Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T14:06:56.770Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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