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Indefinite quadratic polynomials in n variables

Published online by Cambridge University Press:  26 February 2010

D. M. E. Foster
D. M. E. Foster
Affiliation:
Bedford College, London, N.W.I.
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Extract

Let

denote an indefinite quadratic form in n variables with real coefficients and with determinant Δn≠0. Blaney ([1], Theorem 2) proved that for any γ ≥0 there is a number Γ = Γ(γ, n) such that the inequalities

are soluble in integers x1, …, xn for any real α1, …, αn The object of this note is to establish an estimate for Γ as a function of γ. The result obtained, which is naturally only significant if γ is large, is as follows.

Type
Research Article
Information
Mathematika , Volume 3 , Issue 2 , December 1956 , pp. 111 - 116
Copyright
Copyright © University College London 1956

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References

1.Blaney, H. “Indefinite quadratic forms in n variables”, Journal London Math. Soc., 23 (1948), 153–160.Google Scholar
2.Blaney, H. “Some asymmetric inequalities”, Proc. Cambridge Phil. Soc., 46 (1950), 359–376.Google Scholar
3.Macbeath, A. M., “Non-homogeneous lattices in the plane”, Quart. J. of Math. (2), 3 (1952), 268–281.CrossRefGoogle Scholar
4.Davenport, H. and Heilbronn, H., “Asymmetric inequalities for non-homogeneous linear forms”, Journal London Math. Soc., 22 (1947), 53–61.Google Scholar