Let $P$ and $Q$ be monic polynomials with integer coefficients and degrees $n$ and $d$ respectively, where $d\mid n$. Suppose there are infinitely many pairs of positive integers $(a,b)$ for which $P(a)=Q(b)$.
I would like to determine if exists a polynomial $R$ with integer coefficients such that $$P(x)=Q(R(x))$$
The second half of polynomials such that $P(k)=Q(l)$ for all integer $k$ is related though the condition here is weaker. I suspect the answer is yes (for polynomials, I've seen often that if some property occurs infinitely often then it occurs always).
My guess would be that we somehow construct a polynomial related to $P$ and $Q$ that ends up having infinitely many roots because of the infinitely many pairs $(a,b)$, so that we can force $P$ to conform to some sort of polynomial in $Q$. I'm not quite sure what to make of the $d\mid n$ condition; perhaps this could be strengthened? I haven't been able to find a counterexample that forces the divisibility.