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
Full Text: DOI Numdam EuDML EMIS


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.