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 $4$.

Typos corrected