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