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A generalization of Voronoi’s reduction theory and its application. (English) Zbl 1186.11040

Summary: We consider Voronoi’s reduction theory of positive definite quadratic forms, which is based on Delone subdivision. We extend it to forms and Delone subdivisions having a prescribed symmetry group. Even more generally, the theory is developed for forms that are restricted to a linear subspace in the space of quadratic forms. We apply the new theory to complete the classification of totally real, thin algebraic number fields which was recently initiated by E. Bayer-Fluckiger [J. Number Theory 121, No. 2, 305–323 (2006; Zbl 1130.11066)] and E. Bayer-Fluckiger and G. Nebe [J. Théor. Nombres Bordx. 17, No. 2, 437–454 (2005; Zbl 1161.11032)]. Moreover, we apply it to construct new best-known sphere coverings in dimensions \(9,\ldots,15\)

MSC:

11H55 Quadratic forms (reduction theory, extreme forms, etc.)
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)

Software:

Magma; CARAT
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References:

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