How to find all roots of complex polynomials by Newton's method

We investigate Newton's method to find roots of polynomials of fixed degree d, appropriately normalized: we construct a finite set of points such that, for every root of every such polynomial, at least one of these points will converge to this root under Newton's map. The cardinality of such a set can be as small as 1.11 d log² d; if all the roots of the polynomial are real, it can be 1.30 d.