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Bounded representations of the positive values of an indefinite quadratic form

Published online by Cambridge University Press:  24 October 2008

B. Lawton
B. Lawton
Affiliation:
Bedford CollegeLondon, N.W.1
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Extract

Let

be a real quadratic form in n variables (n ≥ 2) with integral coefficients and determinant D = |fij| ≠ 0. Cassels ((1),(2)) has recently proved that if the equation f = 0 is properly soluble in integers x1, …, xn, then there is a solution satisfying

where F = max | fij and cn depends only on n. An example given by Kneser (see (2)) shows that the exponent ½(n – 1) is best possible. A simpler proof of Cassels's result has since been given by Davenport(3), and the theorem has been improved in certain cases by Watson(4). Here I consider the inequality f(x1, …, xn) > 0, where f is an indefinite form, and obtain a result analogous to that of Cassels.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 54 , Issue 1 , January 1958 , pp. 14 - 17
Copyright
Copyright © Cambridge Philosophical Society 1958

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References

REFERENCES

(1)Cassels, J. W. S.Proc. Camb. Phil. Soc. 51 (1955), 262–4.CrossRefGoogle Scholar
(2)Cassels, J. W. S.Proc. Camb. Phil. Soc. 52 (1956), 604.CrossRefGoogle Scholar
(3)Davenport, H.Proc. Camb. Phil. Soc. 53 (1957), 539–40.CrossRefGoogle Scholar
(4)Watson, G. L.Proc. Camb. Phil. Soc. 53 (1957), 541–3.CrossRefGoogle Scholar
(5)Mirsky, L.Introduction to linear algebra (Oxford, 1955).Google Scholar