Weakly Positive 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 positive unit forms, and their sets of positive roots R ⁺(q). A characterization of critical unit forms, those forms that fail to be weakly positive but all their restrictions are, is presented. We give criteria to identify weakly positive forms, for instance finiteness of the set of positive roots R ⁺(q) (due to Drozd and Happel), and Zeldych’s Theorem (looking for properties of principal submatrices of the symmetric matrix associated to q). Ovsienko’s Theorem is also proved, setting 6 as a bound for the entries of all positive roots of a weakly positive unit form. Those forms with a positive root reaching bound 6 are studied, following Ostermann and Pott. A classification of thin forms due to Dräxler, Drozd, Golovachtchuk, Ovsienko and Zeldych is sketched, as well as a procedure to find all weakly positive unit forms starting with (good) thin forms.