论文标题
将毕达哥拉斯的字段数量限制在$ 2^n+1 $
Bounding the Pythagoras number of a field by $2^n+1$
论文作者
论文摘要
给定一个正整数$ n $，在一个字段上，将其毕达哥拉斯号码限制为$ 2^n+1 $。 $ n = 1 $的条件按曲线的功能字段来满足$ \ mathbb {r} $的迭代形式功率系列字段，以及$ \ mathbb {r}（\！（t_0，t_1，t_1）\！）$的有限字段扩展。在这两种情况下，一个人都可以在毕达哥拉斯号码上检索上限$ 3 $。这里介绍的新方法可能有助于建立更普遍的$ 2^n+1 $，作为pythagoras的上限，用于pythagoras的曲线功能字段数量$ \ mathbb {r}（\！（\！（t_1，\ dots，dots，t_n）\！）$，以及$ \ \ m m mathbb {r}（t_ r}（t_ $}（t_ t_ t _ t _ t_t_），
Given a positive integer $n$, a sufficient condition on a field is given for bounding its Pythagoras number by $2^n+1$. The condition is satisfied for $n=1$ by function fields of curves over iterated formal power series fields over $\mathbb{R}$, as well as by finite field extensions of $\mathbb{R}(\!(t_0,t_1)\!)$. In both cases, one retrieves the upper bound $3$ on the Pythagoras number. The new method presented here might help to establish more generally $2^n+1$ as an upper bound for the Pythagoras number of function fields of curves over $\mathbb{R}(\!(t_1,\dots,t_n)\!)$ and for finite field extensions of $\mathbb{R}(\!(t_0,\dots,t_n)\!)$.
