Root separation for polynomials with reducible derivative (Q2057878)

Let \(f=\sum\limits_{i=0}^da_iX^i\in\mathbb{Z}[X]\) be a polynomial of degree \(d\geq 3\) such that the derivative \(f'\) is reducible over \(\mathbb{Q}\) and let \(1\leq m\leq d-1\) be the largest integer such that \(f'\) has an irreducible factor over \(\mathbb{Q}\) of degree \(m\). The aim of the paper is to find a lower bound of the distance between two roots of \(f\) in terms of \(d, m\) and \(H(f):=\max\limits_{0\leq a_i\leq d} \vert a_i\vert \).