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Non-negative values of quadratic forms

Published online by Cambridge University Press:  09 April 2009

R. T. Worley
R. T. Worley
Affiliation:
Monash University, Clayton, Victoria
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In a paper [1] of the same title Barnes considered the problem of finding an upper bound for the infimum m+(f) of the non-negative values1 of an indefinite quadratic form f in n variables, of given determinant det(f) ≠ 0 and of signature s. In particular it was announced (and later proved — see [2]) that m+(f) ≦ (16/5)+ for ternary quadratic forms of determinant 1 and signature — 1. A simple consequence of this result is that m+(f) ≦ (256/135)+ for quaternary quadratic forms of determinant — 1 and signature — 2.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 2 , May 1971 , pp. 224 - 238
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Barnes, E. S., ‘The non-negative values of quadratic forms’, Proc. London. Math. Soc. (3) 5 (1955) 185–196.CrossRefGoogle Scholar
[2]Barnes, E. S. and Oppenheim, A., ‘The non-negative values of a ternary quadratic form’, J. Lond. Math. Soc. 30 (1955) 429–439.Google Scholar
[3]Oppenheim, A., ‘Value of quadratic forms I, Quart. J. Math. (Ox)(2) 4 (1953) 54–59.CrossRefGoogle Scholar
[4]Oppenheim, A., ‘Minima of indefinite quaternary quadratic forms’, Ann. Math. 32 (1931) 271–298.CrossRefGoogle Scholar
[5]Segré, B., Lattice points in infinite domains and asymmetric diophantine approximations’, Duke Math. J. 12 (1945) 337–365.CrossRefGoogle Scholar
[6]Worley, R. T., ‘Asymmetric minima of indefinite ternary quadratic forms’, J. Aust. Math. Soc. 7 (1967) 191–228.CrossRefGoogle Scholar
[7]Worley, R. T., ‘Minimum determinant of asymmetric quadratic forms’, J. Aust. Math. Soc. 7 (1967) 177–190.CrossRefGoogle Scholar