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The minimum determinant of Minkowski-reduced quinary quadratic forms

Published online by Cambridge University Press:  09 April 2009

E. S. Barnes  and
D. W. Trenerry
E. S. Barnes
Affiliation:
The University of Adelaide, Adelaide, 5001, Australia
D. W. Trenerry
Affiliation:
The University of New South Wales, Broken Hill, 2880, Australia
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Abstract

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Minkowski established a lower bound for the determinant D of a Minkowski-reduced quadratic form in terms of the product of its diagonal coefficients aii (i = 1, …n). Oppenheim and Barnes found, for n = 3 and n = 4 respectively, the precise minimum of D in terms of the aii; in each case the minimum is a polynomial in the aii. Here it is shown that no such result exists when n = 5; however a polynomial in all,…, a55 is determined which gives the minimum of D when a55 is sufficiently large.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 32 , Issue 3 , June 1982 , pp. 405 - 411
Copyright
Copyright © Australian Mathematical Society 1982

References

Barnes, E. S. (1978), ‘On Minkowski's fundamental inequality for reduced positive quadratic forms (I)’, J. Austral. Math. Soc. Ser. A 26, 46–52.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.Google Scholar
Van der Waerden, B. L. (1969), ‘Das Minimum von D/f 11f 22…f 55 für reduzierte positive quinäre quadratische Formen’, Aequationes Math. 2, 233–247.Google Scholar