A note on the squarefree density of polynomials

Abstract

The conjectured squarefree density of an integral polynomial $$ in $$ variables is an Euler product $$ which can be considered as a product of local densities. We show that a necessary and sufficient condition for $$ to be 0 when $$ is a polynomial in $$ variables over the integers, is that either there is a prime $$ such that the values of $$ at all integer points are divisible by $$ or the polynomial is not squarefree as a polynomial. We also show that generally the upper squarefree density $$ satisfies $$.