Let $f,g \in \mathbb{Z}[x],\deg(f),\deg(g)>1$ and $f$ is monic.


Assume $f$ and $g$ are coprime.


For integer $a$ is it possible $g(a) \mid f(a)$ many times?


Is it possible unbounded number of times for fixed degree?



$g(a)$ divides monic $f(a)$ many times, is this possible?
If the polynomials $f,g$ are coprime, then there are integer polynomials $f\_1,g\_1$ such that $d=ff\_1+gg\_1$ is a nonzero constant (it's true if we allow $f\_1,g\_1$ to have rational coefficients, then we just clear denominators). If $g(a)\mid f(a)$, then $g(a)\mid d$, from where it's clear that for fixed pair $f,g$ there are only finitely many $a$ like you request.


If we only bound the degree, we can have arbitrarily many divisibilities even in the simplest case $\deg f=\deg g=2$. Namely take $f(x)=x^2+(n^2)!,g(x)=x^2$ for arbitrary $n$. Then for $1\leq a\leq n$, $g(a)=a^2\mid a^2+(n^2)!=f(a)$.


