On a Waring’s problem for integral quadratic and Hermitian forms

Abstract:

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 every positive definite integral quadratic form is equivalent to what we call a balanced Hermite–Korkin–Zolotarev-reduced form and use it to show that 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 on 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 on results of both Conway and Sloane and Kim and Oh on $s$-integrable lattices.