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On the minimum of a positive quadratic form in n ( ≤ 8) variables (verification of Blichfeldt's calculations)

Published online by Cambridge University Press:  24 October 2008

G. L. Watson
G. L. Watson
Affiliation:
University CollegeLondon
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Extract

Let f = f(x1, …, xn) be a positive definite quadratic form in n ( ≤ 8) variables; then we consider the old problem of estimating the minimum of f (its least value for integers xi not all 0) in terms of the determinant Δ(f). Normalizing by supposing the minimum to be 1, the known results may be stated as

each inequality being best possible. Further, for each n, all forms for which equality holds in (1) have integral coefficients and are equivalent to each other.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 4 , October 1966 , pp. 719
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

REFERENCES

(1)Barnes, E. S.Philos. Trans. Roy. Soc. London Ser. A 249 (1956–1957), 461–506.Google Scholar
(2)Blichfeldt, H. F.Math. Z. 39 (1935), 1–15.CrossRefGoogle Scholar
(3)Mordell, L. J.J. London Math. Soc. 19 (1944), 3–6.CrossRefGoogle Scholar