Cyclotomic numerical semigroup polynomials with at most two irreducible factors

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A numerical semigroup S is cyclotomic if its semigroup polynomial mathrm {P}_S is a product of cyclotomic polynomials. The number of irreducible factors of mathrm {P}_S (with multiplicity) is the polynomial length ell (S) of S. We show that a cyclotomic numerical semigroup is complete intersection if ell (S)le 2. This establishes a particular case of a conjecture of Ciolan et al. (SIAM J Discrete Math 30(2):650–668, 2016) claiming that every cyclotomic numerical semigroup is complete intersection. In addition, we investigate the relation between ell (S) and the embedding dimension of S.