A generalization of Voronoi's reduction theory and its application

Details A generalization of Voronoi's reduction theory and its application A generalization of Voronoi's reduction theory and its application

2006-01-04

arxiv.org/abs/math/0601084v4

Duke Math. J. 142 (2008), 127-164

Mathematics

Metric Geometry

31 pages, 2 figures, 2 tables, (v4) minor changes, to appear in Duke Math. J

Scientific paper

10.1215/00127094-2008-003

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 general, the theory is developed for forms which 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 Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known sphere coverings in dimensions 9,..., 15.

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