Voronoï theory over algebraic number fields. (English) Zbl 1139.11321

Martinet, Jacques (ed.), Résaux euclidiens, designs sphériques et formes modulaires. Autour des travaux de Boris Venkov. Genève: L’Enseignement Mathématique (ISBN 2-940264-02-3/pbk). Monogr. Enseign. Math. 37, 147-162 (2001).
Summary: We extend Voronoï’s theorem on extreme forms, i.e. the characterization of quadratic forms achieving a local maximum of Hermite’s invariant in terms of perfection and eutaxy, to Humbert forms, as defined in the work of Icaza and Baeza-Ieaza. Our result strengthens Ieaza’s theorem, in which weaker notions of perfection and eutaxy were defined which were necessary but not sufficient conditions for a Humbert form to be extreme.
For the entire collection see [Zbl 1054.11034].


11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11E12 Quadratic forms over global rings and fields
11H50 Minima of forms