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Low dimensional lattices have a strict Voronoï basis

Part of: Geometry of numbers

Published online by Cambridge University Press:  26 February 2010

Andrew J. Mayer
Andrew J. Mayer
Affiliation:
Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121, U.S.A.
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Abstract

We prove a generalization of a theorem of Ryshkov relating the Voronoï vectors of lattices to the defining conditions for the Minkowski fundamental domain . This is then used to prove that a Minkowski reduced basis of a lattice of dimension n < 7 consists of strict Voronoï vectors.

MSC classification

Secondary: 11H55: Quadratic forms (reduction theory, extreme forms, etc.)
Type
Research Article
Information
Mathematika , Volume 42 , Issue 2 , December 1995 , pp. 229 - 238
Copyright
Copyright © University College London 1995

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References

1.Cassels, J. W. S.. Rational Quadratic Forms (Academic Press, NY, 1978).Google Scholar
2.Conway, J. H. and Sloane, N. J. A.. Low dimensional lattices VI. In Proc. Roy. Soc. London A (1992), p. 55.Google Scholar
3.Conway, J. H. and Sloane, N. J. A.. Sphere Packings, Lattices and Groups (Springer-Verlag, 1988).CrossRefGoogle Scholar
4.Mayer, A. J.. A tiling property of the Minkowski fundamental domain. Ph.D. Thesis (Princeton University, 1993).Google Scholar
5.Minkowski, H.. Diskontinuitàtsbereich für arithmetische aequivalenz. Ges. Abh., 2 (1911), p. 53.Google Scholar
6.Minkowski, H.. Sur la reduction des formes quadratiques positives quaternaires. Ges. Abh., 1 (1911), p. 145.Google Scholar
7.Minkowski, H.. Über positive quadratische formen. Ges. Abh., 1 (1911), p. 149.Google Scholar
8.Minkowski, H.. Zur theorie der positiven quadratischen formen. Ges. Abh., 1 (1911), p. 212.Google Scholar
9.Ryshkov, S. S.. The theory of Hermite-Minkowski reduction of positive definite quadratic forms. Investigations in Number Theory 2, Zap. Nauchn. Ser. Leningrad Otdel. Mat. Inst. Steklov (LOMI), 33 (1973), p. 37. (English Translation: Journal of Soviet Mathematics, Vol. 6 (1976), p. 651.)Google Scholar
10.Siegel, C. L.. Lectures on the Geometry of Numbers (Springer-Verlag, 1989).CrossRefGoogle Scholar
11.Tammela, P.. The Hermite-Minkowski domain of reduction of positive definite quadratic form in six variables. LOMI, 33 (1973), p. 72. (English Translation: Journal of Soviet Mathematics, Vol. 6 (1976), p. 677.)CrossRefGoogle Scholar
12.Tammela, P.. Minkowski's fundamental reduction domain for positive quadratic forms of seven variables. LOMI, 57 (1977), p. 108 and p. 226. (English Translation: Journal of Soviet Mathematics, Vol. 16 (1981), p. 836.)CrossRefGoogle Scholar