Two consequences of Minkowski's 2 n theorem

Abstract

Consider the inequalities (a) ||⩽b,A∈ R r × n r, r < n, b positive vector (here |y| denotes the vector of absolute values of components of the vector y) and x TAx⩽λ,A positive semi-definite∈ R n × n r, r < n, λ>0 Both inequalities are guaranteed a nonzero integer solution x for every positive right-hand side ( b, α respectively). Such solutions will generally have a nonzero orthogonal projection X N( A) on the null space of A. We prove that a nonzero integer solution x exists with | x N( A) | bounded, for (a): ‖X N(A)‖⩽ n−r volA b 1…b r 1 (n−r) for (b): ‖X N(A)‖⩽ 2 n volA λ r 2 K n 1 (n−r) where volA= ϵ det 2A IJ summing over all r × r submatrices A IJ , and K n is the volume of the Euclidean unit ball in R n .