The distance to square-free polynomials
Volume 186 / 2018
Abstract
We consider a variant of Turán’s problem on the distance from a polynomial in $\mathbb Z[x]$ to the nearest irreducible polynomial in $\mathbb Z[x]$. We prove that for any $f \in \mathbb Z[x]$, there exist infinitely many square-free polynomials $g\in \mathbb Z[x]$ such that ${L(f-g) \le 2}$, where $L(f-g)$ denotes the sum of the absolute values of the coefficients of $f-g$. On the other hand, we show that this inequality cannot be replaced by $L(f-g) \le 1$. For this, for each integer $d \geq 15$ we construct infinitely many polynomials $f \in \mathbb Z[x]$ of degree $d$ such that neither $f$ itself nor any $f(x) \pm x^k$, where $k$ is a non-negative integer, is square-free. Polynomials over prime fields and their distances to square-free polynomials are also considered.
