Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type? I'm fairly open about the exact nature of the upper bound, but I don't know where to start looking. If it helps, I'm even willing to assume that $g(x) = x^b$ for some $b$.