Submission date: 30 September 2016

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 Z in 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 ≠ 2 is diophantine. Finally, we show that the set of pairs (x,y) in (K^×)^2 such that x is not a norm in K(\sqrt{y}) is diophantine over K for any global field K of characteristic ≠ 2.

Mathematics Subject Classification: 11U05, 11R37

Keywords and phrases:

Full text arXiv 1609.09787: pdf, ps.