Assume that we have two rational expressions $f,g\in \mathbb{Z}(x,y_1,\ldots,y_n)$ with the property that the variable $x$ can only appear in the numerator of $f$ while only in the denominator of $g$. Is it possible to have
$$f = g$$
as functions over the finite field $\mathbb{F}_p$ of order $p$ for all but finitely many $p$ assuming that $x$ appears in either expression (with the given constraints).
It's trivial that such $f,g$ containing $x$ can't be found when working over $\mathbb{C}$ or $\mathbb{Z}$, which is evident by letting $x\to\infty$. Therefore, I'm assuming that there is some nice theorem that can be used to translate this to a result that holds for almost $p$.
In other words, I assume that for any such $f,g$ containing $x$, there is some $N>0$ s.t. $f\neq g$ as functions over $\mathbb{F}_p$ for $p>N$. To be even more precise, I could even live with for infinitely many $p$ instead of $p>N$ if the stronger condition turns out to be false.