  In this paper, we examine how far a polynomial in $\mathbb{F}_2[x]$ can be
from a squarefree polynomial. For any $\epsilon>0$, we prove that for any
polynomial $f(x)\in\mathbb{F}_2[x]$ with degree $n$, there exists a squarefree
polynomial $g(x)\in\mathbb{F}_2[x]$ such that $\mathrm{deg} (g) \le n$ and
$L_{2}(f-g)<(\ln n)^{2\ln(2)+\epsilon}$ (where $L_{2}$ is a norm to be
defined). As a consequence, the analogous result holds for polynomials $f(x)$
and $g(x)$ in $\mathbb{Z}[x]$.
