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A generalization of Voronoi's reduction theory and its application

by Mathieu Dutour Sikiric, Achill Schürmann, Frank Vallentin.

Series: 2006-22, Preprints

11H55 Quadratic forms (reduction theory, extreme forms, etc.)

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 $9,\dots,15$.

Voronoi reduction, totally real thin number fields, sphere coverings