Year: 2026
Type: preprint
Abstract: For positive integers \(q\) and \(m\) with \(m\geq 2q+3\), consider the symmetric numerical semigroup\[S_{m,q}=\left\langle m,\,m+1,\,qm+2q+2,\,qm+2q+3,\,\ldots,\,qm+m-1\right\rangle.\] Ciolan, García-Sánchez, and Moree asked whether every member of this family with embedding dimension at least \(4\) is noncyclotomic. We answer this question affirmatively for every \(q\geq 1\), including an independent proof of the previously known case \(q=1\). Let \(P_{m,q}\) be the semigroup polynomial. For \(m\geq 2q+4\), an explicit coefficient calculation in its formal logarithm produces a power sum of reciprocal roots whose absolute value exceeds \(\deg P_{m,q}\). This forces \(P_{m,q}\) to have a zero off the unit circle and hence \(S_{m,q}\) to be noncyclotomic. The boundary case \(m=2q+3\) has embedding dimension \(3\) and is cyclotomic. Thus\[S_{m,q}\text{ is cyclotomic}\quad\Longleftrightarrow\quad m=2q+3.\]
Source: arXiv (Cornell University)
Author: Zhang, Zhi-Lin
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213778542