Abstract
The main emphasis of this book has been on the theory of quadratic forms, and we have given special attention to reduction theory. The main question of reduction theory can be formulated in the following way. Let us consider the real-valued quadratic forms in n variables. We look for inequalities for the coefficients such that every form is integrally equivalent to one and only one reduced form, i.e., to a form which satisfies all these inequalities. (From now on, without again stating this explicitly, we will confine ourselves to positive forms.)
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References
H. Minkowski: Gesammelte Abhandlungen. (Particularly: Diskontinuitätsbereich für arithmetische Äquivalenz, Band 2, S. 53–100.)
C. L. Siegel: Gesammelte Abhandlungen,4 vols., Springer-Verlag, Berlin, Heidelberg, New York, 1966, 1979. (Particularly: The Volume of the Fundamental Domain for Some Infinite Groups, Vol. 1, 459–468; A Mean Value Theorem in the Geometry of Numbers, Vol. 3, 39–46.)
A. Weil: Collected Papers, 3 Vols., Springer-Verlag, 1979, Berlin, Heidelberg, New York. (Particularly: Sur quelques résultats de Siegel,Vol. 1, 339–357.)
G. P. L. Dirichlet: Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. Werke II.
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© 1985 Springer Science+Business Media New York
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Scharlau, W., Opolka, H. (1985). Preview: Reduction Theory. In: From Fermat to Minkowski. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-1867-6_10
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DOI: https://doi.org/10.1007/978-1-4757-1867-6_10
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2821-4
Online ISBN: 978-1-4757-1867-6
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