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

Coprime values of polynomials in several variables