Classification of quintic eutactic forms

Batut, Christian

doi:10.1090/S0025-5718-00-01295-3

Classification of quintic eutactic forms
HTML articles powered by AMS MathViewer

by Christian Batut;

Math. Comp. 70 (2001), 395-417

DOI: https://doi.org/10.1090/S0025-5718-00-01295-3

Published electronically: July 21, 2000

PDF | Request permission

Abstract:

From the classical Voronoi algorithm, we derive an algorithm to classify quadratic positive definite forms by their minimal vectors; we define some new invariants for a class, for which several conjectures are proposed. Applying the algorithm to dimension 5 we obtain the table of the 136 classes in this dimension, we enumerate the 118 eutactic quintic forms, and we verify the Ash formula.

References

Avner Ash, On eutactic forms, Canadian J. Math. 29 (1977), no. 5, 1040–1054. MR 491523, DOI 10.4153/CJM-1977-101-2
Avner Ash, On the existence of eutactic forms, Bull. London Math. Soc. 12 (1980), no. 3, 192–196. MR 572099, DOI 10.1112/blms/12.3.192
J.-L. Baril, Autour de l’algorithme de Voronoi: constructions de réseaux euclidiens, Thèse Bordeaux (1996).
C. Bavard, Une formule d’Euler pour les classes minimales de réseaux, preprint.
Anne-Marie Bergé and Jacques Martinet, Sur la classification des réseaux eutactiques, J. London Math. Soc. (2) 53 (1996), no. 3, 417–432 (French). MR 1396707, DOI 10.1112/jlms/53.3.417
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1988. With contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 920369, DOI 10.1007/978-1-4757-2016-7
J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. III. Perfect forms, Proc. Roy. Soc. London Ser. A 418 (1988), no. 1854, 43–80. MR 953277
A. Korkine, G. Zolotareff, Sur les formes quadratiques positives, Math. Ann. 11 (1877), 242–292.
Jacques Martinet, Les réseaux parfaits des espaces euclidiens, Mathématiques. [Mathematics], Masson, Paris, 1996 (French, with French summary). MR 1434803
W. Plesken and B. Souvignier, Computing isometries of lattices, J. Symbolic Comput. 24 (1997), no. 3-4, 327–334. Computational algebra and number theory (London, 1993). MR 1484483, DOI 10.1006/jsco.1996.0130
W.I. Štogrin, Quasi densest lattice packing of spheres., Dokl-Akad-Nauk-SSSR 218 (1974), 62–65.
G. Voronoï, Nouvelles applications des paramètres continus à la théorie des formes quadratiques : 1 Sur quelques propriétés des formes quadratiques positives parfaites, J. Reine Angew. Math 133 (1908), 97–178.
G. L. Watson, On the minimum points of a postive quadratic form, Mathematika 18 (1971), 60–70. MR 289421, DOI 10.1112/S0025579300008378