Bounds for the Representations of Integers by Positive Quadratic Forms

Abstract

SummaryRecent ground-breaking work of Conway, Schneeberger, Bhargava, and Hanke shows that to determine whether a given positive quadratic form F with integer coefficients represents every positive integer (and so is universal), it is only necessary to check that F represents all the integers in an explicitly given finite set S of positive integers. The set contains either nine or twenty-nine integers depending on the parity of the coefficients of the cross-product terms in F and is otherwise independent of F. In this article we show that F represents a given positive integer n if and only if F(y1, …, yk) = n for some integers y1, …, yk satisfying , where the positive rational numbers ci are explicitly given and depend only on F. Let m be the largest integer in S (in fact m = 15 or 290). Putting these results together we have