AI 中文总结
该研究证明对称数值半群族$S_{m,q}$在嵌入维数至少为4时均为非循环的，通过计算半群多项式形式对数的单个系数得到幂和不等式，从而判定循环性仅当$m=2q+3$时成立。
AI 中文摘要
对于正整数$q$和$m$且$m\geq 2q+3$，考虑对称数值半群$S_{m,q}=\langle m,m+1,qm+2q+2,qm+2q+3,\ldots,qm+m-1\rangle$。Ciolan、Garc'ia-S'anchez和Moree曾询问该族中所有嵌入维数至少为4的成员是否都是非循环的。我们对于所有$q\geq 1$肯定地回答了这个问题，包括对先前已知的$q=1$情形给出了独立证明。设$P_{m,q}$为半群多项式，并令$t=m-2q-3$，$L=2(q+1)(m+1)-1$，$D=\deg P_{m,q}=2qm+2q+2$。对于$m\geq 2q+4$，设$\rho_1,\ldots,\rho_D$为$P_{m,q}$的根（按重数计）。对形式对数的单个系数的显式计算给出$\left|\sum_{j=1}^{D}\rho_j^{-L}\right|=L\left|[x^L]\log P_{m,q}(x)\right|\geq Lt-1>D$。严格不等式迫使$P_{m,q}$在单位圆外有根。因此，当$m\geq 2q+4$时，$S_{m,q}$是非循环的。边界情形$m=2q+3$的嵌入维数为3且是循环的。因此，$S_{m,q}$是循环的当且仅当$m=2q+3$。
英文摘要
For positive integers $q$ and $m$ with $m\geq 2q+3$, consider the symmetric numerical semigroup $S_{m,q}=\langle m,m+1,qm+2q+2,qm+2q+3,\ldots,qm+m-1\rangle$. 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 $q\geq 1$, 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 $m\geq 2q+4$, let $ρ_1,\ldots,ρ_D$ be the roots of $P_{m,q}$, counted with multiplicity. An explicit computation of a single coefficient of the formal logarithm gives $\left|\sum_{j=1}^{D}ρ_j^{-L}\right|=L\left|[x^L]\log P_{m,q}(x)\right|\geq Lt-1>D$. The strict inequality forces $P_{m,q}$ to have a root off the unit circle. Hence $S_{m,q}$ is noncyclotomic whenever $m\geq 2q+4$. 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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