Cyclotomic numerical semigroups

Abstract

Given a numerical semigroup $S$, we let ${\mathrm{P}}_S(x)=(1-x){\sum }_{s\in S}{x}^s$ be its semigroup polynomial. We study cyclotomic numerical semigroups; these are numerical semigroups $S$ such that ${\mathrm{P}}_S(x)$ has all its roots in the unit disc. We conjecture that $S$ is a cyclotomic numerical semigroup if and only if $S$ is a complete intersection numerical semigroup and present some evidence for it. Aside from the notion of cyclotomic numerical semigroup we introduce the notion of cyclotomic exponents and polynomially related numerical semigroups. We derive some properties and give some applications of these new concepts.

17 pages, accepted for publication in SIAM J. Discrete Math

References in corpus (2)

• Frobenius vectors, Hilbert series and gluings
• Maximal height of divisors of $x^{p{q}^b}-1$

Cited by in corpus (2)

• numericalsgps, a GAP package for numerical semigroups
• The second Feng-Rao number for codes coming from inductive semigroups