Book Review
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Book Information:
Author:
Jacques Martinet
Title:
Perfect lattices in Euclidean spaces
Additional book information:
Springer-Verlag,
Berlin,
2003,
xxii + 523 pp.,
ISBN 3-540-44236-7,
$99.00$
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Anne-Marie Bergé and Jacques Martinet, Sur un problème de dualité lié aux sphères en géométrie des nombres, J. Number Theory 32 (1989), no. 1, 14–42 (French, with English summary). MR 1002112, DOI 10.1016/0022-314X(89)90095-4
3. H.F. Blichfeldt, The minimum value of positive quadratic forms in six, seven, and eight variables. Math. Z. 39 (1935), 1-15.
4. H. Cohn, A. Kumar, The optimality of the Leech lattice. (preprint, 2003) available via: http://research.microsoft.com/
cohn/Leech/
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1988. With contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 920369, DOI 10.1007/978-1-4757-2016-7
Leonard Eugene Dickson, History of the theory of numbers. Vol. I: Divisibility and primality. , Chelsea Publishing Co., New York, 1966. MR 0245499
Wolfgang Ebeling, Lattices and codes, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1994. A course partially based on lectures by F. Hirzebruch. MR 1280458, DOI 10.1007/978-3-322-96879-1
8. N. Hofreiter, Über Extremformen. Monatshefte f. Math. 40 (1933), 129-152.
David-Olivier Jaquet-Chiffelle, Énumération complète des classes de formes parfaites en dimension $7$, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 1, 21–55 (French, with English and French summaries). MR 1209694
10. A. Korkine, G. Zolotareff, Sur les formes quadratiques. Math. Ann. 5 (1872) 581-583, ibid. 6 (1873) 366-389, ibid. 11 (1877) 242-292.
11. H. Minkowski, Geometrie der Zahlen. Teubner, Leipzig (1910).
Jürgen Opgenorth, Dual cones and the Voronoi algorithm, Experiment. Math. 10 (2001), no. 4, 599–608. MR 1881760
Kaye Stacey, The enumeration of perfect septenary forms, J. London Math. Soc. (2) 10 (1975), 97–104. MR 369272, DOI 10.1112/jlms/s2-10.1.97
Boris Venkov, Réseaux et designs sphériques, Réseaux euclidiens, designs sphériques et formes modulaires, Monogr. Enseign. Math., vol. 37, Enseignement Math., Geneva, 2001, pp. 10–86 (French, with English and French summaries). MR 1878745
15. G.F. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Journal f. d. Reine und Angewandte Mathematik 133 (1908), ibid. 134 (1908), ibid. 136 (1909).
- 1.
- E.S. Barnes, The perfect and extreme senary forms. Canad. J. Math. 9 (1957), 235-242. MR 19:251e
- 2.
- A.-M. Bergé, J. Martinet, Sur un problème de dualité lié aux sphères en géométrie des nombres. J. Number Theory 32 (1989), no. 1, 14-42. MR 1002112
- 3.
- H.F. Blichfeldt, The minimum value of positive quadratic forms in six, seven, and eight variables. Math. Z. 39 (1935), 1-15.
- 4.
- H. Cohn, A. Kumar, The optimality of the Leech lattice. (preprint, 2003) available via: http://research.microsoft.com/cohn/Leech/
- 5.
- J.H. Conway, N.J.A. Sloane, Sphere packings, lattices and groups. Grundlehren 290, Springer (1988). MR 0920369
- 6.
- L.E. Dickson, History of the theory of numbers III, quadratic and higher forms. Chelsea, New York (1966). MR 0245499
- 7.
- W. Ebeling, Lattices and codes. Vieweg (1994). MR 1280458
- 8.
- N. Hofreiter, Über Extremformen. Monatshefte f. Math. 40 (1933), 129-152.
- 9.
- D.-O. Jaquet-Chiffelle, Énumération complète des classes de formes parfaites en dimension
. Ann. Inst. Fourier (Grenoble) 43 (1993), no. 1, 21-55. MR 1209694
- 10.
- A. Korkine, G. Zolotareff, Sur les formes quadratiques. Math. Ann. 5 (1872) 581-583, ibid. 6 (1873) 366-389, ibid. 11 (1877) 242-292.
- 11.
- H. Minkowski, Geometrie der Zahlen. Teubner, Leipzig (1910).
- 12.
- J. Opgenorth, Dual cones and the Voronoi algorithm. Experiment. Math. 10 (2001), no. 4, 599-608. MR 1881760
- 13.
- K.C. Stacey, The enumeration of perfect septenary forms. J. London Math. Soc. (2) 10 (1975), 97-104. MR 0369272
- 14.
- B. B. Venkov, Réseaux et designs sphériques. Réseaux euclidiens, designs sphériques et formes modulaires, 10-86, Monogr. Enseign. Math. 37, Enseignement Math., Geneva (2001). MR 1878745
- 15.
- G.F. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Journal f. d. Reine und Angewandte Mathematik 133 (1908), ibid. 134 (1908), ibid. 136 (1909).
Review Information:
Reviewer:
Gabriele Nebe
Affiliation:
Universität Ulm
Email:
nebe@mathematik.uni-ulm.de
Journal:
Bull. Amer. Math. Soc.
41 (2004), 529-533
Published electronically:
March 24, 2004
Review copyright:
© Copyright 2004
American Mathematical Society
