the gaps of $S,$ the elements of $S$ with unique representations in terms of minimal generators, and the Betti elements $b\in S$ for which the set $\{a \in \operatorname{Betti}(S) : a \le_{S}b\}$ is totally ordered with respect to $\le_S$ (we write $a \le_S b$ whenever $a - b \in S,$ with $a,b\in S$).
This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences.
Our results also apply to \textit{cyclotomic numerical semigroups}, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, Garc\'ia-S\'anchez and Moree (2016) stating that $S$ is cyclotomic if and only if it is a complete intersection.

Item Type: MPIM Preprint
Subjects: 1 Discrete mathematics / algebra > 11-XX Number theory
1 Discrete mathematics / algebra > 20-XX Group theory and generalizations
Divisions: Research > Preprints
Depositing User: Andrea Kohlhuber
Date Deposited: 11 Feb 2021 11:40
Last Modified: 11 Feb 2021 11:41

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