ON SYMMETRIC BUT NOT CYCLOTOMIC NUMERICAL SEMIGROUPS.

A numerical semigroup S is called cyclotomic if its corresponding numerical semi-group polynomial, PS(x) = (1 - x)∑s∊S xs, is expressible as a product of cyclotomic polynomials. Ciolan, García-Sánchez, and Moree [SIAM J. Discrete Math., 30 (2016), pp. 650-668] conjectured that, for every embedding dimension at least 4, there exists a numerical semigroup which is symmetric but not cyclotomic. We prove this conjecture by giving an infinite class of numerical semigroup families Sn,t, which, for every fixed t, are symmetric but not cyclotomic when n ≥ max (8(t+1)3; 40(t+2)). We also verify through a finite case check that the numerical semigroup families Sn,0 and Sn,1 yield noncyclotomic numerical semigroups for every embedding dimension at least 4. [ABSTRACT FROM AUTHOR]