枚举完全形式
Enumerating perfect forms
摘要：
一个正定二次型被称为完美的，如果它是唯一确定的算术最小值和积分向量达到it. In这个自包含的调查，我们解释如何枚举完美的形式在$d$变量的算术等价和缩放。我们把重点放在计算机辅助普查的实际问题。对于Voronoi的必要理论，我们提供了完整的证明基于Ryshkov多面体。这允许非常自然地推广到$T$ -完美形式，其相对于二次型空间中的线性子空间$T$是完美的。重要的例子包括高斯，爱森斯坦和赫尔维茨四元数完美的形式，我们提出了新的分类结果，在尺寸$8,10$和12。
英文摘要：
A positive definite quadratic form is called perfect, if it is uniquely determined by its arithmetical minimum and the integral vectors attaining it. In this self-contained survey we explain how to enumerate perfect forms in $d$ variables up to arithmetical equivalence and scaling. We put an emphasis on practical issues concerning computer assisted enumerations. For the necessary theory of Voronoi we provide complete proofs based on Ryshkov polyhedra. This allows a very natural generalization to $T$-perfect forms, which are perfect with respect to a linear subspace $T$ in the space of quadratic forms. Important examples include Gaussian, Eisenstein and Hurwitz quaternionic perfect forms, for which we present new classification results in dimensions $8,10$ and 12.
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