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For each positive integer $n$, let $g_{\mathbb Z}(n)$ be the smallest integersuch that if an integral quadratic form in $n$ variables can be written as asum 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 $n$ goesto 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 $g_{\mathbbZ}(n)$ which is in the order of an exponential of $n$. We also define ananalogous number $g_{\mathcal O}^*(n)$ for writing hermitian forms over thering of integers $\mathcal O$ of an imaginary quadratic field as sums of normsof integral linear forms, and when the class number of the imaginary quadraticfield is 1, we show that the growth of $g_{\mathcal O}^*(n)$ is at most anexponential of $\sqrt{n}$. We also improve results of Conway-Sloane and Kim-Ohon $s$-integral lattices.

On a Waring's problem for integral quadratic and hermitian forms