  For each positive integer $n$, let $g_{\mathbb Z}(n)$ be the smallest integer
such that if an integral quadratic form in $n$ variables can be written as a
sum of squares of integral linear forms, then it can be written as a sum of
$g_{\mathbb Z}(n)$ squares of integral linear forms. We show that as $n$ goes
to infinity, the growth of $g_{\mathbb Z}(n)$ is at most an exponential of
$\sqrt{n}$. Our result improves the best known upper bound on $g_{\mathbb
Z}(n)$ which is in the order of an exponential of $n$. We also define an
analogous number $g_{\mathcal O}^*(n)$ for writing hermitian forms over the
ring of integers $\mathcal O$ of an imaginary quadratic field as sums of norms
of integral linear forms, and when the class number of the imaginary quadratic
field is 1, we show that the growth of $g_{\mathcal O}^*(n)$ is at most an
exponential of $\sqrt{n}$. We also improve results of Conway-Sloane and Kim-Oh
on $s$-integral lattices.
