论文标题
减少Hermite和一个数字字段的整体二次形式的警告问题
Hermite reduction and a Waring's problem for integral quadratic forms over number fields
论文作者
论文摘要
我们将Hermite-Korkin-Zolotarev（HKZ）还原理论比\ \ Mathbb Q $及其在Beli-chan-icaza-liu最近引入的均衡版本上，以对完全实数的$ k $ k $。我们运用平衡的HKZ还原理论来研究$ k $的整数环的{\ em $ g $ -invariants}的增长。更准确地说，对于每个正整数$ n $，让$ \ Mathcal o $为$ k $的整数和$ g _ {\ Mathcal o}（n）$是最小的整数，以至于每$ n $ n $ -n $ -ar-ary $ -ary $ $ $ linear-linear forms of $ $ - linear forms of $ g g g g _ c _ \ cc s ccal（ $ n $ -ary $ \ Mathcal o $ - 线性表单。我们表明，当$ k $具有1类时，$ g _ {\ mathcal o}（n）$的增长最多是$ \ sqrt {n} $的指数。这扩展了Beli-chan-icaza-liu获得的最新结果，该结果$ g _ {\ Mathbb z}（n）$，并为$ g _ {\ mathcal o}（n）$提供了第一个子指数上限，用于整数的圈环$ \ mathcal $ \ sathcal o $ $ $ $ $ $ $ \ mathbb z $。
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 $\mathbb Z$.
