Bounds for the Representations of Integers by Positive Quadratic Forms

摘要

Recent ground-breaking work of Conway, Schneeberger, Bhargava, and Hanke shows that to de?termine 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（y_1, ..., y_k）=n for some integers y_1, ..., y_k satisfying|y_i|≤ cin, i=1, ..., k, where the positive rational numbers c, 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 F is universal if and only if S ∈ {,F（y_1, ..., y_k） | |y_i|