Abstract
The theory of Hermite and Minkowski reduction of positive n-ary quadratic forms is studied. The question of the equivalence of reduced forms is investigated. It is shown that for n⩾3 the partition of the cone of positivity into precise domains of Hermite-Minkowski reduction is not normal (some domains intersect along pieces of the faces). For n⩽6: 1) contiguous vectors of the Dirichlet domain D(f) are found, where f is a form which is Hermite-Minkowski-reduced; 2) the classical and precise domains of Hermite-Minkowski reduction are computed; 3) a convenient algorithm is developed for the verification of the equivalence of reduced forms. On the basis of this algorithm a precise fundamental domain is constructed for n=3.
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 50, pp. 6–96, 1975.
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Tammela, P.P. Reduction theory of positive quadratic forms. J Math Sci 11, 197–277 (1979). https://doi.org/10.1007/BF01117520
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DOI: https://doi.org/10.1007/BF01117520