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On the minimal points of perfect septenary quadratic forms

Published online by Cambridge University Press:  26 February 2010

G. L. Watson
G. L. Watson
Affiliation:
University College, London.
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Extract

The object of this paper is to prove the following:

Theorem. Every perfect septenary quadratic form assumes its minimum value at a set of 7 points with integer co-ordinates whose determinant is 1.

This is true also, as shown by Rankin [1], with n ≤ 6 in place of 7. The proof will be shortened considerably by using the weaker result obtained in [1] for n = 7, and we shall also use the following classical results, see, e.g., [2], for Hermite's constant γn:

Type
Research Article
Information
Mathematika , Volume 16 , Issue 2 , December 1969 , pp. 170 - 177
Copyright
Copyright © University College London 1969

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References

1.Rankin, R. A., “On the minimal points of perfect quadratic forms”, Math. Zeit., 84 (1964), 228–232.CrossRefGoogle Scholar
2.Korkine, A. et Zolotareff, G., “Sur les formes quadratiques positives”, Math. Annalen, 11 (1877), 242–292.CrossRefGoogle Scholar
3.Mordell, L. J., “Observation on the minimum of a positive quadratic form in eight variables”, Journal London Math. Soc, 19 (1944), 3–6.CrossRefGoogle Scholar
4.Davenport, H. and Watson, G. L., “The minimal points of a positive definite quadratic form”, Mathematika, 1 (1954), 14–17.CrossRefGoogle Scholar