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 .