用沃罗诺伊算法解决9维的格点堆积问题
The lattice packing problem in dimension 9 by Voronoi's algorithm
AI总结：
本研究利用沃罗诺伊算法计算出9维所有完美格点，并证明9维最密集格点堆积为层叠格点Λ_9，同时确定了9维kissing数集。
摘要：
1908年，沃罗诺伊提出了一种能够在有限时间内解决任意维度格点堆积问题的算法。沃罗诺伊证明了任何具有最优堆积密度的格点必须是所谓的完美格点，且他的算法在固定维度下枚举有限多个完美格点。然而，由于该算法的高复杂度，直到现在为止，该枚举工作只完成到8维。在本工作中，我们通过沃罗诺伊算法计算出9维的所有2237251040个完美格点。作为推论，这表明层叠格点Λ_9在9维中给出了最密集的格点堆积。等价地，我们证明了9维的赫尔姆特常数γ_9等于2。此外，我们扩展了沃森（1971）的结果，证明了9维中可能的 kissing 数集恰好是2·{1,……,91,99,120,…,129,136}。
英文摘要：
In 1908, Voronoi introduced an algorithm that solves the lattice packing problem in any dimension in finite time. Voronoi showed that any lattice with optimal packing density must be a so-called perfect lattice, and his algorithm enumerates the finitely many perfect lattices up to similarity in a fixed dimension. However, due to the high complexity of the algorithm this enumeration had, until now, only been completed up to dimension 8. In this work we compute all 2237251040 perfect lattices in dimension 9 via Voronoi's algorithm. As a corollary, this shows that the laminated lattice $Λ_9$ gives the densest lattice packing in dimension 9. Equivalently, we show that the Hermite constant $γ_9$ in dimension 9 equals $2$. Furthermore, we extend a result by Watson (1971) and show that the set of possible kissing numbers in dimension 9 is precisely $2 \cdot \{ 1, \ldots, 91, 99, 120, \ldots, 129, 136 \}$.
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