On the concentration of divisors of powers

For integer n≥ 1 and real u, let ∆(n,u):=|{d:d | n, \e^u < d≤ \e^{u+1}}|. The Erdős--Hooley Delta-function is then defined by ∆(n):=\max_{u \in R}∆(n,u).$ For any fixed integer r ≥ 2 and any irreducible polynomial F\in Z[X], we estimate the normal order of ∆(|F(n)|^r) to within factors that are slowly varying functions of log n. This is then applied to determine with the same precision the normal order of ∆(|F(n)|) for any polynomial F with integer coefficients. We also evaluate some weighted average orders of ∆(n^r) and apply the result to bounding short sums of ∆(|F(n)|) for general F\in Z[X].