Hermite constant and extreme forms for algebraic number fields. (English) Zbl 0874.11047

The classical Hermite constant \(\gamma_n\) is generalized to algebraic number fields \(K\) (notation \(\gamma_{n,k}\)) and a generalization of the Minkowski bound for \(\gamma_n\) to \(\gamma_{n,k}\) is given. Furthermore, the notion of extreme forms is generalized to algebraic number fields. The existence of such extreme forms is established and it is shown that they are perfect and eutactic. In the proofs Humbert’s reduction theory [P. Humbert, Comment. Math. Helv. 23, 50-63 (1949; Zbl 0034.31102)] is used.


11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11E12 Quadratic forms over global rings and fields


Zbl 0034.31102
Full Text: DOI