Abstract
Let f(x
1
..., x
6
)be a positive definite senary quadratic form of determinant
D
. Let
M
be its minimum value for integers x
1
..., x
6
, not all zero. The form is said to be extreme if, for all infinitesimal variations of the coefficients,
M
6
/
D
is maximum. It is proved here for the first time that there are exactly six classes of extreme senary forms, namely, the classes containing the six forms denoted by 0
O
, 0
4
and 0
6
. (Another form 0
5
is shown to be only ‘perfect’, not extreme.) The forms 0
0
0
1
0
2
0
4
are equivalent to A
6
D
6
E
6
E
3/6
, in the notation of Coxeter (1951, p. 394); 0
3
was discovered simultaneously by M. Kneser and the author (1955); 0
6
is new. Although the analogous forms in fewer variables have been known since 1877, the only previous enumeration of extreme forms in six variables was by Hofreiter (1933), who missed 0
3
0
4
0
6
and proposed instead an incorrect form which he called
F
4
.
Reference12 articles.
1. Bachm ann P. 1923 Die Arithmetik der quadratischen Formen 2 . Leipzig: Teubner.
2. Barnes E. S. 1955 Canad.
3. Blichfeldt H . F. 1935 Math.
4. C haundy T . W. 1946 Quart.
5. Goxeter H . S. M . 1951 Canad.
Cited by
37 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献