Isotropic Vectors of Non-negative Integral Quadratic Forms

We say that an integral quadratic formq:Zn → Z of the shape ∑ni = 1qix(i)2 + ∑i < jqijx(i) × (j) emi-unit form ifqi ≤ 1 for every 1 ≤ i ≤ n. A deflation forqis the linear transformationT−ij:Zn → :Zndefined byes ↦ esifs ≠ jandej ↦ ej − eiifqij < 0. We show that for a non-negative semi-unit formq:Zn → Z with corankq = corank+q, there exists an iterated sequence of deflationsTsuch thatq̄ = qTsatisfiesq̄ij ≥ 0 for alli,j. Further we describe the lattice of positive isotropic vectors of the non-negative semi-unit formsqwith corankq = corank+ q = 2.