Cyclotomic numerical semigroup polynomials with at most two irreducible factors

TL;DR

This paper explores the structure of cyclotomic numerical semigroups, showing that those with at most two irreducible factors are complete intersections, and investigates the relationship between polynomial length and embedding dimension.

Contribution

It proves that cyclotomic numerical semigroups with at most two irreducible factors are complete intersections, supporting a broader conjecture about their structure.

Findings

Cyclotomic semigroups with ≤2 factors are complete intersections.

Established partial evidence for a conjecture on cyclotomic semigroups.

Analyzed the link between polynomial length and embedding dimension.

Abstract

A numerical semigroup $S$ is cyclotomic if its semigroup polynomial $P_S$ is a product of cyclotomic polynomials. The number of irreducible factors of $P_S$ (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, Garc\'{i}a-S\'{a}nchez and Moree (2016) claiming that every cyclotomic numerical semigroup is complete intersection. In addition, we investigate the relation between $\ell(S)$ and the embedding dimension of $S.$