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

Abstract

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 $m=2q+3$.