Edge reduction for weakly non-negative quadratic forms
Let \(q: {\mathcal Z}^n \to{\mathcal Z}\) be an integral quadratic form in \(n\) variables. Here \(q\) is a semi-unit form if each coefficient on the square of a single variable is \(\leq 1\), and \(q\) is weakly nonnegative provided \(q(x) \geq 0\) for every nonzero vector \(x\) with nonnegative coordinates. The authors study a process (called edge reduction) of replacing \(q\) with an integral quadratic form \(q'\) in \(n+1\) variables that is a weakly nonnegative semi-unit form if and only if \(q\) is. (There is also a close relationship between the isotropic vectors of \(q\) and those of \(q'\).) This is developed into an algorithm to decide whether or not a given semi-unit form is weakly nonnegative. A computer implementation of the algorithm is working.
scientific article; zbMATH DE number 1185639