Is it possible to show that there is a simple formula, preferably existential, that characterizes a nonstandard model of the ring of integers among the elements of $\prod\_p \mathbb{F}\_p(t)/\mathcal{U}$ where $\mathcal{U}$ is a nonprincipal ultrafilter over the prime numbers?


Thank you



Defining $\mathbb{Z}$ in $\prod_p \mathbb{F}_p(t)/\mathcal{U}$
No, such a formula does not exist. If $R$ is a definable subset of $\prod\_p\mathbb F\_p(t)/\cal U$ which contains $1$ and is closed under addition, then by the argument Joel David Hamkins gave in [his answer](https://mathoverflow.net/questions/86918#86918) to your previous question, we must have $R\supseteq S:=\prod\_p\mathbb F\_p/\cal U$. In $S$, every element is a sum of two squares (because this holds in all finite fields). It follows that any integer (e.g., $-1$) is a sum of two squares in $R$, hence $R$ does not satisfy the universal theory of $\mathbb Z$.


