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

## TL;DR

This paper investigates the proximity of arbitrary polynomials over F_2[x] to squarefree polynomials, establishing bounds on their distance and extending results to integer polynomials.

## Contribution

It proves that any polynomial over F_2[x] can be approximated by a squarefree polynomial within a bound depending on the degree, a novel proximity result.

## Key findings

- Existence of a squarefree polynomial close to any given polynomial
- Bound on the L_2 distance involving ln(n)
- Extension of results to polynomials over Z[x]

## Abstract

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 $\mathbb{Z}[x]$.