The distance to square-free polynomials

We consider a variant of Turán’s problem on the distance from a polynomial in Z[x] to the nearest irreducible polynomial in Z[x]. We prove that for any f∈Z[x], there exist infinitely many square-free polynomials g∈Z[x] such that L(f−g)≤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)≤1. For this, for each integer d≥15 we construct infinitely many polynomials f∈Z[x] of degree d such that neither f itself nor any f(x)±xk, where k is a non-negative integer, is square-free. Polynomials over prime fields and their distances to square-free polynomials are also considered.