The Distance to a Squarefree Polynomial Over $mathbb{F}_2[x]$

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 $f(x)inmathbb{F}_2[x]$ with degree $n$, there exists a squarefree polynomial $g(x)inmathbb{F}_2[x]$ such that $mathrm{deg} (g) le n$ and $L_{2}(f-g)<(ln n)^{2ln(2)+epsilon}$ (where $L_{2}$ is a norm to be defined). As a consequence, the analogous result holds for polynomials $f(x)$ and $g(x)$ in $mathbb{Z}[x]$.