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Let $p(x),q(x)$ be coprime squarefree polynomials with integer coefficients.

For integer $n$ is $\gcd(p(n),q(n))$ bounded by an absolute constant?

In case the answer is negative what is the fastest growing function $f(n,\deg(p(x)),\deg(q(x)))$ such that $\gcd(p(n),q(n)) \ge f(n,\deg(p(x)),\deg(q(x)))$ infinitely often?

According to another question, $q(n) \mid p(n)$ can't happen infinitely often unless $q(x)$ is constant.

For linear polynomials the gcd is bounded.

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    $\begingroup$ Sure: just use Bezout's identity. You have $A(x)p(x) + B(x)q(x) = C$ with $A,B \in \mathbb{Z}[x]$ and $C \in \mathbb{Z} \setminus \{0\}$, and then the GCD of any value has to divide $C$. $\endgroup$
    – Vesselin Dimitrov
    Commented May 29, 2014 at 13:08
  • $\begingroup$ The minimal choice of $C$ is called the resultant of $p(x)$ and $q(x)$. $\endgroup$
    – Jeremy Rouse
    Commented May 29, 2014 at 13:16
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    $\begingroup$ @JeremyRouse No. See: mathoverflow.net/questions/17501/… $\endgroup$
    – Felipe Voloch
    Commented May 29, 2014 at 15:05
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    $\begingroup$ Oops. Certainly the number $C$ can be taken to be the resultant, but the resultant need not be minimal. $\endgroup$
    – Jeremy Rouse
    Commented May 29, 2014 at 15:29

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