Hostname: page-component-6bb9c88b65-spzww Total loading time: 0 Render date: 2025-07-25T17:13:24.882Z Has data issue: false hasContentIssue false
  • English
  • Français

Minkowski's fundamental inequality for reduced positive quadratic forms

Published online by Cambridge University Press:  09 April 2009

E. S. Barnes
E. S. Barnes
Affiliation:
The University of AdelaideAdelaide, 5001Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Forms which are reduced in the sense of Minkowski satisfy the “fundamental inequality” a11a22 hellipann≤λnD; the best possible value of λn is known for n≤5. A more precise result for the minimum value of D in terms of the diagonal coefficients has been stated by Oppenheim for ternary forms. The corresponding precise result for quaternary forms is established here by considering a convex polytope D(α), defined as the intersection of the cone of reduced forms with the hyperplanes aii = αi (i = 1, hellip n).

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 1 , August 1978 , pp. 46 - 52
Copyright
Copyright © Australian Mathematical Society 1978

References

Barnes, E. S. and Cohn, M. J. (1976), “On Minkowski reduction of positive quaternary quadratic forms”, Mathematika 23, 156–158.CrossRefGoogle Scholar
Lekkerkerker, C. G. (1969), Geometry of Numbers (Wolters-Noordhoff, Groningen, 1969).Google Scholar
Mahler, K. (1938), “On Minkowski's theory of reduction of positive quadratic forms’, Quart. J. Math. 9, 259–262.CrossRefGoogle Scholar
Mahler, K. (1940), “On reduced positive definite ternary quadratic forms”, J. London Math. Soc. 15, 193–195.CrossRefGoogle Scholar
Mahler, K. (1946), “On reduced positive definite quaternary quadratic forms”, Nieuw Arch. Wiskunde (2) 22, 207–212.Google Scholar
Nelson, C. E. (1974), “The reduction of positive definite quinary quadratic forms”, Aequationes Math. 11, 163–168.CrossRefGoogle Scholar
Oppenheim, A. (1946), “A positive definite quadratic form as the sum of two positive definite quadratic forms (I)”, J. London Math. Soc. 21, 252–257.CrossRefGoogle Scholar
Van der Waerden, B. L. (1956), “Die Reduktionstheorie der positiven quadratischen Formen”, Acta Math. 96, 265–309.CrossRefGoogle Scholar
Van der Waerden, B. L. (1969), “Das Minimum von D/f 11f 12 hellip f 55 für reduzierte positive quinäre quadratische Formen”, Aequationes Math. 2, 233–247.CrossRefGoogle Scholar