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 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) ∈ K˟ x K˟ such that x is not a norm in K(√y) is diophantine over K for any global field K of characteristic ≠ 2.