Abstract

If S = (Si, . . . , Sm) is an m-tuple of n x n positive definite symmetric real matrices (= Humbert form) and K is a real number field of degree m with ring of integers script O signK, a vector u ? script O signKn is called minimal if S[u] = Min{S[?]| ? ? script O signKn - {0}}, where S[?] = ?Si[?i] and ?i = i-th conjugate of ?. In this note we show that any Humbert form Si has a unimodular minimal vector over K if and only if the class number of K is 1. In [2] we introduced a constant MK,n which measures the non-unimodularity of minimal vectors. We estimate here the constant M K,2 in terms of known constants of K. As a by-product we obtain a lower bound for the classical Hermite constant ?2m.