Two consequences of Minkowski's 2n theorem

AbstractConsider the inequalities (a)||⩽b,A∈Rr × nr, r < n, b positive vector (here |y| denotes the vector of absolute values of components of the vector y) and xTAx⩽λ,Apositive semi-definite∈Rn × nr, r < n, λ>0Both 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 XN(A) on the null space of A. We prove that a nonzero integer solution x exists with |xN(A)| bounded, for (a): ‖XN(A)‖⩽n−rvolAb1…br1(n−r) for (b): ‖XN(A)‖⩽2nvolAλr2Kn1(n−r) where volA=ϵdet2AIJ summing over all r × r submatrices AIJ, and Kn is the volume of the Euclidean unit ball in Rn