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2006-22

A generalization of Voronoi's reduction theory and its application

by Mathieu Dutour Sikiric, Achill Schürmann, Frank Vallentin.


Series: 2006-22, Preprints

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

Abstract:
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 $9,\dots,15$.

Keywords:
Voronoi reduction, totally real thin number fields, sphere coverings