On Symmetric But Not Cyclotomic Numerical Semigroups
Abstract
A numerical semigroup is called cyclotomic if its corresponding numerical semigroup polynomial $P_S(x)=(1x)\sum_{s\in S}x^s$ is expressable as the product of cyclotomic polynomials. Ciolan, GarcíaSánchez, 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:

arXiv eprints
 Pub Date:
 July 2017
 arXiv:
 arXiv:1707.00782
 Bibcode:
 2017arXiv170700782S
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 Typos corrected