Positive Quadratic Forms

Abstract

An integral quadratic form is a unit form if all its diagonal coefficients are equal to one. In this chapter we study positive unit forms, that is, those integral quadratic unit forms \(q:\mathbb {Z}^n \to \mathbb {Z}\) with q(x) > 0 for any nonzero vector x in \(\mathbb {Z}^n\). A unit form q is critical nonpositive if it is not positive, but each proper restriction of q is. A vector v is called radical for q if q(v + u) = q(u) for any vector u in \(\mathbb {Z}^n\). We prove Ovsienko’s Criterion: a unit form in n ≥ 3 variables is critical nonpositive if and only if q is nonnegative with radical generated by a radical vector with no zero among its entries. One of the most important tools in the theory of integral quadratic forms, inflations and deflations, are introduced in this chapter, and are used to provide a classification of positive unit forms in terms of Dynkin types. A combinatorial characterization of such forms in terms of assemblers of graphs is also presented.