Given an integer polynomial $f$, let $L(f)$ be the sum of the absolute values of the coefficients of $f$. In 1960s, Turán asked whether there exists an absolute constant $C$ such that for any integer polynomial $f$ of degree $d$, there is an irreducible integer polynomial $g$ of degree at most d satisfying $L(f-g) < C$. Turán's problem remains open, although a number of partial results have been obtained.
In this talk, I will present some recent work on a variant of Turán's problem. For example, we prove that for any integer polynomial $f$, there exist infinitely many square-free integer polynomials $g$ such that $L(f-g) < 3$. On the other hand, we show that this inequality cannot be replaced by $L(f-g) < 2$. (This is joint work with Artūras Dubickas)