On critical lattices of the unit sphere

Yu. A. Basalov

Čebyševskij sbornik, Tome 23 (2022) no. 5, pp. 20-37.

Résumé

The history of the problem of calculating and estimating the Hermite constant has two centuries. This article provides a brief overview of the history of this problem. Also, this problem is considered from the point of view of critical lattices of the unit sphere. This problem begans from the works of J. L. Lagrange, L. A. Seeber and K. F. Gauss. While developing the theory of reduction of positive definite quadratic forms, they obtained limit forms for which the ratio of the minimum value of these forms at integer points other than 0 to their determinant is maximal. In the middle of the 19th century, Sh. Hermit obtained an estimate for this quantity for an arbitrary dimension. And at the end of the 19th century, A. N. Korkin and E. I. Zolotarev proposed a new method for reducing quadratic forms, which made it possible to obtain exact values of the Hermite constant up to dimension 8. In this paper, we will consider a quantity equivalent to the Hermite constant, the critical determinant of the unit sphere. It should be noted that these quantities are closely connected with other problems in the geometry of numbers, for example, the problems of finding the density of the best packing, finding the shortest lattice vector, and Diophantine approximations. We present critical lattices of dimensions up to 8 and consider some of their metric properties.

Détail

Keywords: critical determinants, lattices, minimum of positive definite quadratic forms.

Bibliographie

[1] Venkov B. A., “K rabote «O nekotorykh svoistvakh polozhitelnykh sovershennykh kvadratichnykh form»”: G. F. Voronoi, Sobr. soch., v. 2, Izd-vo AN USSR, Kiev, 1952

[2] Venkov B. A., “O privedenii polozhitelnykh kvadratichnykh form”, Izv. AN, seriya matem., 4, 1940, 37–52

[3] Venkov B. A., Elementarnaya teoriya chisel, ONTI NKTP SSSR, 1937

[4] Dirikhle P. G. L., Lektsii po teorii chisel, ONTI NKTP SSSR, 1936

[5] Gruber P. M., Lekkerkerker K. G., Geometriya chisel, URSS, 2004

[6] Kassels Dzh. V. S., Vvedenie v geometriyu chisel, Per. s angl., Mir, M., 1965

[7] Kassels Dzh. V. S., Ratsionalnye kvadratichnye formy, Per. s angl., Mir, M., 1982

[8] Konvei Dzh., Sloen N., Upakovki sharov, reshetki i gruppy, Mir, M., 1990

[9] Ryshkov S. S., Baranovskii E. P., “Klassicheskie metody teorii reshetchatykh upakovok”, UMN, 34:4 (1979), 3–63

[10] Barnes E. S., “The complete enumeration of extreme senary forms”, Phil. Trans. Roy. Soc. London, A, 249 (1957), 461–506

[11] Vlishfeldt H. F., “The minimum values of positive quadratic formes in six, seven and eight variables”, Math. Z., 39 (1934), 1–15

[12] Gauss K. F., Trudy po teorii chisel, Izd-vo AN SSSR, 1959

[13] Gauss S. F., Untersuchungen uber die Eigenschaften der positiven ternaren quadratischen, Formen von Ludwig August Seeber, Gottingische gelehrte Anzeigen, 1831

[14] Hermite Ch., “Lettres de m. Hermite a m. Jacobie sur differemts objets de la theorie des Nombres”, J. Reine und Angew. math., 40 (1850), 261–315

[15] Jaquet-Chiffelle D.-O., “Énumération complète des classes de formes parfaites en dimension 7”, Annales de l'Institut Fourier, 43 (1993), 21–55 | DOI

[16] Korkine A., Zolotareff G., “Sur les formes quadratiques positives quaternaires”, Math. Ann., 5 (1872), 581–583

[17] Korkine A., Zolotareff G., “Sur les formes quadratiques”, Math. Ann., 6 (1873), 366–389

[18] Korkine A., Zolotareff G., “Sur les formes quadratiques positives”, Math. Ann., 11 (1877), 242–292

[19] Lagrange J. L., Recherches d'arithmetique, Nouveaux Memoires de 1'Academie royal des Sciences et Belles-Lettres de Berlin, Berlin, 1773

[20] Minkowski N., “Diskontinuitatsbereich fur arithmetische Aquivalenz”, J. Reine und Angew. Math., 129 (1905), 220–274

[21] Minkowski N., “Cher die positiven quadratischen Formen und liber Rettenbruchanliche”, Algorithmen, J. Reine und Angew. Math., 107 (1891), 278–279

[22] Nowak W. G., “Simultaneous Diophantine approximation: Searching for analogues of Hurwitz's theorem”, Essays in mathematics and its applications, eds. T.M. Rassias, P.M. Pardalos, Springer, Switzerland, 2016, 181–197

[23] Seeber L. A., Untersuchungen uber die Eigenschaften der positiven ternaren quadratischen Formen, Freiburg, 1831

[24] Sikiric M., Schuermann A., Vallentin F., “Classification of eight dimensional perfect forms”, Electronic Research Announcements of the American Mathematical Society, 13 (2006), 21–32 | DOI

[25] Stasey K. S., “The enumeration of perfect septenary forms”, J. London Math. Soc., 2:10 (1975), 97–104

[26] Stasey K. C., “The perfect septenary forms with $\delta_{4}=2$”, J. Austral. Math. Soc., 22:2 (1976), 144–164

[27] Voronoi G., “Sur quelques proprietes des formes quadratiques positives parfaites”, J. Reine und Angew. Math., 133 (1907), 97–178