On Symmetric But Not Cyclotomic Numerical Semigroups

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, Garc'ia-S'anchez, 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 $S_{n, t}$,whichforeveryfixed$t$issymmetricbutnotcyclotomicwhen$n\ge \max(8(t+1)^3,40(t+2))$ 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 $4$.