Low-dimensional lattices V. Integral coordinates for integral lattices
Low-dimensional lattices V. Integral coordinates for integral lattices
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DOI：
10.1098/rspa.1989.0124
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发表时间：
1989-12
期刊：
影响因子：
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通讯作者：
J. Conway;N. J. A. Sloane
中科院分区：
文献类型：
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作者：
J. Conway;N. J. A. Sloane
我们说一个n维（经典的）积分格是s-可积的，对于一个整数s，如果它可以由向量s-1/2（x1，...，xk），所有xi <$Z，在维数k <$n的欧几里得空间中。等价地，ε是s-可积的，当且仅当对应于ε的任何二次型f（x）可以写成s-1乘以整系数线性型的k个平方和，或者再次，当且仅当对偶格ε * 包含一个尺度为s的共序星星。本文给出了低维格（如Es格和Leech格）的s-积分方法。一个特殊的结果是任何一维格点都可以在k = 4时1-积分：这是拉格朗日的四平方定理。设n（s）是最小维数n，其中存在不可s-可积的整格。1937年，Ko和Mordell证明了n（1）= 6。我们证明了：<$（2）= 12，<$（3）= 14，21 <$$>（4）<$25，16 <$$>（5）<$22，<$（s）<$4s + 2（s奇），<$（s）<$2 πes（1 + o（1））（s偶）和<$（s）<$2 In s/ln In s（1 + o（1））。
We say that an n-dimensional (classically) integral lattice ⋀ is s-integrable, for an integer s, if it can be described by vectors s-½(x1,...,xk), with all xi ∊ Z, in a euclidean space of dimension k ≽ n. Equivalently, ⋀ is s-integrable if and only if any quadratic form f(x) corresponding to ⋀ can be written as s-1 times a sum of k squares of linear forms with integral coefficients, or again, if and only if the dual lattice ⋀* contains a eutactic star of scale s. This paper gives many techniques for s-integrating low-dimensional lattices (such as Es and the Leech lattice). A particular result is that any one-dimensional lattice can be 1-integrated with k = 4: this is Lagrange’s four-squares theorem. Let ϕ(s) be the smallest dimension n in which there is an integral lattice that is not s-integrable. In 1937 Ko and Mordell showed that ϕ(1) = 6. We prove that ϕ(2) = 12, ϕ(3) = 14, 21 ≼ ϕ(4) ≼ 25, 16 ≼ ϕ(5) ≼ 22, ϕ(s) ≼ 4s + 2 (s odd), ϕ(s) ≼ 2πes(1 + o(1)) (s even) and ϕ(s) ≽ 2In In s/ln In In s(1 + o(1)).