# gcd of polynomial values

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 $$b$$.

• The resultant should give you a first upper bound. – darij grinberg Oct 19 '18 at 0:51
• Have a look at this paper: arxiv.org/abs/1608.07936 – GH from MO Oct 19 '18 at 1:12
• 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 $r$. – José Alejandro Samper Dec 11 '18 at 14:22