Abstract
We prove the “absolute” finiteness of the number of faces (independent of the parameter
) of Venkov's reduction domain
(Izv. Akad. Nauk SSSR, Ser. Mat.4, 37–52 (1940)) ofn-ary positive quadratic forms. The casen=3 is given special consideration. We study the change of the reduction domain
when
changes along a line segment in the space of coefficients.
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E. P. Baranovskii, “Reduction in the sense of Selling of positive quadratic forms in five variables,” Uch. Zap. Ivanovsk. Gos. Univ.,89, 37–64 (1974).
E. P. Baranovskii, “The Selling reduction domain of positive-quadratic forms in five variables,” Tr. Mat. Inst. Akad. Nauk SSSR,152, 5–33 (1980).
B. A. Venkov, “Reduction of positive quadratic forms,” Izv. Akad. Nauk SSSR, Ser. Mat.,4, No. 1, 37–52 (1940), (cf. also B. A. Venkov, Selected Works, Leningrad (1981), pp. 185–200).
B. N. Delone, “Geometry of positive quadratic forms,” Usp. Mat. Nauk,3, 16–62 (1937);4, 103–164 (1938).
A. V. Malyshev, “Quadratic forms — reduction,” Mathematical Encyclopedia [in Russian], Vol. 2, Moscow (1979), pp. 788–791.
S. S. Ryshkov, “Reduction theory of positive-quadratic forms,” Dokl. Akad. Nauk SSSR,198, No. 5, 1028–1031 (1971).
S. S. Ryshkov, “Reduction of positive-quadratic forms inn variables according to Hermite, Minkowski and Venkov,” Dokl. Akad. Nauk SSSR,207, No. 5, 1054–1056 (1972).
S. S. Ryshkov, “Reduction of positive quadratic forms according to Venkov,” Uch. Zap. Ivanovsk. Gos. Univ.,89, 5–36 (1974).
S. S. Ryshkov and E. P. Baranovskii, “C-types ofn-dimensional lattices and five-dimensional primitive parallelohedra (with application to the theory of coverings),” Tr. Mat. Inst. Akad. Nauk SSSR,137 (1976).
P. P. Tammela, “Reduction theory of positive-quadratic forms,” Dokl. Akad. Nauk SSSR,209, 1299–1302 (1973).
P. P. Tammela, “Minkowski reduction domains for positive quadratic forms in seven variables,” J. Sov. Math.,16, No. 1 (1981).
S. N. Chernikov, Linear Inequalities [in Russian], Moscow (1968).
M. I. Shtogrin, “The reduction domains of Voronoi, Venkov, and Minkowski,” Dokl. Akad. Nauk SSSR,207, 1070–1073 (1972).
E. S. Barnes and M. J. Cohn, “On the reduction of positive quaternary quadratic forms,” J. Austral. Math. Soc. Ser. A,22, 54–64 (1976).
A. J. Crisalli, “The fundamental cone and the Minkowski cone,” J. Reine Angew. Math.,277, 74–81 (1975).
M. Koecher, “Beiträge zu einer Reduktionstheorie in Positivitätsbereichen,” Math. Ann.,141, No. 5, 384–432 (1960);144, No. 2, 175–182 (1961).
B. L. van der Waerden, “Die Reduktionstheorie der positiven quadratischen Formen,” Acta Math.,96, 265–309 (1956).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 108–116, 1983.
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Tammela, P.P. Venkov's reduction theory of positive-quadratic forms. J Math Sci 29, 1306–1312 (1985). https://doi.org/10.1007/BF02108244
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DOI: https://doi.org/10.1007/BF02108244