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Cordian Riener
On extreme forms in dimension 8
Journal de théorie des nombres de Bordeaux, 18 no. 3 (2006), p. 677-682, doi: 10.5802/jtnb.565
Article PDF | Reviews MR 2330434 | Zbl 1127.11047

Résumé - Abstract

A theorem of Voronoi asserts that a lattice is extreme if and only if it is perfect and eutactic. Very recently the classification of the perfect forms in dimension $8$ has been completed [5]. There are 10916 perfect lattices. Using methods of linear programming, we are able to identify those that are additionally eutactic. In lower dimensions almost all perfect lattices are also eutactic (for example $30$ out of the $33$ in dimension $7$). This is no longer the case in dimension $8$: up to similarity, there are only $2408$ extreme $8$-dimensional lattices.

Bibliography

[1] D. Avis, The lrs homepage, http://cgm.cs.mcgill.ca/~avis/C/lrs.html.
[2] Ch.  Batut, J. Martinet, A2x-Web-Pages on Lattices, http://math.u-bordeaux.fr/~martinet/.
[3] Ch.  Batut, Classification of quintic eutactic forms. Math. Comp. 70 (2001), 395–417.  MR 1803130 |  Zbl 0971.11034
[4] G. Danzig, Linear Programming and Extensions. Princeton University Press (1963).  MR 201189 |  Zbl 0108.33103
[5] M. Dutour, A. Schürmann, F. Vallentin, Classification of eight-dimensional perfect forms. Electron. Res. Announc. Amer. Math. Soc. 13 (2007).  MR 2300003 |  Zbl 05142879
[6] D.-O. Jaquet-Chiffelle, Énumération complète des classes de formes parfaites en dimension 7. Ann. Inst. Fourier 43 (1993), 21–55. Cedram |  MR 1209694 |  Zbl 0769.11028
[7] J. Martinet, Perfect Lattices in Euclidean Spaces. Springer–Verlag, Heidelberg (2003).  MR 1957723 |  Zbl 1017.11031
[8] G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques: 1. Sur quelques propriétés des formes quadratiques parfaites. J. reine angew. Math. 133 (1908), 97–178.  JFM 38.0261.01