The distance to a squarefree polynomial over $\mathbb F_2[x]$
Volume 193 / 2020
Acta Arithmetica 193 (2020), 419-427
MSC: Primary 11C08; Secondary 11T06.
DOI: 10.4064/aa190618-30-9
Published online: 13 February 2020
Abstract
We examine how far a polynomial in $\mathbb {F}_2[x]$ can be from a squarefree polynomial. For any $\epsilon \gt 0$, 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 $\deg g \le n$ and $L_{2}(f-g) \lt (\ln n)^{2\ln 2+\epsilon }$ (where $L_{2}$ is a norm to be defined). As a consequence, the analogous result holds for polynomials $f(x)$ and $g(x)$ in $\mathbb Z[x]$.
