Squarefree values of multivariable polynomials

Abstract

Given $f\in \mathbf {Z}[x\sb 1,\ldots x\sb n]$, we compute the density of $x\in \mathbf {Z}\sp n$ such that $f(x)$ is squarefree, assuming the abc-conjecture. Given $f,g\in \mathbf {Z}[x\sb 1,\ldots x\sb n]$, we compute unconditionally the density of $x\in \mathbf {Z}\sp n$ such that $\gcd(f(x),g(x))=1$. Function field analogues of both results are proved unconditionally. Finally, assuming the abc-conjecture, given $f\in \mathbf {Z}[x]$, we estimate the size of the image of $f(\{1,2,\ldots n\})$ in $(\mathbf {Q}\sp \ast/\mathbf {Q}\sp {\ast 2})\cup \{0\}$.