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 $b$.
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3$\begingroup$ The resultant should give you a first upper bound. $\endgroup$ – darij grinberg Oct 19 '18 at 0:51

2$\begingroup$ Have a look at this paper: arxiv.org/abs/1608.07936 $\endgroup$ – GH from MO Oct 19 '18 at 1:12

$\begingroup$ You can run the Euclidean algorithm of $\mathbb{Q}$ and clear denominators to get a universal bound: there are integers p,q,r so that pf(x) +qg(x) =r and if you evaluate at $a$ the gcd f(a) and g(a) must divide $r$. $\endgroup$ – José Alejandro Samper Dec 11 '18 at 14:22