Mathematical Research Letters

Volume 25 (2018)

Number 4

Universally and existentially definable subsets of global fields

Pages: 1173 – 1204

DOI: http://dx.doi.org/10.4310/MRL.2018.v25.n4.a6

Authors

Kirsten Eisenträger (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Travis Morrison (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

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 $(x, y) \in K^{\times} \times K^{\times}$ 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