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 {\em $g$-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 $\mathbb Z$.

中文翻译：

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数域上积分二次形式的 Hermite 约简和 Waring 问题

我们将 $\mathbb Q$ 上正定二次型的 Hermite-Korkin-Zolotarev (HKZ) 归约理论及其最近由 Beli-Chan-Icaza-Liu 引入的平衡版本推广到全实数域 $ 上的正定二次型千元。我们应用平衡 HKZ 约简理论来研究 $K$ 整数环的 {\em $g$-invariants} 的增长。更准确地说，对于每个正整数 $n$，令 $\mathcal O$ 是 $K$ 的整数环，$g_{\mathcal O}(n)$ 是最小的整数，使得 $ 的每个平方和n$-ary $\mathcal O$-线性形式必须是 $n$-ary $\mathcal O$-线性形式的 $g_{\mathcal O}(n)$ 平方和。我们表明，当 $K$ 的类数为 1 时，$g_{\mathcal O}(n)$ 的增长至多是 $\sqrt{n}$ 的指数。