A power-sum obstruction to cyclotomicity in a family of symmetric numerical semigroups

Abstract

For positive integers q and m with m≥ 2q+3, consider the symmetric numerical semigroup S_m,q=⟨ m,m+1,qm+2q+2,qm+2q+3,…,qm+m−1⟩. Ciolan, Garc'ia-S'anchez, and Moree asked whether every member of this family with embedding dimension at least $4$ is noncyclotomic. We answer this question affirmatively for every q≥ 1, including an independent proof of the previously known case q=1. Let P_m,q be the semigroup polynomial, and set t=m−2q−3, L=2(q+1)(m+1)−1, and D=°P_m,q=2qm+2q+2. For m≥ 2q+4, let ρ₁,…,ρ_D be the roots of P_m,q, counted with multiplicity. An explicit computation of a single coefficient of the formal logarithm gives |∑_j=1^Dρ_j^−L|=L|[x^L]log P_m,q(x)|≥ Lt−1>D. The strict inequality forces P_m,q to have a root off the unit circle. Hence S_m,q is noncyclotomic whenever m≥ 2q+4. The boundary case m=2q+3 has embedding dimension $3$ and is cyclotomic. Therefore S_m,q is cyclotomic if and only if m=2q+3.

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BibTeX RIS

BibTeX

@misc{zhang2026powersum,
  title = {{A power-sum obstruction to cyclotomicity in a family of symmetric numerical semigroups}},
  author = {Zhi-Lin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29484},
  eprint = {2609.29484},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29484},
}

RIS

TY  - UNPB
TI  - A power-sum obstruction to cyclotomicity in a family of symmetric numerical semigroups
AU  - Zhi-Lin Zhang
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.29484
UR  - https://arxiv.org/abs/2609.29484
AB  - For positive integers q and m with m≥ 2q+3, consider the symmetric numerical semigroup S_m,q=⟨ m,m+1,qm+2q+2,qm+2q+3,…,qm+m−1⟩. Ciolan, Garc'ia-S'anchez, and Moree asked whether every member of this family with embedding dimension at least $4$ is noncyclotomic. We answer this question affirmatively for every q≥ 1, including an independent proof of the previously known case q=1. Let P_m,q be the semigroup polynomial, and set t=m−2q−3, L=2(q+1)(m+1)−1, and D=°P_m,q=2qm+2q+2. For m≥ 2q+4, let ρ_1,…,ρ_D be the roots of P_m,q, counted with multiplicity. An explicit computation of a single coefficient of the formal logarithm gives |∑_j=1^Dρ_j^−L|=L|[x^L]log P_m,q(x)|≥ Lt−1>D. The strict inequality forces P_m,q to have a root off the unit circle. Hence S_m,q is noncyclotomic whenever m≥ 2q+4. The boundary case m=2q+3 has embedding dimension $3$ and is cyclotomic. Therefore S_m,q is cyclotomic if and only if m=2q+3.
ER  -

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