Computational experiences on the distances of polynomials to irreducible polynomials

Abstract:

In this paper we deal with a problem of Turán concerning the ‘distance’ of polynomials to irreducible polynomials. Using computational methods we prove that for any monic polynomial $P\in$ ${\mathbb {Z}}[x]$ of degree $\leq 22$ there exists a monic polynomial $Q\in {\mathbb {Z}}[x]$ with deg($Q$) = deg($P$) such that $Q$ is irreducible over $\mathbb {Q}$ and the ‘distance’ of $P$ and $Q$ is $\leq 4$.