Representations of integral quadratic forms by sums of squares

Abstract

Let g[n] be the minimum number of squares whose sum represents all positive definite integral quadratic forms of rank n which are represented by sums of squares. In this article, we first discuss representations of integers by unimodular lattices. We then estimate the order of magnitude of the diameter of the 2-graph of unimodular lattices of rank n. Combining these results we prove g[n]=O(3 n /2n log n). We also provide a lower bound for g[n]. Finally, we discuss s-integrable lattices as an application of our method.