Universally and existentially definable subsets of global fields

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 $(x, y) \in K^{\times} \times K^{\times}$ such that $x$ is not a norm in $K(\sqrt{y})$ is diophantine over $K$ for any global field $K$ of characteristic $\neq 2$.

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The first author was partially supported by National Science Foundation grant DMS-1056703. The second author was partially supported by National Science Foundation grants DMS-1056703 and CNS-1617802.

Received 30 September 2016

Accepted 25 March 2017

Published 16 November 2018