Weakly Nonnegative Quadratic Forms

Abstract

In this chapter we analyze integral quadratic forms $q:\mathbb {Z}^n \to \mathbb {Z}$ satisfying q(x) ≥ 0 for any positive vector x in $\mathbb {Z}^n$, so-called weakly nonnegative semi-unit forms. Here a prominent role is played by maximal and locally maximal positive roots of q, which can be used to characterize weak nonnegativity. We also describe hypercritical semi-unit forms, those forms not weakly nonnegative such that any proper restriction is weakly nonnegative. Diverse criteria for weak nonnegativity are provided, including Zeldych’s Theorem and a few algorithms using iterated edge reductions, following von Höhne and de la Peña. A generalization of Ovsienko’s Theorem due to Dräxler, Golovachtchuk, Ovsienko and de la Peña is proved in the last section, for which Ringel’s concepts of graphical and semi-graphical forms are essential.