  In this paper we prove that polynomials $F(x_1, \cdots, x_n) \in
\mathbb{Z}[x_1, \cdots, x_n]$ of degree $d \geq 3$, satisfying certain
hypotheses, take on the expected density of $(d-1)$-free values. This extends
the authors' earlier result where a different method implied the similar
statement for polynomials of degree $d\geq 5$.
