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 $f(x)\in\mathbb{F}_2[x]$ 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)