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Martinet, Jacques

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

J. Théor. Nombres Bordx. 9, No. 1, 169-181 (1997).
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.
Reviewer: H.G.Quebbemann (Oldenburg)

Cited in 3 Documents

MSC:

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

Keywords:

extreme lattice; dual-extreme lattice; perfect lattice; eutactic lattice

Citations:

Zbl 0677.10022
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\textit{J. Martinet}, J. Théor. Nombres Bordx. 9, No. 1, 169--181 (1997; Zbl 0889.11023)
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References:

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