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\(k\)-extreme lattices. (Réseaux \(k\)-extrêmes.) (French) Zbl 0861.11040

We study the invariants introduced by R. A. Rankin [J. Lond. Math. Soc. 28, 309-314 (1953; Zbl 0050.27401)] which are defined as a natural generalization of the well-known Hermite constant of a lattice \(L\). \(k\) being an integer, one defines \(\delta_k(L)\) to be the lowest determinant of \(k\)-dimensional subsections in \(L\), and sets \(\gamma_k (L) = \delta_k (L)/ \text{det} (L)^{k/n}\).
We give necessary and sufficient conditions for a lattice to realize a local maximum for the function \(\gamma_k(L)\), analogous to Voronoï’s theorem, in terms of \(k\)-perfection and \(k\)-eutaxy. We also establish the finiteness, up to similarity, of the number of \(k\)-perfect lattices in a given dimension. We finally apply these notions to some classical families of lattices (e.g. root lattices).

MSC:

11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11H06 Lattices and convex bodies (number-theoretic aspects)

Citations:

Zbl 0050.27401
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