Let $k$ be a number fields and let $f,g \in k[x]$. Considering $f:k \to k$ and $g: k \to k$ as functions, what statements can we make about the set $f(k) \cap g(k)$? I had some ideas but I'm hoping there is a nice characterization or criterion for the intersection to be finite/infinite.
When $f$ is linear, $f(k) \cap g(k) = g(k)$.
One can also construct examples where there will be no common solutions: take $a \in K$ with $\sqrt{a} \not \in k$, then $f(x) = ax^2$ and $g(x) = x^2$. I expect this to be the generic condition since one can get finiteness using Faltings' theorem / Mordell conjecture. If $p \in f(k) \cap g(k)$ has $x,y \in k$ such that $f(x) - g(y) = 0$, so if $f(x) - g(y)$ has no low degree irreducible factors, the solution set will be finite by the genus degree formula. Ensuring there are no such factors seems general, but I'd live some sort of criterion.