论文标题
在Henselian离散价值字段上的功能字段中的正方金总和
Sums of squares in function fields over henselian discretely valued fields
论文作者
论文摘要
令$ n \ in \ mathbb {n} $，让$ k $成为一个字段，具有Henselian离散估值的等级$ n $，带有遗传性欧几里得残留物。令$ f/k $为一个变量中的代数函数字段。我们表明，Pythagoras的$ f $为$ 2 $或$ 3 $，我们确定了非零的平方和平方模型总和在$ f $中的两个平方和在$ f $的等价类别的数量上，最多$ f $n。$n。$ n。 Becher和J. Van Geel表明，该商组的顺序由$ 2^{n（g+1）} $界定。我们在示例4.6中显示该界限是最佳的。
Let $n\in\mathbb{N}$ and let $K$ be a field with a henselian discrete valuation of rank $n$ with hereditarily euclidean residue field. Let $F/K$ be an algebraic function field in one variable. We show that the Pythagoras number of $F$ is $2$ or $3$ and we determine the order of the group of nonzero sums of squares modulo sums of two squares in $F$ in terms of the number of equivalence classes of discrete valuations on $F$ of rank at most $n.$ In the case of function fields of hyperelliptic curves of genus $g$, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by $2^{n(g+1)}$. We show in Example 4.6 that this bound is optimal.
