On the Product of Three Homogeneous Linear Forms. IV

Let L 1 , L 2 , L 3 be three homogeneous linear forms in u, v, w with real coefficients and determinant 1. Let M denote the lower bound of for integral values of u, v, w, not all zero. I proved a few years ago ( 1 ) that more precisely, that except when L 1 , L 2 , L 3 are of a special type, in which case If we denote by θ, ø, ψ the roots of the cubic equation t 3 + t 2 -2 t -1 = 0, the special linear forms are equivalent, by an integral unimodular linear transformation, to (in any order), where λ 1 ,λ 2 ,λ 3 are real number whose product is In this case, L 1 L 2 L 3 |λ 1 λ 2 λ 3 is a non-zero integer, and the minimum of its absolute value is 1, giving