A set of dual-extreme lattices. (Une famille de réseaux dual-extrêmes.) (French) Zbl 0889.11023

A lattice \(L\) in \(n\)-dimensional euclidean space is called dual-extreme if the geometric mean of the Hermite numbers of \(L\) and its dual \(L^*\) attains a local maximum. Following the classical method of Voronoi, A.-M. Bergé and J. Martinet [J. Number Theory 32, No. 1, 14-42 (1989; Zbl 0677.10022)] have characterized such lattices as being dual-perfect and dual-eutactic in the appropriate sense. As a consequence, if \(L\) is extreme and \(L^*\) is eutactic, then \(L\) is dual-extreme. However, for any even \(n\geq 8\), this paper gives an example where \(L\) is dual-extreme, neither \(L\) nor \(L^*\) is perfect, and only one of them is eutactic.


11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11H50 Minima of forms


Zbl 0677.10022
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[1] Bergé, A-M., Minimal vectors of pairs of dual lattices, J. Number Theory52 (1995), 284-298. · Zbl 0829.11036
[2] Bergé, A-M., Martinet, J., Sur un problème de dualité lié aux sphères en géométrie des nombres, J. Number Theory32 (1989), 14-42. · Zbl 0677.10022
[3] Conway, J.H., Sloane, N.J.A., Sphere Packings, Lattices and Groups, Springer-Verlag, Grundlehren n°290, Heidelberg, 1988, (seconde édition: 1993). · Zbl 0634.52002
[4] Conway, J.H., Sloane, N.J.A., Low-dimensional lattices. III. Perfect forms, Proc. Royal Soc. London A 418 (1988), 43-80. · Zbl 0655.10022
[5] Conway, J.H., Sloane, N.J.A., On Lattices Equivalent to Their Duals, J. Number Theory48 (1994), 373-382. · Zbl 0810.11041
[6] Laïhem, M., Thèse, Bordeaux, 1992.
[7] Martinet, J., Les réseaux parfaits des espaces euclidiens, Masson, Paris, 1996. · Zbl 0869.11056
[8] Watson, G.L., On the minimum points of a positive quadratic form, Mathematika18 (1971), 60-70. · Zbl 0219.10032
[9] Zahareva, N.V., Centerings of 8-dimensional lattices that preserve a frame of successive minima, Proc. Steklov Inst. math.152 (1982), 107-134, (original en russe: 1980). · Zbl 0501.10030
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