## A generalization of the Voronoi algorithm for quadratic forms. (Une généralisation de l’algorithme de Voronoi pour les formes quadratiques.)(French)Zbl 0812.11037

Coray, Daniel (ed.) et al., Journées arithmétiques. Exposés présentés aux dix-septièmes congrès à Genève, Suisse, 9-13 septembre 1991. Paris: Société Mathématique de France, Astérisque. 209, 137-158 (1992).
G. Voronoi [J. Reine Angew. Math. 133, 97-156 (1907; JFM 38.0261.01)] has given an algorithm to determine all perfect positive $$n$$-ary quadratic forms. Earlier work together with the use of this algorithm resulted in the enumeration of all such forms for $$n\leq 7$$ [see E. S. Barnes, Philos. Trans. R. Soc. Lond., Ser. A 249, 461-506 (1957; Zbl 0077.266) and D.-O. Jacquet, Thèse, Univ. Neuchâtel (1991; Zbl 0748.11033)] but for $$n\geq 8$$ the calculations are prohibitively complicated. It thus makes sense to look for variants on more general versions of Voronoi’s algorithm which perhaps may help to determine further perfect forms.
In the present article the authors propose an interesting and natural generalization of Voronoi’s algorithm and study it thoroughly. The results are too numerous to be cited.
For the entire collection see [Zbl 0772.00022].
Reviewer: P.Gruber (Wien)

### MSC:

 11H55 Quadratic forms (reduction theory, extreme forms, etc.) 11H50 Minima of forms 11H56 Automorphism groups of lattices

### Citations:

Zbl 0077.266; Zbl 0748.11033; JFM 38.0261.01