• DOI:10.1137/17M1138479
• Corpus ID: 51881315

On Symmetric But Not Cyclotomic Numerical Semigroups

@article{Sawhney2018OnSB,
  title={On Symmetric But Not Cyclotomic Numerical Semigroups},
  author={Mehtaab Sawhney and David Stoner},
  journal={SIAM J. Discret. Math.},
  year={2018},
  volume={32},
  pages={1296-1304}
}

• Mehtaab Sawhney, David Stoner
• Published 3 July 2017
• Mathematics
• SIAM J. Discret. Math.

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, Garcia-Sanchez, 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 $4$.