  Given two polynomials $P(\underline x)$, $Q(\underline x)$ in one or more
variables and with integer coefficients, how does the property that they are
coprime relate to their values $P(\underline n), Q(\underline n)$ at integer
points $\underline n$ being coprime? We show that the set of all $\gcd
(P(\underline n), Q(\underline n))$ is stable under gcd and under lcm. A
notable consequence is a result of Schinzel: if in addition $P$ and $Q$ have no
fixed prime divisor (i.e., no prime dividing all values $P(\underline n)$,
$Q(\underline n)$), then $P$ and $Q$ assume coprime values at "many" integer
points. Conversely we show that if "sufficiently many" integer points yield
values that are coprime (or of small gcd) then the original polynomials must be
coprime. Another noteworthy consequence of this paper is a version over the
ring of integers of Hilbert's irreducibility theorem.
