论文标题
立方议法
Cubic sublattices
论文作者
论文摘要
如果存在元素是成对的正交且长度相等的sublattice的基础，则三维整数晶格$ \ mathbb z^3 $的sublatice称为立方sublattice。我们表明，对于整数矢量$ \ mathbf v \ in \ mathbb z^3 $，其平方长度可除以$ d^2 $，存在一个包含$ \ mathbf v $的立方sublattice，带有边缘长度$ d $。这改善了Goswick等人的论文[Arxiv：0806.3943]的主要结果之一，在该论文中，通过使用Hurwitz积分季节的分解理论证明了类似的定理。我们使用跨产品进行大量的基本证明。这种方法使我们能够表征立方议法。
A sublattice of the three-dimensional integer lattice $\mathbb Z^3$ is called cubic sublattice if there exists a basis of the sublattice whose elements are pairwise orthogonal and of equal lengths. We show that for an integer vector $\mathbf v\in\mathbb Z^3$ whose squared length is divisible by $d^2$, there exists a cubic sublattice containing $\mathbf v$ with edge length $d$. This improves one of the main result of a paper [arxiv:0806.3943] of Goswick et al., where similar theorem was proved by using the decomposition theory of Hurwitz integral quaternions. We give an elementary proof heavily using cross product. This method allows us to characterize the cubic sublattices.
