Universally and existentially definable subsets of global fields

Abstract

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 $\neq 2$.