AI 中文总结
研究\(\mathbb{Z}^4\)中秩为三的正格的可允许基，通过特定条件刻画了一类格向量，构造了含可允许基的正格\(L(m)\)，其相关格理想是一般的，得到有七个最小二项式生成元的一般格理想无限族，证明刻画精确。
AI 中文摘要
我们引入了\(\mathbb{Z}^4\)中秩为三的正格的可允许基的概念，即满足关于基向量坐标的三个明确符号条件（I）、（II）和（III）的\(\mathbb{Z}\)-基\(\{{\bf c}_1,{\bf c}_2,{\bf c}_3\}\)。研究了形如\({\bf u}^{(\mu,\lambda)}=\mu{\bf c}_1+{\bf c}_2+\lambda{\bf c}_3\)的格向量，其中\(\mu\)和\(\lambda\)为正整数。在条件（I）、（II）和（III）下，得到了第一个和第四个坐标为正且是原点邻点的向量\({\bf u}^{(\mu,\lambda)}\)的完整刻画：这样一个向量是邻点当且仅当\(\mu = 1\)且\(\lambda\leq2\)。作为应用，对于每个整数\(m\geq1\)，构造了一个包含可允许基的正格\(L(m)\subseteq \mathbb{Z}^4\)，其相关格理想是一般的。这产生了一个具有恰好七个最小二项式生成元的一般格理想的无限族。该族表明该刻画是精确的，因为\({\bf u}^{(1,1)}\)和\({\bf u}^{(1,2)}\)都是原点的邻点。
英文摘要
We introduce the notion of an admissible basis for rank-three positive lattices in $\mathbb{Z}^4$, namely a $\mathbb{Z}$-basis $\{{\bf c}_1,{\bf c}_2,{\bf c}_3\}$ satisfying three explicit sign conditions \textup{(I)}, \textup{(II)}, and \textup{(III)} on the coordinates of the basis vectors. We study lattice vectors of the form ${\bf u}^{(μ,λ)}=μ{\bf c}_1+{\bf c}_2+λ{\bf c}_3,$ where $μ$ and $λ$ are positive integers. Under conditions \textup{(I)}, \textup{(II)}, and \textup{(III)}, we obtain a complete characterization of the vectors ${\bf u}^{(μ,λ)}$ with positive first and fourth coordinates that are neighbors of the origin: such a vector is a neighbor if and only if $μ=1$ and $λ\le2$. As an application, for every integer $m\ge1$ we construct a positive lattice $L(m)\subseteq \mathbb{Z}^4$ admitting an admissible basis whose associated lattice ideal is generic. This yields an infinite family of generic lattice ideals with exactly seven minimal binomial generators. This family shows that the characterization is sharp, since both ${\bf u}^{(1,1)}$ and ${\bf u}^{(1,2)}$ occur as neighbors of the origin.
Comments17 pages
