Root separation for polynomials with reducible derivative

Abstract

Abstract Suppose f is a degree d polynomial with integer coefficients whose derivative f′ is a polynomial reducible over ℚ. We give a lower bound for the distance between two distinct roots of f in terms of d, the height H(f) of f, and the degree m of the irreducible factor of f′ with largest degree. The exponent (d + m − 1)/2 that appears as the power of H(f) is smaller than the corresponding exponent d − 1 in Mahler’s bound.