Isotropic vectors of non-negative integral quadratic forms
An integral quadratic form \[ q(x)=\sum_{i=1}^nq_ix(i)^2 + \sum_{i<j}q_{ij}x(i)x(j) \] is said to be a unit or semi-unit form if \(q_i=1\) or \(q_i\leq 1\), respectively, for every \(1\leq i\leq n\). The linear transformation \(T_{ij}^- :{\mathbb Z}^n \rightarrow {\mathbb Z}^n\) defined by \(e_s \mapsto e_s\) if \(s\neq j\), \(e_j\mapsto e_j-e_i\) is a deflation for \(q\) if \(q_{ij}<0\). A non-negative unit form \(q\) with \(\text{corank} q=t\) is said to have positive corank \(\text{corank}^+ q=s\) if \(s\leq t\) is maximal such that there are isotropic vectors \(v_1\), \dots, \(v_s\) of \(q\) with non-negative coordinates and which are linearly independent. It is shown that for a non-negative semi-unit form \(q\) with \(\text{corank} q=\text{corank}^+ q\), there exists an iterated sequence of deflations \(T\) such that \(\overline{q}=qT\) satisfies \(\overline{q}_{ij}\geq 0\) for all \(i\), \(j\). In particular, if \(\text{corank} q=\text{corank}^+ q=s\), then the above mentioned form \(\overline{q}\) is the orthogonal sum \(\overline{q}^0\oplus \overline{q}^1\), where \(\overline{q}^0\) is the zero form in \(s\) variables and \(\overline{q}^1\) is a positive unit form in \(n-s\) variables. Further the authors describe the lattice of positive isotropic vectors of the non-negative semi-unit forms \(q\) with \(\text{corank} q=\text{corank}^+ q=2\).
scientific article; zbMATH DE number 1199947