Fundamental Concepts

Abstract

Throughout the chapter we introduce important concepts as well as basic results: given an integral quadratic form $q:\mathbb {Z}^n \to \mathbb {Z}$, numbers of the form q(v) for a vector v in $\mathbb {Z}^n$ are said to be represented by q, and the form q is said to be universal if every positive integer is represented by q. We sketch the proof of Conway and Schneeberger’s Fifteen Theorem, which states that a positive integral form with associated symmetric matrix having integer coefficients is universal if and only if it represents all positive integers up to 15. We also survey the theory of binary integral quadratic forms (due originally to Gauss), and apply it to specific examples referred to as Kronecker and Pell forms. Finally, we find general conditions for a real quadratic form $q_{\mathbb {R}}:\mathbb {R}^n \to \mathbb {R}$ to be positive (that is, $q_{\mathbb {R}}(x)>0$ for any nonzero vector x) or nonnegative ($q_{\mathbb {R}}(x)\geq 0$ for any vector x in $\mathbb {R}^n$).