Coprime values of polynomials in several variables

TL;DR

This paper explores the relationship between polynomial coprimality over multiple variables and their integer values, revealing stability properties and implications for Hilbert's irreducibility theorem.

Contribution

It establishes new stability results for gcd and lcm of polynomial values and connects coprimality at integer points with the polynomials' intrinsic coprimality.

Findings

Set of gcds of polynomial values is stable under gcd and lcm.

Polynomials with no fixed prime divisor often produce coprime values at many points.

Sufficient coprimality at integer points implies the polynomials are coprime.

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.