Royal Society Publishing

On the Product of Three Homogeneous Linear Forms and Indefinite Ternary Quadratic Forms

J. W. S. Cassels, H. P. F. Swinnerton-Dyer

Abstract

Isolation theorems for the minima of factorizable homogeneous ternary cubic forms and of indefinite ternary quadratic forms of a new strong type are proved. The problems whether there exist such forms with positive minima other than multiples of forms with integer coefficients are shown to be equivalent to problems in the geometry of numbers of a superficially different type. A contribution is made to the study of the problem whether there exist real $\phi $, $\psi $ such that x|$\phi $x-y||$\psi $x-z| has a positive lower bound for all integers x > O, y, z. The methods used have wide validity.

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