全局域的泛在和存在可定义子集
Universally and existentially definable subsets of global fields
摘要：
证明了奇特征的整体函数域$K$的$S$ -整数环在$K$中是一阶泛可定义的.这扩展了Koenigsmann和Park的工作，他们分别在$\mathbb{Q}$和数域中的整数环中对$\mathbb{Z}$进行了同样的证明。本文还证明了Poonen的一个定理，并证明了特征为$\neq 2$的整体域上的非方集是丢番图.最后，我们证明了在$(K^{\times})^2$中的对$(x,y)$的集合，使得$x$不是$K(\sqrt{y})$中的范数，对于任何具有特征$\neq 2$的全局域$K$，$(x,y)$在$K$上是丢番图。
英文摘要：
We show that rings of $S$-integers of a global function field $K$ of odd characteristic are first-order universally definable in $K$. This extends work of Koenigsmann and Park who showed the same for $\mathbb{Z}$ in $\mathbb{Q}$ and the ring of integers in a number field, respectively. We also give another proof of a theorem of Poonen and show that the set of non-squares in a global field of characteristic $\neq 2$ is diophantine. Finally, we show that the set of pairs $(x,y)$ in $(K^{\times})^2$ such that $x$ is not a norm in $K(\sqrt{y})$ is diophantine over $K$ for any global field $K$ of characteristic $\neq 2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
