Hermite reduction and a Waring’s problem for integral quadratic forms over number fields

We generalize the Hermite-Korkin-Zolotarev (HKZ) reduction theory of positive definite quadratic forms over $\mathbb Q$ and its balanced version introduced recently by Beli-Chan-Icaza-Liu to positive definite quadratic forms over a totally real number field $K$. We apply the balanced HKZ-reduction theory to study the growth of the $g$-invariants of the ring of integers of $K$. More precisely, for each positive integer $n$, let $\mathcal {O}$ be the ring of integers of $K$ and $g_{\mathcal {O}}(n)$ be the smallest integer such that every sum of squares of $n$-ary $\mathcal {O}$-linear forms must be a sum of $g_{\mathcal {O}}(n)$ squares of $n$-ary $\mathcal {O}$-linear forms. We show that when $K$ has class number 1, the growth of $g_{\mathcal {O}}(n)$ is at most an exponential of $\sqrt {n}$. This extends the recent result obtained by Beli-Chan-Icaza-Liu on the growth of $g_{\mathbb Z}(n)$ and gives the first sub-exponential upper bound for $g_{\mathcal {O}}(n)$ for rings of integers $\mathcal {O}$ other than $\mathbb Z$.