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Criteria for extreme forms

Published online by Cambridge University Press:  09 April 2009

E. S. Barnes
E. S. Barnes
Affiliation:
University of Adelaide
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A positive quadratic form , of determinant and minimum M for integral , is said to be extreme if the ratio is a (local) maximum for small variations in the coefficients .

Minkowski [3] has given a criterion for extreme forms in terms of a fundamental region (polyhedral cone) in the coefficient space. This criterion, however, involves a complete knowledge of the edges of the region and is therefore of only theoretical value.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 1 , Issue 1 , August 1959 , pp. 17 - 20
Copyright
Copyright © Australian Mathematical Society 1959

References

[1]Barnes, E. S., “On a theorem of Voronoi”, Proc. Camb. Phil. Soc. 53 (1957), 537–539.CrossRefGoogle Scholar
[2]Coxeter, H. S. M., “Extreme forms”, Canad. J. Math. 3 (1951), 391–441.CrossRefGoogle Scholar
[3]Minkowski, H., “Diskontinuitätsbereich für arithmetische Äquivalenz”, J. reine angew. Math. 129 (1905), 220–274.CrossRefGoogle Scholar
[4]Voronoï, G., “Sur quelques propriétés des formes quadratiques positive parfaites”, J. reine angew. Math. 133 (1907), 97–178.Google Scholar