Cyclotomic numerical semigroup polynomials with at most two irreducible factors

Abstract

A numerical semigroup S is cyclotomic if its semigroup polynomial $$\mathrm {P}_S$$ P S is a product of cyclotomic polynomials. The number of irreducible factors of $$\mathrm {P}_S$$ P S (with multiplicity) is the polynomial length $$\ell (S)$$ ℓ ( S ) of S. We show that a cyclotomic numerical semigroup is complete intersection if $$\ell (S)\le 2$$ ℓ ( S ) ≤ 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 $$\ell (S)$$ ℓ ( S ) and the embedding dimension of S.

Graded algebras with cyclotomic Hilbert series

“…Moreover, we prove that complete intersections and cyclotomic algebras coincide also under the assumption that R is standard graded and its hpolynomial is irreducible over Q. This is in line with a result in the forthcoming article [4]: see Question 4.1 and the discussion preceding it. Recall that the Kronecker polynomials which are irreducible over Q are precisely the cyclotomic polynomials Φ m (x).…”

“…Let R be a cyclotomic graded algebra and assume that N R is irreducible over Q. This means that N R (x) = Φ m (x) for some m ∈ N. Under this condition, in the case when R = [S] for some numerical semigroup S (and hence N R equals the semigroup polynomial P S ), it is proved in [4] that then S = p, q for some primes p = q, and consequently m = pq. Since each numerical semigroup of the form p, q is a complete intersection, this implies in particular that Conjecture 1.1 holds true when P S is irreducible.…”

“…Cyclotomic graded algebras have been considered in lattice polytope theory, where they are related to Ehrhart-positivity [17,6], and numerical semigroup theory [10,20,15,4,22]. Moreover, every graded complete intersection is cyclotomic (see the discussion before Proposition 2.4 for a more precise statement).…”