×

Cyclotomic quadratic forms. (English) Zbl 0977.11029

Let \(G\) be the companion matrix of the \(m\)th cyclotomic polynomial, of degree \(n = \phi(m)\), generating the standard \(n\)-dimensional representation of the cyclic group \(C_m\). A cyclotomic form is a real quadratic form \(q(x)\) in \(n\) variables, invariant under this action of \(C_m\): if \(A\) is the symmetric matrix of \(q\), then \(G^tAG = G\). Voronoi’s algorithm is a long-known method providing an explicit complete classification of perfect quadratic forms, and thus a means of computing the densest quadratic forms in each dimension, albeit with tremendous computational complexity.
This article generalizes Voronoi’s algorithm to cyclotomic forms (which substantially reduces the computational complexity), and provides a complete classification of perfect cyclotomic forms for \(\phi(m) < 16\) and for \(m = 17\). In the process, precise upper bounds for the Hermite invariant of cyclotomic forms in this range are obtained. In many cases, these bounds improve the best-known or conjectured values of the Hermite constant for the corresponding dimensions.

MSC:

11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11E99 Forms and linear algebraic groups
11H50 Minima of forms
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML EMIS

References:

[1] Bachoc, C., Batut, C., Etude algorithmique de réseaux construits avec la forme trace. J. Exp. Math.1 (1992), 183-190. · Zbl 0787.11024
[2] Bayer-Fluckiger, E., Martinet, J., Formes quadratiques liées aux algèbres semi-simples. J. reine angew. Math.451 (1994), 51-69. · Zbl 0801.11020
[3] Bergé, A.-M., Martinet, J., Réseaux extrêmes pour un groupe d’automorphismes. Astérisque198-200 (1992), 41-66. · Zbl 0753.11026
[4] Bergé, A.-M., Martinet, J., Sigrist, F., Une généralisation de l’algorithme de Voronoï pour les formes quadratiques. Astérisque209 (1992), 137-158. · Zbl 0812.11037
[5] Conway, J.H., Sloane, N.J.A., Sphere Packings, Lattices, and Groups. Springer-Verlag (1992). · Zbl 0785.11036
[6] Jaquet-Chiffelle, D.-O., Enumération complète des classes de formes parfaites en dimension 7. Ann. Inst. Fourier43 (1993), 21-55. · Zbl 0769.11028
[7] Jaquet-Chiffelle, D.-O., Trois théorèmes de finitude pour les G-formes. J. Théor. Nombres Bordeaux7 (1995), 165-176. · Zbl 0843.11032
[8] Martinet, J., Les réseaux parfaits des espaces euclidiens. Masson (1996). · Zbl 0869.11056
[9] Voronoï, G., Sur quelques propriétés des formes quadratiques positives parfaites. J. reine angew. Math.133 (1908), 97-178. · JFM 38.0261.01
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.