Sums of squares of integral linear forms (Q2707061)

For any positive integer \(n\) let \(g_{\mathbb{Z}}(n)\) be the smallest positive integer such that every positive definite \(n\)-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of \(g_{\mathbb{Z}}(n)\) squares. In the 30's Mordell and Ko showed that \(g_{\mathbb{Z}}(n)= n+3\) for \(1\leq n\leq 5\). \textit{M.-H. Kim} and \textit{B.-K. Oh} proved \(g_{\mathbb{Z}}(6)= 10\) [J. Number Theory 63, 89-100 (1997; Zbl 0873.11029)]. In the paper under review, the author shows \(g_{\mathbb{Z}}(7)\leq 25\) (improving an earlier estimate by Oh) and \(g_{\mathbb{Z}}(n)\leq 2\cdot 3^n+ n+6\) for \(n=12\) and 13 and \(g_{\mathbb{Z}}(n)\leq 3\cdot 4^n+ n+3\) for \(14\leq n\leq 20\).