Coprime values of polynomials in several variables

Abstract

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.