On Symmetric But Not Cyclotomic Numerical Semigroups
Abstract
A numerical semigroup is called cyclotomic if its corresponding numerical semigroup polynomial $P_S(x)=(1-x)\sum_{s\in S}x^s$ is expressable as the product of cyclotomic polynomials. Ciolan, García-Sánchez, and Moree conjectured that for every embedding dimension at least $4$, there exists some numerical semigroup which is symmetric but not cyclotomic. We affirm this conjecture by giving an infinite class of numerical semigroup families $S_{n, t}$, which for every fixed $t$ is symmetric but not cyclotomic when $n\ge \max(8(t+1)^3,40(t+2))$ and then verify through a finite case check that the numerical semigroup families $S_{n, 0}$, and $S_{n, 1}$ yield acyclotomic numerical semigroups for every embedding dimension at least $4$.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2017
- DOI:
- 10.48550/arXiv.1707.00782
- arXiv:
- arXiv:1707.00782
- Bibcode:
- 2017arXiv170700782S
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- Typos corrected